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# Traveling and pinned fronts in bistable reaction-diffusion systems on
networks
Nikos E. Kouvaris1,∗, Hiroshi Kori2, Alexander S. Mikhailov1
( 1 Department of Physical Chemistry, Fritz Haber Institute of the Max Planck
Society, Faradayweg 4-6, D-14195 Berlin, Germany $2$ Department of Information
Sciences, Ochanomizu University, Tokyo 112-8610, Japan ∗ E-mail: nkoub@fhi-
berlin.mpg.de )
## Abstract
Traveling fronts and stationary localized patterns in bistable reaction-
diffusion systems have been broadly studied for classical continuous media and
regular lattices. Analogs of such non-equilibrium patterns are also possible
in networks. Here, we consider traveling and stationary patterns in bistable
one-component systems on random Erdös-Rényi, scale-free and hierarchical tree
networks. As revealed through numerical simulations, traveling fronts exist in
network-organized systems. They represent waves of transition from one stable
state into another, spreading over the entire network. The fronts can
furthermore be pinned, thus forming stationary structures. While pinning of
fronts has previously been considered for chains of diffusively coupled
bistable elements, the network architecture brings about significant
differences. An important role is played by the degree (the number of
connections) of a node. For regular trees with a fixed branching factor, the
pinning conditions are analytically determined. For large Erdös-Rényi and
scale-free networks, the mean-field theory for stationary patterns is
constructed.
## Introduction
Studies of pattern formation in reaction-diffusion systems far from
equilibrium constitute a firmly established research field. Starting from the
pioneering work by Turing [1] and Prigogine [2], self-organized structures in
distributed active media with activator-inhibitor dynamics have been
extensively investigated and various non-equilibrium patterns, such as
rotating spirals, traveling pulses, propagating fronts or stationary
dissipative structures could be observed [3, 4]. Recently, the attention
became turned to network analogs of classical reaction-diffusion systems,
where the nodes are occupied by active elements and the links represent
diffusive connections between them. Such situations are typical for
epidemiology where spreading of diseases over transportation networks takes
place [5]. The networks can also be formed by diffusively coupled chemical
reactors [6] or biological cells [7]. In distributed ecological systems, they
consist of individual habitats with dispersal connections between them [8].
Detailed investigations of synchronization phenomena in oscillatory systems
[9] and of infection spreading over networks [10] have been performed. Turing
patterns in activator-inhibitor network systems have also been considered
[11].
The analysis of bistable media is of principal importance in the theory of
pattern formation in reaction-diffusion systems. Traveling fronts which
represent waves of transition from one stable state to another are providing a
classical example of self-organized wave patterns; they are also playing an
important role in understanding of more complex self-organization behavior in
activator-inhibitor systems and excitable media (see, e.g., [4, 12]). The
velocity and the profile of a traveling front are uniquely determined by the
properties of the medium and do not depend on initial conditions. Depending on
the parameters of a medium, either spreading or retreating fronts can
generally be found. Stationary fronts, which separate regions with two
different stable states, are not characteristic for continuous media; they are
found only at special parameter values where a transition from spreading to
retreating waves takes place. When discrete systems, formed by chains or
fractal structures of diffusively coupled bistable elements, are considered,
traveling fronts can however become pinned if diffusion is weak enough, so
that stable stationary fronts, which are found within entire parameter
regions, may arise [13, 14, 15, 16].
In the present study, pattern formation in complex networks formed by
diffusively coupled bistable elements is numerically and analytically
investigated. Our numerical simulations, performed for random Erdös-Rényi (ER)
or scale-free networks and for irregular trees, reveal a rich variety of time-
dependent and stationary patterns. The analogs of spreading and retreating
fronts are observed. Furthermore, stationary patterns, localized on subsets of
network nodes, are found. To understand such phenomena, an approximate
analytical theory for the networks representing regular trees is developed.
The theory yields the bifurcation diagram which determines pinning conditions
for trees with different branching factors and for different diffusion
constants. Its results are used to interpret the behavior found in irregular
trees and for ER networks. Statistical properties of stationary patterns in
large random networks are moreover analyzed in the framework of the mean-field
approximation, which has been originally proposed for spreading-infection
problems [17, 18, 19] and has also been used in the analysis of Turing
patterns on the networks [11].
### Bistable systems on networks
Classical one-component reaction-diffusion systems in continuous media are
described by equations of the form
$\dot{u}(\mathbf{x},t)=f(u)+D\nabla^{2}u(\mathbf{x},t)$ (1)
where $u(\mathbf{x},t)$ is the local activator density, function $f(u)$
specifies local bistable dynamics (see Methods) and $D$ is the diffusion
coefficient. Depending on the particular context, the activator variable $u$
may represent concentration of a chemical or biological species which
amplifies (i.e. auto-catalyzes) its own production.
In the present study, we consider analogs of the phenomena described by the
model (1), which are however taking place on networks. In network-organized
systems, the activator species occupies the nodes of a network and can be
transported over network links to other nodes. The connectivity structure of
the network can be described in terms of its adjacency matrix $\mathbf{T}$
whose elements are $T_{ij}=1$, if there is a link connecting the nodes $i$ and
$j$ ($i,j=1,...,N$), and $T_{ij}=0$ otherwise. We consider processes in
undirected networks, where the adjacency matrix T is symmetric
($T_{ij}=T_{ji}$). Generally, the network analog of system (1) is given by
$\dot{u}_{i}=f(u_{i})+D\sum_{j=1}^{N}\\!\left(T_{ij}u_{j}-T_{ji}u_{i}\right)$
(2)
where $u_{i}$ is the amount of activator in network node $i$ and $f(u_{i})$
describes the local bistable dynamics of the activator. The last term in Eq.
(2) takes into account diffusive coupling between the nodes. Parameter $D$
characterizes the rate of diffusive transport of the activator over the
network links.
Instead of the adjacency matrix, it is convenient to use the Laplacian matrix
$\mathbf{L}$ of the network, whose elements are defined as
$L_{ij}=T_{ij}-k_{i}\delta_{ij}$, where $\delta_{ij}=1$ for $i=j$, and
$\delta_{ij}=0$ otherwise. In this definition, $k_{i}$ is the degree, or the
number of connections, of node $i$ given by $k_{i}=\sum_{j}T_{ji}$. In new
notations Eq. (2) takes the form
$\dot{u}_{i}=f(u_{i})+D\sum_{j=1}^{N}\\!L_{ij}u_{j}\,.$ (3)
When the considered network is a lattice, its Laplacian matrix coincides with
the finite-difference expression for the Laplacian differential operator after
discretization on this lattice.
A classical example of a one-component system exhibiting bistable dynamics is
the Schlögl model [20]. This model describes a hypothetical trimolecular
chemical reaction which exhibits bistability (see Methods). In the Schlögl
model, the nonlinear function $f(u)$ is a cubic polynomial
$f(u)=-\frac{\partial V}{\partial u}=-(u-r_{1})(u-r_{2})(u-r_{3})$ (4)
so that $V(u)$ has one maximum at $r_{2}$ and two minima at $r_{1}$ and
$r_{3}$. We have performed numerical simulations and analytical investigations
of the reaction-diffusion system (3) for different kinds of networks using the
Schlögl model.
## Results
### Numerical simulations
In this section we report the results of numerical simulations of the bistable
Schlögl model (3) for random ER networks and for trees (the results for random
scale-free networks are given in the Supporting Information S1). The ER
networks with the mean degree $\langle k\rangle=7$ and sizes $N=150$ or
$N=500$ are considered. The trees have several components with different
branching factors. The model (3) with the parameters $r_{1}=1$ and $r_{3}=3$
is chosen; the parameter $r_{2}$ and the diffusion constant $D$ were varied in
the simulations. The parameter $r_{2}$ was restricted to the interval
$1<r_{2}<2$.
Figure 1: Traveling front in an Erdös-Rényi network. The network size is
$N=500$ and the mean degree is $\langle k\rangle=7$. Three consequent
snapshots of activity patterns at times $t=0,10,21$ are shown. Quantity
$\rho_{h}$ is the average value of the activator density $u$ in the subset of
network nodes located at distance $h$ from the node which was initially
activated. Other parameters are $r_{1}=1,r_{2}=1.2,r_{2}=3$; the diffusion
constant is $D=0.1$.
Traveling activation fronts were observed in ER networks. To initiate such a
front, a node at the periphery (with the minimum degree $k$) could be chosen
and set into the active state $u=r_{3}$, whereas all other nodes were in the
passive state $u=r_{1}$. This configuration was found to generate a wave of
transition from the passive to the active states. The wave spreads from the
initially active node to the rest of the system and reaches equidistant nodes,
located at the same distance (the shortest path length) from the initial node,
at about the same time.
Front propagation is illustrated in Fig. 1, where the nodes are grouped
according to their distance from the first activated node and the average
value $\rho_{h}$ of the activator density $u$ in each group is plotted as a
function of the distance $h$. Three snapshots of the traveling front at
different times are displayed. As we see, for increasing time the front moves
into the subsets of nodes with the larger distances. At the end, all nodes are
in the active state $r_{3}$.
Figure 2: Stationary pattern in an Erdös-Rényi network. The network size is
$N=150$ and the mean degree is $\langle k\rangle=7$. The nodes with higher
degrees are located closer to the center. The nodes are colored according to
their activation level, as indicated in the bar. The other parameters are
$r_{1}=1$, $r_{2}=1.4$ and $r_{3}=3$; the diffusion constant is $D=0.01$.
Not all initial conditions lead, however, to spreading fronts. If for example,
for the same model parameters as in Fig. 1, a hub node was initially
activated, a spreading activation front could not be produced. Retreating
fronts were found at these parameter values if the initial activation was set
in a few neighbor nodes with large degrees. Under weak diffusive coupling,
stationary localized patterns were furthermore observed. If the initial
activation was set on the nodes with moderate degrees, the activation could
neither spread nor retreat, thus staying as a stationary localized structure.
On the other hand, traveling fronts could also become pinned when some nodes
were reached, so that the activation could not spread over the entire network
and stationary patterns with coexistence of the two states were established.
Degrees of the nodes play an important role in front pinning. In the
representation used in Fig. 2, the nodes with higher degrees lie in the
center, whereas the nodes with small degrees are located in the periphery of
the network. To produce the stationary pattern shown in this figure, some of
the central hub nodes were set into the active state $r_{3}$, while all other
nodes were in the passive state $r_{1}$. The activation front started to
propagate towards the periphery, but the front became pinned and a stationary
pattern was formed. Figure 3 shows another example of a stationary pattern.
Here, we have sorted network nodes according to their degrees, so that the
degree of a node becomes higher as its index $i$ is increased (the stepwise
red curve indicates the degrees of the nodes). Localization on a subset of the
nodes with high degrees is observed.
Figure 3: Nodes activation levels for a stationary pattern in an Erdös-Rényi
network. Dependence of the activation level $u_{i}$ on the degrees $k_{i}$ of
the nodes $i$ is presented for a stationary pattern in the ER network of size
$N=500$ and mean degree $\langle k\rangle=7$. The nodes are ordered according
to their increasing degrees, shown by the stepwise red curve. The other
parameters are $r_{1}=1$, $r_{2}=1.4$ and $r_{3}=3$; the diffusion constant is
$D=0.01$.
The importance of the degrees of the nodes becomes particularly clear when
front propagation in the trees with various branching factors is considered
(in a tree, the branching factor of a node with degree $k$ is $k-1$). The
networks shown in Fig. 4 consist of the component trees with the branching
factors $2,3,4$ and $5$ which are connected at their origins. If the
activation is initially applied to the central node, it spreads for $D=0.1$
through the trees with branching factors $2$ and $3$, but cannot propagate
through the trees with higher branching factors (Fig. 4A). If we choose a
larger diffusion constant $D=0.35$ and apply activation to a subset of nodes
inside the tree with the branching factor $5$, the activation retreats and
dies out (Fig. 4B). When diffusion is weak ($D=0.03$), the application of
activation inside the component trees leads to its spreading towards the roots
of the trees. The activation cannot however propagate further and pinned
stationary structures are formed (Fig. 4C).
Thus, we see that both traveling fronts and pinned stationary structures can
be observed in the networks. Our numerical simulations suggest that degrees of
the nodes (and the related branching factors in the trees) should play an
important role in such phenomena. The observed behavior is however complex and
seems to depend on the architecture of the networks and on how the initial
activation was applied. Below, it is analytically investigated for regular
trees with fixed branching factors. The approximate mean-field description for
stationary patterns in large random networks is moreover constructed. Using
analytical results, complex behavior observed in numerical simulations can be
understood.
Figure 4: Spreading, retreating and pinning of activation fronts in trees. A)
For $D=0.1$, the fronts spread to the periphery through the nodes with the
degrees $k=2,3,4$, while they are pinned at the nodes with the larger degrees.
B) For $D=0.35$, the front is retreated from nodes with degree $k=6$. C) For
$D=0.03$, the fronts propagate towards the root, but not towards the
periphery. In each row, the initial configuration (left) and the final
stationary pattern (right) are displayed. The same color coding for node
activity as in Fig. 2 is used. Other parameters are $r_{1}=1$, $r_{2}=1.4$ and
$r_{3}=3$.
### Front dynamics in regular trees
Let us consider the model (3) for a regular tree with the branching factor
$k-1$. In such a tree, all nodes, lying at the same distance $l$ from the
origin, can be grouped into a single shell and front propagation along the
sequence of the shells $l=1,2,3,...$ can be studied. Suppose that we have
taken a node which belongs to the shell $l$. This node should be diffusively
coupled to $k-1$ nodes in the next shell $l+1$ and to just one node in the
previous shell $l-1$. Introducing the activation level $u_{l}$ in the shell
$l$, the evolution of the activator distribution on the tree can therefore be
described by the equation
$\dot{u}_{l}=f(u_{l})+D(u_{l-1}-u_{l})+D(k-1)(u_{l+1}-u_{l})\,.$ (5)
Note that for $k=2$, Eq. (5) describes front propagation in a one-dimensional
chain of coupled bistable elements. Propagation failure and pinning of fronts
in chains of bistable elements have been previously investigated [13, 14, 15].
The approximate analytical theory for front pinning in the trees, which is
presented below, represents an extension of the respective theory for the
chains [15]. Note furthermore that model (5) can be formally considered for
any values of $k>2$ of the parameter $k$ (but actual trees correspond only to
the integer values of this parameter).
Comparing the situations for the chains of coupled single elements and for
coupled shells in a tree (Eq. (5)), an important difference should be
stressed. In a chain, both propagation directions (left or right) are
equivalent, because of the chain symmetry. In contrast to this, an activation
front propagating from the root to the periphery of a tree is physically
different from the front propagating in the opposite direction, i.e. towards
the tree root. As we shall soon see, one of such fronts can be spreading while
the other can be pinned or retreating for the same set of model parameters.
The approximate analytical theory of front pinning can be constructed (cf.
[15]) if diffusion is weak and the fronts are very narrow. A pinned front is
found by setting $\dot{u}_{l}=0$ in Eq. (5), so that we get
$f(u_{l})+D(u_{l-1}-u_{l})+D(k-1)(u_{l+1}-u_{l})=0\,.$ (6)
Suppose that the pinned front is located at the shell $l=m$ and it is so
narrow that the nodes in the lower shells $l<m$ are all approximately in the
active state $r_{3}$, whereas the nodes in the higher shells are in the
passive state $r_{1}$. Then, the activation level $u_{m}$ in the interface
$l=m$ should approximately satisfy the condition
$g(u_{m})=f(u_{m})+D\left[(k-1)r_{1}-ku_{m}+r_{3}\right]=0\,.$ (7)
Figure 5: Functions $g(u_{m})$ for three different values of $D$. The other
parameters are $r_{1}=1$, $r_{2}=1.4,r_{3}=3$ and $k=4$.
Thus, the problem becomes reduced to finding the solutions of Eq. (7). When
$D=0$, we have $g(u_{m})=f(u_{m}$) and, therefore, Eq. (7) has three roots
$u_{m}=r_{1},r_{2},r_{3}$; the front is pinned then. Equation (7) has also
three roots if $D$ is small enough (see Fig. 5 for $D=0.03$). In this
situation, the front continues to be pinned. Under further increase of the
diffusion constant (see Fig. 5 for $D=0.1$), the two smaller roots merge and
disappear, so that only one (larger) root remains. As previously shown for
one-dimensional chains of diffusively coupled elements [15], such transition
corresponds to the disappearance of pinned stationary fronts.
The transition from pinned to traveling fronts takes place through a saddle-
node bifurcation. When $k$ is fixed, the bifurcation occurs when some critical
value of $D$ is exceeded (see Fig. 6A). If the diffusion constant is fixed,
pinned fronts are found inside an interval of degrees $k$ (see Fig. 6B).
Figure 6: The roots of Eq. (7). The roots $u_{m}$ are plotted as functions (A)
of the diffusion constant $D$ for $k=4$ and (B) of the degree $k$ for $D=0.1$.
Pinned fronts correspond to red parts of the curves. The model parameters are
$r_{1}=1,r_{2}=1.4$ and $r_{3}=3$.
Generally, the bifurcation boundary can be determined from the conditions
$g(u_{m})=0$ and $g^{\prime}(u_{m})=0$, which can be written in the parametric
form as
$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{\left(2u_{m}-r_{2}-r_{3}\right)\left(r_{1}-u_{m}\right)^{2}}{r_{1}-r_{3}}$
$\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{-3u_{m}^{2}+2\left(r_{1}+r_{2}+r_{3}\right)u_{m}-r_{1}r_{2}-r_{1}r_{3}-r_{2}r_{3}}{D}\,.$
(8)
Equations (Front dynamics in regular trees) determine boundaries between
regions II and III or II and IV in the bifurcation diagram in Fig. 7. The two
boundaries merge in the cusp point, which is defined by the conditions
$g(u_{m})=g^{\prime}(u_{m})=g^{\prime\prime}(u_{m})=0$ and is located at
$\displaystyle D^{\text{cusp}}$ $\displaystyle=$
$\displaystyle\frac{\left(r_{3}+r_{2}-2r_{1}\right)^{3}}{27\left(r_{3}-r_{1}\right)}$
$\displaystyle k^{\text{cusp}}$ $\displaystyle=$
$\displaystyle\frac{r_{1}^{2}+r_{2}^{2}+r_{3}^{2}-r_{1}r_{2}-r_{1}r_{3}-r_{2}r_{3}}{3D^{\text{cusp}}}\,$
(9)
in the parameter plane.
Above the cusp point, there should be a boundary line separating regions III
and IV. Indeed, fronts propagate in opposite directions in these two regions
and, to go from one to another, one needs to cross a line on which the
propagation velocity vanishes. This boundary can be identified by using the
arguments given below.
Figure 7: The bifurcation diagram. Regions with different dynamical regimes
are shown in the parametric plane $k-D$. Black curves indicate the saddle-node
bifurcations given by the Eq. (Front dynamics in regular trees), while the
blue curve stands for the saddle-node bifurcations given by Eq. (Front
dynamics in regular trees). The green dot indicates the cusp point given in
Eq. (Front dynamics in regular trees), the red curve shows the boundary
determined by Eq. (10). The model parameters are $r_{1}=1,r_{2}=1.4$ and
$r_{3}=3$.
Suppose that the diffusion constant is fixed and $D<D^{\text{cusp}}$. Then the
pinned fronts are found inside an interval of degrees $k$, where equation
$g(u_{m})=0$ has three roots, as in Fig. 8A for $k=7$. Outside this interval,
equation $g(u_{m})=0$ has a single root, which corresponds to spreading fronts
if it is close to $r_{3}$ (Fig. 8A for $k=4$) or to retreating fronts if it is
close to $r_{1}$ (Fig. 8A for $k=9$). Thus, if we traverse the bifurcation
diagram in Fig. 7 below $D^{\text{cusp}}$ by increasing $k$, function
$g(u_{m})$ will change its form as shown in Fig. 8, having three zeroes within
an entire interval of degrees $k$ that corresponds to the pinning region II.
When the the diffusion constant is increased and the cusp at
$D=D^{\text{cusp}}$ is approached, such interval shrinks to a point. If we
traverse the bifurcation diagram in Fig. 7 above the cusp, the function
$g(u_{m})$ changes as shown in Fig. 8B. For a given diffusion constant $D$,
there is only one degree $k$, such that the function $g(u_{m)}$ has an
inflection point coinciding with its zero. The boundary separating regions III
and IV is determined by the conditions $g(u_{m})=0$ and
$g^{\prime\prime}(u_{m})=0$. In the parameter plane, these conditions yield
the curve
$D=\frac{\left(r_{1}+r_{2}-2r_{3}\right)\left(r_{2}+r_{3}-2r_{1}\right)\left(r_{1}+r_{3}-2r_{2}\right)}{9\left[3(r_{3}-r_{1})-\left(r_{3}+r_{2}-2r_{1}\right)k\right]}\,$
(10)
where the propagation velocity of the fronts is changing its sign.
Figure 8: The typical form of functions $g(u_{m})$ in different regions of the
parameter plane. Functions $g(u_{m})$ are shown below (A, $D=0.1$) and above
(B, $D=0.32$) the cusp point. The green curve in part (B) corresponds to the
boundary between regions III and IV, where the front propagation velocity
vanishes. The other parameters are $r_{1}=1$, $r_{2}=1.4$ and $r_{3}=3$.
The above results refer to the first kind of fronts, where the nodes in the
lower shells ($l<m$) of the tree are in the active state $u=r_{3}$ and the
nodes in the periphery are in the passive state $u=r_{1}$. A similar analysis
can furthermore be performed for the second kind of fronts, where the nodes in
the periphery are in the active state and the nodes in the lower shells are in
the passive state. Such pinned fronts are again determined by equation (7),
where however the parameters $r_{3}$ and $r_{1}$ should be exchanged. The
pinning boundary for them can be obtained from equations (Front dynamics in
regular trees) under the exchange of $r_{1}$ and $r_{3}$. This yields
$\displaystyle D$ $\displaystyle=$
$\displaystyle\frac{\left(r_{1}+r_{2}-2u_{m}\right)\left(r_{3}-u_{m}\right)^{2}}{r_{1}-r_{3}}$
$\displaystyle k$ $\displaystyle=$
$\displaystyle\frac{-3u_{m}^{2}+2\left(r_{1}+r_{2}+r_{3}\right)u_{m}-r_{1}r_{2}-r_{1}r_{3}-r_{2}r_{3}}{D}\,.$
(11)
Fronts of the second kind are pinned for sufficiently weak diffusion, inside
region I in the bifurcation diagram in Fig. 7. The boundary of this region is
determined in the parametric form by Eqs. (Front dynamics in regular trees).
Thus, our approximate analysis has allowed us to identify regions in the
parameter plane ($k,D$) where the fronts of different kinds are pinned or
propagate in specific directions. Predictions of the approximate analytical
theory agree well with numerical simulations for regular trees. Figure 9 shows
traveling and pinned fronts found by direct integration of Eq. (5) in
different regions of the parameter plane. For each region, the behavior of two
kinds of the fronts, with the activation applied to the nodes of the lower
shells ($l\leq 6$) or periphery nodes ($l>6$), is illustrated. When the
parameters $D$ and $k$ are chosen within region I of the bifurcation diagram,
both kinds of fronts are pinned (Figs. 9A(I),B(I)). In region II, the front
initiated from the tree origin is pinned (Fig. 9A(II)), whereas the front
initiated in the periphery propagates towards the root (Fig. 9B(II)).
Activation fronts which propagate in both directions, towards the root and the
periphery, are found in region III (Fig. 9A(III),B(III)). In region IV, the
activation front initiated at the root is retreating (Fig. 9A(IV)), whereas
the front initiated at the periphery is spreading (Fig. 9B(IV)).
Figure 9: Stationary and traveling fronts in regular trees. The arrows show
the propagation direction for traveling fronts. The labels refer to different
regions of the bifurcation diagram in Fig. 7. They correspond to the parameter
values (I) $k=3,D=0.01$, (II) $k=6,D=0.03$, (III) $k=3,D=0.1$ and (IV)
$k=12,D=0.1$. The other parameters are $r_{1}=1,r_{2}=1.4$ and $r_{3}=3$.
In addition to providing examples of the front behavior, Fig. 9 also allows us
to estimate the accuracy of approximations used in the derivation of the
bifurcation diagram. In this derivation, we have assumed (similar to Ref.
[15]) that diffusion is weak and the fronts are so narrow that only in a
single point the activation level differs from its values $r_{1}$ and $r_{3}$
in the two uniform stable states. Examining Fig. 9, we can notice that this
assumption holds well for the lowest diffusion constant $D=0.01$ in region I,
whereas deviations can be already observed for the faster diffusion in regions
II, III and IV. Still, the deviations are relatively small and the
approximately analytical theory remains applicable.
Figure 10: Dependence of the front velocity on node degree. The blue curve
corresponds to the first kind of fronts, shown in Fig. 9A; the red curve is
for the second kind of fronts shown in Fig. 9B. The diffusion constant is
$D=0.1$; the other parameters are $r_{1}=1,r_{2}=1.4$ and $r_{3}=3$.
Figure 10 shows the numerically determined propagation velocity of both kinds
of activation fronts for different degrees $k$ at the same diffusion constant
$D=0.1$. The blue curve corresponds to the fronts of the first kind, with the
activation applied at the tree root. Such front is spreading towards the
periphery for small degrees $k$ (region III), is pinned in an interval of the
degrees corresponding to region II, and retreats towards the root for the
larger values of $k$ (region IV). The red curve displays the propagation
velocity for the second kind of fronts, with the activation applied at the
periphery. For the chosen value of the diffusion constant, such front is
always spreading, i.e. moving towards the root. We can notice that, for the
same parameter values, the absolute propagation velocity of the second kind of
fronts is always higher than that for the fronts of the first kind.
Figure 11: Nonlinear evolution of local perturbations. The evolution of
different local perturbations (A,B) is shown in various regions of the
bifurcation diagram. The arrows show the propagation direction. The parameter
values are (I) $k=3,D=0.01$, (II) $k=6,D=0.03$, (III) $k=3,D=0.1$ and (IV)
$k=12,D=0.1$. The other parameters are $r_{1}=1,r_{2}=1.4$ and $r_{3}=3$.
Using Fig. 9, we can consider evolution of various localized perturbations in
different parts of the bifurcation diagram (Fig. 7). Inside region I, all
fronts are pinned. Therefore, any localized perturbation (see Fig.
11A(I),B(I)) is frozen in this region. If the activation is locally applied
inside region II, it spreads towards the root, but cannot spread in the
direction to the periphery (Fig. 11A(II)). On the other hand, if a local
“cold” region is created in region II on the background of the “hot” active
state, it shrinks and disappears (Fig. 11B(II)). In region III, local
activation spreads in both directions, eventually transferring the entire tree
into the hot state (Fig. 11A(III)), whereas the local cold region on the hot
background shrinks and disappears (Fig. 11 B(III)). An interesting behavior is
found in region IV. Here, both kinds of fronts are traveling in the same
direction (towards the root), but the velocity of the second of them is higher
(cf. Fig. 10). Therefore, the hot domain would gradually broaden while
traveling in the root direction (Fig. 11A(IV)). The local cold domain (Fig.
11B(IV)) would be however shrinking while traveling in the same direction.
With these results, complex behavior observed in numerical simulations for the
trees with varying branching factors (Fig. 4) can be understood. In the
simulation shown in Fig. 4A, the diffusion constant was $D=0.1$ and, according
to the bifurcation diagram in Fig. 7, the nodes with degrees $k=2,3,4$ should
correspond to region III, while the nodes with the higher degrees $k=5,6$ are
in the region II. Indeed, we can see in Fig. 4A that activation can propagate
from the root over the subtrees with the small branching factors, but the
front fails to propagate through the subtrees with node degrees $5$ and $6$.
In the simulation for $D=0.35$ shown in Fig. 4B, the activation has been
initially applied to a group of nodes with degree $k=6$ corresponding to
region IV. In accordance with the behavior illustrated in Fig. 11A(IV), such
local perturbations broaden while traveling towards the root of the tree, but
get pinned and finally disappear. In Fig. 4C we have $D=0.03$ and, therefore,
we are in region II for all degrees $k$. According to Fig. 11A(II), local
activation in any component tree should spread towards the root, but cannot
propagate towards the periphery, in agreement with the behavior illustrated in
Fig. 4C.
Although our study has been performed for the trees, its results can also be
used in the analysis of front propagation in large random Erdös-Rényi
networks. Indeed, it is known [21] that the ER networks are locally
approximated by the trees. Hence, if the initial perturbation has been applied
to a node and starts to spread over the network, its propagation is
effectively taking place on a tree formed by the node neighbors. Previously,
we have used this property in the analysis of oscillators entrainment by a
pacemaker in large ER networks [22, 23]. Only when the activation has already
covered a sufficiently high fraction of the network nodes, loops start to play
a role. When this occurs, the activation may arrive at a node along different
pathways and the tree approximation ceases to hold. In this opposite
situation, a different theory employing the mean-field approximation can
however be applied.
### Mean-field approximation
Random ER networks typically have short diameters and diffusive mixing in such
networks should be fast. Under the conditions of ideal mixing, the mean-field
approximation is applicable; it has previously been used to analyze epidemic
spreading [17, 18, 19], limit-cycle oscillations and turbulence [24], or
Turing patterns [11] on large random networks. In this approximation, details
of interactions between neighbors are neglected and each individual node is
viewed as being coupled to a global mean field which is determined by the
entire system. The network nodes contribute to the mean field according to
their degrees $k$. The strength of coupling of a node to such global field and
also the amount of its contribution to the field are not the same for all
nodes and are proportional to their degrees. Thus, a node with a higher number
of connections is more strongly affected by the mean field, generated by the
rest of the network, and it also contributes stronger to such field. The mean-
field approximation is applied below to analyze statistical properties of
stationary activation patterns which are well spread over a network and
involve a relatively large fraction of nodes.
Similar to publications [24, 11], we start by introducing the local field
$q_{i}=\sum_{j=1}^{N}\\!T_{ij}u_{j}$ (12)
determined by the activation of the first neighbors of a network node $i$.
Then, the evolution equation (2) can be written in the form
$\dot{u}_{i}=f(u_{i})+D(q_{i}-k_{i}u_{i})\,$ (13)
so that it describes the interaction of the element at node $i$ with the local
field $q_{i}$.
The mean-field approximation consists of the replacement of the local fields
$q_{i}$ by $q_{i}=k_{i}Q$, where the global mean field is defined as
$Q=\sum_{j=1}^{N}\\!w_{j}u_{j}\,.$ (14)
Here, the weights
$w_{j}=\frac{k_{j}}{\sum_{n=1}^{N}\\!k_{n}}\,$ (15)
guarantee that the nodes with higher degrees $k_{i}$ contribute stronger to
the mean field. After such replacement, Eq. (13) yields
$\dot{u}=f(u)+\beta(Q-u)\,$ (16)
where $\beta=Dk$. Note that the index $i$ could be removed because the same
equation holds for all network nodes.
Equation (16) describes bistable dynamics of an element coupled to the mean
field $Q$. The coupling strength is determined by the parameter $\beta$ which
is proportional to the degree $k$ of the considered node. According to Eq.
(16), behavior of the elements located in the nodes with small degrees (and
hence small $\beta$) is mostly determined by local bistable dynamics, whereas
behavior of the elements located in the nodes with large degrees (and large
$\beta$) is dominated by the mean field.
Figure 12: The stationary activity pattern and the mean-field bifurcation
diagram. (A) The bifurcation diagram of Eq. (16) for the mean field $Q=1.5$.
(B) Activity distribution in the stationary pattern in the ER network of size
$N=500$ and mean degree $\langle k\rangle=7$ at $D=0.01$ is compared with the
activator levels $u$ predicted by the mean-field theory for $Q=1.5$. Blue
crosses show the simulation data. Black and red curves indicate stable and
unstable fixed points of the mean-field equation (16). The other parameters
are $r_{1}=1,r_{2}=1.4,r_{3}=3$.
The fixed points of Eq. (16) yield activator levels $u$ in single nodes
coupled with strength $\beta$ to the mean field $Q$. Self-organized stationary
patterns on a random ER network can be analyzed in terms of this mean-field
equation. Indeed, the activator level in each node $i$ of a pattern can be
calculated from Eq. (16), assuming that the node is coupled to the mean field
determined by the entire network. In Fig. 12, the mean-field approximation is
applied to analyze the stationary pattern shown in Fig. 3. This pattern has
developed in the ER network of size $N=500$ and mean degree $\langle
k\rangle=7$ when the diffusion constant was fixed at $D=0.01$. The mean field
corresponding to such pattern was computed in direct numerical simulations and
is equal to $Q=1.5$. Substituting this value of $Q$ into Eq. (16), activator
levels $u$ in single node, coupled to this mean field can be obtained. In Fig.
12A, the activator level $u$ is plotted as a function of the parameter
$\beta$. When a node is decoupled ($\beta=0$), Eq. (16) has three fixed points
$r_{1},r_{2},r_{3}$. As $\beta$ is increased, the system undergoes a saddle-
node bifurcation beyond which only one stable fixed point remains.
According to the definition of the parameter $\beta$, each node $i$ with
degree $k_{i}$ is characterized by its own value $\beta_{i}=Dk_{i}$ of this
bifurcation parameter. Therefore, the fixed points of Eq. (16) can be used to
determine the activation levels for each node $i$, if its degree $k_{i}$ is
known. The stationary activity distributions, predicted by the mean-field
theory and found in direct numerical simulations, are displayed in Fig. 12B,
where the nodes are ordered according to their increasing degrees. Note that
the value $Q=1.5$ of the mean field, used to determine the activity levels,
has been taken here from the numerical simulation. As we see, data points
indeed lie on the two stable branches of the bifurcation diagram, indicating a
good agreement with the mean-field approximation. In the Supporting
Information S1, a similar mean-field analysis is performed for self-organized
stationary activity patterns on scale-free networks.
## Discussion
Traveling fronts represent classical examples of non-equilibrium patterns in
bistable reaction-diffusion media. As shown in our study, such patterns are
also possible in networks of diffusively coupled bistable elements, but their
properties are significantly different. In addition to spreading or retreating
activation fronts, stationary fronts are found within large parameter regions.
The behavior of the fronts is highly sensitive to network architecture and
degrees of network nodes play an important role here.
In the special case of regular trees, an approximate analytical theory could
be constructed. The theory reveals that branching factors of the trees and,
thus, the degrees of their nodes, are essential for front propagation
phenomena. By using this approach, front pinning conditions could be derived
and parameter boundaries, which separate pinned and traveling fronts, could be
determined. As we have found, propagation conditions are different for the
fronts traveling from the tree root to the periphery or in the opposite
direction. Generally, all fronts become pinned as the diffusion constant is
gradually reduced. While the theory has been developed for regular trees,
where the branching factor is fixed, it is also applicable to irregular trees
where node degrees are variable. Indeed, at sufficiently weak diffusion the
front pinning occurs locally and its conditions are effectively determined
only by the degrees of the nodes at which a front becomes pinned.
The results of such analysis are relevant for understanding the phenomena of
activation spreading and pinning in large random networks. It is well known
(see, e.g., [21]) that, in the large size limit, random networks are locally
approximated by the trees. If the number of connections (the degree) of a node
is much smaller than the total number of nodes in a network, the probability
that a neighbor of a given node is also connected to another neighbor of the
same node is small, implying that the local pattern of connections in the
vicinity of a node has a tree structure. This property holds as long as the
number of nodes in the considered neighborhood is still much smaller that the
total number of nodes in the network. Previously, the local tree approximation
has been successfully used in the analysis of pacemakers in large random
oscillatory networks [22, 23].
When activation is applied to a node in a large random network, it spreads
through a subnetwork of its neighbors and, at sufficiently short distances
from the original node, such subnetwork should be a tree. Hence, our study of
front propagation on the trees is also providing a theory for the initial
stage of front spreading from a single activated node in large random
networks. Depending on the diffusion constant and other parameters, the fronts
may become pinned while the activation has not yet spread far away from the
original node. Whenever this takes place, the approximate pinning theory,
constructed for the trees, is applicable.
On the other hand, if the activation spreads far from the origin and a large
fraction of network nodes become thus affected, the patterns can be well
understood with the mean-field approximation. This approximation, proposed in
the analysis of infection spreading on networks [17], has also been applied to
analyze Turing patterns in network-organized activator-inhibitor systems [11]
and effects of turbulence in oscillator networks [24]. In this paper, we have
applied this approximation to the analysis of stationary activity
distributions in random Erdös-Rényi and scale-free networks of diffusively
coupled bistable elements. We could observe that, within the mean-field
approximation, statistical properties of network activity distributions are
well reproduced. It should be noted that, similar to previous studies [24,
11], the mean-field values used in the theory were taken from direct numerical
simulations and were not obtained through the solution of a consistency
equation. Hence, we could only demonstrate that such an approximation is
applicable for the statistical description of the emerging stationary
patterns, but did not use it here for the prediction of such patterns.
Thus, our investigations have shown that a rich behavior involving traveling
and pinned fronts is characteristic for networks of diffusively coupled
bistable elements. In the past, pinned fronts were observed in the experiments
using weakly coupled bistable chemical reactors on a ring [25, 26]. It will be
interesting to perform similar experiments for the networks of coupled
chemical reactors. Recent developments in nanotechnology allow to design
chemical reactors at the nanoscale and couple them by diffusive connections to
build networks [6]. It should be also noted that, while the chemical Schlögl
model has been used in our numerical simulations, the results are general and
applicable to any networks formed by diffusively coupled bistable elements.
The phenomena of front spreading and pinning should be possible for
diffusively coupled ecological populations and similar effects may be involved
when epidemics spreading under bistability conditions is considered.
## Methods
Bistable dynamics. The Schlögl model [20] corresponds to a hypothetical
reaction scheme
$\displaystyle A+2X$
$\displaystyle\overset{c_{1}}{\underset{c_{2}}{\rightleftarrows}}$
$\displaystyle 3X$ $\displaystyle X$
$\displaystyle\overset{c_{3}}{\underset{c_{4}}{\rightleftarrows}}$
$\displaystyle B\,.$ (17)
If concentrations of reagents $A$ and $B$ are kept fixed, the rate equation
for the concentration $u$ of the activator species $X$ reads
$\dot{u}(t)=-c_{2}u^{3}(t)+c_{1}au^{2}(t)-c_{3}u(t)+c_{4}b$ (18)
where the coefficients $c_{1},c_{2},c_{3},c_{4}$ are rate constants of the
reactions; $a=[A]$, $b=[B]$ and $u=[X]$ are concentrations of chemical
species. By choosing appropriate time units, we can set $c_{2}=1$. Then, the
right side of Eq. (18), can be written as
$f(u)=-(u-r_{1})(u-r_{2})(u-r_{3})$ (19)
where the parameters $r_{1},r_{2},r_{3}$ satisfy the conditions
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{r_{1}+r_{2}+r_{3}}{a}$ $\displaystyle c_{3}$
$\displaystyle=$ $\displaystyle r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}$
$\displaystyle c_{4}$ $\displaystyle=$
$\displaystyle\frac{r_{1}r_{2}r_{3}}{b}\,.$ (20)
The cubic polynomial $f(u)$ has three real roots which correspond to the
steady states (fixed points) of the dynamical system (18).
Networks. Erdös-Rényi networks were constructed by taking a large number $N$
of nodes and randomly connecting any two nodes with some probability $p$. This
construction algorithm yields a Poisson degree distribution with the mean
degree $\langle k\rangle=pN$ [27]. In our study we have considered the largest
connected component network, namely, we have removed the nodes with the degree
$k=0$.
Tree networks with branching factor $k-1$ were constructed by a simple
iterative method. We start with a single root node and at each step add $k-1$
nodes to each existing node with the degree $k=1$. After $L$ steps this
algorithm leads to a tree network with the size $N=\sum_{l=1}^{L}(k-1)^{l-1}$,
where the root node has degree $k-1$, the last added nodes have degree $1$ and
all other nodes have degree $k$. In our numerical simulations we have also
used complex trees consisting of component trees with different fixed
branching factors which are connected at their origins.
Scale-free networks, considered in the Supporting Information S1, were
constructed by the preferential attachment algorithm of Barábasi and Albert
[27]. Starting with a small number of $m$ nodes with $m$ connections, at each
next time step a new node is added, with $m$ links to $m$ different previous
nodes. The new node will be connected to a previous node $i$, which has
$k_{i}$ connections, with the probability $k_{i}/\sum_{j}\\!k_{j}\,$. After
many time steps, this algorithm leads to a network composed by $N$ nodes with
the power-law degree distribution $P(k)\sim k^{-3}$ and the mean degree
$\left<k\right>=2m$.
To display the networks in Figs. 2, 4 and S2B we have used the Fruchterman-
Reingold force-directed algorithm which is available in the open-source Python
package NetworkX [28]. This network visualization algorithm places the nodes
with close degrees $k$ near one to another in the network projection onto a
plane.
Numerical methods. For networks of coupled bistable elements, simulations were
carried out by numerical integration of Eq. (3) using the explicit Euler
scheme
$u_{i}^{(t+1)}=u_{i}^{(t)}+dt\left[f\left(u_{i}^{(t)}\right)+D\sum_{j=1}^{N}\\!L_{ij}u_{j}^{(t)}\right]$
(21)
with the time step $dt=10^{-3}$. The integration was performed for $5\times
10^{5}$ steps. The initial conditions were $u_{i}=1$ for all network nodes
$i$, except a subset of nodes to which initial activation was applied and
where we had $u_{i}=3$. The explicit Euler scheme with the same time step $dt$
was also used to integrate Eq. (5) which describes patterns on regular trees.
## Supporting Information
Supporting Information S1. The results of the numerical simulations of the
bistable Schlögl model (3) for scale-free networks are provided. Traveling
fronts and stationary localized patterns are reported for networks with mean
degree $\langle k\rangle=6$ and sizes $N=150$ or $N=500$ nodes. The observed
stationary pattern is compared with the mean-field bifurcation diagram.
## Acknowledgments
Financial support from the DFG Collaborative Research Center SFB910 “Control
of Self-Organizing Nonlinear Systems” and from the Volkswagen Foundation in
Germany is gratefully acknowledged.
## Author Contribution
Designed the study: NK ASM. Performed the simulations: NK. Conceived the
analytical approximations: NK HK ASM. Wrote the article: NK HK ASM.
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## Supporting Information S1
In this supporting information, results of numerical simulations of the
bistable Schlögl model (3) for scale-free networks with mean degree $\langle
k\rangle=6$ and sizes $N=150$ or $N=500$ are reported. The model (3) with the
parameters $r_{1}=1$ and $r_{3}=3$ is chosen; the parameter $r_{2}$ and the
diffusive constant $D$ were varied in the simulations.
Both traveling and pinned fronts were observed. To initiate a traveling front,
a node at the periphery with the degree $k=3$ was set into the active state
$r_{3}$, whereas the rest of the nodes were in the passive state $r_{1}$. This
initial configuration generated a front which spread over the entire network.
Front propagation is seen in Fig. S1, where the nodes are grouped according to
their distance from the first activated node and the average value $\rho_{h}$
of the activator density $u$ in each group is plotted as a function of the
distance $h$. Three snapshots of the traveling front at different times are
displayed. At $t=0$, the activation is localized on one node. By time $t=10$,
it spreads to the second neighbors of the original node. At $t=21$, the
activation extends to the fifth neighbors of the original node, covering
almost the entire network. Note that a definite traveling front is observed
only while the activation is still close to the origin. At the final stage,
the front rapidly broadens and the transition to the final uniform active
state is quickly taking place.
Figure S1. Activation front in a scale-free network with mean degree $\langle
k\rangle=6$ and size $N=500$. Three consequent snapshots of activity patterns
at times $t=0,10,21$ are displayed. Quantity $\rho_{h}$ is the average value
of the activator density $u$ in the subset of network nodes located at
distance $h$ from the node which was initially activated. The model parameters
are $r_{1}=1,r_{2}=1.2,r_{2}=3$ and the diffusion constant is $D=0.1$.
When one of the hub nodes was initially activated, a spreading activation
front could not be produced. In this case, retracting fronts were observed, if
a compact group of nodes with large degrees was initially activated. For weak
diffusive coupling, stationary localized patterns were also found, either by
appropriate choosing initial conditions, or when traveling fronts was getting
pinned at some nodes so that the spreading activation could not reach all
network nodes. Two examples of such stationary patterns are shown in Fig. S2.
Figure S2. (A) Dependence of the activation level $u_{i}$ on the degrees
$k_{i}$ of the nodes $i$ for a stationary pattern in the scale-free network of
size $N=500$ and mean degree $\langle k\rangle=6$. The red curve shows the
degrees of the nodes. (B) Stationary pattern in the scale-free network of size
$N=150$ and mean degree $\langle k\rangle=6$. The nodes with higher degrees
are located closer to the center. The nodes are colored according to their
activation level, as indicated in the bar. The parameters are
$r_{1}=1,r_{2}=1.4$ and $r_{3}=3$; the diffusion constant is $D=0.01$. Figure
S3. (A) The bifurcation diagram of Eq. (16) for the mean field $Q=1.68$. (B)
Activity distribution in the stationary pattern in the scale-free network of
size $N=500$ and mean degree $\langle k\rangle=6$ at $D=0.01$ is compared with
the activator levels $u$ predicted by the mean-field theory for the mean field
$Q=1.68$ of the numerically computed pattern. Blue crosses show the simulation
data. Black and red curves indicate stable and unstable fixed points of the
mean-field equation (16). The other parameters are
$r_{1}=1,r_{2}=1.4,r_{3}=3$.
The mean-field approximation could be used to describe self-organized
stationary patterns on the scale-free networks. The mean-field computed in the
numerical simulations for the stationary pattern shown in Fig. S2A was equal
to $Q=1.68$. Substituting this value into the Eq. (16) we obtain the
bifurcation diagram of a single node coupled to this mean-field. The
stationary pattern is compared with the mean-field bifurcation diagram in Fig.
S3B. The curves are predictions of the mean-field approximation and the
crosses show the simulation data. We see that the data points are distributed
along the two stable branches of the bifurcation diagram, indicating good
agreement with the mean-field approximation.
|
arxiv-papers
| 2012-06-20T10:28:15 |
2024-09-04T02:49:31.994861
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nikos E. Kouvaris (1), Hiroshi Kori (2) and Alexander S. Mikhailov (1)\n ((1) Department of Physical Chemistry, Fritz Haber Institute of the Max\n Planck Society, Berlin, Germany (2) Department of Information Sciences,\n Ochanomizu University, Tokyo, Japan)",
"submitter": "Nikos Kouvaris N.K.",
"url": "https://arxiv.org/abs/1206.4447"
}
|
1206.4779
|
††institutetext: Department of Physics and Astronomy, Rutherford Building,
University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
# CP violatingTri-bimaximal-Cabibbo mixing
D. V. Ahluwalia dharamvir.ahluwalia@canterbury.ac.nz
###### Abstract
In view of the new data from the Daya Bay and RENO collaborations, King has
presented a very natural deformation of tri-bimaximal mixing. Here we show
that $L/E$ flatness of the $e$-like event ratio in the atmospheric neutrino
data, when coupled with King’s observation that the smallest neutrino mixing
angle, $\theta_{13}$, seems to be related to the largest quark mixing angle
(the Cabibbo angle $\theta_{C}$), leads to a CP violating tri-bimaximal-
Cabibbo mixing. King’s tri-bimaximal-Cabibbo mixing follows as a leading order
approximation from our result.
###### Keywords:
Neutrino physics, CP violation
The precise form of the neutrino mixing matrix, $U$, that defines the
relationship between the flavour and mass eigenstates, $|\nu_{\ell}\rangle$
and $|\nu_{j}\rangle$ respectively Chau:1984fp ; Beringer:2012bj , reads
$|\nu_{\ell}\rangle=\sum_{j}U^{\ast}_{\ell
j}|\nu_{j}\rangle,\quad\ell=e,\mu,\tau,\quad j=1,2,3,$ (1)
and the knowledge of the masses for the underlying mass eigenstates, arise
from yet unknown physics. Nevertheless, once the parameters that determine the
mixing matrix and the mass-squared differences are deciphered from the data
one can derive their phenomenological consequences on supernova explosions
Ahluwalia:2004dv ; Lunardini:2007vn ; Duan:2006an ; Duan:2007sh , on the
synthesis of elements Yoshida:2006sk , on the cosmic microwave background and
the distribution of large-scale structure Lesgourgues:2006nd . In particular,
if the neutrino mixing angle $\theta_{13}\neq 0$ then one can obtain CP
violation in the neutrino sector with many interesting physical consequences
Khlopov:1981nq ; Frampton:2002qc ; Balantekin:2007es .
The T2K, MINOS, and Double CHOOZ indications that the smallest neutrino mixing
angle $\theta_{13}$ may be non-zero Abe:2011ph ; Adamson:2011qu ; Abe:2011fz
has now been confirmed by the results of the Daya Bay and RENO collaborations
An:2012eh ; Ahn:2012nd . King has made the observation King:2012vj that the
smallest neutrino mixing angle $\theta_{13}$, seems to be related to the
largest quark mixing angle, the Cabibbo angle $\theta_{C}$ Cabibbo:1963yz , or
equivalently to the Wolfenstein parameter, $\lambda=0.2253\pm 0.0007$
Wolfenstein:1983yz ; Beringer:2012bj :111It is worth noting that Mohapatra and
Smirnov had earlier conjectured King’s observation (Mohapatra:2006gs, , Sec.
3.1).
$\theta_{13}~{}\mbox{(or,
}\theta_{reac}\mbox{)}=\arcsin\left(\frac{\sin\theta_{C}}{\sqrt{2}}\right)=\arcsin\left(\frac{\lambda}{\sqrt{2}}\right).$
(2)
To this observation we now add that the $L/E$ — where $L$ is the neutrino
source-detector distance and $E$ is the neutrino energy — flatness of the
$e$-like event ratio observed for atmospheric neutrinos Fukuda:1998mi
requires that
$\theta_{23}~{}\mbox{(or,
}\theta_{atm}\mbox{)}=\frac{\pi}{4},\quad\delta=\pm\frac{\pi}{2}.$ (3)
This observation was first made in reference Ahluwalia:2002tr . The $\delta$
obtained in Ahluwalia:2002tr was also introduced recently as an Ansatz in
Ref. Zhang:2012ys .
Global analysis of neutrino oscillation data by two independent groups shows:
(a) $\delta$ to be $\left(0.83^{+0.54}_{-0.64}\right)\pi$ for the normal mass
hierarchy while allowing for the full $[0,2\pi]$ range for the inverted mass
hierarchy Tortola:2012te , (b) $\delta\approx\pi$ with no significant
difference between the normal and inverted mass hierarchies Fogli:2012ua . A
detailed study of these two papers reveals that there is no statistically
significant indication which disfavours $\delta=\pm\pi/2$. Regarding
$\theta_{23}$: (a) the first of the mentioned groups obtains
$\sin^{2}\theta_{23}=0.49^{+0.08}_{-0.05}$ for the normal mass hierarchy, and
$\sin^{2}\theta_{23}=0.53^{+0.05}_{-0.07}$ for the inverted mass hierarchy
(these values are consistent with $\theta_{23}=\pi/4$), while (b) the second
group finds a slight preference for $\theta_{23}<\pi/4$.
Both groups agree with the tri-bimaximal mixing value for the remaining angle
Tortola:2012te ; Fogli:2012ua
$\theta_{12}~{}\mbox{(or,
}\theta_{\odot}\mbox{)}=\arcsin\left(\frac{1}{\sqrt{3}}\right).$ (4)
With all the angles and phases thus fixed, the neutrino mixing matrix for the
choice $\delta=\pi/2$ in equation (3) takes the form
$U^{+}=\begin{pmatrix}\sqrt{\frac{2}{3}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}&\sqrt{\frac{1}{3}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}&i\frac{1}{\sqrt{2}}\lambda\\\
-\frac{1}{\sqrt{6}}\left(1-i\lambda\right)&\frac{1}{\sqrt{3}}\left(1+i\frac{1}{2}\lambda\right)&\frac{1}{\sqrt{2}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}\\\
\frac{1}{\sqrt{6}}\left(1+i\lambda\right)&-\frac{1}{\sqrt{3}}\left(1-i\frac{1}{2}\lambda\right)&\frac{1}{\sqrt{2}}\left(1-\frac{\lambda^{2}}{2}\right)^{1/2}\end{pmatrix}.$
(5)
Its counterpart, $U^{-}$, for $\delta=-\pi/2$ is obtained by letting $i\to-i$
in $U^{+}$. As a measure of CP violation, following Beringer:2012bj , we
define the asymmetries
$A_{CP}^{(\ell^{\prime}\ell)}\colonequals
P(\nu_{\ell}\to\nu_{\ell^{\prime}})-P(\bar{\nu}_{\ell}\to\bar{\nu}_{\ell^{\prime}}),$
(6)
and find
$\displaystyle A_{CP}^{(\mu e)}=-A^{(\tau e)}_{CP}=A_{CP}^{(\tau\mu)}$
$\displaystyle=\mp\frac{1}{3}\lambda\left(2-\lambda^{2}\right)\left(\sin\frac{\Delta
m^{2}_{32}}{2p}L+\sin\frac{\Delta m^{2}_{21}}{2p}L+\sin\frac{\Delta
m^{2}_{13}}{2p}L\right)$ $\displaystyle\approx\mp 0.146\left(\sin\frac{\Delta
m^{2}_{32}}{2p}L+\sin\frac{\Delta m^{2}_{21}}{2p}L+\sin\frac{\Delta
m^{2}_{13}}{2p}L\right),$ (7)
where all symbols have their usual meaning. The $\mp$ sign holds for
$\delta=\pm\frac{\pi}{2}$. For $\lambda=0$, or equivalently $\theta_{13}=0$,
the $U^{\pm}$ reduce to the standard tri-bimaximal mixing matrix
Harrison:2002er .222This may be compared with (Stancu:1999ct, , Eq. (26)) that
gives an interpolating matrix with $\theta_{\odot}$ as a variable. In one
limit the interpolating matrix gives the bimaximal mixing Vissani:1997pa ;
Ahluwalia:1998xb ; Barger:1998ta and in another it yields tri-bimaximal
mixing Harrison:2002er .
The result (7) is modified by matter effects Wolfenstein:1977ue ;
Mikheev:1986gs . Its general features are studied in detail by various authors
Gava:2008rp ; Balantekin:2007es ; Kneller:2009vd ; Kisslinger:2012se . In
gravitational environments the following argument suggests that one must
expect a significant modification to the result (7). Neutrino oscillations
provide us with a set of flavour oscillation clocks. These clocks must
redshift according to the general expectations of the theory of general
relativity. In gravitational environments of neutron stars the dimensionless
gravitational potential is $\Phi^{NS}_{grav}\approx 0.2$ (cf. for Earth,
$\Phi^{\oplus}_{grav}\approx 6.95\times 10^{-10}$). For a given source-
detector distance, and a given energy, the asymmetries $A_{CP}$ for supernovae
modeling must be accordingly modified Ahluwalia:1996ev ; Ahluwalia:1998jx ;
Konno:1998kq ; Wudka:2000rf ; Mukhopadhyay:2005gb ; Singh:2003sp at the
$20\%$ level, or thereabouts.
An examination of the $U^{\pm}$ immediately shows that the expectation values
of the $\nu_{\mu}$ and $\nu_{\tau}$ masses are identical. To
$\mathcal{O}(\lambda^{2})$ the $U^{-}$ obtained above reproduces to King’s
result (King:2012vj, , Eq. (8)) for $\delta=\pi/2$. The presented $U^{\pm}$
not only accommodate the implications of the Daya Bay and RENO collaborations,
but also the L/E flatness of the $e$-like event ratio seen in the atmospheric
neutrino data while respecting all other known data on neutrino oscillations.
###### Acknowledgements.
The result presented here was obtained on 10 May 2012, and was presented the
next day at a MatScience Seminar. The author thanks Institute of Mathematical
Sciences (“MatScience”, Chennai, India) for its hospitality and for its
vibrant scholarly environment.
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|
arxiv-papers
| 2012-06-21T05:42:50 |
2024-09-04T02:49:32.014002
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "D. V. Ahluwalia",
"submitter": "D. V. Ahluwalia",
"url": "https://arxiv.org/abs/1206.4779"
}
|
1206.4790
|
# Torus Actions and the Halperin-Carlsson Conjecture
Yoshinobu Kamishima Department of Mathematics
Tokyo Metropolitan University
Minami-Ohsawa 1-1,Hachioji, Tokyo 192-0397, JAPAN kami@tmu.ac.jp and Mayumi
Nakayama Department of Mathematics and Information of Sciences
Tokyo Metropolitan University
Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, JAPAN nakayama-
mayumi@ed.tmu.ac.jp
###### Abstract.
We give an affirmative answer to the Halperin-Carlsson conjecture for the
homologically injective torus actions on closed manifolds.
This class contains _holomorphic torus actions on compact Kähler manifolds_ ,
_torus actions on compact Riemannian flat manifolds_.
###### Key words and phrases:
Torus action, Halperin-Carlsson Conjecture, Seifert fiber space, Riemannian
flat manifold, Kähler manifold.
###### 2000 Mathematics Subject Classification:
53C55, 57S25, 51M10
## 1\. Introduction
Recently the real Bott tower and its generalization have been studied by
several people ([6], [13], [18], [15],[12]). A real Bott manifold is
originally defined to be the set of real points in the Bott manifold [9].
Among several characterizations by group actions, the Halperin-Carlsson
conjecture is true for real Bott manifolds. The Halperin-Carlsson torus
conjecture says that if there is an almost free torus action $T^{k}$ on a
closed $n$-manifold $M$, the following inequality holds:
(1.1) $2^{k}\leq\mathop{\sum}_{j=0}^{n}b_{j}.$
Here $b_{j}={\rm rank}\,H_{j}(M;{\mathbb{Z}})$ is the j-th Betti number of
$M$. See [19] for details and the references therein, see also [10].
Another characterization is that a real Bott manifold $M$ is a euclidean space
form (Riemannian flat manifold) admitting a torus action $T^{k}$ with $k={\rm
rank}\,H_{1}(M)$. It is conceivable whether the _Halperin-Carlsson conjecture_
holds for compact euclidean space forms more generally.
By this motivation we revisit the classical results of the Calabi construction
of euclidean space forms with nonzero $b_{1}$ [4] and the Conner-Raymond’s
injective torus actions [8]. In this direction, we shall introduce _injective-
splitting action_ of a torus $T^{k}$ on closed manifolds more generally. Our
purpose of this paper is to prove the Halperin-Carlsson conjecture for such
torus actions affirmatively.
Let $T^{k}$ be a $k$-dimensional torus $(k\geq 1)$. Given an _effective_
$T^{k}$-action on a closed manifold $M$, the orbit map at $x\in M$ is defined
to be ${\rm ev}(t)=tx$ $({}^{\forall}\,t\in T^{k})$. Put
$\pi_{1}(T^{k})=H_{1}(T^{k};{\mathbb{Z}})={\mathbb{Z}}^{k}$ and
$\pi_{1}(M)=\pi$. The map ${\rm ev}$ induces a homomorphism $\displaystyle{\rm
ev}_{\\#}:{\mathbb{Z}}^{k}\rightarrow\pi$ and $\displaystyle{\rm
ev}_{*}:{\mathbb{Z}}^{k}\rightarrow H_{1}(M;{\mathbb{Z}})$ respectively.
According to the definition of Conner-Raymond [8], if ${\rm ev}_{\\#}$ is
_injective_ , the action $(T^{k},M)$ is said to be _injective_. ( Refer to
[14, Theorem 2.4.2, also Subsection 11.1] for the definition to be independent
of the choice of the base point $x\in M$.) Classically it is known that
$\displaystyle{\rm ev}_{\\#}$ is injective for closed _aspherical_ manifolds
[7]. On the other hand, if $\displaystyle{\rm
ev}_{*}:{\mathbb{Z}}^{k}\rightarrow H_{1}(M;{\mathbb{Z}})$, the $T^{k}$-action
is said to be _homologically injective_ [8]. We have shown
###### Theorem A.
If $T^{k}$ is a homologically injective action on a closed $n$-manifold $M$,
then
(1.2) ${}_{k}C_{j}\leq b_{j}.$
In particular the Halperin-Carlsson conjecture is true.
To prove Theorem A we revisit the Conner and Raymond’s work [8]. Then this is
a consequence from _injective-splitting torus actions_ more generally. See
Theorem 2.3. We verify that the following actions are in fact injective-
splitting torus actions.
###### Corollary B.
Every effective $T^{k}$-action on a compact $n$-dimensional euclidean space
form $M$ is homologically injective. Thus $\displaystyle{}_{k}C_{j}\leq
b_{j}$, the Halperin-Carlsson conjecture (1.1) holds.
We obtain a characterization of _holomorphic_ torus actions originally
observed by Carrell [5].
###### Corollary C.
Every holomorphic action of the complex torus $T^{k}_{\mathbb{C}}$ on a
compact Kähler manifold is homologically injective. In particular,
$\displaystyle{}_{2k}C_{j}\leq b_{j}$, the Halperin-Carlsson conjecture holds.
In Section 2, we introduce _injective-splitting actions_ on closed manifolds
and prove our main theorem 2.3. Using this theorem, we show Corollaries B and
C. In Section 3 we shall give a proof concerning the existence of torus
actions common to both the Calabi’s theorem and the Conner-Raymond’s theorem
as our motivation (cf. Theorem 3.2).
###### Theorem E.
A compact $n$-dimensional euclidean space form $M$ admits a homologically
injective action of $T^{k}$ with $k={\rm rank}\,H_{1}(M)$ in which ${\rm
rank}\,C(\pi)={\rm rank}\,H_{1}(M)$.
## 2\. Injective torus actions
Suppose $(T^{k},M)$ is an _injective action_ on a closed manifold $M$. Let
$\tilde{M}$ be the universal covering space of $M$ and denote $N_{{\rm
Diff}(\tilde{M})}(\pi)$ the normalizer of $\pi$ in ${\rm Diff}(\tilde{M})$.
The conjugation homomorphism $\displaystyle\mu:N_{{\rm
Diff}(\tilde{M})}(\pi)\rightarrow{\rm Aut}(\pi)$ defined by
$\mu(\tilde{f})(\gamma)=\tilde{f}\circ\gamma\circ\tilde{f}^{-1}$
$({}^{\forall}\,\gamma\in\pi)$ induces a homomorphism $\varphi$ which has a
commutative diagram:
(2.1) $\begin{CD}111\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\ C(\pi)@>{}>{}>\pi
@>{\mu}>{}>{\rm Inn}(\pi)\\\ @V{}V{}V@V{}V{}V@V{}V{}V\\\ C_{{\rm
Diff}(\tilde{M})}(\pi)@>{}>{}>N_{{\rm Diff}(\tilde{M})}(\pi)@>{\mu}>{}>{\rm
Aut}(\pi)\\\ @V{\nu}V{}V@V{\nu}V{}V@V{}V{}V\\\ {\rm Diff}(M)^{0}\leq{\rm
ker}\,\varphi @>{}>{}>{\rm Diff}(M)@>{\varphi}>{}>{\rm Out}(\pi)\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ 111\\\ \end{CD}$
(Compare [16].) Here $C_{{\rm Diff}(\tilde{M})}(\pi)$ is the centralizer of
$\pi$ in ${\rm Diff}(\tilde{M})$. As $T^{k}\leq{\rm Diff}(M)^{0}$, we have a
lift $\tilde{T}^{k}\leq C_{{\rm Diff}(\tilde{M})}(\pi)$ to $\tilde{M}$. Note
$\tilde{T}^{k}={\mathbb{R}}^{k}$. For this, suppose some $S^{1}\leq T^{k}$
lifts to $S^{1}$ (but not ${\mathbb{R}}$) on $\tilde{M}$. If
$p:\tilde{M}\rightarrow M$ is the covering map which is equivariant;
$p(tx)=t^{m}p(x)$ $(t\in S^{1})$ for some $m\in{\mathbb{Z}}$, chasing a
commutative diagram
$\begin{CD}\pi_{1}(S^{1})@>{{\rm ev}_{\\#}}>{}>\pi_{1}(\tilde{M})=1\\\
@V{{m\cdot}}V{}V@V{p_{\\#}}V{}V\\\ \pi_{1}(S^{1})@>{{\rm
ev}_{\\#}}>{}>\pi_{1}(M),\end{CD}$
it follows ${\rm ev}_{\\#}(m{\mathbb{Z}})=1$. This contradicts the injectivity
of $S^{1}\leq T^{k}$. We have a lift of groups from (2.1):
(2.2) $\begin{CD}C(\pi)@>{}>{}>C_{{\rm Diff}(\tilde{M})}(\pi)@>{}>{}>{\rm
Diff}(M)^{0}\leq{\rm ker}\,\varphi\\\ @A{}A{}A@A{}A{}A@A{}A{}A\\\
{\mathbb{Z}}^{k}@>{}>{}>{\mathbb{R}}^{k}@>{}>{}>T^{k}.\end{CD}$
Since ${\mathbb{Z}}^{k}\leq C(\pi)$, letting $Q=\pi/{\mathbb{Z}}^{k}$, there
is a central group extension:
(2.3) $1\rightarrow{\mathbb{Z}}^{k}\rightarrow\pi\longrightarrow Q\rightarrow
1.$
Now ${\mathbb{R}}^{k}$ acts properly and freely on $\tilde{M}$ such that
$\tilde{M}={\mathbb{R}}^{k}\times W$ where $W=\tilde{M}/{\mathbb{R}}^{k}$ is a
simply connected smooth manifold. The group extension (2.3) represents a
$2$-cocycle $f$ in $H^{2}(Q;{\mathbb{Z}}^{k})$ in which $\pi$ is viewed as the
product ${\mathbb{Z}}^{k}\times Q$ with group law:
$(n,\alpha)(m,\beta)=(n+m+f(\alpha,\beta),\alpha\beta).$
Let $Map(W,{\mathbb{R}}^{k})$ (respectively $Map(W,T^{k})$) be the set of
smooth maps of $W$ into ${\mathbb{R}}^{k}$ (respectively $T^{k}$) endowed with
a $Q$-module structure in which there is an exact sequence of $Q$-modules [7]:
$1\rightarrow{\mathbb{Z}}^{k}\rightarrow
Map(W,{\mathbb{R}}^{k})\stackrel{{\scriptstyle\exp}}{{\longrightarrow}}Map(W,T^{k})\rightarrow
1.$
As $Q$ acts properly discontinuously on $W$ with compact quotient, it follows
from [7, Lemma 8.5] (also [14]):
(2.4) $H^{i}(Q;Map(W,{\mathbb{R}}^{k}))=0\,\ (i\geq 1)$
so that the connected homomorphism
$\displaystyle\delta:H^{1}(Q;Map(W,T^{k}))\rightarrow
H^{2}(Q;{\mathbb{Z}}^{k})$ is an isomorphism. From this, there exists a map
$\chi:Q\rightarrow Map(W,{\mathbb{R}}^{k})$ such that $\delta^{1}\chi=f$. Then
the $\pi$-action on $\tilde{M}$ can be described as
(2.5) $\begin{split}(n,\alpha)(x,w)&=(n+x+\chi(\alpha)(\alpha w),\alpha w)\\\
&\ \ \ \
({}^{\forall}(n,\alpha)\in\pi,{}^{\forall}(x,w)\in{\mathbb{R}}^{k}\times
W).\end{split}$
The action of $\pi$ may depend on the choice of $\chi^{\prime}$ such that
$\delta^{1}\chi^{\prime}=f$. However, the vanishing of (2.4) shows
###### Proposition 2.1.
Such $\pi$-actions are equivalent to each other.
Let $(T^{k},M)$ be an injective $T^{k}$-action on a closed manifold $M$ which
induces a central group extension (2.3) as above.
###### Definition 2.2.
A $T^{k}$-action is said to be _injective-splitting_ if there exists a finite
index normal subgroup $Q^{\prime}$ of $Q$ such that the induced extension
splits;
$\pi^{\prime}={\mathbb{Z}}^{k}\times Q^{\prime}.$
Here is a key result concerning _injective-splitting torus actions_.
###### Theorem 2.3.
Suppose that a closed manifold $M$ admits an injective-splitting
$T^{k}$-action. Then the following hold.
* (i)
$\displaystyle{}_{k}C_{j}\leq b_{j}$. In particular the Halperin-Carlsson
conjecture is true.
* (ii)
The $T^{k}$-action is homologically injective.
###### Proof.
Algebraic part. (2.3) induces a commutative diagram:
(2.6) $\begin{CD}1@>{}>{}>{\mathbb{Z}}^{k}@>{}>{}>\pi @>{}>{}>Q@>{}>{}>1\\\
||@A{}A{\iota}A@A{}A{\iota^{\prime}}A\\\
1@>{}>{}>{\mathbb{Z}}^{k}@>{}>{}>\pi^{\prime}@>{}>{}>Q^{\prime}@>{}>{}>1\\\
\end{CD}$
Here $Q/Q^{\prime}$ is a finite group by Definition 2.2. For the cocycle $f$
representing the upper group extension, it follows $\iota^{\prime*}[f]=0\in
H^{2}(Q^{\prime};{\mathbb{Z}}^{k})$ by the hypothesis. We may assume
(2.7) $f|_{Q^{\prime}}=0.$
On the other hand, if $\tau:H^{2}(Q^{\prime};{\mathbb{Z}}^{k})\rightarrow
H^{2}(Q;{\mathbb{Z}}^{k})$ is the transfer homomorphism (cf. [3], [2]), then
$\tau\circ\iota^{\prime*}=|Q:Q^{\prime}|:H^{2}(Q;{\mathbb{Z}}^{k})\rightarrow
H^{2}(Q;{\mathbb{Z}}^{k})$ so that $[f]$ is a torsion in
$H^{2}(Q;{\mathbb{Z}}^{k})$. There exists an integer $\ell$ such that
$\ell\cdot f=\delta^{1}\tilde{\lambda}$ for some function
$\tilde{\lambda}:Q\rightarrow{\mathbb{Z}}^{k}$. Put
$\displaystyle\lambda=\frac{\tilde{\lambda}}{\ell}:Q\rightarrow{\mathbb{R}}^{k}$.
Then
(2.8) $f=\delta^{1}\lambda.$
(2.7) shows $\displaystyle[\lambda|_{Q^{\prime}}]\in
H^{1}(Q;{\mathbb{R}}^{k})$. Viewed ${\mathbb{R}}^{k}\leq
Map(W,{\mathbb{R}}^{k})$ as constant maps, $[\lambda|_{Q^{\prime}}]\in
H^{1}(Q;Map(W,{\mathbb{R}}^{k}))=0$ by (2.4). So there is an element $h\in
Map(W,{\mathbb{R}}^{k})$ such that $\lambda|_{Q^{\prime}}=\delta^{0}h$. The
equality $\lambda(\alpha^{\prime})=\delta^{0}h(\alpha^{\prime})(w)$
$({}^{\forall}\,\alpha^{\prime}\in Q^{\prime},{}^{\forall}\,w\in W)$ implies
(2.9) $h(w)=h(\alpha^{\prime}w)+\lambda(\alpha^{\prime}).$
Geometric part. Noting Proposition 2.1, the $\pi$-action (2.5) on $\tilde{M}$
is equivalent with
(2.10) $(n,\alpha)(x,w)=(n+x+\lambda(\alpha),\alpha w)\ \
({}^{\forall}\,(x,w)\in{\mathbb{R}}^{k}\times W).$
Recall that $\pi$ has the splitting subgroup
$\pi^{\prime}={\mathbb{Z}}^{k}\times Q^{\prime}$. Obviously we have the
product action of ${\mathbb{Z}}^{k}\times Q^{\prime}$ on
${\mathbb{R}}^{k}\times W$ such that ${\mathbb{R}}^{k}\times
W/{\mathbb{Z}}^{k}\times Q^{\prime}=T^{k}\times W/Q^{\prime}$. Define a
diffeomorphism $\tilde{G}:{\mathbb{R}}^{k}\times
W\rightarrow{\mathbb{R}}^{k}\times W$ to be
$\displaystyle\tilde{G}(x,w)=(x+h(w),w)$. Using (2.9), it is easy to check
that $\tilde{G}:(\pi^{\prime},{\mathbb{R}}^{k}\times
W)\rightarrow({\mathbb{Z}}^{k}\times Q^{\prime},{\mathbb{R}}^{k}\times W)$ is
an equivariant diffeomorphism with respect to the action (2.10) and the
product action. Putting $\displaystyle{\mathbb{R}}^{k}\times
W/\pi^{\prime}=T^{k}\mathop{\times}_{Q^{\prime}}W$ as a quotient space,
$\tilde{G}$ induces a diffeomorphism $\displaystyle
G:T^{k}\mathop{\times}_{Q^{\prime}}W\rightarrow T^{k}\times W/Q^{\prime}$. Let
$\displaystyle q:T^{k}\times W\rightarrow T^{k}\mathop{\times}_{Q^{\prime}}W$
be the covering map ($q(t,w)=[t,w]$). Then
(2.11) $G\circ q(t,w)=G([t,w])=(t\exp 2\pi\mathbf{i}h(w),[w]).$
Noting (2.10), $\pi$ induces an action of $Q$ on
$\tilde{M}/{\mathbb{Z}}^{k}=T^{k}\times W$ such that
(2.12) $\alpha(t,w)=(t\exp 2\pi\mathbf{i}\lambda(\alpha),\alpha w)\,\
({}^{\forall}\,\alpha\in Q).$
$F=Q/Q^{\prime}$ has an induced action on $\displaystyle
T^{k}\mathop{\times}_{Q^{\prime}}W$ by $\displaystyle\hat{\alpha}[t,w]=[t\exp
2\pi\mathbf{i}\lambda(\alpha),\alpha w]$ $({}^{\forall}\,\hat{\alpha}\in F)$
which gives rise to a covering map:
(2.13) $F\rightarrow
T^{k}\mathop{\times}_{Q^{\prime}}W\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}T^{k}\mathop{\times}_{Q}W=M.$
For any $\alpha\in Q$, consider the commutative diagram:
(2.14) $\begin{CD}H_{j}(T^{k}\times W)@>{\alpha_{*}}>{}>H_{j}(T^{k}\times
W)\\\ @V{}V{q_{*}}V@V{}V{q_{*}}V\\\
H_{j}(T^{k}\mathop{\times}_{Q^{\prime}}W)@>{\hat{\alpha}_{*}}>{}>H_{j}(T^{k}\mathop{\times}_{Q^{\prime}}W)\end{CD}$
in which $H_{j}(T^{k})\otimes H_{0}(W)\leq H_{j}(T^{k}\times W)$. By the
formula (2.12), the $Q$-action on the $T^{k}$-summand is a translation by
$\exp 2\pi\mathbf{i}\lambda(\alpha)\in T^{k}$ so the homology action
$\alpha_{*}$ on $H_{j}(T^{k})\otimes H_{0}(W)$ is trivial. If $\displaystyle
H_{j}(T^{k}\mathop{\times}_{Q^{\prime}}W)^{F}$ denotes the subgroup left fixed
under the homology action for every element $\hat{\alpha}\in F$, it follows
(2.15) $q_{*}(H_{j}(T^{k})\otimes H_{0}(W))\leq
H_{j}(T^{k}\mathop{\times}_{Q^{\prime}}W)^{F}.$
Using the transfer homomorphism ([2, 2.4 Theorem, III], [3]), $\nu$ of(2.13)
induces an isomorphism:
$\displaystyle\nu_{*}:H_{j}(T^{k}\mathop{\times}_{Q^{\prime}}W;{\mathbb{Q}})^{F}\longrightarrow
H_{j}(M;{\mathbb{Q}})$. In particular,
$\displaystyle\nu_{*}:q_{*}(H_{j}(T^{k};{\mathbb{Q}})\otimes
H_{0}(W;{\mathbb{Q}}))\rightarrow H_{j}(M;{\mathbb{Q}})$ is injective.
On the other hand, let $\displaystyle q^{\prime}:W\rightarrow W/Q^{\prime}$ be
the projection $q^{\prime}(w)=[w]$. Define a homotopy
$\Psi_{\theta}:T^{k}\times W\rightarrow T^{k}\times W/Q^{\prime}$
$(\theta\in[0,1])$ to be
$\Psi_{\theta}(t,w)=(t\exp 2\pi\mathbf{i}(\theta\cdot h(w)),[w]).$
Then $\Psi_{0}={\rm id}\times q^{\prime}\simeq G\circ q$ from (2.11). As
$G_{*}\circ q_{*}={\rm id}\times
q^{\prime}_{*}:H_{j}(T^{k};{\mathbb{Q}})\otimes
H_{0}(W;{\mathbb{Q}})\rightarrow H_{j}(T^{k};{\mathbb{Q}})\otimes
H_{0}(W/Q^{\prime};{\mathbb{Q}})$ is obviously isomorphic, it implies that
$\displaystyle q_{*}:H_{j}(T^{k};{\mathbb{Q}})\otimes
H_{0}(W;{\mathbb{Q}})\longrightarrow
H_{j}(T^{k}\mathop{\times}_{Q^{\prime}}W;{\mathbb{Q}})$ is injective. If
$\displaystyle p=\nu\circ q:T^{k}\times W\rightarrow M$ is the projection,
then $\displaystyle p_{*}:H_{j}(T^{k};{\mathbb{Q}})\otimes
H_{0}(W;{\mathbb{Q}})\longrightarrow H_{j}(M;{\mathbb{Q}})$ is injective.
As $p:T^{k}\times W\rightarrow M$ is $T^{k}$-equivariant, letting $p(1,w)=x\in
M$, it follows $p(t,w)=tx={\rm ev}(t)$ $({}^{\forall}\,t\in T^{k})$. Define an
embedding $\tilde{\rm ev}:T^{k}\rightarrow T^{k}\times W$ to be $\tilde{\rm
ev}(t)=(t,w)$. Obviously $\tilde{\rm
ev}_{*}:H_{j}(T^{k};{\mathbb{Q}})\rightarrow H_{j}(T^{k};{\mathbb{Q}})\otimes
H_{0}(W;{\mathbb{Q}})$ is an isomorphism. Since $p\circ\tilde{\rm ev}(t)={\rm
ev}(t)$, chasing a commutative diagram:
(2.16) $\begin{CD}H_{j}(T^{k};{\mathbb{Q}})@>{\tilde{\rm
ev}_{*}}>{}>H_{j}(T^{k};{\mathbb{Q}})\otimes H_{0}(W;{\mathbb{Q}})\\\ {\rm
ev}_{*}\searrow\swarrow p_{*}\ \ \\\ &\ \ H_{j}(M;{\mathbb{Q}})&\\\ \end{CD}$
$\displaystyle{\rm ev}_{*}:H_{j}(T^{k};{\mathbb{Q}})\rightarrow
H_{j}(M;{\mathbb{Q}})$ is injective. As $H_{j}(T^{k};{\mathbb{Z}})$ has no
torsion, ${\rm ev}_{*}:H_{j}(T^{k};{\mathbb{Z}})\rightarrow
H_{j}(M;{\mathbb{Z}})$ turns out to be injective. This proves (i)
$\displaystyle{}_{k}C_{j}\leq b_{j}$. In particular, ${\rm
ev}_{*}:{\mathbb{Z}}^{k}\rightarrow H_{1}(M;{\mathbb{Z}})$ is injective for
$j=1$, i.e. the $T^{k}$-action is homologically injective by the definition.
This shows (ii). ∎
Any homologically injective action is obviously injective. Theorem A is
obtained from the following corollary.
###### Corollary 2.4.
If $T^{k}$ is a homologically injective action on a closed manifold $M$, then
$\displaystyle{}_{k}C_{j}\leq b_{j}$. Thus the Halperin-Carlsson conjecture is
true.
###### Proof.
The proof is essentially the same as [8, 2.2. Lemma]. Let $\displaystyle
1\rightarrow{\mathbb{Z}}^{k}\rightarrow\pi\longrightarrow Q\rightarrow 1$ be
the central group extension. As ${\rm
ev}_{*}:H_{1}(T^{k};{\mathbb{Z}})={\mathbb{Z}}^{k}\rightarrow
H_{1}(M;{\mathbb{Z}})={\mathbb{Z}}^{\ell}\oplus F$ is injective, ${\rm
ev}_{*}({\mathbb{Z}}^{k})\leq{\mathbb{Z}}^{k}$ such that ${\rm
ev}_{*}({\mathbb{Z}}^{k})\oplus{\mathbb{Z}}^{\ell-k}\leq{\mathbb{Z}}^{\ell}$.
If $q:\pi\rightarrow H_{1}(M;{\mathbb{Z}})$ is a canonical projection, then
$\pi^{\prime}=q^{-1}({\rm
ev}_{*}({\mathbb{Z}}^{k})\oplus{\mathbb{Z}}^{\ell-k}\oplus F)$ is a finite
index normal splitting subgroup of $\pi$. ∎
###### Remark 2.5.
As a consequence of this corollary and ${\rm(ii)}$ of Theorem 2.3, _injective-
splitting_ action is equivalent with _homologically injective_ action.
Let $(M,g)$ be a $2n$-dimensional Kähler manifold with Kähler form $\Omega$.
###### Corollary 2.6 ([5]).
Every _holomorphic_ action of a complex torus $T^{k}_{\mathbb{C}}$ on a
compact Kähler manifold manifold $(M,\Omega)$ is homologically injective. Thus
the Halperin-Carlsson conjecture holds.
###### Proof.
Averaging the Kähler metric by $T^{k}_{\mathbb{C}}$, we may assume that
$T^{k}_{\mathbb{C}}$ acts as Kähler isometries on $M$. $T^{k}_{\mathbb{C}}$
induces the Killing vector fields $\xi_{i},J\xi_{i}$ on $M$ $(i=1,\dots,k)$.
Note that each $\xi_{i}$ is a non-vanishing vector field by the maximum
principle. In fact, if $G$ is the connected component of the stabilizer of
$T^{k}_{\mathbb{C}}$ at $x\in M$, then the action induces a holomorphic
representation $\rho:G\rightarrow{\rm GL}(n,{\mathbb{C}})$. As $G$ is compact,
$\rho$ is trivial so that $G=\\{1\\}$.
Put $\theta_{i}=\iota_{\xi_{i}}\Omega$ and
$\theta_{i+k}=\iota_{J\xi_{i}}\Omega$ $(i=1,\dots,k)$. By the Cartan formula,
$d\Omega=0$ implies $d\theta_{i}=0$ $(i=1,\dots,2k)$. We have $2k$-number of
$1$-cocycles $[\theta_{i}]\in H^{1}(M;{\mathbb{R}})$. As ${\rm ev}(t)=t\cdot
x\in M$, it follows ${\rm ev}_{*}((\xi_{i})_{1})=(\xi_{i})_{x}$. Since ${\rm
ev}^{*}\theta_{i}((\xi_{k+i})_{1})=\Omega((\xi_{i})_{x},(J\xi_{i})_{x})=g(\xi_{i},\xi_{i})>0$
on $T^{k}_{\mathbb{C}}$, ${\rm ev}^{*}:H^{1}(M;{\mathbb{R}})\rightarrow
H^{1}(T^{k}_{\mathbb{C}};{\mathbb{R}})$ is surjective. So ${\rm
ev}_{*}:H_{1}(T^{k}_{\mathbb{C}};{\mathbb{Z}})\rightarrow
H_{1}(M;{\mathbb{Z}})$ is injective. ∎
For example, any _holomorphic_ action of $T^{k}_{\mathbb{C}}$ on a compact
complex euclidean space form is homologically injective. Recall that any
effective $T^{s}$-action on a closed aspherical manifold is injective. We
prove Corollary B.
###### Theorem 2.7.
Any effective $T^{s}$-action on a compact euclidean space form $M$ is
homologically injective. Thus the Halperin-Carlsson conjecture is true.
###### Proof.
Given a $T^{s}$-action for some $s\geq 1$, there is a central group extension:
$\displaystyle
1\rightarrow{\mathbb{Z}}^{s}\rightarrow\pi=\pi_{1}(M)\longrightarrow
Q\rightarrow 1$. Since $\pi$ has a unique maximal normal finite index abelian
subgroup ${\mathbb{Z}}^{n}$ (cf. [20]), it follows
${\mathbb{Z}}^{s}\leq{\mathbb{Z}}^{n}$. $Q$ has a finite index subgroup
$Q^{\prime}={\mathbb{Z}}^{n}/{\mathbb{Z}}^{s}\cong G\times{\mathbb{Z}}^{n-s}$
where $G$ is a finite abelian group. The inclusion
$\iota:{\mathbb{Z}}^{n-s}\rightarrow Q^{\prime}$ induces a group (extension)
${\mathsf{A}}$ of finite index in ${\mathbb{Z}}^{n}$. As ${\mathsf{A}}$ is
isomorphic to ${\mathbb{Z}}^{s}\times{\mathbb{Z}}^{n-s}$, ${\mathsf{A}}$ is a
finite index normal splitting subgroup of $\pi$. The $T^{s}$-action is
injective-splitting so apply Theorem 2.3. ∎
## 3\. Calabi construction and torus actions
In [8, § 7], Conner and Raymond have stated that the Calabi’s theorem [4]
shows the existence of a $T^{k}$-action with $k={\rm
rank}\,H_{1}(M;{\mathbb{Z}})>0$. We agree the existence of such actions in
view of the Calabi construction. However when we look at a proof of the
Calabi’s theorem ([20, p.125]), it is not easy to find such $T^{k}$-actions.
In fact, let $\nu:\pi\rightarrow{\mathbb{Z}}^{k}$ be the projection onto the
direct summand ${\mathbb{Z}}^{k}$ of $H_{1}(M;{\mathbb{Z}})$. Then there is a
group extension $\displaystyle
1\rightarrow\Gamma\rightarrow\pi\rightarrow{\mathbb{Z}}^{k}\rightarrow 1$ in
which $\Gamma$ is the fundamental group of a euclidean space form
$M^{n-k}={\mathbb{R}}^{n-k}/\Gamma$. In general an element $\gamma\in\pi$ has
the form
$\left(\left[\begin{array}[]{c}a\\\ b\\\
\end{array}\right],\left(\begin{array}[]{lr}A&B\\\
0&I\end{array}\right)\right)\ \
(a\in{\mathbb{R}}^{n-k},b\in{\mathbb{R}}^{k}).$
The holonomy group $\displaystyle L(\pi)=\\{\left(\begin{array}[]{lr}A&B\\\
0&I\end{array}\right)\\}$ does not necessarily leave the subspace
$0\times{\mathbb{R}}^{k}$ invariant. (In particular,
${\mathbb{Z}}^{n}\cap(0\times{\mathbb{R}}^{k})$ is not necessarily uniform in
$0\times{\mathbb{R}}^{k}$.) So we have to find another decomposition to get a
$T^{k}$-action on $M$.
###### Lemma 3.1.
Let $\pi$ be a Bieberbach group such that ${\rm rank}\,\pi/[\pi,\pi]=k>0$.
Then there exists a faithful representation $\rho:\pi\rightarrow{\rm E}(n)$
such that the euclidean space form ${\mathbb{R}}^{n}/\rho(\pi)$ admits an
effective $T^{k}$-action.
###### Proof.
By the hypothesis, there is a group extension $\displaystyle
1\rightarrow\Gamma\rightarrow\pi\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}{\mathbb{Z}}^{k}\rightarrow
1$. Since $\pi$ is a Bieberbach group, it admits a maximal normal finite index
abelian subgroup ${\mathbb{Z}}^{n}$. Put $\nu({\mathbb{Z}}^{n})=A$. Consider
the commutative diagram of the group extensions:
(3.1) $\begin{CD}1@>{}>{}>\Gamma @>{\iota}>{}>\pi
@>{\nu}>{}>{\mathbb{Z}}^{k}@>{}>{}>1\\\ @A{}A{}A@A{}A{}A@A{}A{}A\\\
1@>{}>{}>\Gamma\cap{\mathbb{Z}}^{n}@>{\iota}>{}>{\mathbb{Z}}^{n}@>{\nu}>{}>A@>{}>{}>1.\\\
\end{CD}$
Since
$\pi/{\mathbb{Z}}^{n}\stackrel{{\scriptstyle\hat{\nu}}}{{\rightarrow}}{\mathbb{Z}}^{k}/A$
is surjective, $A$ is a free abelian subgroup of rank $k$. By the embedding
$\displaystyle\hat{\iota}:\Gamma/\Gamma\cap{\mathbb{Z}}^{n}\leq\pi/{\mathbb{Z}}^{n}$,
$\Gamma\cap{\mathbb{Z}}^{n}$ is a finite index subgroup of $\Gamma$. It
follows easily that $\Gamma\cap{\mathbb{Z}}^{n}$ is a maximal normal abelian
subgroup of $\Gamma$. We may put
$\displaystyle\Gamma\cap{\mathbb{Z}}^{n}={\mathbb{Z}}^{n-k}$ so that
${\mathbb{Z}}^{n}={\mathbb{Z}}^{n-k}\times A$. Putting
$Q=\pi/{\mathbb{Z}}^{n-k}$ and $F=\Gamma/{\mathbb{Z}}^{n-k}$ is a finite
group, we have the group extensions:
(3.2) $\begin{CD}1@>{}>{}>{\mathbb{Z}}^{n-k}@>{i}>{}>\pi
@>{\mu}>{}>Q@>{}>{}>1,\\\ \end{CD}$
where
(3.3)
$\begin{CD}1@>{}>{}>F@>{\hat{\iota}}>{}>Q@>{\hat{\nu}}>{}>{\mathbb{Z}}^{k}@>{}>{}>1\end{CD}$
is also a group extension. Consider the commutative diagram of group
extensions:
(3.4) $\begin{CD}1@>{}>{}>{\mathbb{Z}}^{n-k}@>{\iota}>{}>\pi
@>{\mu}>{}>Q@>{}>{}>1\\\ ||@A{}A{}A@A{}A{\iota^{\prime}}A\\\
1@>{}>{}>{\mathbb{Z}}^{n-k}@>{\iota}>{}>{\mathbb{Z}}^{n}@>{\mu}>{}>B@>{}>{}>1\\\
\end{CD}$
where we put $B=\mu({\mathbb{Z}}^{n})$. As
$\hat{\nu}(B)=\nu({\mathbb{Z}}^{n})=A$ from (3.2), it follows
${\mathbb{Z}}^{n}={\mathbb{Z}}^{n-k}\times B$. Thus ${\mathbb{Z}}^{n}$ is a
splitting subgroup of $\pi$. Since (3.2) is not necessarily central, let
$\phi:Q\rightarrow{\rm Aut}({\mathbb{Z}}^{n-k})$ be the conjugation
homomorphism. If $[f]\in H^{2}_{\phi}(Q;{\mathbb{Z}}^{k})$ is the
representative cocycle of (3.2), then $\iota^{\prime*}[f]=0$ in
$H^{2}(B;{\mathbb{Z}}^{n-k})$. Then $[f]$ is a torsion because
$\tau\circ\iota^{\prime*}=|Q/B|:H^{2}_{\phi}(Q;{\mathbb{Z}}^{k})\rightarrow
H^{2}_{\phi}(Q;{\mathbb{Z}}^{k})$ still holds for the transfer homomorphism
$\tau:H^{2}(B;{\mathbb{Z}}^{n-k})\rightarrow
H^{2}_{\phi}(Q;{\mathbb{Z}}^{n-k})$. (Compare [3].) Similarly as in the proof
of Theorem 2.3 there is a function $\lambda:Q\rightarrow{\mathbb{R}}^{n-k}$
such that $f=\delta^{1}\lambda$. Note from (2.8) that
(3.5) $\ell\cdot\lambda(Q)\leq{\mathbb{Z}}^{n-k}.$
Let ${\mathbb{Z}}^{k}$ act on ${\mathbb{R}}^{k}$ by translations and by (3.3)
$\hat{\nu}:Q\rightarrow{\mathbb{Z}}^{k}\leq{\rm E}(k)$ defines a properly
discontinuous action of $Q$ on ${\mathbb{R}}^{k}$;
$\alpha(w)=\hat{\nu}(\alpha)+w\ \,({}^{\forall}\,\alpha\in
Q,{}^{\forall}\,w\in{\mathbb{R}}^{k}).$
As in (2.5) we have a properly discontinuous action of $\pi$ on
${\mathbb{R}}^{n}={\mathbb{R}}^{n-k}\times{\mathbb{R}}^{k}$:
$\begin{split}(n,\alpha)\left[\begin{array}[]{c}x\\\
w\end{array}\right]&=\left[\begin{array}[]{c}n+\bar{\phi}(\alpha)(x)+\lambda(\alpha)\\\
\hat{\nu}(\alpha)+w\end{array}\right]\\\
&=\left(\left[\begin{array}[]{c}n+\lambda(\alpha)\\\
\hat{\nu}(\alpha)\end{array}\right],\left(\begin{array}[]{lr}\bar{\phi}(\alpha)&\\\
&I_{k}\end{array}\right)\right)\left[\begin{array}[]{c}x\\\
w\end{array}\right].\end{split}$
Since $\phi|_{B}={\rm id}$ from (3.4), the image $\phi(Q)$ is finite in ${\rm
Aut}({\mathbb{Z}}^{n-k})$ which implies $\phi(Q)\leq{\rm O}(n-k)$ up to
conjugate. As $\pi$ is torsionfree and acts properly discontinuously, we
obtain a faithful homomorphism $\rho:\pi\rightarrow\rho(\pi)\leq{\rm E}(n)$
defined by
(3.6) $\rho(n,\alpha)=\left(\left[\begin{array}[]{c}n+\lambda(\alpha)\\\
\hat{\nu}(\alpha)\end{array}\right],\left(\begin{array}[]{lr}\bar{\phi}(\alpha)&\\\
&I_{k}\end{array}\right)\right).$
Therefore we have a compact euclidean space form ${\mathbb{R}}^{n}/\rho(\pi)$.
We prove that ${\mathbb{R}}^{n}/\rho(\pi)$ admits a $T^{k}$-action. Noting
(3.5), we define a subgroup of ${\mathbb{Z}}^{n}$ by
(3.7)
$\tilde{B}=\\{(-\ell\cdot\lambda(\beta),\ell\cdot\beta)\in{\mathbb{Z}}^{n}\,|\,\beta\in
B\\}.$
It is isomorphic to $B\cong{\mathbb{Z}}^{k}$. As $\phi|_{B}={\rm id}$,
(3.8)
$\rho(-\ell\cdot\lambda(\beta),\ell\cdot\beta)=\left(\left[\begin{array}[]{c}0\\\
\ell\cdot\hat{\nu}(\beta)\end{array}\right],I_{n}\right)\in
0\times{\mathbb{R}}^{k}.$
Thus $\rho(\tilde{B})$ is a translation subgroup with rank $k$:
(3.9) $\rho(\tilde{B})\leq(0\times{\mathbb{R}}^{k})\cap\rho(\pi).$
Since $(0\times{\mathbb{R}}^{k})/\rho(\tilde{B})$ is compact, so is
$(0\times{\mathbb{R}}^{k})/(0\times{\mathbb{R}}^{k})\cap\rho(\pi)$. We may put
$\displaystyle(0\times{\mathbb{R}}^{k})/(0\times{\mathbb{R}}^{k})\cap\rho(\pi)=T^{k}$.
Moreover, from (3.6) a calculation shows that
(3.10) $\rho(n,\alpha)\cdot\left(\left[\begin{array}[]{c}0\\\
y\end{array}\right],I_{n}\right)\\\ =\left(\left[\begin{array}[]{c}0\\\
y\end{array}\right],I_{n}\right)\cdot\rho(n,\alpha),$
i.e. each $y\in{\mathbb{R}}^{k}$ centralizes $\rho(\pi)$;
(3.11) $0\times{\mathbb{R}}^{k}\leq C_{{\rm E}(n)}(\rho(\pi)).$
Let ${\rm Isom}({\mathbb{R}}^{k}/\rho(\pi))^{0}$ denote the identity component
of euclidean isometries of ${\mathbb{R}}^{k}/\rho(\pi)$. From (3.11) we have
the following covering groups (cf. (2.2)):
$\begin{CD}1@>{}>{}>C(\rho(\pi))@>{}>{}>C_{{\rm E}(n)}(\rho(\pi))@>{}>{}>{\rm
Isom}({\mathbb{R}}^{k}/\rho(\pi))^{0}\\\ @A{}A{}A@A{}A{}A@A{}A{}A\\\
1@>{}>{}>(0\times{\mathbb{R}}^{k})\cap\rho(\pi)@>{}>{}>(0\times{\mathbb{R}}^{k})@>{}>{}>T^{k}\\\
\end{CD}$
Hence ${\mathbb{R}}^{k}/\rho(\pi)$ admits a $T^{k}$-action. ∎
When the center $C(\pi)$ of $\pi=\pi_{1}(M)$ for a closed manifold $M$ is
finitely generated, ${\rm rank}\,C(\pi)$ denotes the rank of a free abelian
subgroup. Recall from [17] that an effective $T^{s}$-action on $M$ is said to
be _maximal_ if $s={\rm rank}\,C(\pi)$. If $M$ is a closed aspherical manifold
admitting an effective $T^{s}$-action, then $s\leq{\rm rank}\,C(\pi)$ (cf.
[7]). Let $M$ be a euclidean space form $M={\mathbb{R}}^{n}/\pi$. If $C(\pi)$
has rank $k$, then it is easy to see that $M$ admits a $T^{k}$-action. In
particular, ${\rm rank}\,C(\pi)\leq{\rm rank}\,H_{1}(M;{\mathbb{Z}})$ by
Theorem 2.3.
The Bieberbach theorem implies that $M$ is affinely diffeomorphic to
${\mathbb{R}}^{n}/\rho(\pi)$. Combined Lemma 3.1 with Theorem 2.7, we obtain
###### Theorem 3.2.
Let $M$ be a compact $n$-dimensional euclidean space form with ${\rm
rank}\,H_{1}(M;{\mathbb{Z}})=k>0$. Then $M$ admits a homologically injective
$T^{k}$-action. In particular, ${\rm rank}\,C(\pi)={\rm
rank}\,H_{1}(M;{\mathbb{Z}})$.
Acknowledgement. We thank Professor M. Masuda who called our attention to the
Halperin-Carlsson conjecture.
## References
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|
arxiv-papers
| 2012-06-21T07:09:48 |
2024-09-04T02:49:32.020138
|
{
"license": "Public Domain",
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"submitter": "Mayumi Nakayama",
"url": "https://arxiv.org/abs/1206.4790"
}
|
1206.4876
|
¡html¿¡head¿ ¡meta http-equiv=”content-type” content=”text/html;
charset=ISO-8859-1”¿
¡title¿CERN-2012-001¡/title¿
¡/head¿
¡body¿
¡h1¿40th Anniversary of the First Proton-Proton Collisions in the CERN
Intersecting Storage Rings (ISR)¡/h1¿
¡h2¿CERN, Geneva, Switzerland, 18 Jan 2011¡/h2¿
¡h2¿CERN Yellow Report ¡a
href=”https://cdsweb.cern.ch/record/1456765”¿CERN-2012-004¡/a¿¡/h2¿
¡h3¿authors: U. Amaldi, P. J. Bryant, P. Darriulat and K. Hubner¡/h3¿
¡h2¿Lectures¡/h2¿
¡p¿ LIST:arXiv:1206.3948¡br¿
LIST:arXiv:1206.3950¡br¿
LIST:arXiv:1206.3954¡br¿
LIST:arXiv:1206.4131¡/p¿ ¡/body¿¡/html¿
|
arxiv-papers
| 2012-06-21T13:42:05 |
2024-09-04T02:49:32.028420
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "U. Amaldi, P.J. Bryant, P. Darriulat and K. Hubner",
"submitter": "Scientific Information Service CERN",
"url": "https://arxiv.org/abs/1206.4876"
}
|
1206.4992
|
Fermilab Lattice and MILC Collaborations
# Refining new-physics searches in $B\to D\tau\nu$ decay with lattice QCD
Jon A. Bailey Department of Physics and Astronomy, Seoul National University,
Seoul, South Korea A. Bazavov Physics Department, Brookhaven National
Laboratory, Upton, NY, USA C. Bernard Department of Physics, Washington
University, St. Louis, Missouri, USA C.M. Bouchard Department of Physics,
The Ohio State University, Columbus, Ohio, USA C. DeTar Physics Department,
University of Utah, Salt Lake City, Utah, USA Daping Du ddu@illinois.edu
Physics Department, University of Illinois, Urbana, Illinois, USA A.X. El-
Khadra Physics Department, University of Illinois, Urbana, Illinois, USA J.
Foley Physics Department, University of Utah, Salt Lake City, Utah, USA E.D.
Freeland Department of Physics, Benedictine University, Lisle, Illinois, USA
E. Gámiz CAFPE and Departamento de Fìsica Teórica y del Cosmos, Universidad
de Granada, Granada, Spain Steven Gottlieb Department of Physics, Indiana
University, Bloomington, Indiana, USA U.M. Heller American Physical Society,
Ridge, New York, USA Jongjeong Kim Department of Physics, University of
Arizona, Tucson, Arizona, USA A.S. Kronfeld Fermi National Accelerator
Laboratory, Batavia, Illinois, USA J. Laiho SUPA, School of Physics and
Astronomy, University of Glasgow, Glasgow, UK L. Levkova Physics Department,
University of Utah, Salt Lake City, Utah, USA P.B. Mackenzie Fermi National
Accelerator Laboratory, Batavia, Illinois, USA Y. Meurice Department of
Physics and Astronomy, University of Iowa, Iowa City, Iowa, USA E.T. Neil
Fermi National Accelerator Laboratory, Batavia, Illinois, USA M.B. Oktay
Physics Department, University of Utah, Salt Lake City, Utah, USA Si-Wei Qiu
Physics Department, University of Utah, Salt Lake City, Utah, USA J.N. Simone
Fermi National Accelerator Laboratory, Batavia, Illinois, USA R. Sugar
Department of Physics, University of California, Santa Barbara, California,
USA D. Toussaint Department of Physics, University of Arizona, Tucson,
Arizona, USA R.S. Van de Water ruthv@bnl.gov Physics Department, Brookhaven
National Laboratory, Upton, NY, USA Ran Zhou Department of Physics, Indiana
University, Bloomington, Indiana, USA
###### Abstract
The semileptonic decay channel $B\to D\tau\nu$ is sensitive to the presence of
a scalar current, such as that mediated by a charged-Higgs boson. Recently the
BaBar experiment reported the first observation of the exclusive semileptonic
decay $B\to D\tau^{-}\overline{\nu}$, finding an approximately 2$\sigma$
disagreement with the Standard-Model prediction for the ratio
$R(D)=\text{BR}(B\to D\tau\nu)/\text{BR}(B\to D\ell\nu)$, where $\ell=e,\mu$.
We compute this ratio of branching fractions using hadronic form factors
computed in unquenched lattice QCD and obtain $R(D)=0.316(12)(7)$, where the
errors are statistical and total systematic, respectively. This result is the
first Standard-Model calculation of $R(D)$ from ab initio full QCD. Its error
is smaller than that of previous estimates, primarily due to the reduced
uncertainty in the scalar form factor $f_{0}(q^{2})$. Our determination of
$R(D)$ is approximately 1$\sigma$ higher than previous estimates and, thus,
reduces the tension with experiment. We also compute $R(D)$ in models with
electrically charged scalar exchange, such as the type II two-Higgs doublet
model. Once again, our result is consistent with, but approximately 1$\sigma$
higher than, previous estimates for phenomenologically relevant values of the
scalar coupling in the type II model. As a byproduct of our calculation, we
also present the Standard-Model prediction for the longitudinal polarization
ratio $P_{L}(D)=0.325(4)(3)$.
###### pacs:
13.20.He, 12.38.Gc
††preprint: FERMILAB-PUB-12/333-T
Motivation. – The third generation of quarks and leptons may be particularly
sensitive to new physics associated with electroweak symmetry breaking due to
their larger masses. For example, in the minimal supersymmetric extension of
the Standard Model, charged-Higgs contributions to tauonic $B$ decays can be
enhanced if $\tan\beta$ is large. Thus the semileptonic decay $B\to D\tau\nu$
is a promising new-physics search channel Grzadkowski:1992qj ; Tanaka:1994ay ;
Garisto:1994vz ; Tsai:1996ps ; Wu:1997uaa ; Kiers:1997zt ; Miki:2002nz ;
Itoh:2004ye ; Chen:2006nua ; Nierste:2008qe ; Kamenik:2008tj ; Tanaka:2010se ;
Fajfer:2012vx .
The BaBar experiment recently measured the ratios $R(D^{(*)})=\text{BR}(B\to
D^{(*)}\tau\nu)/\text{BR}(B\to D^{(*)}\ell\nu)$, where $\ell=e,\mu$, and
reported excesses in both channels which, when combined, disagree with the
Standard Model by $3.4\sigma$ :2012xj . BaBar also interpreted these
measurements in terms of the type-II two-Higgs-doublet model (2HDM II) and
claimed to exclude the theory at 99.8% confidence-level. In this work, we
update the prediction for $R(D)$ in the Standard-Model and in new-physics
theories with a scalar current (such as the 2HDM II) using unquenched lattice-
QCD calculations of the $B\to D\ell\nu$ form factors $f_{0}(q^{2})$ and
$f_{+}(q^{2})$ by the Fermilab Lattice and MILC collaborations Bailey:2012rr .
This is the first determination of $R(D)$ from ab initio full QCD.
With lepton helicity defined in the rest frame of the virtual $W$ boson, the
general expressions for the differential rates for semileptonic $B\to
D\ell\nu$ decay are given by
$\displaystyle\frac{d\Gamma_{-}}{dq^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{24\pi^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}|\bm{p}_{D}|^{3}\left|{G}_{V}^{\ell
cb}f_{+}(q^{2})-\frac{m_{\ell}}{M_{B}}G_{T}^{\ell
cb}f_{2}(q^{2})\right|^{2}\,,$ (1) $\displaystyle\frac{d\Gamma_{+}}{dq^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi^{3}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}\frac{|\bm{p}_{D}|}{q^{2}}\left\\{\frac{1}{3}|\bm{p}_{D}|^{2}\left|m_{\ell}{G}_{V}^{\ell
cb}f_{+}(q^{2})-\frac{q^{2}}{M_{B}}G_{T}^{\ell
cb}f_{2}(q^{2})\right|^{2}\right.$ $\displaystyle+$
$\displaystyle\left.\frac{\left(M_{B}^{2}-M_{D}^{2}\right)^{2}}{4M_{B}^{2}}\left|\left(m_{\ell}{G}_{V}^{\ell
cb}-\frac{q^{2}}{m_{b}-m_{c}}G_{S}^{\ell
cb}\right)f_{0}(q^{2})\right|^{2}\right\\}\,,$
where the subscript denotes helicity, and $q=(p_{\ell}+p_{\nu})$ is the
momentum carried by the charged lepton-neutrino pair. The total semileptonic
width is the sum of the partial widths,
$\Gamma_{\text{tot}}=(\Gamma_{+}+\Gamma_{-})$. At tree-level of the Standard
Model electroweak interaction, the scalar- and tensor-exchange couplings are
$G_{S}=G_{T}=0$, while the vector coupling is ${G}_{V}^{\ell ij}=G_{F}V_{ij}$.
In the infinite heavy-quark-mass limit, the form factors $f_{+}(q^{2})$ and
$f_{2}(q^{2})$ are related via
$f_{2}(q^{2})=-f_{+}(q^{2})-\frac{M_{B}^{2}-M_{D}^{2}}{q^{2}}\left[f_{+}(q^{2})-f_{0}(q^{2})\right]\,.$
(3)
The form factor $f_{2}(q^{2})$ is only relevant for theories with tensor
currents, however, which we do not consider here.
Because the Standard-Model positive-helicity contribution to semileptonic
$B\to D$ decay is proportional to the lepton mass-squared, it can be neglected
for the light leptons $\ell=e,\mu$; thus experimental measurements of $B\to
D\ell\nu$ decays are sensitive only to the vector form factor $f_{+}(q^{2})$.
On the other hand, the differential rate for $B\to D\tau\nu$ is sensitive also
to the positive-helicity contribution and, hence, $f_{0}(q^{2})$. Existing
Standard-Model estimates of $d\Gamma_{-}/dq^{2}$ for $B\to D\tau\nu$ have
relied on the kinematic constraint $f_{0}(0)=f_{+}(0)$, dispersive bounds on
the shape Caprini:1997mu , relations from heavy-quark symmetry, and quenched
lattice QCD (neglecting $u$, $d$, and $s$ quark loops) deDivitiis:2007ui ;
deDivitiis:2007uk . See Refs. Kamenik:2008tj ; Nierste:2008qe ; Tanaka:2010se
; Fajfer:2012vx for details. In this letter, we replace quenched QCD and
heavy-quark estimates with a full, 2+1-flavor QCD calculation. In particular,
we determine the following ratios within the Standard Model (where
$\ell=e,\mu$):
$\displaystyle R(D)$ $\displaystyle=$ $\displaystyle\text{BR}(B\to
D\tau\nu)/\text{BR}(B\to D\ell\nu)\,,$ (4) $\displaystyle P_{L}(D)$
$\displaystyle=$ $\displaystyle\left(\Gamma_{+}^{B\to
D\tau\nu}-\Gamma_{-}^{B\to D\tau\nu}\right)/\Gamma_{\rm tot}^{B\to
D\tau\nu}\,.$ (5)
These quantities enable particularly clean tests of the Standard Model and
probes of new physics because the CKM matrix elements and many of the hadronic
uncertainties cancel between the numerator and denominator.
Lattice-QCD calculation. – Here we briefly summarize the lattice-QCD
calculation of the $B\to D\ell\nu$ semileptonic form factors $f_{+}(q^{2})$
and $f_{0}(q^{2})$ Bailey:2012rr . Our calculation is based on a subset of the
(2+1)-flavor ensembles generated by the MILC Collaboration Bazavov:2009bb . We
use two lattice spacings $a\approx 0.12$ and 0.09 fm, and two light-quark
masses at each lattice spacing with (Goldstone) pion masses in the range
315–520 MeV; the specific numerical simulation parameters are given in Table I
of Ref. Bailey:2012rr .
This relatively small data set is sufficient for ratios such as those studied
here and in Ref. Bailey:2012rr , given the mild chiral and continuum
extrapolations. We use the Fermilab action ElKhadra:1996mp for the heavy
quarks (bottom and charm) and use the asqtad-improved staggered action
Bazavov:2009bb for the light valence and sea quarks ($u,d,s$). We minimize
the systematic error due to contamination from radial excitations in 2-point
and 3-point correlation functions by employing fits including their
contributions, as described in Sec. III of Ref. Bailey:2012rr . We renormalize
the lattice vector current $\bar{c}\gamma^{\mu}b$ (and other heavy-heavy
currents) using a mostly nonperturbative method ElKhadra:2001rv in which we
determine the flavor-conserving normalizations nonperturbatively. The
remaining correction is close to unity and can be calculated in one-loop
tadpole-improved lattice perturbation theory Lepage:1992xa . When
extrapolating the lattice simulation results to the physical light-quark
masses and the continuum limit, we carefully account for the leading
nonanalytic dependence on the light-quark masses at nonzero but small momentum
transfer Chow:1993hr including the effects of lattice artifacts (generic
discretization errors and taste-symmetry breaking introduced by the staggered
action) Aubin:2005aq ; Laiho:2005ue . The chiral-continuum extrapolation
results are plotted in the left panels of Figs. 6 and 7 of Ref. Bailey:2012rr
; for details see Eqs. (4.1) and (4.2) of the same work and the surrounding
text.
Figure 1: The form factors $f_{+}$ (top, red) and $f_{0}$ (bottom, blue) from
lattice QCD. The range of simulated recoil values is to the left of the
vertical line. The filled colored bands show the interpolation/extrapolation
of the numerical lattice data over the full kinematic range using the $z$
parameterization. For comparison, the experimental measurement from BaBar
Aubert:2009ac is shown as solid filled circles (using $|V_{cb}|=41.4\times
10^{-3}$ deDivitiis:2007ui ; Aubert:2009ac ).
Figure 1 shows the results for $f_{+}(q^{2})$ and $f_{0}(q^{2})$ Bailey:2012rr
. The simulated data are in the range $w<1.17$ (to the left of the dashed
vertical line), where $w=(M_{B}^{2}+M_{D}^{2}-q^{2})/(2M_{B}M_{D})$. In this
region, we parameterize the $w$ dependence of the form factors by a quadratic
expansion about $w=1$, which works well for small recoil. To extend the form-
factor results beyond the simulated recoil values (to the right of the dashed
vertical line) we reparameterize the form factors in terms of the variable $z$
Boyd:1997kz , and then extrapolate to large recoil using a model-independent
fit function based on general quantum-theory-principles of analyticity and
crossing-symmetry. The functional forms used to extrapolate $f_{+}(q^{2})$ and
$f_{0}(q^{2})$ are defined in Eqs. (5.1)–(5.6) of Ref. Bailey:2012rr . The
fits are plotted in the left panel of Fig. 9, and the results are given in the
upper panel of Table V of the same work. As seen in Fig. 1, our result for
$f_{+}(q^{2})$ agrees very well with experimental measurements Aubert:2009ac
over the full kinematic range. This nontrivial check gives confidence in the
extrapolation of $f_{0}(q^{2})$, which cannot be obtained experimentally and
for which lattice-QCD input is crucial. In particular, lattice-QCD
uncertainties are smallest near $q^{2}=(M_{B}-M_{D})^{2}$, so the discussion
below hinges principally on our calculation of $f_{0}(q^{2})$ near this point,
the validated $f_{0}(0)=f_{+}(0)$, and a smooth connection between the two
limits.
We calculate the Standard-Model $B\to D\ell\nu$ partial decay rates into the
three generations of leptons using these form factors and Eqs. (1) and
(Refining new-physics searches in $B\to D\tau\nu$ decay with lattice QCD) with
$G_{S}=G_{T}=0$, $G_{V}=G_{F}V^{*}_{cb}$. The resulting distributions are
plotted in Fig. 2. To illustrate the role of the scalar form factor
$f_{0}(q^{2})$, we also show the rates with only the contributions from
$f_{+}(q^{2})$. Due to the significant helicity suppression, the differential
decay rates into light leptons are well-approximated by a single contribution
from the form factor $f_{+}(q^{2})$. For $B\to D\tau\nu$, however, the
contribution from the scalar form factor $f_{0}(q^{2})$ comprises half of the
Standard-Model rate.
Figure 2: Differential decay rates for $B\to De\nu$ (green), $B\to D\mu\nu$
(blue), and $B\to D\tau\nu$ (red) in the Standard Model. The black dot-dashed
curves show the rates calculated with $f_{0}(q^{2})=0$.
Given the lattice-QCD determinations of $f_{+}(q^{2})$ and $f_{0}(q^{2})$ we
can obtain the Standard-Model values for $R(D)$ and $P_{L}(D)$. These are the
primary results of this letter, and we now discuss the sources of systematic
uncertainty. In Ref. Bailey:2012rr , many statistical and several systematic
errors cancelled approximately or exactly in the ratio $f_{0}^{B_{s}\to
D_{s}\ell\nu}/f_{0}^{B\to D\ell\nu}$ studied there. Some of these do not
cancel (as well) in $R(D)$ and $P_{L}(D)$, however, because they affect
$f_{+}(q^{2})$ and $f_{0}(q^{2})$ differently.
Table 1 shows the error budgets for $R(D)$ and $P_{L}(D)$. The statistical
error in $R(D)$ is significant (3.7%) due to the different phase-space
integrations in the numerator and denominator, whereas for $P_{L}(D)$ the
correlated statistical fluctuations largely cancel. For the same reason, the
errors in $R(D)$ arising from the extrapolation to the physical light-quark
masses and the continuum limit ($1.4\%$) and to the full $q^{2}$ range
($1.5\%$), are much larger than for $P_{L}(D)$. We estimate the error from the
chiral-continuum extrapolation by comparing the results for fits with and
without next-to-next-to-leading order analytic terms in the chiral expansion.
We estimate the error from the $z$ extrapolation by varying the range of
synthetic data used in the $z$ fit, including an additional pole in the fit
function, and including higher powers of $z$. The specific chiral and $z$-fit
variations considered are enumerated in Table VI of Ref. Bailey:2012rr and
discussed in detail in the surrounding text. The remaining sources of
uncertainty in Table 1 do not contribute significantly to the quantities
studied in Ref. Bailey:2012rr , so we describe them in greater detail below.
We determine the bare heavy-quark masses in our simulations by tuning the
parameters $\kappa_{b}$ and $\kappa_{c}$ in the heavy-quark action such that
the kinetic masses of the pseudoscalar $B_{s}$ and $D_{s}$ mesons match the
experimentally-measured values ElKhadra:1996mp . In practice, it is easier to
work with the form factors $h_{\pm}(w)$ on the lattice, which are linear
combinations of $f_{+,0}(q^{2})$ Bailey:2012rr . We study how the form factors
$h_{\pm}(w)$ depend on $\kappa_{b,c}$ by re-computing the form factors on some
ensembles at values of $\kappa_{b,c}$ slightly above and below the default
ones, and extracting the slopes with respect to $\kappa_{b,c}$. We use these
slopes to correct our results for $R(D)$ and $P_{L}(D)$ slightly from the
simulated $\kappa$ values to the physical ones, and conservatively take the
full size of the shift as the error due to $\kappa$-tuning.
We remove the leading taste-breaking light-quark discretization errors in the
form factors with the chiral-continuum extrapolation, and estimate the
remaining discretization errors from the heavy-quark action with power
counting Kronfeld:2000ck . We compute both the coefficient of the dimension 5
operator in the Fermilab action $c_{SW}$ and the rotation parameter $d_{1}$
for the heavy-quark fields at tree level in tadpole-improved lattice
perturbation theory ElKhadra:1996mp . Then the leading heavy-quark errors in
$h_{+}(1)$ are of $\mathrm{O}\left(\alpha_{s}(\Lambda/2m_{Q})^{2}\right)$ and
$\mathrm{O}\left((\Lambda/2m_{Q})^{3}\right)$, where $\Lambda$ is a typical
hadronic scale. Using the values $\alpha_{s}=0.3$, $\Lambda=500$ MeV, and
$m_{c}=1.2$ GeV, we estimate that heavy-quark discretization errors in
$h_{+}(1)$ are $\sim$1–2%. At nonzero recoil, $w>1$, there are corrections to
$h_{+}(w)$ of $\mathrm{O}\left(\alpha_{s}\Lambda/2m_{Q}\right)$, but these are
suppressed by $(1-w)$ because they vanish in the limit $w=1$ by Luke’s theorem
Luke:1990eg . We expect them to be largest at our highest recoil point
$w=1.2$, and estimate their size to be $\sim$ 1%. Thus we estimate the
uncertainty in $h_{+}(w)$ from heavy-quark discretization errors to be 2%,
which leads to negligible errors in $R(D)$ and $P_{L}(D)$. The leading heavy-
quark error in $h_{-}(w)$ is of
$\mathrm{O}\left(\alpha_{s}\Lambda/2m_{Q}\right)$, which we estimate with the
input parameters above to be $\sim$ 6%. To be conservative, we take the error
in the ratio $h_{-}(w)/h_{+}(w)$ to be 10%, which leads to small errors in
$R(D)$ and $P_{L}(D)$.
Our methods for computing $B\to D$ transitions incorporate the bulk of the
matching of the lattice vector current to continuum automatically, leaving a
factor $\rho_{V_{cb}^{\mu}}$ close to unity Harada:2001fj . For $R(D)$ and
$P_{L}(D)$, only the relative matching of the spatial and temporal components
of the current matters, $\rho_{V_{cb}^{i}}/\rho_{V_{cb}^{0}}$. Although we
have a one-loop calculation of $\rho_{V_{cb}^{0}}$ in hand, no nontrivial
result for $\rho_{V_{cb}^{i}}$ is available. We take
$\rho_{V_{cb}^{i}}/\rho_{V_{cb}^{0}}=1.0\pm 0.2$ to estimate the uncertainty
from this source. In similar calculations, we have never seen $\rho$ factors
that differ from unity by more than 5%, so this range is extremely
conservative. The uncertainty in the current renormalization factors leads to
a small error in $R(D)$, but is the second-largest source of error in
$P_{L}(D)$, after statistics.
We also consider the systematic uncertainties from tuning the light-quark
masses and determining the absolute lattice scale $r_{1}$, but these produce
negligible errors in both $R(D)$ and $P_{L}(D)$.
Table 1: Error budgets for the branching fraction and longitudinal polarization ratios discussed in the text. Errors are given as percentages. Source | $R(D)$ | $P_{L}(D)$
---|---|---
Monte-Carlo statistics | 3.7 | 1.2
Chiral-continuum extrapolation | 1.4 | 0.1
$z$-expansion | 1.5 | 0.1
Heavy-quark mass ($\kappa$) tuning | 0.7 | 0.1
Heavy-quark discretization | 0.2 | 0.3
Current $\rho_{V_{cb}^{i}}/\rho_{V_{cb}^{0}}$ | 0.4 | 0.7
total | 4.3% | 1.5%
Results and Conclusions. – We obtain the following determinations for the
branching-fraction and longitudinal-polarization ratios for $B\to D\ell\nu$
semileptonic decay:
$\displaystyle R(D)$ $\displaystyle=$ $\displaystyle 0.316(12)(7)\,,$ (6)
$\displaystyle P_{L}(D)$ $\displaystyle=$ $\displaystyle 0.325(4)(3)\,,$ (7)
where the errors are statistical and total systematic, respectively. The value
of $R(D)$ is approximately 1$\sigma$ larger than the recent Standard-Model
values obtained using estimates of $f_{0}(q^{2})$ from Refs. Kamenik:2008tj ;
Fajfer:2012vx , but it is still 1.7$\sigma$ lower than the recent BaBar
measurement, $R(D)=0.440\pm 0.058\pm 0.042$ :2012xj . The results for $R(D)$
from Belle Adachi:2009qg agree with those of BaBar, but have larger
uncertainties. Current experimental measurements of $R(D)$ are statistics-
limited, so the luminosities available at Belle II and SuperB should enable
significant improvement on $R(D)$ and possibly a determination of $P_{L}(D)$.
We also re-examine the interpretation of the BaBar measurement of $R(D)$ as a
constraint on the 2HDM II; the result is plotted in Fig. 3.
Figure 3: Lattice-QCD calculation (red) and experimental measurement :2012xj
(blue) of $R(D)$ vs. $\tan\beta/M_{H^{\pm}}$ in the 2HDM II. For the BaBar
result, the dark- (light-) blue regions denote the 1$\sigma$ (2$\sigma$) error
bands. For comparison, the 2HDM II curve based on Refs. Kamenik:2008tj ;
Fajfer:2012vx and used in Ref. :2012xj is shown in green.
For this theory, the scalar-exchange coupling in Eq. (Refining new-physics
searches in $B\to D\tau\nu$ decay with lattice QCD) is given by
$G_{S}^{\ell
cb}=G_{F}V_{cb}\frac{m_{\ell}(m_{c}+m_{b}\tan^{2}\beta)}{M_{H^{\pm}}^{2}}.$
(8)
As soon as $\tan\beta\gtrsim 4$, $m_{c}$ in Eq. (8) can be neglected. On the
other hand, evaluating Eq. (Refining new-physics searches in $B\to D\tau\nu$
decay with lattice QCD) depends sensitively on $m_{c}/m_{b}$ via the prefactor
multiplying $f_{0}(q^{2})$. We take $m_{c}/m_{b}=0.22$, following Refs.
Fajfer:2012vx ; :2012xj . Our improved calculation of the scalar form factor
$f_{0}(q^{2})$ increases the prediction for $R(D)$ by about $1\sigma$ for all
$\tan\beta/M_{H^{+}}<0.6~{}\text{GeV}^{-1}$. This sensitivity of $R(D)$ to
differences in $f_{0}(q^{2})$ suggests that one should be cautious in using
indirect estimates of the form factors to constrain new-physics models in
other decay channels such as $B\to D^{*}\tau\nu$.
Inspection of the general formulas for the differential decay rates, Eqs. (1)
and (Refining new-physics searches in $B\to D\tau\nu$ decay with lattice QCD),
shows that many new-physics explanations for the $\sim 3\sigma$ tension
between measurements of $\\{R(D),R(D^{*})\\}$ and the Standard Model are
possible. Even in the context of 2HDM, variants other than type II can have
different phenomenology Crivellin:2012ye . Lattice-QCD calculations of
$f_{+}(q^{2})$ and $f_{0}(q^{2})$ can be used to provide reliable predictions
for $R(D)$ in any model given values for the couplings
$\\{G_{S},G_{V},G_{T}\\}$. We note, however, that the Dalitz distribution for
the lepton and $D$-meson energies in the $B$-meson rest frame may be more
sensitive to tensor interactions than $R(D)$ Kronfeld:2008gu .
The largest source of uncertainty in our determinations of $R(D)$ and
$P_{L}(D)$ is statistical errors, which we expect to reduce with an analysis
of our full data set in a future work Qiu:2011ur . The ratio $R(D)$ is
correlated with $P_{L}(D)$ as well as other observables such as $R(D^{*})$ or
$R(D_{s})$ in many new-physics models; thus the pattern of experimental
results for these quantities can help distinguish between new-physics
scenarios, such as those with and without a charged-Higgs boson Tanaka:2010se
. We will present lattice-QCD results for $R(D^{*})$ and $P_{L}(D^{*})$ in a
future paper on the $B\to D^{*}\ell\nu$ form factors, and also note that we
could easily obtain Standard-Model predictions for $R(D_{s})$ and
$P_{L}(D_{s})$ if measurements of these quantities were possible with an
$\Upsilon(5S)$ run at a $B$ factory.
Given the present tensions between experimental measurements and Standard-
Model predictions for both $\\{R(D),R(D^{*})\\}$ and the leptonic branching
fraction $\text{BR}(B\to\tau\nu)$ Bona:2009cj ; Lunghi:2010gv ; Charles:2011va
; Laiho:2012ss , lattice-QCD calculations of $B\to D\tau\nu$ form factors and
other hadronic weak matrix elements can play a key role in revealing whatever
theory beyond the Standard Model is realized in Nature.
Acknowledgements. – We thank Manuel Franco Sevilla, Jernej Kamenik, and Bob
Kowaleski for valuable discussions. We thank Yang Bai for emphasizing the
importance of the choice of scale for $m_{c}$ in Eq. (Refining new-physics
searches in $B\to D\tau\nu$ decay with lattice QCD). Computations for this
work were carried out with resources provided by the USQCD Collaboration, the
Argonne Leadership Computing Facility, the National Energy Research Scientific
Computing Center, and the Los Alamos National Laboratory, which are funded by
the Office of Science of the United States Department of Energy; and with
resources provided by the National Institute for Computational Science, the
Pittsburgh Supercomputer Center, the San Diego Supercomputer Center, and the
Texas Advanced Computing Center, which are funded through the National Science
Foundation’s Teragrid/XSEDE Program. This work was supported in part by the
U.S. Department of Energy under Grants No. DE-FG02-91ER40628 (C.B.), No. DOE
FG02-91ER40664 (Y.M.), No. DE-FC02-06ER41446 (C.D., J.F., L.L., M.B.O.), No.
DE-FG02-91ER40661 (S.G., R.Z.), No. DE-FG02-91ER40677 (D.D., A.X.K.), No. DE-
FG02-04ER-41298 (J.K., D.T.); by the National Science Foundation under Grants
No. PHY-1067881, No. PHY-0757333, No. PHY-0703296 (C.D., J.F., L.L., M.B.O.),
No. PHY-0757035 (R.S.); by the Science and Technology Facilities Council and
the Scottish Universities Physics Alliance (J.L.); by the MICINN (Spain) under
grant FPA2010-16696 and Ramón y Cajal program (E.G.); by the Junta de
Andalucía (Spain) under Grants No. FQM-101, No. FQM-330, and No. FQM-6552
(E.G.); by European Commission (EC) under Grant No. PCIG10-GA-2011-303781
(E.G.); and by the Creative Research Initiatives program (3348-20090015) of
the NRF grant funded by the Korean government (MEST) (J.A.B.). This manuscript
has been co-authored by employees of Brookhaven Science Associates, LLC, under
Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy. Fermilab is
operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359
with the U.S. Department of Energy.
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|
arxiv-papers
| 2012-06-21T19:25:19 |
2024-09-04T02:49:32.034982
|
{
"license": "Public Domain",
"authors": "Jon A. Bailey, A. Bazavov, C. Bernard, C. M. Bouchard, C. DeTar,\n Daping Du, A. X. El-Khadra, J. Foley, E. D. Freeland, E. Gamiz, Steven\n Gottlieb, U. M. Heller, Jongjeong Kim, A. S. Kronfeld, J. Laiho, L. Levkova,\n P. B. Mackenzie, Y. Meurice, E. T. Neil, M. B. Oktay, Si-Wei Qiu, J. N.\n Simone, R. Sugar, D. Toussaint, R. S. Van de Water, Ran Zhou",
"submitter": "Ruth Van de Water",
"url": "https://arxiv.org/abs/1206.4992"
}
|
1206.5022
|
# Toward a Unified AGN Structure
Demosthenes Kazanas11affiliation: Email: Demos.Kazanas@nasa.gov 22affiliation:
Astrophysics Science Division, NASA/Goddard Space Flight Center, Greenbelt, MD
20771 , Keigo Fukumura22affiliation: Astrophysics Science Division,
NASA/Goddard Space Flight Center, Greenbelt, MD 20771 33affiliation:
University of Maryland, Baltimore County (UMBC/CRESST), Baltimore, MD 21250 ,
Ehud Behar44affiliation: Research Center for Astronomy, Academy of Athens,
Athens 11527, Greece ,
Ioannis Contopoulos55affiliation: Department of Physics, Technion, Haifa
32000, Israel
and
Chris Shrader22affiliation: Astrophysics Science Division, NASA/Goddard Space
Flight Center, Greenbelt, MD 20771 66affiliation: Universities Space Research
Association
###### Abstract
We present a unified model for the structure and appearance of accretion
powered sources across their entire luminosity range from galactic X-ray
binaries to luminous quasars, with emphasis on AGN and their phenomenology.
Central to this model is the notion of MHD winds launched from the accretion
disks that power these objects. These winds provide the matter that manifests
as blueshifted absorption features in the UV and X-ray spectra of a large
fraction of these sources; furthermore, their density distribution in the
poloidal plane determines the “appearance” (i.e. the column and velocity
structure of these absorption features) as a function of the observer
inclination angle. This work focuses on just the broadest characteristics of
these objects; nonetheless, it provides scaling laws that allow one to
reproduce within this model the properties of objects spanning a very wide
luminosity range and viewed at different inclination angles, and trace them to
a common underlying dynamical structure. Its general conclusion is that the
AGN phenomenology can be accounted for in terms of three parameters: The wind
mass flux in units of the Eddington value, $\dot{m}$, the observer’s
inclination angle $\theta$ and the logarithmic slope between the O/UV and
X-ray fluxes $\alpha_{OX}$. However, because of a significant correlation
between $\alpha_{OX}~{}$and UV luminosity, we conclude that the AGN structure
depends on only two parameters. Interestingly, the correlations implied by
this model appear to extend to and consistent with the characteristics of
galactic X-ray sources, suggesting the presence of a truly unified underlying
structure for accretion powered sources.
accretion, accretion disks — galaxies: active — methods: numerical — quasars:
absorption lines — X-rays: galaxies
## 1 Introduction
The notion of AGN as an astronomical object of solar system dimensions and
luminosity surpassing that of a galaxy has been with us for about half a
century now. Since then, the advent of novel observational techniques, the
accumulation of data and theoretical modeling has refined and advanced our
notions as to what constitutes an AGN, with accretion onto a black hole as the
source of the observed radiation now being universally accepted. At the same
time, the discovery of galactic bright X-ray binary (XRB) sources, powered
also by accretion onto compact objects (neutron stars and stellar size black
holes) has extended the notion of accretion powered source to the stellar
domain. Indeed, the general similarity of the X-ray spectral properties of AGN
and galactic black hole candidates (GBHC) and XRBs in general, including their
broad Fe K$\alpha$ fluorescence features (Miller, 2007), argues for near
horizon structures which are very similar, despite the huge disparity in the
objects’ scales. This structure is thought to consist of a Shakura-Sunyaev
(Shakura & Sunyaev, 1973) disk that extends to the ISCO (innermost stable
circular orbit) of the corresponding flow, supplemented by an overlying hot,
X-ray emitting corona.
Even though it is generally accepted that the AGN radiant energy is released
by the accretion of matter onto a black hole (or in certain cases by
extraction of the hole’s rotational energy) in a region comparable to its
horizon, there is plenty of evidence that a significant fraction of the AGN
power is emitted, after reprocessing, at much larger radii. [One should note
however, that accretion energy can also be transported outward not only
radiatively but also mechanically by the viscous stresses that transport the
accretion flow’s angular momentum (Blandford & Begelman, 1999)]. Thus, the UV
and optical lines that constitute, typically, a fraction $f\sim 10\%$ of the
AGN bolometric luminosity, are emitted presumably by clouds at distances $\sim
0.1-10$ pc that cover a fraction $f$ of the AGN solid angle. In addition to
the line emission, the AGN ionizing continuum is also reprocessed into IR and
far–IR radiation by matter at even larger distances, which apparently subtends
an even larger fraction of the AGN solid angle ($\sim 50\%$). The geometry of
this component is thought to be cylindrical (rather than spherical) with a
column density that depends strongly on the angle $\theta$ of the observers’
line of sight (LoS) with the symmetry axis. It was proposed that such a
geometry nicely unifies the Seyfert-1 and Seyfert-2 AGN subclasses (Antonucci
& Miller, 1985) and also those of the broad and narrow line radio galaxies
(BLRG - NLRG) (Barthel, 1989), according to the angle $\theta$: Thus,
Seyfert-1s (or BLRG) are AGN in which the observer’s LoS makes a small angle
with their axis of symmetry, the column of the intervening cold gas is small
($N_{H}<10^{21}{\rm cm}^{-2}$) and the continuum source and its surrounding
broad line emission (concentrated in the inner AGN regions) are directly
visible. Seyfert-2s (or NLRG) on the other hand, represent the same objects
viewed at a large inclination angle, along which the column density to the
source is much larger ($N_{H}>10^{23}{\rm cm}^{-2}$), obscuring the continuum
source and allowing the view of only the large distance (hence narrow
component) of the emission lines. This obscuring structure is referred to as
the “AGN molecular torus”, considering that it must consist of gas in
molecular state, given its low effective temperature ($T\sim 10-100$ K).
Statistics of Seyfert-1 and Seyfert-2 AGN imply that the height $h$ of these
torii must be comparable to their distance $R$ from the AGN center, i.e.
$h/R\simeq 1$. However, the value of this ratio is in conflict with that
implied by hydrostatic equilibrium and the ratio of their thermal ($v_{\rm
th}\sim 1\,(T/{\rm 100K})$ km/s) and Keplerian ($v_{\rm K}\sim 300-500$ km/s)
velocities, namely $h/R\simeq v_{\rm th}/v_{\rm K}\sim 10^{-3}$, thus
presenting us with a conundrum concerning the physics of these structures.
These spectroscopically inferred components, along with observations of narrow
radio jets along the AGN symmetry axis, led to the now well known AGN picture
of (Urry & Padovani, 1995), which consists simply of their arrangement at the
appropriate positions in the AGN vicinity. Compelling as this picture might be
observationally, it includes very little, if any, of the underlying physics.
The AGN constituent components are independent of each other with physical
properties assigned as needed by the observations of the specific objects.
However, more recent observational developments suggest that such a picture is
rather incomplete. To begin with, Boroson & Green (1992) have shown the
existence of interrelations among AGN the line properties and also relations
to other bands of the spectrum (notably the X-rays). Then, the increase in UV
spectral resolution afforded by HST has shown that roughly 50% of Seyfert-1s
exhibit UV absorption troughs due to plasma outflowing at $v\simeq 300-1000$
km/s, too narrow to have been discerned by the earlier IUE observations
(Crenshaw et al., 1999), which did detect some, but in a much smaller fraction
of the overall AGN population. To these flows one must also include those of
the so-called BAL QSOs, which reach velocities along the observer’s LoS in
excess of $10^{4}$ km/s (Weymann et al., 1991). These are observed in about
$\simeq 10\%$ of high luminosity quasars, implying that they subtend a similar
fraction of the continuum source solid angle in these objects.
In addition to these UV absorption features, outflowing components were also
found in the AGN X-ray spectra. The increase in spectral resolution provided
by ASCA showed that approximately $50\%$ of Seyfert-1s exhibit also blue-
shifted absorption features in their X-ray spectra (George et al., 2000),
indicative of outflowing plasma, but of different ionization state than that
responsible for the UV absorption features. More recently, Tombesi et al.
(2010a) have shown that Fe-K absorption features at velocities $v\sim 0.1c$
are rather common in nearby Seyfert galaxies and coined for them the term
ultra-fast outflows (UFO). The simultaneous presence of both UV and X-ray
absorbers in the same objects implies they belong to the same outflowing
plasma (see e.g. Gabel et al., 2003). However, despite a large number of
studies supporting this hypothesis, (Mathur et al., 1994; Mathur, Elvis &
Wilkes, 1995; Collinge et al., 2001; Crenshaw et al., 2003; Brandt et al.,
2009), an understanding of the underlying gas dynamics is lacking. A common
origin for the plasma of these components as features of a common, radiatively
driven flow would be hard to reconcile with their different velocities and
ionization properties.
An account of the observed AGN outflows, in particular of the most challenging
high velocity ones of BAL QSOs was put forward semi-analytically by Murray et
al. (1995). These authors, in analogy with the winds of O-stars, proposed that
they are driven off the inner regions of the QSO accretion disks by UV and
optical line radiation pressure to achieve velocities consistent with those
observed. The same issue was taken up in more detail in 2D numerical
calculations by Proga, Stone, & Kallman (2000) who included in these
calculations the detailed photoionization of the line driven wind by the QSO
X-ray radiation. As shown in this work, efficient wind driving by line
pressure requires that the line driven wind material be shielded from the
ionizing effects of the X-rays, otherwise line driving becomes ineffective.
Their calculations showed that the “failed wind” from the highly ionized
innermost regions of the AGN accretion disk did provide the required
shielding. The fact that BAL QSOs are weak X-ray emitters appears to advocate
for such a point of view.
The ubiquity of AGN outflows implies that they should be included in the AGN
structure schematic of Urry & Padovani (1995); however, the broad range of
observed velocities and their different values in the UV and X-ray bands make
such a construct complicated in the absences of an underlying unifying
principle. However, such attempts have been made. Thus, Elvis (2000),
motivated by the velocity fields produced by Proga, Stone, & Kallman (2000) in
modeling the BAL QSO outflows, proposed a scheme that would supplement the AGN
picture of Urry & Padovani (1995) with outflow components consistent with
observed phenomenology. By limiting the fast ($v\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}10^{4}$ km/s),
radiatively driven flow to a narrow angular sliver ($\Delta\theta\simeq
6^{\circ}$) around $\theta_{s}\simeq 50^{\circ}$ (Proga, Stone, & Kallman,
2000), he accounted for the observed fraction of BAL QSOs in the overall QSO
population. He then attributed the lower velocities of the typical X-ray and
UV absorption features to the projection effects of viewing this flow at a
larger angle $(\theta>\theta_{s})$ and the absence of absorption features in
fraction of the objects to the low column and high ionization of the wind at
$\theta<\theta_{s}$. He also postulated that the angular position $\theta_{s}$
and the opening angle $\Delta\theta$ of the high velocity radiatively driven
radial stream would vary with source luminosity in a way that could account
the variation of source properties with luminosity. However, his approach
ignored the outflows seen in the AGN X-ray spectra, which at the time were not
as well documented.
The AGN picture proposed herein is in the same spirit as that of Elvis (2000)
in that, employing a well defined wind dynamical model, it provides a
framework of systematizing the multitude of observational facts, in particular
the more recent high resolution X-ray spectroscopy observations. However, it
does more than that; it provides, in addition, scaling arguments similar to
those put forward by Boroson (2002), which allow one to incorporate within
this single framework the ionization properties of Seyferts, BAL QSOs and
XRBs, the structure of the AGN molecular torii and the corresponding IR
spectra. This is possible because the underlying dynamical models, which span
many decades in radius, are to a large extent independent of the mass of the
accreting object and, as such, they can be applied to objects over a very wide
range of luminosity. As we will discuss in the ensuing sections, the models we
present provide the possibility of a broader classification of the structure
of accreting sources in terms of a small number of parameters (2) thus
providing an opportunity for a unified treatment of all accretion powered
sources. The present work will concentrate on the structure of AGN, however it
will be argued that the structures of XRBs are quite similar, their appearance
being different only because of their very different ionization.
In §2 we provide a brief review of AGN outflow phenomenology with emphasis on
the more recent high spectral resolution observations of X-ray absorbers by
Chandra and XMM-Newton. In §3 we present our model and its general scalings.
In §4 the model is applied to produce the absorber properties of galactic and
extragalactic objects as specific cases of its parameters along with its
general structure properties, depicted in two diagrams that relate the
absorber column, velocity and observation angle. In §5 we focus on the
emission properties of these winds and provide an account of the observed
linear relation between the H$\alpha$ and bolometric AGN luminosities as well
as their IR–to–far-IR spectra. Finally in §6 the results are reviewed and
directions for future research are outlined.
## 2 X-Ray Spectroscopy and Warm Absorbers
Following the discovery of the ubiquitous nature of X-ray absorption features
(Warm Absorbers) in the ASCA spectra, the launch of Chandra and XMM-Newton
ushered a new era in the study of these features. Their superior sensitivity
and resolution compared with those of previous missions made clear the
presence of a plethora of transitions (see e.g. Behar et al., 2003) in the
spectra of numerous AGN. For example, the X-ray bright QSO IRAS 13349+2438
($z=0.10764$) and the Seyfert-1 galaxy (e.g. MCG-6-30-15 at $z=0.007749$)
observed extensively with ROSAT (Brandt, Fabian & Pounds, 1996) and ASCA
(Brandt et al., 1997) gave indications of absorption features in the sub-keV
regime. Their Chandra and XMM-Newton spectra confirmed this fact and exhibited
a wealth of transitions such as N vii, Ne vii–Ne x, Mg v–Mg xii, Na xi, Si
vi–Si xiv, and Ni xix while successfully identifying almost all charge states
of Fe and O in some cases (Behar et al., 2003; Holczer et al., 2007, hereafter
HBK07). Furthermore, these observations showed the various transitions to be
blueshifted relative to the host galaxy, indicating that the warm absorber
plasma is, in fact, outflowing.
Though the majority of the X-ray absorber features in the Chandra and XMM-
Newton archives are associated with the spectra of Seyfert galaxies,
absorption features have been also seen in the spectra of quasars; these are
mainly Fe-K features usually at velocities higher than those seen in Seyferts,
in the range of $v/c\sim 0.1-0.8$, suggesting launching of these winds from
regions very close to the compact object. These include both BAL [see Chartas
et al. (2003) for PG 1115+080, Chartas et al. (2007) for H 1413+117, Chartas
et al. (2002, 2009) for APM 08279+5255; as such these also exhibit high
velocity UV absorption features] as well as non-BAL quasars [see Pounds et al.
(2003) for PG 0844+349, Reeves et al. (2003) for PDS 456, Pounds & Page (2006)
for PG 1211+143]. The columns of these features are $N_{H}\sim
10^{23}-10^{24}$ cm-2 which are typically at least $\sim 10$ times higher than
that of UV absorbers (Srianand & Petitjean, 2000; Chartas et al., 2009).
Finally, as discussed above, Fe-K features at velocities $v>10^{4}$ km/s, have
been detected in a large fraction ($\sim 1/3$) of observed AGN. Tombesi et al.
(2010a) refer to these as ultra-fast outflows (UFO).
Absorption features of varying ionization states have also been observed in
galactic sources (GBHC, XRB) (e.g. Miller et al., 2008; Neilsen, Remillard &
Lee, 2011; Brandt & Schulz, 2000). Considering that the presence of the ionic
species observed in these sources, like in AGN, is due to photoionization of
an outflow by the photons of the continuum source, their distribution is
determined by the photoionization parameter $\xi=L/nr^{2}$, where $L$ is the
ionizing luminosity, $n$ the local gas density and $r$ the distance of this
gas from the source. The higher S/N ratios of galactic sources afforded better
determination of the velocities of the different ions as well as the densities
of the outflowing wind at a given radius $r$. Based on these parameters, it
was concluded (see e.g. §3.1) that the wind of GRO 1655-40 (Miller et al.,
2008) cannot be driven by the radiation pressure or by the X-ray heating of
this source and therefore it must be driven by the action of magnetic fields
(e.g. Blandford & Payne, 1982; Contopoulos & Lovelace, 1994). Similar analysis
of the X-ray spectra of GRS 1915+105 by Neilsen, Remillard & Lee (2011)
concluded that the wind mass flux at the outer edge of its accretion disk
(associated with the lowest ionization ions) could be as much as twenty times
larger than the mass flux needed to power the source’s X-ray luminosity, while
a similar conclusion was reached by Behar et al. (2003) concerning the wind of
NGC 3783. Finally, it should be noted here that the wind velocities of the
highest ionization species (Fe-K) are significantly smaller in GBHC than in
Seyferts, which in turn are smaller than those of BAL QSOs, a fact that should
be accounted within the general framework of any wind model.
### 2.1 The Absorption Measure Distribution
The very detection in the AGN X-ray spectra of species of such diverse states
of ionization as Fe xxv, Mg v and O i is an important fact in itself. It
implies a wind with density distribution that produces ionic columns,
sufficiently large to be detected, for ions that “live” at widely different
values of $\xi$ and, most likely, also at widely different distances from the
ionizing source. This alone is a significant constraint on outflow models.
Taking this argument into consideration, Holczer et al. (2007) and
subsequently Behar (2009) fit the ensemble of the absorber data in the X-ray
spectra of a number of AGN with a continuous distribution in $\xi$ (rather
than adding components of different $\xi$ until a sufficiently low $\chi^{2}$
is achieved). In doing so, they developed a statistical measure of the
plethora of the transitions in the Chandra/XMM-Newton X-ray spectra, the
absorption measure distribution (AMD). This is the hydrogen equivalent column
density of specific ions, $N_{H}$, per decade of ionization parameter $\xi$,
as a function of $\xi$, i.e. $AMD(\xi)=dN_{\rm H}/d\log\xi$. For a monotonic
distribution of the wind density $n(r)$ with the radius $r$, determination of
AMD is tantamount to determining the ionized wind’s density dependence on $r$
(along the observers’ LoS). Therefore, considering that $AMD\propto N_{H}$, a
power law dependence on $\xi$ of the form $AMD\propto
N_{H}\propto\xi^{\alpha}$, implies also a power law dependence for the wind
density on $r$ i.e. $n(r)\propto r^{-s}$ with $s=(2\alpha+1)/(\alpha+1)$. So
for $\alpha\simeq 0$ ($AMD$ independent of $\xi$), $s\simeq 1$, consistent
with the conclusion reached by Behar (2009).
As noted in Behar (2009), an $n(r)\propto 1/r$ density profile is inconsistent
with the asymptotic dependence of a mass conserving spherical wind which
obtains $n(r)\propto r^{-2}$ and whose ionization parameter “freezes” to a
constant value $\xi_{\infty}$. On the other hand, considering that the
velocity of such winds is achieved only asymptotically, with the velocity
having the general behavior $v(r)=v_{\infty}(1-r_{*}/r)$ (Murray et al.,
1995), the $AMD$ would exhibit in this case too a dependence on $\xi$ at radii
$r\mathrel{\raise 1.29167pt\hbox{$<$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}r_{*}$. Assuming a mass conserving spherical wind with
the above velocity structure, i.e. $n(r)v(r)r^{2}=\dot{m}=const.$, one can
easily deduce that
$\frac{dN_{H}}{d{\rm log}\,\xi}\sim
N_{H}(\xi)=\frac{\dot{m}}{v_{\infty}r_{*}}\left[\frac{\xi_{\infty}}{\xi}-1\right]$
(1)
where $\xi_{\infty}$ is the ionization parameter at infinity. Under the above
velocity field, near $r_{*}$, the wind base, $\xi$ (and also $v$) approaches
zero and the gas column diverges so that $AMD\propto
N_{H}\simeq(\dot{m}/v_{\infty}r_{*})(\xi_{\infty}/{\xi})$. At large distances,
with $\xi=\xi_{\infty}-\epsilon,~{}(\epsilon/\xi_{\infty}\ll 1$),
$N_{H}=(\dot{m}/v_{\infty}r_{*})\,\epsilon$, so that the absorption measure
distribution has the form $AMD\propto(\epsilon/\xi_{\infty})\propto r_{*}/r$,
i.e. the wind column decreases like $r^{-1}$, as expected. The arguments
leading to the above scalings are certainly overly simple, however they show
that for radiatively driven winds, the column decreases with increasing
ionization and increasing velocity, in significant disagreement with the
dependence found by Holczer et al. (2007) and Behar (2009); they also suggest
that in attempting to account for the AMD behavior, one should search for wind
models with density and velocity profiles asymptotically different from those
driven by radiation pressure.
While it is hard to produce radiation pressure driven winds with the density
profile implied by the AMD, i.e. $n(r)\propto r^{-1}$ (even an extended source
of radiation asymptotically appears point-like and the winds acquire density
$n(r)\propto r^{-2}$), MHD winds off accretion disks were known to provide
such density profiles for some time now (Contopoulos & Lovelace, 1994,
hereafter, CL94). These are generalizations of the two dimensional MHD winds
enunciated by Blandford & Payne (1982) in that they allow also for mass flux
rates which depends on the radius. Given that the velocities of these winds
asymptotically scale like the disk Keplerian velocity at the point of
launching, i.e. $v\propto r^{-1/2}$, one can then easily see that for
$n(r)\propto r^{-1}$, $\dot{m}(r)\propto r^{2}n(r)v(r)\propto r^{1/2}$, i.e.
the wind mass flux $\dot{m}$ increases with distance. In this respect these
winds follow the ADIOS prescription discussed by Blandford & Begelman (1999).
The specific density profile inferred from the AMD is interesting in that it
provides equal column per decade of radius. As such, it is consistent with the
density needed to account for the lags in the power spectra of XRB and AGN
(Kazanas et al., 1997; Papadakis et al., 2001) in terms of Compton scattering.
Finally, this same profile for the dust density of AGN winds, at radii beyond
the dust sublimation radius, was employed by Königl & Kartje (1994) and Rowan-
Robinson (1995) to produce Seyfert-2 and QSO IR-spectra in agreement with
observations. The fact that this wind density profile can accommodate in a
natural way several diverse and independent aspects of AGN phenomenology under
the same framework is suggestive of its central role in the AGN structure and
appearance.
## 3 The Model MHD Winds
Motivated by the straightforward interpretation of the AMD in terms of density
profiles associated with MHD winds, we outline in this section some properties
of these wind models with emphasis on their scalings with the black hole mass,
mass flux rate and distance. We pay particular attention on the scalings of
the winds’ ionization and columns, as these are related to the observations of
X-ray absorbers.
The MHD winds considered here are launched by accretion disks threaded by
poloidal magnetic fields under the combined action of rotation, gravity and
magnetic stresses as discussed in Blandford & Payne (1982). To simplify the
treatment it is assumed that the wind is axisymmetric and as such one need
only solve the field geometry and fluid flow in the poloidal
($r,\theta$)-plane. To simplify further the problem one looks for solutions
separable in $r$ and $\theta$ with power law $r-$dependence for all magnetic
field and fluid variables. After consideration of all conserved quantities in
the poloidal plane (mass flux per magnetic field flux, angular momentum,
magnetic line angular velocity and Bernoulli integral) one is left with the
force balance in the $\theta-$direction (the so-called Grad-Safranov
equation). The solution of this equation provides the angular dependence of
all the fluid and magnetic field variables with their initial values given on
the surface of the disk at ($r_{o},90^{\circ}$), where $r_{o}$ is a fiducial
radius that sets the radial scale of the system and it is of order of the
Schwarzschild radius, $r_{S}$. The Grad-Safranov equation has the form of a
wind equation with several critical points, the most important of which for
our purposes the Alfvén point; this can be crossed by appropriately choosing
the flow boundary conditions on the disk surface (see CL94, BP82).
The scalings of the magnetic field, velocity, pressure and density are given
by
$\displaystyle\textbf{\em B}(r,\theta)$ $\displaystyle\equiv$
$\displaystyle(r/r_{o})^{-(s+1)/2}\tilde{\textbf{\em B}}(\theta)B_{o}\ ,$ (2)
$\displaystyle\textbf{\em v}(r,\theta)$ $\displaystyle\equiv$
$\displaystyle(r/r_{o})^{-1/2}\tilde{\textbf{\em v}}(\theta)v_{o}\ ,$ (3)
$\displaystyle p(r,\theta)$ $\displaystyle\equiv$
$\displaystyle(r/r_{o})^{-(s+1)}{\cal P}(\theta)B_{o}^{2}\ ,$ (4)
$\displaystyle n(r,\theta)$ $\displaystyle\equiv$
$\displaystyle(r/r_{o})^{-s}\tilde{n}(\theta)B_{o}^{2}v_{o}^{-2}\ .$ (5)
The density normalization is given in terms of the poloidal field on the disk
$B_{o}$, however, it is more instructive to express it in terms of the
accretion or wind outflow rate $\dot{m}$ as discussed in FKCB10; then the
density normalization at the inner edge of the disk at $\theta=90^{\circ}$ is
given on setting $\tilde{n}(90^{\circ})=1$ by
$n_{o}=\frac{f_{W}{\dot{m}_{o}}}{2\sigma_{T}r_{S}}\ ,$ (6)
where $f_{W}$ is the ratio of of the mass-outflow rate in the wind to the
mass-accretion rate $\dot{m_{o}}$, assumed here to be $f_{W}\simeq 1$, and
$\sigma_{T}$ is the Thomson cross-section. It is important to note here that
because the mass flux in these winds depends in general on the radius, the
normalized parameter used throughout this work, $\dot{m}_{o}$, always refers
to the mass flux at the innermost value of the flow radius, i.e. at
$r/r_{o}\equiv x\simeq 1$. With the scalings given above we then have
$\dot{m}(x)=\dot{m}_{o}x^{-s+3/2}$, so for $s=1$, as inferred from the fits of
HBK07, $\dot{m}\propto x^{1/2}$ (the Blandford & Payne (1982) solution has
$s=3/2$, so $n(r)\propto x^{-3/2}$ and therefore $\dot{m}\propto x^{0}$ or
$d\dot{m}/dlogx=const.$).
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=209.58255pt,angle={-0}]{density.eps}&\includegraphics[keepaspectratio={false},width=209.58255pt,angle={-0}]{v_los.eps}\end{array}$
Figure 1: (a) Wind density as a function of the polar angle $\theta$
normalized to its value on the disk surface ($\theta=90^{\circ}$). (b) The
line of sight wind velocity on a given field line, normalized to the Keplerian
velocity of the foot point of the specific field line $v_{o}$.
### 3.1 Scaling Laws
The reason that the winds we present here can be applied to AGN as well as to
GBHC, as mentioned above, is that winds and flows in general are known to be
self-similar when: The radius $r$ is normalized to the Schwarzschild radius
$r_{S}$ ($x=r/r_{S}$, $r_{S}=3M$ km, where $M={\rm M/M}_{\odot}$ is the mass
of the accreting object M in units of the solar mass M⊙); the mass flux
$\dot{M}$ is expressed in units of the Eddington accretion rate
$\dot{M}_{E}=L_{E}/c^{2}$ ($L_{E}$ is the Eddington luminosity, $L_{E}\simeq
1.3\cdot 10^{38}\,M$ erg s-1) as
$\dot{m}=\dot{M}/\dot{M}_{E}\propto\dot{M}/M$; their velocities are Keplerian,
i.e. $v\simeq x^{-1/2}\,c$.
While the wind scalings of eqs. (2) - (5) were introduced for the solution of
the MHD equations one can write quite general scaling laws that incorporate
those of the above equations as special cases. We do so now both for accretion
flows and winds and use them to produce scalings for expressions of the wind
density, column density, ionization parameter and AMD as a function of the
dimensionless radius $x$, the dimensionless mass flux rate $\dot{m}$ and black
hole mass $M$.
1\. Accretion: In applying the scalings of eqs. (2) - (5) to spherical
accretion one obtains for the density $n(x)\propto x^{-3/2}\dot{m}M^{-1}$ and
for the column density $N_{H}(x)\propto x^{-1/2}\,\dot{m}$, i.e. an expression
independent of $M$; so, flows onto objects of widely different masses but of
the same $\dot{m}=\dot{M}/\dot{M}_{E}$ have the same column at the same
distance $x$ from the accreting object; its normalization is such to yield
Thomson depth $\tau_{T}\simeq 1$ for $x\simeq 1$ at $\dot{m}\simeq 1$ (this is
the similarity of Advection Dominated Accretion Flows (ADAF) of (Narayan & Yi,
1994)).
2\. Winds: Analogous considerations of similarity apply equally well to winds,
given that, generally, their asymptotic velocity, $v_{\infty}$, is
proportional to the Keplerian velocity at the wind launching radius, $r_{l}$,
i.e. $v_{\infty}\propto x_{l}^{-1/2}$, with $x_{l}=r_{l}/r_{S}$. So mass
conservation in physical and dimensionless units reads respectively
$\dot{M}\sim n\,R^{2}\,v_{\infty}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}\dot{m}_{W}\,M\propto n(x)\,x^{2}M^{2}\,x_{l}^{-1/2}$ (7)
with $\dot{m}_{W}$ denoting in this case the mass outflow rate in units of the
Eddington rate $\dot{M}_{E}$. Therefore, the wind density and local column
density are given in dimensionless units correspondingly by the expressions
(dropping the subscript $W$ from $\dot{m}_{W}$)
$n(x)\propto\frac{\dot{m}}{M}\frac{x_{l}^{1/2}}{x^{2}}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}N_{H}(x)\propto\dot{m}\,\frac{x_{l}^{1/2}}{x}~{}~{}.$ (8)
with the object mass $M$ again dropping out of the expression for the column
density $N_{H}$. For 1D winds, e.g. stellar winds or winds driven by the
intense radiation pressure near a compact object, $x_{l}$ is roughly constant
and $N_{H}(x)\propto\dot{m}/x$, i.e. the wind column decreases inversely
proportionally to the distance. However, in the more general case of 2D winds
(e.g. those of BP82 and CL94) where the wind is launched from a wide range of
radii in an accretion disk with, generally, different mass flux rates at each
radius, $x_{l}\sim x,\,\dot{m}=\dot{m}(x)$ and
$N_{H}(x)\propto\dot{m}(x)x^{-1/2}$ and numerically $N_{H}(x)\simeq
10^{24}\dot{m}(x)x^{-1/2}\,{\rm cm}^{-2}$ (it should be understood that
$n(x),\,N_{H}(x)$ have an additional angular dependence discussed in the
previous section). As noted in Blandford & Begelman (1999) winds with position
dependent $\dot{m}$ are the means by which 2D accretion flows dispose of the
excess energy and angular momentum transferred mechanically by the viscous
stresses from the smaller to the larger radii, so that finally only a small
fraction of the available mass accretes on the compact object (BB99). For wind
density of the form $n(x)\propto x^{-s}$, the mass flow rate varies with $x$
like (BB99)
$\dot{m}_{W}(x)=\dot{m}_{o}\,x^{-s+3/2},\;s\leq 3/2,$ (9)
with $\dot{m}_{o}$ being the mass outflow from the smallest radius $x\sim
1.5-3$; for $s=1,~{}\dot{m}\propto r^{1/2}$ and $N_{H}$ constant per decade of
$r$.
3\. Photoionization: The wind ionization is determined by the local ratio of
photons to electrons, the ionization parameter $\xi=L/n(r)r^{2}$, ($L$ is the
source’s ionizing luminosity – different from the total luminosity, $n(r)$ the
local density and $r$ the distance from the ionizing source). This can also be
expressed in dimensionless units: If $\eta\,(\simeq 10\%)$ is the radiative
efficiency of the accretion process, then the luminosity $L$ can be written as
$L\propto\eta\,\dot{m}_{a}\,M$ ($\dot{m}_{a}$ is the accretion rate that
reaches the compact object to produce the luminoisty $L$), or
$L\propto\eta\,\dot{m}_{a}^{2}\,M$ in the case of ADAF (Narayan & Yi, 1994)
[i.e for $\dot{m}_{a}\mathrel{\raise 1.29167pt\hbox{$<$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}\alpha^{2}$ with $\alpha$ the disk viscosity
parameter], yielding the for $\xi$ an expression also independent of $M$,
implying similarity in wind ionization, whether in AGN or XRB (FKCB10)
$\displaystyle\xi(x)\simeq\frac{L}{n(r)r^{2}}\simeq\left\\{\begin{array}[]{ll}\displaystyle\frac{\eta\dot{m}_{a}}{N_{H}(x)x}\simeq
10^{8}\displaystyle\frac{\eta}{f_{W}}\displaystyle\frac{1}{x^{-s+2}}{}{}{}&\mbox{
for $\dot{m}_{a}>\alpha^{2}$ (non-ADAF)}\\\
\displaystyle\frac{\eta\dot{m}_{a}^{2}}{N_{H}(x)x}\simeq
10^{8}\displaystyle\frac{\eta}{f_{W}}\displaystyle\frac{\dot{m}_{a}}{x^{-s+2}}{}{}{}&\mbox{
for $\dot{m}_{a}<\alpha^{2}$ (ADAF)}\\\ \end{array}\right.$ (12)
where $f_{W}=\dot{m}_{o}/\dot{m}_{a}\,(\sim 1)$ is the ratio of mass flux in
wind and accretion at the smallest radii and $s$ the index of the density
dependence on $r$. These relations are also independent of $M$, implying a
similarity in the ionization structure of winds, whether in AGN of any type or
GBHC. For a spherical wind, ($s=2$) the ionization parameter is independent of
$x$ (asymptotically), while for $s=1$ it decreases linearly with distance from
the ionizing source.
4\. The AMD: Writing Eq. (12) as $N_{H}(x)\propto\eta\,\dot{m}_{a}/\xi(x)x$
one can form the expression for AMD (HBK07), namely
$AMD=\frac{dN_{H}(x)}{d{\rm
log}\xi(x)}\simeq\frac{\eta\,\dot{m}_{a}}{\xi(x)x}$ (13)
The fact that AMD is largely independent of $\xi(x)$ (HBK07) implies
$\xi(x)\propto 1/x$ or $N_{H}(x)\propto log(x)$, $n(x)\propto 1/x$ and
$\dot{m}\propto x^{1/2}$, i.e. the wind mass flux increases with radius, as
discussed in BB99. In this case, both the ionization parameter and the wind
density decrease like $1/r$ while the column of the ions found at a given
$\xi$ remains roughly constant, in broad agreement with Behar (2009). So, in
disks of sufficiently large radial extent, the wind launched at their larger
radii will be of sufficiently low ionization and temperature to conform with
the properties of the AGN unification torus. Finally, one should bear in mind
that the above scaling laws do not include the $\theta-$dependence of the
density and the column density (see fig. 1). At a given (radial) distance $r$
from the source, this leads to much higher ionization of the plasma flowing
along the symmetry axis and even smaller neutral absorption column than along
the higher column density, less ionized equatorial directions, in agreement
with the unification considerations (Königl & Kartje, 1994).
Under the above scaling laws, the ionization, velocity and AMD of the winds
discussed in this work are independent of the mass of the accreting object.
Therefore these expressions should hold equally well for AGN and GBHC or XRB
in general, a fact in obvious disagreement with observation, given their very
different X-ray absorber properties. This would indeed be the case if the
ionizing spectrum were also independent of the mass of the accreting object.
However, the similarity of wind ionization is finally broken by the dependence
of the ionizing (X-ray) flux on the mass of the accreting object and leads to
the different wind ionization properties not only between XRB and AGN, but
also amongst AGN: Both in AGN and XRB a major fraction of their bolometric
luminosity is emitted in quasithermal features referred to as the Big Blue
Bump (BBB) in AGN and the Multi Color Disk (MCD) in XRB. Both are thought to
represent emission of the accretion luminosity in black body form by a
geometrically thin optically thick disk of size a few Schwarzschild radii. The
characteristic temperature of this feature scales as $M^{-1/4}$, being in the
X-ray band for solar mass objects and in the UV for AGN of $M\sim 10^{8}$ M⊙.
This dominance of the ionizing luminosity (X-rays) as a fraction of the
bolometric one in XRB over AGN is, according to our model, the main cause for
their differences in their wind ionization properties as it will be discussed
in detail in the next section. Even within AGN, the ionizing flux is in fact
not a constant fraction of the bolometric luminosity but it depends on the
absolute source luminosity. A measure of their ionizing luminosity is given by
the index $\alpha_{OX}$, the logarithmic slope of the flux between 2500
$\buildrel{}_{\circ}\over{\mathrm{A}}$ and 2 keV. This quantity has a strong
dependence on $L(2500\buildrel{}_{\circ}\over{\mathrm{A}})$ (Steffen et al.,
2006; Brandt et al., 2009), a fact that bears relation to the corresponding
AGN absorber properties, as it will be discussed in the next section.
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=158.99377pt,angle={-90},clip={false}]{fe-30deg.eps}&\includegraphics[trim=0.0pt
0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=158.99377pt,angle={-90},clip={false}]{fe-60deg.eps}\end{array}$
Figure 2: The ionization structure of iron appropriate for a Seyfert galaxy
with $\dot{m}_{o}\simeq 0.1$ and an ionizing photon spectrum with
$dF_{\nu}/d\nu\propto\nu^{-\Gamma},~{}\Gamma=1.9$, as seen at two different
inclination angles $\theta=30^{\circ}$ (left) and $\theta=60^{\circ}$ (right).
The ionization decreases with increasing distance $r$ along a given LoS and
with increasing $\theta$ for a given value of $r$. The column of each ion is
given in the left ordinate, while the velocity at each position along the LoS
in the right one. The LoS velocity at each $\xi$ (lower abscissa) and the
corresponding radius (upper abscissa) is given by the black triangles.
## 4 The Ionization Structure of MHD Winds
With the wind structure set by the solution of the Grad-Safranov equation, one
can now proceed to study its ionization structure. The details are given in
Fukumura et al. (2010a) so we simply restrict ourselves to a broad description
of this procedure. Given the (approximately) $1/r$ density profile, we have
split the radius along a given LoS from $r_{o}$ to $10^{6}-10^{8}\,r_{o}$ in
approximately 40 segments, spaced equally in log$r$, so that each has the same
column density. Then we assume a value for $\dot{m}_{o}$ which provides the
normalization of the wind density through Eq. (6) (we generally assume
$f_{W}=1$) and a luminosity $L$ related to $\dot{m}_{o}$ either as $L=1.3\cdot
10^{38}\,\dot{m}_{o}M$ erg/s or $L=1.3\cdot 10^{38}\,\dot{m}_{o}^{2}M$ erg/s
(for $\dot{m}_{o}<\alpha^{2}$, according to the ADAF prescription) with a
specific spectrum; then we invoke the photoionization code `XSTAR` to compute
the ionization, opacity and emissivity of the first zone. Using the derived
opacities and emissivity we compute the spectrum exiting this zone, which is
used as the input for computing the ionization opacity and emissivity of the
next zone. The procedure is repeated to compute the ionization of the wind
along a given line of sight and then for different values of the polar angle
$\theta$, to produce the ionization structure of the wind over the entire
poloidal plane.
Fig. 3 of Fukumura et al. (2010a) depicts the density and ionization structure
of such winds out to $10^{8}\,r_{o}$ in linear coordinates. One can see that
the wind ionization depends both on the distance from the source $r$ and on
the angle $\theta$, being highly ionized at small values ($\xi\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}10^{3}$ near
$\theta=0$) even at these large distances, while it decreases to $\xi<1$ at
$\theta\sim 80^{\circ}$. It is also clear that for a given $\theta$, a harder
X-ray spectrum will ionize the wind out to larger distances. Since the wind
velocity along a given LoS depends mainly on the radius of launching the
specific parcel of gas (but also the inclination angle $\theta$), absorption
features associated with a given ion should appear at smaller velocities the
harder the ionizing spectra. This issue is in fact a little more complicated
because we only observe the wind velocity along our line of sight which
depends also on the local shape of the poloidal magnetic field lines.
### 4.1 Ionization of XRB Winds
The high X-ray content of the spectra of XRB, implies that they should be, in
general, much more ionized than those of AGN. Nonetheless, X-ray absorption
features have also been observed in XRBs, as indeed seems to be the case. So
P-Cygni Fe-K features were first detected in Cir X-1, which is (presumably) an
accreting neutron star in a 16.6 day binary orbit around a main sequence
companion (Brandt & Schulz, 2000; Schulz & Brandt, 2002). The high ionization
of the inner sections of these outflows, render them essentially devoid of any
absorption features. Therefore, the highest ionization species Fe xxvi, Fe xxv
should occur at radii such that $\xi\mathrel{\raise
1.29167pt\hbox{$<$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}10^{4}$, which,
as the authors suggest sets them at distances $r\simeq 10^{5}$ km and
corresponding velocities of only $v\simeq 1,000$ km/s.
There are several other XRB in which similar features have been detected, the
best known being GRO 1655-40 (Miller et al., 2006, 2008) and GRS 1915+105
(Neilsen, Remillard & Lee, 2011) both of which show Fe xxvi and Fe xxv
absorption features at velocities similar to those of Cir X-1. GRO 1655-40
presents a multitude of absorption features similar to those of AGN, however,
because of the higher S/N ratio their velocities and EW are better determined
and thus more contraining to the outflow models. Following detailed modeling
it was concluded (Miller et al., 2006, 2008) that the wind of GRO 1655-40 is
not consistent with being driven by either line radiation pressure or
thermally by the X-ray heating of the outer regions of the disk. The authors
then concluded that the only reasonable means of driving the outflow in this
object is magnetic, via an MHD wind similar to those outlined in this work.
GRS 1915+105 affords a smilar analysis (Neilsen, Remillard & Lee, 2011, and
references therein). Detailed photoinization modeling of this source provided
both the column and velocity associated with specific transitions. Assuming
that the line absorbing medium has thickness $\Delta r\simeq r$, and that the
observed velocities are effectively Keplerian, i.e.
$v/c\simeq(r/r_{S})^{-1/2}$, one can obtain simultaneous estimates of both $r$
and $v$; these along with the measured absorption column $N_{H}\simeq nr$
yield an estimate of the mass flux of the wind, which it was found that it
could be as much as several times larger than the mass accretion rate needed
to power the observed X-ray emission of this source. In fact, one is led to a
similar conclusion concerning the mass flux in the wind of Cir X-1 by using
the parameters provided in Schulz & Brandt (2002), indicating that a wind mass
flux increasing with $r$ may not be an uncommon feature in accretion powered
sources.
Finally, it is worth noting that the disks of all three of the above sources
are apparently at high inclinations, i.e. directions that favor observations
of such features (less ionization, higher column densities). For lower
inclination systems, the winds are likely far more highly ionized, yielding no
apparent absorption features.
### 4.2 Ionization Structure of Seyfert Winds
The procedure outlined in §4 describing the detailed calculation of the
ionization of MHD winds was applied to winds associated with Seyfert galaxies
(Fukumura et al., 2010a). As noted above, given the scale invariance of the
wind column and ionization parameter, the main difference between the
ionization winds off the disks of different classes of accretion powered
sources (e.g. XRB, Seyferts, BAL QSOs) is the spectrum of the ionizing
radiation. Since for Seyfert galaxies the X-ray and UV-optical luminosities
are approximately the same, we have approximated the ionizing spectra with a
single power law from $1-1,000$ Rydberg of photon index $\Gamma=1.9$ and for
the accretion rate we used the value of $\dot{m}_{o}=0.1$.
The 2D density and poloidal plane ionization structure of a wind for the above
parameters is given in Fig. 3 of Fukumura et al. (2010a). Here, we present in
Figure 2 the ionic species distribution of just one element of this wind,
namely iron, as a function of the distance from the continuum source (upper
abscissa) and the corresponding value of $\xi$ (lower abscissa), for two
different values of the observer inclination angle
$\theta=30^{\circ},\;60^{\circ}$ (left and right panels respectively). The
column of each ionic species is given in the left ordinate, while the
corresponding velocity along the LoS is given in the right ordinate and
represented by the black triangles.
We can see that increasing $\theta$ increases the column of a specific ionic
species. Also, because the radius at which it now forms is smaller, the
velocity at the peak abundance of this ion is correspondingly higher. Finally,
one can obtain the AMD of this flow by adding the columns of all ions of Fe at
a given value of $\xi$. As expected, and shown explicitly in (Fukumura et al.,
2010a), this is constant (i.e. independent of the value of $\xi$ or the value
of $r/r_{S}$), reflecting the column per decade of radius of the underlying
wind.
The ionization properties of Seyfert galaxies are consistent with the models
presented in this figure. The conclusions of Behar (2009), based on the AMDs
of the five AGN he has analyzed are broadly consistent with ours as presented
in this section. Some of them require slightly steeper density profiles
($s\simeq 1.25$) whose ionization structure and velocity properties should be
similar to those presented in this section.
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=195.12877pt,angle={-0},clip={false}]{logden3.eps}&\includegraphics[trim=0.0pt
0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=238.49121pt,angle={-0},clip={false}]{sed.eps}\end{array}$
Figure 3: (a) 2D poloidal plane density structure of an MHD wind of
$\dot{m}=0.1$ and $M=10^{9}M_{\odot}$, ionized by the spectrum shown in (b).
The density, $\log(n[\textrm{cm}^{-3}])$, is shown in color with the dotted
lines being the iso-density contours, with the log of the corresponding
density indicated by the associated numbers. The $r,z$ coordinates are in
logarithmic scale, so lines of increasing $\theta$ are lines at 45∘
inclination of increasing abscissa intercepts. Also shown are the magnetic
field lines (black solid) and the loci (white dashed lines) of the positions
of formation of Fe xxv (circles) and C iv (squares). (b) The form of the SED
used to produce the wind ionization of (a) consisting of (i) a thermal MCD (as
BBB) spectrum of innermost temperature $kT_{\rm in}=5$ eV and (ii) a power-law
continuum of photon index $\Gamma$. The X-ray flux is normalized relative to
that of BBB by $\alpha_{\rm ox}$. [See the electronic edition of the Journal
for a color version of this figure.]
### 4.3 Ionization Structure of BAL QSO Winds
If the tenets of the proposed model are correct, it should be able to
reproduce the velocity and ionization structure of the absorption features of
accretion powered objects other than Seyfert galaxies through judicious
choices of the values of the available parameters. Should such an enterprise
be successful, it could delineate the regions of parameter space occupied by
individual AGN classes and hence provide their systematization on the basis of
an underlying physical model.
Such an attempt was made by Fukumura et al. (2010b), who employed the specific
MHD winds to model the absorption features in the spectra of BAL QSOs, in
particular that of APM 08279+0255 (Chartas et al., 2002, 2003). This AGN class
was chosen because the properties of absorption features of its members are
most different from those of Seyferts, since they exhibit Fe xxv at velocities
$v\simeq 0.5c$ and C iv at $v\simeq 0.1c$. Because these objects were selected
on the basis of their broad UV absorption features, any such model should be
able to provide an account of the combined properties of both their X-ray and
UV features.
The property that most clearly separates Seyferts and BAL QSOs is the relative
weakness of their X-ray to their UV luminosity. This is reflected in the
corresponding value of their $\alpha_{OX}$ parameter, which in BAL QSOs has
values $\alpha_{OX}\simeq-2$ compared to $-1$ to $-1.2$ for Seyferts. To model
the ionization of BAL QSO winds, Fukumura et al. (2010b) have chosen as
ionizing spectrum that shown in Fig. 3b. This consists of a BBB component of
maximum temperature 5 eV along with a power law component of photon index
$\Gamma$, normalized at $E=2$ keV relative to the BBB so that it results in
$\alpha_{OX}=-2$, consistent with that of APM 08279+0255. The decrease in the
value of $\alpha_{OX}$ is significant, because it implies that the ratio of
ionizing photon density (given by the X-rays) to the matter density
(represented by the luminosity of the BBB that peaks in the UV) is quite
small, yielding a value for the ionization parameter at $r\simeq r_{o}$ much
smaller than that of Seyfert galaxies. Furthermore, the cooling effects of the
prominent UV emission further contributes to lowering the gas temperature at
the smallest radii and hence its ionization. It is then possible that some of
the heavy elements (e.g. Fe) will not be fully ionized even at wind radii
$r\simeq r_{o}$, leading to a high outflow velocity for Fe xxv, Fe xxvi.
The presence of non-fully ionized Fe at radii $r\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}r_{o}$, along
with the very steep (unobservable) EUV source spectrum, depletes severely the
photons at energies $E\mathrel{\raise 2.15277pt\hbox{$>$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}50$ eV needed to produce the ion $C^{+3}$ of carbon.
Since this ion forms at a specific value of $\xi$, this suppression of
ionizing photons is made up by a decrease in the radius $r$ at which it forms
(and an increase in the local wind velocity), leading to C iv absorption
features at velocities much higher than those of Seyfert galaxies. Figure 3a
shows in logarithmic scale the density distribution of a wind of
$\dot{m}_{o}=0.1,~{}{\rm and}~{}{\rm M}=10^{9}\,{\rm M}_{\odot}$. Dashed lines
are the isodensity contours with the logarithm of the corresponding density
denoted by the assigned number, while the thick black lines are the lines of
the poloidal magnetic field. The wind flow in the poloidal plane is along the
magnetic field lines, while the observers’ LoS (radial coordinates) correspond
to lines of 45 degree inclination; an increasing inclination angle $\theta$
corresponds to increasing abscissa intercepts for these lines. The white
circles and squares along with their connecting curves indicate
correspondingly the positions of formation of the Fe xxv and C iv transitions
for observers at different values of $\theta$. The ionization structure of the
wind can thus be translated into the spatial localization of specific ionic
species. With a judicious choice of $\theta\;(\simeq 50^{\circ})$ one can
obtain $r($Fe xxv$)\sim 5-40\;r_{S}$ and $r($C iv$)\sim 300-900\;r_{S}$
(assuming a single LoS) to produce absorption velocities for these transitions
consistent those of the BAL QSO APM 08279+0255.
In Fig. 4 we show the hydrogen equivalent columns of several ions of iron and
carbon as a function of the ionization parameter $\xi$ for a LoS at an angle
$\theta=50^{\circ}$ and for the ionizing spectrum given in Fig. 3b. This is a
figure equivalent to those for Seyferts shown in Fig. 2 but appropriate for a
BAL QSO. The figure shows also the corresponding wind velocity along this LoS
(red dashed line), while the shaded regions indicate the positions of maximum
column for Fe xxv and C iv, which are indicative of the observed velocity of
the corresponding ion.
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=224.03743pt,angle={-0}]{C-Fe_5_2_2_obs50.eps}&\end{array}$
Figure 4: Simulated distribution of local column densities $\Delta N_{H}$ (on
the left ordinate) and the outflow LoS velocity $v/c$ (on the right ordinate)
as a function of ionization parameter $\xi$ for (a) C and (b) Fe with
$\theta=50^{\circ}$. Vertical lines denote the ionization parameter where the
local emergent column is dominated primarily by C iv in (a) and Fe xxv in (b),
respectively, also showing the corresponding outflow velocity $v/c$. [See the
electronic edition of the Journal for a color version of this figure.]
### 4.4 The Characteristic Curves of Ionized MHD Winds
The previous two subsections have shown that, while the underlying velocity
structure of MHD winds may be very similar and their overall column depending
only on the dimensionless mass flux $\dot{m}_{o}$, their ionization structure
and the location of specific ions in these winds can be quite different
depending on their ionizing spectra. Under these (admittedly oversimplified)
assumptions one can present the kinematic properties of specific ionic species
of these winds into two diagrams that encapsulate their essential features
(namely velocity and column) as a function of the ionizing radiation spectral
properties and the observers’ LoS.
Such diagrams for the transitions C iv($\lambda\lambda 1548,1551$Å) and Fe xxv
are shown in Figs. 5a,b. The C iv characteristic curves are shown in blue
while those of Fe xxv in red. In Fig. 5a, appropriate for an observer at
$\theta=50^{\circ}$, we show: (i) The correlations between the C iv, Fe xxv
velocities and the X-ray photon index $\Gamma$ for $\alpha_{OX}=-2$ (solid
lines, left ordinate). (ii) The correlation between the C iv, Fe xxv
velocities and the source $\alpha_{OX}$ for an X-ray photon slope $\Gamma=2$
(dashed lines, right ordinate). The pairs in parentheses indicate the values
of ($\Gamma,\alpha_{OX}$) at each point.
We observe that the velocities of both C iv and Fe xxv depend more strongly on
$\alpha_{OX}$ than on $\Gamma$. However, for $\alpha_{OX}=-2$, as is the case
with APM 08279+0255, the Fe xxv velocity changes significantly with $\Gamma$
and in a way that is actually in agreement with the observations of Chartas et
al. (2009), given by the squares along the solid red curve. The C iv velocity
appears to be much less sensitive to $\Gamma$, however it is quite sensitive
to the value of $\alpha_{OX}$, ranging between $v\simeq 0.15c~{}{\rm
and}~{}0.01c$ as $\alpha_{OX}$ varies from $-2.1$ to $-1.6$, nicely covering
the observed range of the velocity of this transition between BAL QSO and
Seyferts.
In Fig. 5b we show the correlation between the LoS velocity of the specific
ions (i.e. C iv, Fe xxv) for a wind ionized by a source with
$\alpha_{OX}=2,~{}\Gamma=2$ (values appropriate for a BAL QSO), and the
corresponding hydrogen equivalent columns, $N_{H}$, with the LoS inclination
angle as a parameter along these curves. For these specific spectral
parameters, we see that both velocities increase with $\theta$ up to a
critical value different for each transition and then level-off as the
corresponding ions are produced in parts of the wind that it is still
accelerating. It is also worth noting that even for the specific spectral
distribution, appropriate for BAL QSOs, the velocity of C iv is quite small
for sufficiently small observer inclination angle, while at the same time the
corresponding column may be too small for the detection of such a feature.
Diagrams such as that of Fig. 5b for different ionizing spectra and wind
parameters will be useful for the determination of the wind parameter space
($N_{H},v$) at which specific features become observable. We expect to provide
several such diagrams for the most important such transitions in future works.
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=209.58255pt,angle={-0}]{cor.eps}&\includegraphics[keepaspectratio={false},width=209.58255pt,angle={-0}]{los1.eps}\end{array}$
Figure 5: (a) The dependence of the model wind LoS velocity (for
$\theta=50^{\circ}$) on $\Gamma$ (solid lines; left ordinate) and
$\alpha_{OX}$ (dashed lines; right ordinate). The ordered pairs at each point
are the values of ($\Gamma,\alpha_{OX}$) for C iv (blue lines) and Fe xxv (red
lines). The red squares indicate the change in the Fe-K absorption velocity
variations with $\Gamma$ for APM 08259+5255. (b) The model MHD wind
correlation between the LoS velocity of a given transition (blue for C iv, red
for Fe xxv) and the corresponding hydrogen equivalent column $\Delta
N_{H}$(ion) with the observer inclination angle $\theta$ as a parameter along
these curves (indicated by the numbers in degrees). [See the electronic
edition of the Journal for a color version of this figure.]
## 5 Wind Emission Properties
The models and computational setup outlined in the previous were motivated by
and dealt mainly with the absorption feature properties of accretion powered
sources, both galactic and extragalactic. However, besides the absorber
properties, the reprocessing of radiation onto these winds provides also
emission characteristics, both in continuum and in lines. These will be given
a first brief treatment in this section, which serves mainly as the road map
of future work. So in section 5.1 we discuss the effects of dust reprocessing
into continuum while in 5.2 reprocessing into lines and specifically that of
H$\alpha$.
### 5.1 Torus Structure and Infrared Emission
The issue of AGN torii and their importance in unification schemes was
discussed in the introduction. Also outlined there was the conundrum of the
disparity between their (statistically inferred) height-to-radius ratios
($h/R\simeq 1$) and the values implied by the ratio of their thermal to
Keplerian velocities ($h/R\simeq v_{\rm th}/v_{\rm K}\simeq 10^{-3}$). The
proposal that resolves this conundrum in a straightforward fashion is that of
Königl & Kartje (1994) who suggested that the torii are not structures in
hydrostatic equilibrium, but dynamical objects, in particular MHD winds
driven-off the outer environs of the AGN disks. Furthermore, in fitting the
observed IR emission as disk UV radiation reprocessed in these winds, they
concluded that the corresponding density of the reprocessing gas had to be
proportional to $1/r$ as in the winds discussed in §3.
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=224.03743pt,angle={-0}]{SpectrumIR2.eps}&\end{array}$
Figure 6: Model spectra resulting from reprocessing the BBB AGN continuum
(with peak at $\nu\simeq 10^{15}$ Hz) by winds with three different density
profiles $s=0.9$ (black), $s=1.0$ (red), $s=1.1$ (blue), that extend roughly
two orders of magnitude beyond the dust sublimation radius. [See the
electronic edition of the Journal for a color version of this figure.]
Similarly, Rowan-Robinson (1995) has shown that reprocessing the AGN continuum
luminosity by dust can provide good fits to their (nearly flat in $\nu
F_{\nu}$) IR spectra, provided that the dust density has a profile similar to
that used herein, namely proportional to $1/r$. There is no mystery here: Dust
distribution with the specific density profile reprocesses the same amount of
continuum flux per decade of radius and re-emits it in (roughly) black body
form; since the dust temperature in regions of increasing radius $r$ falls-off
like $T\propto r^{-1/2}$ the peak emission will also shift to lower
frequencies while emitting the same luminosity; the resulting spectrum would
then be “flat” in $\nu F_{\nu}$ units, consistent with the Spitzer
observations of a survey of QSO spectra (Netzer et al., 2007).
The precise spectrum determination depends, of course, on the details of dust
formation and properties and also on the details of the reprocessed spectrum
and radiative transfer, as the emission is not necessarily black body and as
the outer layers of the MHD wind reprocess not only the continuum luminosity
but also the IR emission of the inner “torus” regions. One of the more
important aspects of the IR and far-IR emission is the frequency at which the
spectrum turns over from “flat” to the Rayleigh-Jeans form ($\nu
F_{\nu}\propto\nu^{3}$), as this would provide an estimate of the outer torus
radius. Combined space (Herschel) and ground (ALMA) observations will be
instrumental in this respect. One should simply note here that the radial dust
temperature dependence, $T\propto r^{-1/2}$, implies that the number of
decades in radius over which dust reprocessing takes place is twice the number
of decades over which the AGN $\nu F_{\nu}$ spectrum is “flat”. In this
respect, there should be a synergy between AGN X-ray spectroscopy and far-IR
observations: The regions at which the far-IR emission turns over to the
Rayleigh-Jeans form (the “egde” of the torus) are comparable to those at which
the lowest $\xi$ ions should be located; the “edge” of the torus should then
manifest then also in the X-ray spectra as a steep drop in the X-ray column
with decreasing $\xi$. Finally, one should note that IR spectra similar to
those of Seyfert-2 galaxies are not expected in XRB, despite their similarity
of their winds; this is because the distance at which the dust sublimation
temperature ($T\sim 1000$ K) is reached in the wind of an XRB of $L=10^{37}$
erg s-1, is larger than $10^{12}$ cm, the typical XRB binary separation
distance.
Fig. 6 presents an oversimplified example of such a spectrum. In there we
present the reprocessing of the BBB spectrum of an AGN (the component that
peaks near $\nu\sim 10^{15}$ Hz), by dust which extends from the radius of
dust sublimation to the “edge” of the torus, a region that spans roughly two
decades in radius. The dust density in this region has a form $n(r)\propto
r^{-s}$ with $s=0.9,1,1.1$ (black, red and blue curves respectively) to
indicate the effect that different density profiles on the far-IR AGN spectra.
In this specific case the AGN is viewed “face-on” and the reprocessing is
effectively limited to $\theta\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}50^{\circ}$. The
details of the X-ray obscuration and its relation to IR emission, bearing
relation to the angular distribution of our solutions, can be improved with
combined X-ray – IR surveys such as that of (Rowan-Robinson et al., 2009).
### 5.2 H$\alpha$ Emission of MHD Winds
The issue of line emission in AGN is a subject far too broad to be included in
any detail in the present work. Nonetheless, to our surprise, we found that
the basic model we have been discussing so far is consistent with well known
features and correlations concerning the emission line properties of AGN and
XRB. So we do make a brief foray into this subject to touch upon two issues on
which our model seems to provide a novel insight.
One of the fundamental characteristics of the winds outlined in section 3 are
that their velocity fields are mainly Keplerian at low latitudes
($\theta\mathrel{\raise 2.15277pt\hbox{$>$}\mkern-14.0mu\lower
2.58334pt\hbox{$\sim$}}70^{\circ}$) while becoming (almost) radial at larger
ones (Contopoulos & Lovelace, 1994; Fukumura et al., 2010a). This particular
property allows for the possibility of exhibiting emission line features
associated both with disks and winds. In fact, there are observations of a
“double horn” H$\alpha$ line profile in certain AGN (Eracleous & Halpern,
1994), the tell-tale sign of disk emission. These have been modeled as
emission excited by the reprocessing of the AGN X-ray continuum on a ring-like
structure, whose radii and illumination were adjusted to provide agreement
with observations. The MHD wind models in their most simplistic incarnation,
namely that described §3, allow for such profiles for the H$\alpha$
transition, precisely because of the disk-like geometry of the flow at low
latitudes. Furthermore, the extent of the wind in the $\theta-$direction
allows also the calculation of the EW of this transition, which is found to be
consistent with observations (Fukumura & Kazanas in preparation).
The calculations outlined in §4 provide the photonionization of the entire
wind, from its smallest ($x\mathrel{\raise
2.15277pt\hbox{$>$}\mkern-14.0mu\lower 2.58334pt\hbox{$\sim$}}1$) to its
largest ($x\sim 10^{5}-10^{7}$) scales, set by the size of the accretion disk
that powers the source. With the wind ionization given, one can also produce
scalings and correlations involving properties of the corresponding emission
of lines of the photoionized plasma. While such an in-depth study is beyond
the scope of the present paper, some straightforward scalings can be produced
even at this early stage. With this in mind, we compute below the scaling of
one of the prominent transitions AGN, namely H$\alpha$ with the source
luminosity, given that such a relation has already been well documented
(Osterbrok, 1989) (we refrain focusing on the Ly$\alpha$ and C iv transitions
since, being resonant ones, demand a more careful treatment within the
specific model).
Quite generally, the luminosity of the H$\alpha$ transition produced in the
MHD winds discussed in the previous sections can be written as
$L_{{\rm H}\alpha}\simeq\alpha(T)\,n^{2}V\,\epsilon_{{\rm H}\alpha}$ (14)
where $\alpha(T)$ is the recombination coefficient, $n^{2}V$ the wind emission
measure and $\epsilon_{{\rm H}\alpha}\simeq 2$ eV, is the H$\alpha$ energy. As
noted in Eq. (5), the wind density is given by the expression
$n(r,\theta)\simeq n_{o}\tilde{n}(\theta)x^{-1}\dot{m}/M$, where
$\tilde{n}(\theta)\simeq e^{(\theta-\pi/2)/0.22}$. The normalization $n_{o}$
is set so that $n_{o}r_{o}\sigma_{T}\simeq 1$ for $\dot{m}\simeq 1$ in
spherical geometry. For the same value of the total mass flux, since the flow
in the MHD winds considered herein is concentrated near the equatorial plane,
the density normalization must be higher by a factor $A\simeq
1/\int_{0}^{\pi/2}\tilde{n}(\theta)\,{\rm sin}\theta\,d\theta\simeq 1/0.21$
with the density profile now reading $n(r,\theta)\simeq
n_{o}\,A\,\tilde{n}(\theta)x^{-1}\dot{m}/M$. Then the emission measure of the
wind reads
$\displaystyle n^{2}V$ $\displaystyle=$ $\displaystyle
2\pi\,n_{o}^{2}\;\frac{\dot{m}_{o}^{2}}{M^{2}}\;r_{o}^{3}\,M^{3}\,x_{m}\;2\int_{0}^{\pi/2}A^{2}\,\tilde{n}(\theta)^{2}\,{\rm
sin}\theta\,d\theta$ (15) $\displaystyle\simeq$ $\displaystyle
10\pi\,n_{o}^{2}r_{o}^{2}\sigma_{T}^{2}\;\frac{r_{o}}{\sigma_{T}^{2}}\;x_{m}\,\dot{m}_{o}^{2}\,M\simeq
7.1\times 10^{49}r_{o}\,\dot{m}^{2}\,M\,x_{m}$ (16)
where the extra factor of 2 in front of the integral takes care of the
emission by both hemispheres, the numerical value of the integral is $\simeq
2.5$, $\sigma_{T}$ is the Thomson cross section and we have taken for the
(normalized in units of $r_{S}$) maximum radius of the disk the value
$x_{m}\sim 10^{6}$, with $r_{o}\simeq r_{S}\simeq 3\times 10^{5}$ cm, the
Schwarzschild radius of one solar mass. Then, the H$\alpha$ line luminosity
will be
$L_{{\rm H}\alpha}\simeq\alpha(T)\,n^{2}V\,\epsilon_{{\rm H}\alpha}\simeq
2\cdot 10^{36}\,\dot{m}_{o}^{2}\,M~{}{\rm erg~{}s}^{-1}~{}.$ (17)
At the same time, the accretion luminosity will be
$L\simeq 10^{38}\;\eta\,\dot{m}_{o}\,M$ (18)
where $\eta\simeq 0.1$ is the accretion disk radiative efficiency ($\eta\simeq
0.3$ for neutron stars and $L\propto\dot{m}_{o}^{2}\,M$ in the case of ADAF).
The obvious point to note between Eqs (17) and (18) is that they both are
proportional to the object’s mass $M$; also for $\dot{m}_{o}<\alpha^{2}$
($\alpha$ is the accretion disk viscosity parameter) implying that the
accretion flow is in the ADAF regime, the ratio $R=L_{{\rm H}\alpha}/L$ of the
line to the bolometric accretion luminosity is independent of both
$\dot{m}_{o}$ and the mass $M$. This ratio depends only on the recombination
coefficient $\alpha(T)({\rm with~{}assumed~{}value}\sim 10^{-13}~{}{\rm
cm}^{3}~{}{s}^{-1}$ in Eq. (17) above) and the disk efficiency $\eta$ and has
a value $R\simeq 1/20-1/50$. The data indicating the correlation between the
H$\alpha$ line and the continuum luminosity at $\lambda 4800$ Å is shown in
Fig. 7. The data shown are those of 11.6 of Osterbrok (1989) with the abscissa
converted from flux to luminosity; this figure includes in addition the
H$\alpha$ luminosity of Cir X-1 (filled square; Johnston et al., 1999), a
galactic XRB; inclusion of this point in the figure is important because it
almost doubles the dynamic range of this relation which now covers eight
decades in the line luminosity $L_{{\rm H}\alpha}$. The H$\alpha$ luminosity
of this source falls slightly below the extrapolation of a linear relation
from AGN to XRB. A likely reason for this could be the reduced emissivity at
small values of $x$ (small radii) due to the higher gas temperature there.
Incidentally, for winds with the density profile of the Blandford & Payne
(1982) scaling, i.e. $n(x)\propto x^{-3/2}$, Eq. (17) would be proportional to
${\rm log}x_{m}$ instead of $x_{m}$ and the line luminosity would be roughly
$10^{5}$ times smaller than observed. We consider the straightforward, minimal
assumption way that this model accounts for the linear relation between the
H$\alpha$ and bolometric luminosities an indication of the validity of its
fundamental premises.
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=224.03743pt,angle={-0}]{L_Halpha-
L_bol.eps}&\end{array}$
Figure 7: The H$\alpha$ luminosity of a number of AGN (triangles) as a
function of their bolometric luminosity (taken to be their monochromatic
luminosity at $\lambda=4680$ Å, as given in Osterbrock 1989). The square
denotes the H$\alpha$ luminosity of Cir X-1 given in Johnston et al. (1999)
assuming the bolometric luminosity to be that of Eddington for a 1.4 solar
mass neutron star.
## 6 Summary, Discussion
In the previous sections we presented a broad strokes picture of the 2D AGN
structure, one that encompasses many decades in radius and frequency and
supplements the well known schematic of Urry & Padovani (1995) with an outflow
launched across the entire disk area and velocity roughly equal to the local
Keplerian velocity at each launch radius. These outflows are responsible for
the absorption features first detected in the AGN UV spectra and more recently
established also in their X-ray ones. As noted in the introduction, the goal
and the spirit of this paper is not to reproduce the detailed phenomenology of
specific AGN spectral bands, but provide an account of their most general and
robust trends; however, it purports to do so on the basis of a single, well
founded, semi-analytic model for outflows off accretion disks that involves a
small number of free parameters. In this same spirit, Fig. 8 presents a
schematic of our model; the radius is in logarithmic space and the shading is
indicative of the local column density in the spherical$-r$ direction, which
is constant, but has a strong dependence on the angle $\theta$. The gray lines
are illustrative of the magnetic field geometry with the corresponding scaling
of the wind velocity on each one relative to the fiducial one $v_{o}$ shown in
the figure.
As shown in section §3, these outflows have the interesting property that
their ionization and dynamical structures scale mainly with one parameter, the
dimensionless accretion rate $\dot{m}$. As such, they are applicable, in
principle, to all accretion powered sources from galactic accreting black
holes to the most luminous quasars. The mass of the object, $M$, simply
provides the overall scale of an object’s luminosity and size (and also a
characteristic temperature for the BBB), in a way similar to that proposed by
Boroson (2002).
$\begin{array}[]{cc}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,keepaspectratio={false},width=224.03743pt,angle={-0}]{schematic.eps}&\end{array}$
Figure 8: A schematic of the model presented in §3. The radius is shown in
logarithmic scale, with the solid gray lines representing the poloidal field
at each radius shown and the corresponding wind velocity relative to the
fiducial one $v_{o}$. The shading is proportional to the local column in the
$r-$direction which does not depend on $r$ but has a strong
$\theta-$dependence as required by AGN unification. At sufficiently large
distances, $r\mathrel{\raise 1.93748pt\hbox{$>$}\mkern-14.0mu\lower
2.32501pt\hbox{$\sim$}}10^{4}\,r_{o}$ the flow is sufficiently cool to be
molecular and acts as the torus required by AGN unification.
However, despite this economy of parameters, because of the inherently 2D
character of these winds, their appearance (and that of the AGN central
regions) depends quite significantly on the observer’s inclination angle
$\theta$, a desirable feature and in agreement with our notions of AGN
unification. Furthermore, and most importantly, as noted in section §4, the
ionization structure of these flows depends also on the spectrum of the
ionizing radiation. This dependence complicates the situation because it
breaks the overall wind flow scale invariance on the mass $M$. As noted
earlier, it is effectively the dependence of $\alpha_{OX}$ on luminosity which
is responsible for the differences in the absorption feature properties
between Seyferts and BAL QSOs. This dependence of $\alpha_{OX}$ on $L(2500$Å)
(and effectively on $\dot{M}$) suggests that eventually the wind ionization
properties may constitute a two rather than three parameter family. Finally,
the high ionization of the inner regions of these flows in XRB, naturally
accounts for the low velocities of the Fe-K features observed in galactic
sources.
The crucial and fundamental aspect of the underlying MHD wind model that
allows the broad consolidation of the very diverse observational phenomenology
of the previous sections “under the same roof” is their ability to produce
density profiles that decreases like $\sim 1/r$ with the radius. It is this
property that allows the ionization parameter to decrease with distance while
still providing sufficient column to allow the detection of both high and low
ionization ions in the AGN X-ray spectra. It is also this property that
controls the velocities $v$ of the Fe-K transitions in galactic sources,
Seyferts and BAL QSO, due to the relation between $v$ and $\xi$
($v^{2}\propto\xi$ for the scaling proposed herein; the BP82 scaling leads to
$v\propto\xi$ and therefore to much smaller velocity for a given ion, i.e. a
given value of $\xi$ – and also to a much smaller column). Furthermore, it is
this specific density profile of the wind that allows us to incorporate the
physics of the AGN torii within the context of the broader physics of
accretion onto the compact object, while at the same time producing IR spectra
in broad agreement with observation. This confluence of the AGN spectral
properties with the wind spacial structure indicates that AGN and XRB are
objects that span many decades in radius and frequency, despite the fact that
most of their luminosity is released within a few Schwarzschild radii.
The price to pay for a density distribution such as that proposed above is the
need to invoke winds whose mass flux increases with distance from the source
(and more specifically like $\sim r^{1/2}$). While this feature can be
accommodated by the choice of a parameter in the models of Contopoulos &
Lovelace (1994), it was given a rather transparent explanation in Blandford &
Begelman (1999), in terms of the dynamics of accretion. Interestingly, recent
Chandra spectroscopy (Brandt & Schulz, 2000; Behar et al., 2003; Neilsen,
Remillard & Lee, 2011) appear to support such a notion. At this point, it is
not obvious how theoretically compelling is this particular feature in the
general scheme of accretion properties. Why is this mass loss preferred to one
that would result in, for instance, the BP82 density profile? Are there AGN
with winds/accretion flows consistent with $n(r)\propto r^{-3/2}$? What
parameter determines which one is chosen by nature? These are pressing
questions for which we currently have no answers. However, the importance of
the specific mass flux dependence on $r$ in the interpretation of the AGN
X-ray and UV absorber properties will likely attract the attention of future
studies on this issue.
Clearly a presentation as broad as that above by necessity ignores a large
number of issues, both observational and theoretical each of which is in fact
a separate branch in the study of AGN physics. As such, we have ignored the
effects of radiation pressure, considered in much detail numerically by Proga
and collaborators (Proga, Stone, & Kallman, 2000; Proga, 2003; Proga &
Kallman, 2004) and semi-analytically by Murray et al. (1995). The effects of
radiation pressure in combination with those of the MHD winds discussed here
have been considered by Everett (2005) and also by Königl & Kartje (1994) in
discussing the AGN IR spectra. These works have shown the effects of radiation
pressure to be local, as they depend on the local flow opacity, thus breaking
the similarity of the angular dependence of the solutions. It was shown in
these works that radiation pressure “pushes” to open up the field lines to
produce a density dependence on $\theta$ different from that given in Fig. 2a
and hence would influence the population ratios of objects observed at a given
column density. However, the effects of radiation pressure do not affect the
dependence of wind mass flux on the distance ($\dot{m}\propto r^{1/2}$), which
is set by conditions on the disk; it is this property that determines the
radial dependence of the wind density, the property necessary to account for
their observed phenomenology.
As noted above, a radiation driven wind, while 2D in the region of launch, it
will appear radial at sufficiently large distance producing density profiles
$\simeq 1/r^{2}$ there. As appealing and compelling as the notion of radiation
driven winds is, there is little evidence for them, at least in the AGN and
XRB X-ray absorber spectra. Advocating radiation driven 2D winds across the
entire accretion disk, in a fashion similar to those of §3, appears difficult
because the photon field does not have enough momentum at these large
distances to drive a wind with the required mass flux. Interestingly the
$\phi-$component of the magnetic field, decreasing like $B_{\phi}\propto 1/r$
has precisely the momentum needed to drive a wind with $\dot{m}\propto
r^{1/2}$. In this respect one should bear in mind that these winds produce
most of the kinetic energy at small radii, most of the mass flux at large
radii and equal momentum per decade of radius.
Another issue that was only briefly touched upon in section 5, is that of line
emission. It is generally thought that the line emission in AGN comes from
clouds in pressure equilibrium with a hot intercloud medium, the result of the
X-ray heating thermal instability (Krolik, McKee & Tarter, 1981). Our simple
estimates, even though they have ignored this possibility, they nonetheless
provide an account for the observed correlation of the H$\alpha$ (a transition
with minimal radiative transfer nuances) with the AGN bolometric luminosity.
However, one should bear in mind that our model winds do allow for the
formation of such clouds at sufficiently small latitudes (below the Alfvén
surface) where the flow is close to hydrostatic equilibrium, i.e. under
conditions of a given pressure. We expect that past the Alfvén point, where
the flows are under conditions of given density, the formation of these clouds
will be less forthcoming. The issue of cloud or wind AGN line emission is an
issue that deserves more attention and study, given the smoothness of the AGN
lines profiles that implies a very large number clouds involved in this
process (Arav et al., 1997), but certainly beyond the scope of the present
paper.
Our treatment of the AGN IR emission has also glossed over much of what it
constitutes an altogether distinct subfield of AGN study. It has been
suggested that clouds are also involved in this component of the AGN spectra,
however with different properties and at radii larger than those of the UV and
optical line emitting clouds (Nenkova et al., 2008). This is clearly a point
that will have to be looked upon with greater care. A complicating factor in
this direction is that of star formation in the AGN environment, whose IR
contribution introduces additional parameters in such a study. Finally, with
respect to the IR and far-IR AGN spectra, we would like to point to the
synergy between X-ray spectroscopy and far-IR observations discussed in §5.1
that would help establish the consistency of this scheme across two very
different frequency bands.
Finally, we close with a few words on the “feedback” of our winds on the
surrounding medium, which likely provides the mass that eventually “feeds” the
AGN. First, the angular distribution of the flow, as determined by the
poloidal field structure, due to its collimation, interacts with only part of
the surrounding stellar cluster. Second, the energy flux of the winds
considered here, despite their increasing mass flux with radius, is still
dominated by the flux at small radii ($\dot{E}=\dot{m}v^{2}/2\propto
r^{-1/2}$); however, they do carry equal momentum per decade of radius
($\dot{P}=\dot{m}v\propto r^{0}$), a fact with potential feedback effects,
perhaps the subject of a future publication. Finally, the radiation effects of
the AGN are also limited in $\theta$ due precisely to the winds’ column
increase with this parameter. However, as shown in Tueller et al. (2008), the
fraction of Swift-BAT selected AGN at galactic latitude $|b|>15^{\circ}$ with
X-ray column $N_{H}>10^{22}\,{\rm cm}^{-2}$ decreases from $\simeq 0.5$ to
close to zero above luminosity $L\simeq 10^{44}\;{\rm erg\,s}^{-1}$.
Therefore, considering that the observed column pertains not only to the AGN
wind but also to the entire matter distribution along the LoS, the radiative
effects of high luminosity AGN may affect significantly the evolution of their
environment. Such constraints will have to be taken into account producing a
global evolutionary sequence for our models, which are beyond the scope of the
present paper.
Authors are grateful to Tim Kallman for insightful discussions and help with
running `XSTAR`. K.F. and D.K. would also like to thank G. Chartas, J. Turner,
L. Miller, S. Kraemer, T. Misawa for their constructive comments. D.K. would
like to thank R. Mushotzky, C. Reynolds and C. Miller for discussions and the
Astronomy Department of the University of Maryland for the hospitality during
his sabbatical visit there.
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|
arxiv-papers
| 2012-06-21T21:53:47 |
2024-09-04T02:49:32.043949
|
{
"license": "Public Domain",
"authors": "Demosthenes Kazanas (NASA/GSFC), Keigo Fukumura (NASA/CRESST), Ehud\n Behar (Technion, Israel), Ioannis Contopoulos (Academy of Athens, Greece),\n Chris Shrader (NASA/USRA)",
"submitter": "Keigo Fukumura",
"url": "https://arxiv.org/abs/1206.5022"
}
|
1206.5135
|
11institutetext: School of Computing Science, Newcastle University, Newcastle-
upon-Tyne, UK 22institutetext: Bioinformatics Support Unit, Newcastle
University, Newcastle-upon-Tyne, UK 33institutetext: School of Computer
Science, University of Manchester, UK 33email: phillip.lord@newcastle.ac.uk
# Three Steps to Heaven: Semantic Publishing in a Real World Workflow
Phillip Lord 11 Simon Cockell and Robert Stevens 2233
###### Abstract
Semantic publishing offers the promise of computable papers, enriched
visualisation and a realisation of the linked data ideal. In reality, however,
the publication process contrives to prevent richer semantics while
culminating in a ‘lumpen’ PDF. In this paper, we discuss a web-first approach
to publication, and describe a three-tiered approach which integrates with the
existing authoring tooling. Critically, although it adds limited semantics, it
does provide value to all the participants in the process: the author, the
reader and the machine.
License: This work is licensed under a Creative Commons Attribution 3.0
Unported License. http://creativecommons.org/licenses/by/3.0/. It is also
available at http://www.russet.org.uk/blog/2012/04/three-steps-to-heaven/
## 1 Introduction
The publishing of both data and narratives on those data are changing
radically. Linked Open Data and related semantic technologies allow for
semantic publishing of data. We still need, however, to publish the narratives
on that data and that style of publishing is in the process of change; one of
those changes is the incorporation of semantics [1, 2, 3]. The idea of
semantic publishing is an attractive one for those who wish to consume papers
electronically; it should enhance the richness of the computational component
of papers [2]. It promises a realisation of the vision of a next generation of
the web, with papers becoming a critical part of a linked data environment [1,
4], where the results and naratives become one.
The reality, however, is somewhat different. There are significant barriers to
the acceptance of semantic publishing as a standard mechanism for academic
publishing. The web was invented around 1990 as a light-weight mechanism for
publication of documents. It has subsequently had a massive impact on society
in general. It has, however, barely touched most scientific publishing; while
most journals have a website, the publication process still revolves around
the generation of papers, moving from Microsoft Word or LaTeX [5], through to
a final PDF which looks, feels and is something designed to be printed onto
paper111This includes conferences dedicated to the web and the use of web
technologies.. Adding semantics into this environment is difficult or
impossible; the content of the PDF has to be exposed and semantic content
retro-fitted or, in all likelihood, a complex process of author and publisher
interaction has to be devised and followed. If semantic data publishing and
semantic publishing of academic narratives are to work together, then academic
publishing needs to change.
In this paper, we describe our attempts to take a commodity publication
environment, and modify it to bring in some of the formality required from
academic publishing. We illustrate this with three exemplars—different kinds
of knowledge that we wish to enhance. In the process, we add a small amount of
semantics to the finished articles. Our key constraint is the desire to add
value for all the human participants. Both authors and readers should see and
recognise additional value, with the semantics a useful or necessary byproduct
of the process, rather than the primary motivation. We characterise this
process as our “three steps to heaven”, namely:
* •
make life better for the machine to
* •
make life better for the author to
* •
make life better for the reader
While requiring additional value for all of these participants is hard, and
places significant limitations on the level of semantics that can be achieved,
we believe that it does increase the likelihood that content will be generated
in the first place, and represents an attempt to enable semantic publishing in
a real-world workflow.
## 2 Knowledgeblog
The knowledgeblog project stemmed from the desire for a book describing the
many aspects of ontology development, from the underlying formal semantics, to
the practical technology layer and, finally, through to the knowledge domain
[6]. However, we have found the traditional book publishing process
frustrating and unrewarding. While scientific authoring is difficult in its
own right, our own experience suggests that the _publishing_ process is
extremely hard-work. This is particularly so for multi-author collected works
which are often harder for the editor than writing a book “solo”. Finally, the
expense and hard copy nature of academic books means that, again in our
experience, few people read them.
This contrasts starkly with the web-first publication process that has become
known as blogging. With any of a number of ready made platforms, it is
possible for authors with little or no technical skill, to publish content to
the web with ease. For knowledgeblog (“kblog”), we have taken one blogging
engine, WordPress [7], running on low-end hardware, and used it to develop a
multi-author resource describing the use of ontologies in the life sciences
(our main field of expertise). There are also kblogs on
bioinformatics222http://bioinformatics.knowledgeblog.org and the Taverna
workflow environment333http://taverna.knowledgeblog.org [8]. We have
previously described how we addressed some of the social aspects, including
attribution, reviewing and immutablity of articles[6].
As well as delivering content, we are also using this framework to investigate
_semantic academic publishing_ , investigating how we can enhance the machine
interpretability of the final paper, while living within the key constraint of
making life (slightly) better for machine, author and reader without adding
complexity for the human participants.
Scientific authors are relatively conservative. Most of them have well-
established toolsets and workflows which they are relatively unwilling to
change. For instance, within the kblog project, we have used workshops to
start the process of content generation. For our initial meeting, we gave
little guidance on authoring process to authors, as a result of which most
attempted to use WordPress directly for authoring. The WordPress editing
environment is, however, web-based, and was originally designed for editing
short, non-technical articles. It appeared to not work well for most
scientists.
The requirements that authors have for such ‘scientific’ articles are
manifold. Many wish to be able to author while offline (particularly on trains
or planes). Almost all scientific papers are multi-author, and some degree of
collaboration is required. Many scientists in the life sciences wish to author
in Word because grant bodies and journals often produce templates as Word
documents. Many wish to use LaTeX, because its idiomatic approach to
programming documents is unreplicable with anything else. Fortunately, it is
possible to induce WordPress to accept content from many different authoring
tools, including Word and LaTeX[6].
As a result, during the kblog project, we have seem many different workflows
in use, often highly idiosyncratic in nature. These include:
Word/Email:
Many authors write using MS Word and collaborate by emailing files around.
This method has a low barrier to entry, but requires significant social
processes to prevent conflicting versions, particularly as the number of
authors increases.
Word/Dropbox:
For the taverna kblog, authors wrote in Word and collaborated with
Dropbox.444http://www.dropbox.com This method works reasonably well where many
authors are involved; Dropbox detects conflicts, although it cannot prevent or
merge them.
Asciidoc/Dropbox:
Used by the authors of this paper.
Asciidoc555http://www.methods.co.nz/asciidoc/ is relatively simple, somewhat
programmable and accessible. Unlike LaTeX which can be induced to produce HTML
with effort, asciidoc is designed to do so.
Of these three approaches probably the Word/Dropbox combination is the the
most generally used.
From the readers perspective, a decision that we have made within
knowledgeblog is to be “HTML-first”. The initial reasons for this were
entirely practical; supporting multiple toolsets is hard, particularly if any
degree of consistency is to be maintained; the generation of the HTML is at
least partly controlled by the middleware – WordPress in kblog’s case. As well
as enabling consistency of presentation, it also, potentially, allows us to
add additional knowledge; it makes semantic publication a possibility.
However, we are aware that knowledgeblog currently scores rather badly on what
we describe as the “bath-tub test”; while exporting to PDF or printing out is
possible, the presentation is not as “neat” as would be ideal. In this regard
(and we hope only in this regard), the knowledgeblog experience is limited.
However, increasingly, readers are happy and capable of interacting with
material on the web, without print outs.
From this background and aim, we have drawn the following requirements:
1. 1.
The author can, as much as possible, remain within familiar authoring
environments;
2. 2.
The representation of the published work should remain extensible to, for
instance, semantic enhancements;
3. 3.
The author and reader should be able to have the amount of “formal” academic
publishing they need;
4. 4.
Support for semantic publishing should be gradual and offer advantages for
author and reader at all stages.
We describe how we have achieved this with three exemplars, two of which are
relatively general in use, and one more specific to biology. In each case, we
have taken a slightly different approach, but have fulfilled our primary aim
of making life better for machine, author and reader.
## 3 Representing Mathematics
The representation of mathematics is a common need in academic literature.
Mathematical notation has grown from a requirement for a syntax which is
highly expressive and relatively easy to write. It presents specific
challenges because of its complexity, the difficulty of authoring and the
difficulty of rendering, away from the chalk board that is its natural home.
Support for mathematics has had a significant impact on academic publishing.
It was, for example, the original motivation behind the development of TeX
[9], and it still one of the main reasons why authors wish to use it or its
derivatives. This is to such an extent that much mathematics rendering on the
web is driven by a TeX engine somewhere in the process. So MediaWiki (and
therefore Wikipedia), Drupal and, of course, WordPress follow this route. The
latter provides plugin support for TeX markup using the wp-latex plugin [10].
Within kblog, we have developed a new plugin called mathjax-latex [11]. From
the kblog author’s perspective these two offer a similar interface –
differences are, therefore, described later.
Authors write their mathematics directly as TeX using one of the four markup
syntaxes. The most explicit (and therefore least likely to happen
accidentally) is through the use of
“shortcodes”.666http://codex.wordpress.org/Shortcode These are a HTML-like
markup originating from some forum/bulletin board systems. In this form an
equation would be entered as `[latex]e=mc^2[/latex]`, which would be rendered
as “$e=mc^{2}$”. It is also possible to use three other syntaxes which are
closer to math-mode in TeX: `$$e=mc^2$$`, `$latex e=mc^2$`, or `\[e=mc^2\]`.
From the authorial perspective, we have added significant value, as it is
possible to use a variety of syntaxes, which are independent of the authoring
engine. For example, a TeX-loving mathematician working with a Word-using
biologist can still set their equations using TeX syntax; although Word will
not render these at authoring time but, in practice, this causes few problems
for such authors, who are experienced at reading TeX. Within an LaTeX workflow
equations will be renderable both locally with source compiled to PDF, and
published to WordPress.
There is also a W3C recommendation, MathML for the representation and
presentation of mathematics. The kblog environment also supports this. In this
case, the equivalent source appears as follows:
<math>
<mrow>
<mi>E</mi>
<mo>=</mo>
<mrow>
<mi>m</mi>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</math>
One problem with the MathML representation is obvious: it is very long-winded.
A second issue, however, is that it is hard to integrate with existing
workflows; most of the publication workflows we have seen in use will on
recognising an angle bracket turn it into the equivalent HTML entity. For some
workflows (LaTeX, asciidoc) it is _possible_ , although not easy, to prevent
this within the native syntax.
It is also possible to convert from Word’s native OMML (“equation editor”) XML
representation to MathML, although this does not integrate with Word’s native
blog publication workflow. Ironically, it is because MathML shares an XML
based syntax with the final presentation format (HTML) that the problem
arises. The shortcode syntax, for example, passes straight-through most of the
publication frameworks to be consumed by the middleware. From a pragmatic
point of view, therefore, supporting shortcodes and TeX-like syntaxes has
considerable advantages.
For the reader, the use of mathjax-latex has significant advantages. The
default mechanism within WordPress uses a math-mode like syntax `$latex
e=mc^2$`. This is rendered using a TeX engine into an image which is then
incorporated and linked using normal HTML capabilities. This representation is
opaque and non-semantic; it has significant limitations for the reader. The
images are not scalable – zooming in cases severe pixalation; the background
to the mathematics is coloured inside the image, so does not necessarily
reflect the local style.
Kblog, however, uses the MathJax library[12]; this has a number of significant
advantages for the reader. First, where the browser supports them, MathJax
uses webfonts to render the images; these are scalable, attractive and
standardized. Where they are not available, MathJax can fall-back to bitmapped
fonts. The reader can also access additional functionality: clicking on an
equation will raise a zoomed in popup; while the context menu allows access to
a textual representation either as TeX or MathML irrespective of the form that
the author used. This can be cut-and-paste for further use. Kblog uses the
MathJax library[12] to render the underlying TeX directly on the client.
Our use of MathJax provides no significant disadvantages to the middleware
layers. It is implemented in JavaScript and runs in most environments.
Although, the library is fairly large ($>$100Mb), but is available on a CDN so
need not stress server storage space. Most of this space comes from the bit-
mapped fonts which are only downloaded on-demand, so should not stress web
clients either. It also obviates the need for a TeX installation which wp-
latex may require (although this plugin can use an external server also).
At face value, mathjax-latex necessarily adds very little semantics to the
maths embedded within documents. The maths could be represented as
`$$E=mc^2$$`, `\(E=mc^2\)]` or
<math> <mrow> <mi>E</mi> <mo>=</mo> <mrow> <mi>m</mi>
<msup> <mi>c</mi><mn>2</mn> </msup>
</mrow> </mrow> </math>
So, we have a heterogenous representation for identical knowledge. However, in
practice, the situation is much better than this. The author of the work
created these equations and has then read them, transformed by MathJax into a
rendered form. If MathJax has failed to translate them correctly, in line with
the author’s intention, or if it has had some implications for the text in
addition to setting the intended equations (if the TeX style markup appears
accidentally elsewhere in the document), the author is likely to have seen
this and fixed the problem. Someone wishing, for example, to extract all the
mathematics as MathML from these documents computationally, therefore, knows:
* •
that the document contains maths as it imports MathJax
* •
that MathJax is capable of identifying this maths correctly
* •
that equations can be transformed to MathML using MathJax777This is assuming
MathJax works correctly in general. The authors and readers are checking the
rendered representation. It is possible that an equation would render
correctly on screen, but be rendered to MathML inaccurately.
So, while our publication environment does not result directly in lower level
of semantic heterogeneity, it does provide the data and the tools to enable
the computational agent to make this transformation. While this is imperfect,
it should help a bit. In short, we provide a practical mechanism to identify
text containing mathematics and a mechanism to transform this to a single,
standardised representation.
## 4 Representing References
Unlike mathematics, there is no standard mechanism for reference and in-text
citation, but there are a large number of tools for authors such as BibTeX,
Mendeley [13] or EndNote. As a result of this, the integration with existing
toolsets is of primary importance, while the representation of the in-text
citations is not, as it should be handled by the tool layer anyway.
Within kblog, we have developed a plugin called
kcite.888http://wordpress.org/extend/plugins/kcite/ For the author, citations
are inserted using the syntax:
`[cite]10.1371/journal.pone.0012258[/cite]`.
The identifier used here is a DOI, or digital object identifier and, is widely
used within the publishing and library industry. Currently, kcite supports
DOIs minted by either CrossRef999http://wordpress.org/extend/plugins/kcite/ or
DataCite101010http://www.datacite.org/ (in practice, this means that we
support the majority of DOIs). We also support identifiers from
PubMed111111http://www.ncbi.nlm.nih.gov/pubmed/ which covers most biomedical
publications and arXiv,121212http://arxiv.org/ the physics (and other
domains!) preprints archive, and we now have a system to support arbitrary
URLs. Currently, authors are required to select the identifier where it is not
a DOI.
We have picked this “shortcode” format for similar reasons as described for
maths; it is relatively unambiguous, it is not XML based, so passes through
the HTML generation layer of most authoring tools unchanged and is explicitly
supported in WordPress, bypassing the need for regular expressions and later
parsing. It would, however, be a little unwieldy from the perspective of the
author. In practice, however, it is relatively easy to integrate this with
many reference managers. For example, tools such as Zotero [14] and Mendeley
use the Citation Style Language, and so can output kcite compliant citations
with the following slightly elided code:
<citation>
<layout prefix="[cite]" suffix="[/cite]"
delimiter="[/cite] [cite]">
<text variable="DOI"/>
</layout>
</citation>
We do not yet support LaTeX/BibTeX citations, although we see no reason why a
similar style file should not be supported. We do, however, support BibTeX-
formatted files: the first author’s preferred editing/citation environment is
based around these with Emacs, RefTeX, and asciidoc. While this is undoubtedly
a rather niche authoring environment, the (slightly elided) code for
supporting this demonstrates the relative ease with which tool chains can be
induced to support kcite:
(defadvice reftex-format-citation (around phil-asciidoc-around activate)
(if phil-reftex-citation-override
(setq ad-return-value (phil-reftex-format-citation entry format))
ad-do-it))
(defun phil-reftex-format-citation( entry format )
(let ((doi (reftex-get-bib-field "doi" entry)))
(format "pass:[[cite source=’doi’\\]%s[/cite\\]]" doi)))
The key decision with kcite from the authorial perspective is to ignore the
reference list itself and focus only on in-text citations, using public
identifiers to references. This simplifies the tool integration process
enormously, as this is the only data that needs to pass from the author’s
bibliographic database onward. The key advantage for authors here is two-fold:
they are not required to populate their reference metadata for themselves, and
this metadata will update if it changes. Secondly, the identifiers are
checked; if they are wrong, the authors will see this straightforwardly as the
entire reference will be wrong. Adding DOIs or other identifiers moves from
becoming a burden for the author to becoming a specific advantage.
While supporting multiple forms of reference identifier (CrossRef DOI,
DataCite DOI, arXiv and PubMed ID) provides a clear advantage to the author,
it comes at considerable cost. While it is possible to get metadata about
papers from all of these sources, there is little commonality between them.
Moreover, resolving this metadata requires one outgoing HTTP request131313In
practice, it is often more; DOI requests, for instance, use 303 redirects. per
reference, which browser security might or might not allow.
So, while the presentation of mathematics is performed largely on the client,
for reference lists the kcite plugin performs metadata resolution and data
integration on the server. A caching functionality is provided, storing this
metadata in the WordPress database. The bibliographic metadata is finally
transferred to the client encoded as JSON, using asynchronous call-backs to
the server.
Finally, this JSON is rendered using the citeproc-js library on the client. In
our experience, this performs well, adding to the readers’ experience; in-text
citations are initially shown as hyperlinks; rendering is rapid, even on aging
hardware, and finally in-text citations are linked both to the bibliography
and directly through to the external source. Currently, the format of the
reference list is fixed, however, citeproc-js is a generalised reference
processor, driven using CSL141414http://citationstyles.org/. This makes it
straight-forward to change citation format, at the option of the reader,
rather than the author or publisher. Both the in-text citation and
bibliography support outgoing links direct to the underlying
resources151515Where the identifier allows – PubMed IDs redirect to PubMed..
As these links have been used to gather metadata, they are likely to be
correct. While these advantages are relatively small currently, we believe
that the use of JavaScript rendering over a linked references can be used to
add further reader value in future.
For the computational agent wishing to consume bibliographic information, we
have added significant value compared to the pre-formatted HTML reference
list. First, all the information required to render the citation is present in
the in-text citation next to the text that the authors intended. A
computational agent can, therefore, ignore the bibliography list itself
entirely. These primary identifiers are, again, likely to be correct because
the authors now need them to be correct for their own benefit.
Should the computational agent wish, the (denormalised) bibliographic data
used to render the bibliography is actually available, present in the
underlying HTML as a JSON string. This is represented in a homogeneous format,
although, of course, represents our (kcite’s) interpretation of the primary
data.
A final, and subtle, advantage of kcite is that the authors can only use
public metadata, and not their own. If they use the correct primary
identifier, and still get an incorrect reference, it follows that the public
metadata must be incorrect161616Or, we acknowledge, that kcite is broken!.
Authors and readers therefore must ask the metadata providers to fix their
metadata to the benefit of all. This form of data linking, therefore, can even
help those who are not using it.
### 4.1 Microarray Data
Many publications require that papers discussing microarray experiments lodge
their data in a publically available resource such as ArrayExpress [15].
Authors do this placing an ArrayExpress identifier which has the form
`E-MEXP-1551`. Currently, adding this identifier to a publication, as with
adding the raw data to the repository is no direct advantage to the author,
other than fulfilment of the publication requirement. Similarly, there is no
existing support within most authoring environments for adding this form of
reference.
For the knowledgeblog-arrayexpress
plugin,171717http://knowledgeblog.org/knowledgeblog-arrayexpress therefore, we
have again used a shortcode representation, but allowed the author to
automatically fill metadata, direct from ArrayExpress. So a tag such as:
`[aexp id="E-MEXP-1551"]species[/aexp]`
will be replaced with Saccharomyces cerevisiae, while:
`[aexp id="E-MEXP-1551"]releasedate[/aexp]`
will be replaced by “2010-02-24”. While the advantage here is small, it is
significant. Hyperlinks to ArrayExpress are automatic, authors no longer need
to look up detailed metadata. For metadata which authors are likely to know
anyway (such as Species), the automatic lookup operates as a check that their
ArrayExpress ID is correct. As with references(see Section References), the
use of an identifier becomes an advantage rather than a burden to the authors.
Currently, for the reader there is less significant advantage at the moment.
While there is some value to the author of the added correctness stemming from
the ArrayExpress identifier. However, knowledgeblog-arrayexpress is currently
under-developed, and the added semantics that is now present could be used
more extensively. The unambiguous knowledge that:
`[aexp id="E-MEXP-1551"]species[/aexp]`
represents a species would allow us, for example, to link to the NCBI taxonomy
database.181818http://www.ncbi.nlm.nih.gov/Taxonomy/
Likewise, advantage for the computational agent from knowledgeblog-
arrayexpress is currently limited; the identifiers are clearly marked up, and
as the authors now care about them, they are likely to be correct. Again,
however, knowledgeblog-arrayexpress is currently under developed for the
computational agent. The knowledge that is extracted from ArrayExpress could
be presented within the HTML generated by knowledgeblog-arrayexpress, whether
or not it is displayed to the reader for, essentially no cost. By having an
underlying shortcode representation, if we choose to add this functionality to
knowledgeblog-arrayexpress, any posts written using it would automatically
update their HTML. For the text-mining bioinformatician, even the ability to
unambiguously determine that a paper described or used a data set relating to
a specific species using standardised nomenclature191919the standard
nomenclature was only invented in 1753 and is still not used universally.
would be a considerable boon.
## 5 Discussion
Our approach to semantic enrichment of articles is a measured and evolutionary
approach. We are investigating how we can increase the amount of knowledge in
academic articles presented in a computationally accessible form. However, we
are doing so in an environment which does not require all the different
aspects of authoring and publishing to be over-turned. More over, we have
followed a strong principle of semantic enhancement which offers advantages to
both reader and author immediately. So, adding references as a DOI, or other
identifier, ‘automagically’ produces an in text citation and a nicely
formatted reference list: that the reference list is no longer present in the
article, but is a visualisation over linked data; that the article itself has
become a first class citizen of this linked data environment is a happy by-
product.
This approach, however, also has disadvantages. There are a number of semantic
enhancements which we could make straight-forwardly to the knowledgeblog
environment that we have not; the principles that we have adopted requires
significant compromise. We offer here two examples.
First, there has been significant work by others on CiTO [16] – an ontology
which helps to describe the relationship between the citations and a paper.
Kcite lays the ground-work for an easy and straight-forward addition of CiTO
tags surrounding each in-text citation. Doing so, would enable increased
machine understandability of a reference list. Potentially, we could use this
to the advantage to the reader also: we could distinguish between reviews and
primary research papers; highlight the authors’ previous work; emphasise older
papers which are being refuted. However, to do this requires additional
semantics from the author. Although these CiTO semantic enhancements would be
easy to insert directly using the shortcode syntax, most authors will want to
use their existing reference manager which will not support this form of
semantics; even if it does, the author themselves gain little advantage from
adding these semantics. There are advantages for the reader, but in this case
not for both author and reader. As a result, we will probably add such support
to kcite; but, if we are honest, find it unlikely that when acting as content
authors, we will find the time to add this additional semantics.
Second, our presentation of mathematics could be modified to automatically
generate MathML from any included TeX markup. The transformation could be
performed on the server, using MathJax; MathML would still be rendered on the
client to webfonts. This would mean that any embedded maths would be
discoverable because of the existence of MathML, which is a considerable
advantage. However, neither the reader nor the author gain any advantage from
doing this, while paying the cost of the slower load times and higher server
load that would result from running JavaScript on the server. More over, they
would pay this cost regardless of whether their content were actually being
consumed computationally. As the situation now stands, the computational user
needs to identify the insert of MathJax into the web page, and then transform
the page using this library, none of which is standard. This is clearly a
serious compromise, but we feel a necessary one.
Our support for microarrays offers the possibility of the most specific and
increased level of semantics of all of our plugins. Knowledge about a species
or a microarray experimental design can be precisely represented. However,
almost by definition, this form of knowledge is fairly niche and only likely
to be of relevance to a small community. However, we do note that the
knowledgeblog process based around commodity technology does offer a
publishing process that can be adapted, extended and specialised in this way
relatively easily. Ultimately the many small communities that make up the
long-tail of scientific publishing adds up to one large one.
## 6 Conclusion
Semantic publishing is a desirable goal, but goals need to be realistic and
achievable. to move towards semantic publishing in kblog, we have tried to put
in place an approach that gives benefit to readers, authors and computational
interpretation. As a result, at this stage, we have light semantic publishing,
but with small, but definite benefits for all.
Semantics give meaning to entities. In kblog, we have sought benefit by
“saying” within the kblog environment that entity _x_ is either maths, a
citation or a microarray data entity reference. This is sufficient for the
kblog infra-structure to “know what to do” with the entity in question.
Knowing that some publishable entity is a “lump” of maths tells the infra-
structure how to handle that entity: the reader has benefit from it looking
like maths; the author has benefit by not having to do very much; and the
infra-structure knows what to do. In addition, this approach leaves in hooks
for doing more later.
It is not necessarily easy to find compelling examples that give advantages
for all steps. Adding in CiTO attributes to citations, for instance, has
obvious advantages for the reader, but not the author. However, advantages may
be indirect; richer reader semantics may give more readers and thus more
citations—the thing authors appreciate as much as the act of publishing
itself. It is, however, difficult to imagine how such advantages can be
conveyed to the author at the point of writing. It is easy to see the
advantages of semantic publishing for readers, as a community we need to pay
attention to advantages to the authors. Without these “carrots”, we will only
have “sticks” and authors, particularly technically skilled ones, are highly
adept at working around sticks.
## References
* [1] Shadbolt, N., Hall, W., Berners-Lee, T.: The semantic web revisited. Intelligent Systems, IEEE 21(3) (2006) 96–101
* [2] Shotton, D.: Semantic publishing: the coming revolution in scientific journal publishing. Learned Publishing 22(2) (2009) 85–94
* [3] Shotton, D., Portwin, K., Klyne, G., Miles, A.: Adventures in semantic publishing: exemplar semantic enhancements of a research article. PLoS computational biology 5(4) (2009) e1000361
* [4] Bizer, C., Heath, T., Berners-Lee, T.: Linked data-the story so far. International Journal on Semantic Web and Information Systems (IJSWIS) 5(3) (2009) 1–22
* [5] Landport, L.: The LaTeX book. Adison wesley, Reading, MA (1984)
* [6] Lord, P., Cockell, S., Swan, D.C., Stevens, R.: The ontogenesis knowledgeblog: Lightweight publishing about semantics, with lightweight semantic publishing. In: Semantic Web Technologies for Libraries and Readers. (2011)
* [7] Wordpress: http://www.wordpress.org.
* [8] Hull, D., Wolstencroft, K., Stevens, R., Goble, C., Pocock, M.R., Li, P., Oinn, T.: Taverna: a tool for building and running workflows of services. Nucleic Acids Res 34(Web Server issue) (Jul 2006) 729–732
* [9] Knuth, D.E.: The TeX Book. 3rd edition edn. Adison Wesley, Reading, MA (1986)
* [10] WP Latex: http://wordpress.org/extend/plugins/wp-latex/.
* [11] Mathjax-Latex: http://wordpress.org/extend/plugins/mathjax-latex/.
* [12] Mathjax: http://www.mathjax.org.
* [13] Mendeley: http://www.mendeley.org.
* [14] Zotero: http://www.zotero.org.
* [15] Brazma, A., Parkinson, H., Sarkans, U., Shojatalab, M., Vilo, J., Abeygunawardena, N., Holloway, E., Kapushesky, M., Kemmeren, P., Lara, G.G., Oezcimen, A., Rocca-Serra, P., Sansone, S.A.: ArrayExpress- public repository for microarray gene expression data at the EBI. Nucleic Acids Research 31(1) (2003) 68–71
* [16] Shotton, D.: CiTO, the Citation Typing Ontology. Journal of Biomedical Semantics 1(Suppl 1) (2010) S6
|
arxiv-papers
| 2012-06-22T12:56:58 |
2024-09-04T02:49:32.059854
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Phillip Lord and Simon Cockell and Robert Stevens",
"submitter": "Phillip Lord Dr",
"url": "https://arxiv.org/abs/1206.5135"
}
|
1206.5158
|
# Direct CP Violation in Charm Decays due to Left-Right Mixing
Chuan-Hung Chen1,2111Email: physchen@mail.ncku.edu.tw, Chao-Qiang
Geng3,2222Email: geng@phys.nthu.edu.tw and Wei Wang4333Email:wwang@ihep.ac.cn
1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
2National Center for Theoretical Sciences, Hsinchu 300, Taiwan
3 Department of Physics, National Tsing Hua University, Hsinchu 300, Taiwan
4 Department of Physics, Huazhong University of Science and Technology, Wuhan
430074, China
###### Abstract
Motivated by the $3.8\sigma$ deviation from no CP violation hypothesis for the
CP asymmetry (CPA) difference between $D^{0}\to K^{+}K^{-}$ and
$D^{0}\to\pi^{+}\pi^{-}$, reported recently by LHCb and CDF, we investigate
the CP violating effect due to the left-right (LR) mixing in the general LR
symmetric model. In particular, in the non-manifest LR model we show that the
large CPA difference could be explained, while the constraints from
$(\epsilon^{\prime}/\epsilon)_{K}$ and $D^{0}$-$\bar{D}^{0}$ are satisfied.
In the standard model (SM), we expect that the CP asymmetries (CPAs) in
$D^{0}$ decays, defined by
$A_{CP}(D^{0}\to f)\equiv\frac{\Gamma(D^{0}\to f)-\Gamma(\bar{D}^{0}\to
f)}{\Gamma(D^{0}\to f)+\Gamma(\bar{D}^{0}\to
f)}\,,~{}~{}~{}(f=K^{+}K^{-}\,,~{}\pi^{+}\pi^{-})$ (1)
should be vanishingly small, and therefore an observation of a large CPA in
the charm sector clearly indicates physics beyond the SM. Recently, both LHCb
Aaij:2011in and CDF CDF collaborations have seen a large difference between
the time-integrated CPAs in the decays $D^{0}\to K^{+}K^{-}$ and
$D^{0}\to\pi^{+}\pi^{-}$, $\Delta A_{CP}\equiv A_{CP}(D^{0}\to
K^{+}K^{-})-A_{CP}(D^{0}\to\pi^{+}\pi^{-})$, given by
$\displaystyle\Delta A_{CP}$ $\displaystyle=$ $\displaystyle(-0.82\pm
0.21(\text{stat.})\pm 0.11(\text{sys.}))\%\,~{}~{}(\text{LHCb})\,,$ (2)
$\displaystyle=$ $\displaystyle(-0.62\pm 0.21(\text{stat.})\pm
0.10(\text{sys.}))\%\,~{}~{}(\text{CDF})\,,$
based on 0.62 fb-1 and 9.7 fb-1 of data, respectively. By combing the above
results with fully uncorrelated uncertainties, one obtains the average value
CDF
$\displaystyle\Delta A^{\rm avg}_{CP}$ $\displaystyle=$
$\displaystyle(-0.67\pm 0.16)\%\,,$ (3)
which is about $3.8\sigma$ away from zero.
As the time dependent CPA involves both direct and indirect parts, $i.e.$
$A_{CP}^{dir}(D^{0}\to f)$ and $A_{CP}^{ind}(D^{0}\to f)$, one gets
Aaij:2011in
$\displaystyle\Delta A_{CP}$ $\displaystyle\simeq$ $\displaystyle\Delta
A_{CP}^{dir}+(9.8\pm 0.3)\%A_{CP}^{ind}\,,$ (4)
where $\Delta A^{dir}_{CP}\equiv A_{CP}(D^{0}\to
K^{+}K^{-})-A_{CP}(D^{0}\to\pi^{+}\pi^{-})$ and $A_{CP}^{ind}\equiv
A_{CP}^{ind}(D^{0}\to f)$ is universal for $f=K^{+}K^{-}$ and $\pi^{+}\pi^{-}$
and less than $0.3$% due to the mixing parameters. It is clear that the
average value in Eq. (3) is dominated by the difference of the direct CP
asymmetries, $\Delta A_{CP}^{dir}$.
In order to have a nonzero direct CPA, two amplitudes $A_{1}$ and $A_{2}$ with
both nontrivial weak and strong phase differences, $\theta_{W}$ and
$\delta_{S}$, are essential, giving the CPA
$\displaystyle A_{CP}(D^{0}\to f)$ $\displaystyle=$
$\displaystyle{-2|A_{1}||A_{2}|\sin\theta_{W}\sin\delta_{S}\over|A_{1}|^{2}+|A_{2}|^{2}+2|A_{1}||A_{2}|\cos\theta_{W}\cos\delta_{S}}\,.$
(5)
The SM description of the direct CPA for $D^{0}\to f$ arises from the
interference between tree and penguin contributions, in which decay amplitudes
have the generic expressions
$\displaystyle A^{q}_{SM}(D^{0}\to
f)=V_{cq}^{*}V_{uq}\left(T^{q}_{SM}+E^{q}_{SM}e^{i\delta^{q}_{S}}\right)-V_{cb}^{*}V_{ub}P^{q}_{\rm
SM}e^{i\phi^{q}_{S}},$ (6)
where $q=(d,s)$ represents $f=(\pi^{+}\pi^{-},K^{+}K^{-})$, respectively,
$V_{q^{\prime}q}$ is the Cabibbo-Kobayashi-Maskawa (CKM) matrix element,
$T^{\prime}_{SM}(P^{\prime}_{SM})$ denotes the tree (penguin) contribution in
the SM, $E^{\prime}_{SM}$ stands for the contributions of W-exchange topology,
and $\delta^{q}_{S}(\phi^{q}_{S})$ is the associated CP-even phase. Due to the
hierarchy in the CKM matrix elements $V_{cq}^{*}V_{uq}\gg V_{cb}^{*}V_{ub}$,
the direct CPA could be estimated by
$\displaystyle A^{dir}_{CP}(D^{0}\to f)$ $\displaystyle\sim$ $\displaystyle
Im\left(\frac{V^{*}_{cb}V_{ub}}{V^{*}_{cq}V_{uq}}\right)\frac{2P^{q}_{SM}}{|T^{q}_{SM}+E^{q}_{SM}e^{i\delta^{q}_{S}}|^{2}}\left(T^{q}_{SM}\sin\phi^{q}_{S}+E^{q}_{SM}\sin(\delta^{q}_{S}-\phi^{q}_{S})\right)\,.$
(7)
With ${\rm Im}(V^{*}_{cb}V_{ub}/V^{*}_{cq}V_{uq})\approx\pm
A^{2}\lambda^{4}\eta$, $E^{q}_{SM}\sim T^{q}_{SM}$, and
$\sin\phi^{q}_{S}\sim\sin(\delta^{q}_{S}-\phi^{q}_{S})\sim O(1)$, we could
have
$\displaystyle A_{CP}(K^{-}K^{+})$ $\displaystyle\sim$ $\displaystyle-
A_{CP}(\pi^{-}\pi^{+})\sim-A^{2}\lambda^{4}\eta{P^{q}_{SM}\over
T^{q}_{SM}}\,.$ (8)
Unless $P^{q}_{SM}$ could be enhanced to several orders larger than
$T^{q}_{SM}$ by some unknown QCD effects, normally the predicted $\Delta
A_{CP}$ in the SM is far below the central value in Eq. (3). The detailed
analysis by various approaches in the SM can be referred to Refs. Bigi:2011 ;
Cheng:2012wr ; Feldmann:2012js ; Li:2012cf ; Franco:2012ck . Clearly, a
solution to the large $\Delta A_{CP}$ in Eq. (3) is to introduce some new CP
violating mechanism beyond the CKM.
To understand the LHCb and CDF data, many theoretical studies Bigi:2011 ;
Cheng:2012wr ; Feldmann:2012js ; Li:2012cf ; Franco:2012ck ; Isidori:2011 ;
Brod:2011re ; Wang:2011uu ; Gersabeck:2011xj ; arXiv:1111.6949 ;
Hochberg:2011ru ; Pirtskhalava:2011va ; Chang:2012gn ; Giudice:2012qq ;
Bhattacharya:2012ah ; Altmannshofer:2012ur ; Chen:2012am ; Lodone:2012kp ;
Inguglia:2012ik ; Brod:2012ud ; Hiller:2012wf ; Mannel:2012hb ;
Grossman:2012eb ; Cheng:2012xb ; Isidori:2012yx ; KerenZur:2012fr ;
Barbieri:2012bh ; HYCheng have been done. Since the mixing induced CPA in
$D$-meson now is limited to be less than around $0.3\%$ and no significant
evidence shows a non-vanishing CPA, if a large $\Delta A_{CP}$ indicates some
new physics effects, the same mechanism contributing to $A^{ind}_{CP}$ should
be small or negligible. To satisfy the criterion of a small $A^{ind}_{CP}$, it
is interesting to explore the tree induced new CP violating effects in which
the loop contributions are automatically suppressed. In this paper, we examine
the new CP source associated with right-handed charged currents and the left-
right (LR) mixing angle, $\xi$, in a general $SU(2)_{L}\times SU(2)_{R}\times
U(1)$ model Pati:1973rp ; Langacker:1989xa . It is known that the unitarity of
the CKM matrix gives a strict limit on $\xi$ Wolfenstein:1984ay . However, it
was found that the allowed value of the mixing angle indeed could be as large
as of order of $10^{-2}$ when the right-handed mixing matrix has a different
pattern from the CKM and carries large CP phases Langacker:1989xa . The
constraints from rare $B$ decays could be referred to Refs. Grzadkowski:2008mf
; Crivellin:2011ba . Based on the possible large new CP phases and sizable
$\xi$, we study the impact on the direct CPAs in $D^{0}\to f$ decays.
In terms of the notations in Ref. Langacker:1989xa , we first write the mass
eigenstates of charged gauge bosons as
$\displaystyle\left(\begin{array}[]{c}W^{\pm}_{L}\\\
W^{\pm}_{R}\end{array}\right)=\left(\begin{array}[]{cc}\cos\xi&-\sin\xi\\\
e^{i\omega}\sin\xi&e^{i\omega}\cos\xi\\\
\end{array}\right)\left(\begin{array}[]{c}W^{\pm}_{1}\\\
W^{\pm}_{2}\end{array}\right)\,.$ (15)
The phase $\omega$ arises from the complex vacuum expectation values (VEVs) of
bidoublet scalars which are introduced to generate the masses of fermions.
Since $m_{W}\ll m_{W_{R}}$, it is more useful to take the approximation of
$\cos\xi\approx 1$ and $\sin\xi\approx\xi$ . Accordingly, the charged current
interactions in the flavor space can be expressed by
$\displaystyle-{\cal L}_{CC}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bar{U}\gamma_{\mu}\left(g_{L}V^{L}P_{L}+g_{R}\xi\bar{V}^{R}P_{R}\right)DW^{+}_{1}$
(16) $\displaystyle+$
$\displaystyle\frac{1}{\sqrt{2}}\bar{U}\gamma_{\mu}\left(-g_{L}\xi
V^{L}P_{L}+g_{R}\bar{V}^{R}P_{R}\right)DW^{+}_{2}+h.c.$
where the flavor indices are suppressed, $V^{L}$ is the CKM matrix,
$\bar{V}^{R}=e^{i\omega}V^{R}$ and $V^{R}$ is the flavor mixing matrix for
right-handed currents. Consequently, the four-Femi interactions for $c\to
uq\bar{q}$ induced by the LR mixing are given by
$\displaystyle{\cal H}^{q}_{\chi\chi^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{4G_{F}}{\sqrt{2}}\frac{g_{R}}{g_{L}}\xi\left[V^{\chi^{\prime}}_{uq}V^{\chi^{*}}_{cq}\left(C^{\prime}_{1}(\mu)(\bar{u}q)_{\chi^{\prime}}(\bar{q}c)_{\chi})+C^{\prime}_{2}(\mu)(\bar{u}_{\alpha}q_{\beta})_{\chi^{\prime}}(\bar{q}_{\beta}c_{\alpha})_{\chi}\right)\right.$
(17) $\displaystyle+$
$\displaystyle\left.V^{\chi}_{uq}V^{\chi^{\prime*}}_{cq}\left(C^{\prime}_{1}(\mu)(\bar{u}q)_{\chi}(\bar{q}c)_{\chi^{\prime}})+C^{\prime}_{2}(\mu)(\bar{u}_{\alpha}q_{\beta})_{\chi}(\bar{q}_{\beta}c_{\alpha})_{\chi^{\prime}}\right)\right]\,,$
where $\chi=L(R)$ and $\chi^{\prime}=R(L)$ while $q=s(d)$, and
$(\bar{q}q^{\prime})_{L(R)}=\bar{q}\gamma^{\mu}P_{L(R)}q^{\prime}$. The Wilson
coefficients $C^{\prime}_{1}=\eta_{+}$ and
$C^{\prime}_{2}=-(\eta_{+}-\eta_{-})/3$ with QCD corrections could be
estimated by Cho:1993zb ; Chang:1994wk
$\displaystyle\eta_{+}$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha_{s}(\mu)}{\alpha_{s}(m_{c})}\right)^{-3/27}\left(\frac{\alpha_{s}(m_{c})}{\alpha_{s}(m_{b})}\right)^{-3/25}\left(\frac{\alpha_{s}(m_{b})}{\alpha_{s}(m_{W})}\right)^{-3/23}\,,$
$\displaystyle\eta_{-}$ $\displaystyle=$ $\displaystyle\eta_{+}^{-8}\,.$ (18)
Due to the suppression of $g^{2}_{R}/m^{2}_{R}$, as usual we neglect the
$W_{R}$ itself contributions Langacker:1989xa ; Chang:1994wk .
Based on the decay constants and transition form factors, defined by
$\displaystyle\langle 0|\bar{q}^{\prime}\gamma^{\mu}\gamma_{5}q|P(p)\rangle$
$\displaystyle=$ $\displaystyle if_{P}p^{\mu}\,,$ $\displaystyle\langle
P(p_{2})|\bar{q}\gamma_{\mu}c|D(p_{1})\rangle$ $\displaystyle=$ $\displaystyle
F^{DP}_{+}(k^{2})\Big{\\{}Q_{\mu}-\frac{Q\cdot k}{k^{2}}k_{\mu}\Big{\\}}$ (19)
$\displaystyle+$ $\displaystyle\frac{Q\cdot
k}{k^{2}}F^{DP}_{0}(k^{2})\,k_{\mu}\,,$
respectively, with $Q=p_{1}+p_{2}$ and $k=p_{1}-p_{2}$, the decay amplitude
for $D^{0}\to f$ in the QCD factorization approach is found to be
$\displaystyle A^{q}_{LR}(D^{0}\to f)$ $\displaystyle=$
$\displaystyle\left(\bar{V}^{R^{*}}_{cq}V^{L}_{uq}-V^{L^{*}}_{cq}\bar{V}^{R}_{uq}\right)T^{q}_{LR}$
(20)
with
$\displaystyle T^{d}_{LR}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\frac{g_{R}}{g_{L}}\xi
a^{\prime}_{1}f_{\pi}F^{D\pi}_{0}(m^{2}_{D}-m^{2}_{\pi})\,,$ $\displaystyle
T^{s}_{LR}$ $\displaystyle=$
$\displaystyle\frac{f_{K}}{f_{\pi}}\frac{F^{DK}_{0}}{F^{D\pi}_{0}}\frac{m^{2}_{D}-m^{2}_{K}}{m^{2}_{D}-m^{2}_{\pi}}T^{d}_{RL}\,,$
and $a^{\prime}_{1}=C^{\prime}_{1}+C^{\prime}_{2}/N_{c}$. The associated
branching ratio could be obtained by ${\cal B}(D^{0}\to
f)=\tau_{D}|\vec{p_{f}}|A^{q}(D^{0}\to f)|^{2}/8\pi m_{D}^{2}$, where
$\tau_{D}$ is the lifetime of the $D^{0}$ meson, $|p_{f}|$ is the magnitude of
the $\pi(K)$ momentum and $A^{q}=A^{q}_{SM}+A^{q}_{LR}$. With
$V^{L}_{us}\approx-V^{L}_{cd}\approx\lambda$, the squared amplitude
differences between $D^{0}\to f$ and its CP conjugate are
$\displaystyle|A^{d}|^{2}-|\bar{A}^{d}|^{2}$ $\displaystyle=$
$\displaystyle-4E^{d}_{SM}T^{d}_{LR}\sin\delta^{d}_{S}\frac{a_{1}^{\prime}}{a_{1}}\xi\left(\lambda^{2}ImV^{R}_{ud}-\lambda
ImV^{R}_{cd}\right)\,,$ $\displaystyle|A^{s}|^{2}-|\bar{A}^{s}|^{2}$
$\displaystyle=$
$\displaystyle-4E^{s}_{SM}T^{s}_{LR}\sin\delta^{s}_{S}\frac{a_{1}^{\prime}}{a_{1}}\xi\left(\lambda
ImV^{R}_{us}+\lambda^{2}ImV^{R}_{cs}\right)\,.$ (21)
Clearly, the direct CPA in $D^{0}\to f$ decay will strongly depend on the CP
violating phases in $V^{R}_{cq,uq}$. Since the (pseudo) manifest LR model,
denoted by $V^{L}=V^{R^{(*)}}$, has a strict limit on $\xi$, in this paper, we
only focus on the non-manifest LR model, where except the unitarity, the
elements in $V^{R}$ are arbitrary free parameters.
In the numerical calculations, the input values of the SM are listed in Table
1 Cheng:2012wr ; Cheng:2010ry ; PDG2010 , where the resulting branching ratios
(BRs) for $D^{0}\to(\pi^{-}\pi^{+},K^{-}K^{+})$ are estimated as
$(1.38,3.96)\times 10^{-3}$, while the current data are ${\cal
B}(D^{0}\to\pi^{-}\pi^{+})=(1.400\pm 0.026)\times 10^{-3}$ and ${\cal
B}(D^{0}\to K^{-}K^{+})=(3.96\pm 0.08)\times 10^{-3}$ PDG2010 .
Table 1: Numerical inputs for the parameters in the SM. $T^{d}_{SM}$ | $T^{s}_{SM}$ | $E^{d}_{SM}$ | $E^{s}_{SM}$ | $\delta^{d}_{S}$ | $\delta^{s}_{S}$
---|---|---|---|---|---
$3.0\times 10^{-6}$GeV | $4.0\times 10^{-6}$GeV | $1.3\times 10^{-6}$GeV | $1.6\times 10^{-6}$GeV | $145^{\circ}$ | $108^{\circ}$
$V^{L}_{us}$ | $m_{\pi(K)}$ | $m_{D}$ | $f_{\pi(K)}$ | $F^{D\pi(K)}_{0}$ | $m_{t}$
$0.22$ | $0.139(0.497)$GeV | $1.863$GeV | $0.13(0.16)$GeV | $0.666(0.739)$ | 162.8 GeV
Although the QCD related SM inputs are extracted from the Cabibbo allowed
decays, the influence of the new effects on these decays is small. Due to the
W-exchange topology dominated by the final state interactions, the short-
distance effects could be ignored. It is known that the box diagrams with
$W_{L}$ and $W_{R}$ yield important contributions to the $K^{0}$-$\bar{K}^{0}$
mixing Beall:1981ze . However, due to the quarks in the diagrams for the
D-system being down-type ones, we find the enhancement on the
$D^{0}$-$\bar{D}^{0}$ oscillation is small. Hence, the constraint from $\Delta
m_{D}$ could be ignored. Since the CPAs involve $V^{R}_{ud,us}$, we need to
consider the constraint from the direct CPA in $K\to\pi\pi$ decays. Using the
result in Ref. He:1988th , we know $(\epsilon^{\prime}/\epsilon)_{K}\sim
1.25\times 10^{-3}g_{R}/g_{L}\xi
Im(\bar{V}^{R}_{us}-\lambda\bar{V}^{R^{*}}_{ud})$. Therefore, to avoid the
constraint from $(\epsilon^{\prime}/\epsilon)_{K}$, we adopt two cases: (I)
$Im(V^{R}_{us,ud})\to 0$ and (II) $Im(\bar{V}^{R}_{us})\approx\lambda
Im(\bar{V}^{R^{*}}_{ud})$ He:1988th . We investigate the two cases separately
as follows:
Case I: In this case, Eq. (21) is simplified as
$\displaystyle|A^{d}|^{2}-|\bar{A}^{d}|^{2}$ $\displaystyle=$ $\displaystyle
4E^{d}_{SM}T^{d}_{LR}\sin\delta^{d}_{S}\frac{a_{1}^{\prime}}{a_{1}}\xi\lambda
Im\bar{V}^{R}_{cd}\,,$ $\displaystyle|A^{s}|^{2}-|\bar{A}^{s}|^{2}$
$\displaystyle=$
$\displaystyle-4E^{s}_{SM}T^{s}_{LR}\sin\delta^{s}_{S}\frac{a_{1}^{\prime}}{a_{1}}\xi\lambda^{2}Im\bar{V}^{R}_{cs}\,.$
(22)
In general, $V^{R}_{cd}$ and $V^{R}_{cs}$ are free parameters. In order to
illustrate the impact of the LR mixing effects on $\Delta A_{CP}$ and make the
CPAs of $\pi^{+}\pi^{-}$ and $K^{+}K^{-}$ modes to be more correlated, an
interesting choice is $\bar{V}^{R}_{cd}\approx-\lambda e^{i\theta}$ and
$\bar{V}^{R}_{cs}\approx e^{i\theta}$. Hence, the involving free parameters
for the CPAs are the CP phase $\theta$ and the mixing angle $\xi$. Using Eqs.
(20) and $A^{q}=A^{q}_{SM}+A^{q}_{LR}$, BRs for $D^{0}\to\pi^{+}\pi^{-}$
(dashed) and $D^{0}\to K^{+}K^{-}$ (dash-dotted) as functions of $\xi$ and
$\theta$ are shown in Fig. 1, where $1\sigma$ errors of data in BRs with units
of $10^{-3}$ are taken. From this figure, we constrain the free parameters as
$\displaystyle-5.3\times 10^{-2}<\xi<-3.\times 10^{-2},\;\;1.47<\theta<1.87.$
(23)
Figure 1: BRs for $D^{0}\to\pi^{+}\pi^{-}$ (dashed) and $D^{0}\to K^{+}K^{-}$
(dash-dotted) and $\Delta A_{CP}$ (solid), where the shaded band represents
the allowed region.
Case II: In this case, Eq. (21) becomes
$\displaystyle|A^{s}|^{2}-|\bar{A}^{s}|^{2}$ $\displaystyle=$
$\displaystyle-4E^{s}_{SM}T^{s}_{LR}\sin\delta^{s}_{S}\frac{a_{1}^{\prime}}{a_{1}}\xi\lambda^{2}\left(ImV^{R*}_{ud}+ImV^{R}_{cs}\right)\,.$
(24)
Without further limiting the pattern of $V^{R}$, apparently the situation in
Case II is more complicated. It was pointed out that one can have a weaker
constraint on the mass of $W_{R}$ when the right-handed flavor mixing matrix
is centered around the following two forms Langacker:1989xa :
$\displaystyle V^{R}_{A}(\alpha)=\left(\begin{array}[]{ccc}1&0&0\\\
0&c_{\alpha}&\pm s_{\alpha}\\\ 0&s_{\alpha}&\mp c_{\alpha}\\\
\end{array}\right)\,,\ \ \
V^{R}_{B}(\alpha)=\left(\begin{array}[]{ccc}0&1&0\\\ c_{\alpha}&0&\pm
s_{\alpha}\\\ s_{\alpha}&0&\mp c_{\alpha}\\\ \end{array}\right)\,,$ (31)
where $c_{\alpha}=\cos\alpha$, $s_{\alpha}=\sin\alpha$ and $\alpha$ is an
arbitrary angle. We note that the null elements denote the values that are
smaller than $O(\lambda^{2})$, thus their effects could be ignored in the
analysis. We will focus on the implication of the two special forms. In
$V^{R}_{A}(\alpha)$, since $V^{R}_{cd}\to 0$ and $Im(V^{R}_{ud})\to 0$ due to
$(\epsilon^{\prime}/\epsilon)_{K}$, only the CPA for $D^{0}\to K^{+}K^{-}$
could be compatible with the current data, while the CPA for
$D^{0}\to\pi^{+}\pi^{-}$ decay is small, i.e. $\Delta A_{CP}\approx
A_{CP}(D^{0}\to K^{+}K^{-})$. With $\alpha\approx 0$ which satisfies the
constraint from $b\to d\gamma$ Crivellin:2011ba , we present the constraint of
${\cal B}(D^{0}\to K^{+}K^{-})$ and $\Delta A_{CP}$ as functions of
$\bar{\xi}=g_{R}/g_{L}\xi$ and $\theta=$arg$(\bar{V}^{R}_{cs})$ in Fig. 2,
where the shaded band shows the allowed region for the parameters,
corresponding to
$\displaystyle 0.7\times 10^{-2}<\xi<1.4\times 10^{-2},\;\;0.56<\theta<2.61.$
(32)
Figure 2: BR for $D^{0}\to K^{+}K^{-}$ (dash-dotted) and $\Delta A_{CP}$
(solid), where the shaded band stands for the allowed region.
For $V^{R}_{B}(\alpha)$, due to $V^{R}_{ud,cs}\to 0$, the CPA for $D^{0}\to
K^{+}K^{-}$ is small and only $A_{CP}(D^{0}\to\pi^{+}\pi^{-})$ could be
compatible with the data. As a result, we have $\Delta A_{CP}\approx-
A_{CP}(D^{0}\to\pi^{+}\pi^{-})$. Similar to $V^{R}_{A}(0)$ with $\alpha=0$, we
display ${\cal B}(D^{0}\to\pi^{+}\pi^{-})$ (dashed) and $\Delta A_{CP}$
(solid) as functions of $\bar{\xi}$ and the phase $\theta$ defined as
$\bar{V}^{R}_{cd}=-\lambda e^{-i\theta}$ in Fig. 3. In this case, the allowed
$\xi$ is negative
$\displaystyle-1.6\times 10^{-2}<\xi<-0.6\times 10^{-2},\;\;1.12<\theta<2.76.$
(33)
Figure 3: The Legend is the same as Fig. 2 but for $D^{0}\to\pi^{+}\pi^{-}$.
In summary, we have studied the impact of the LR mixing in the general LR
model on the CPA difference between $D^{0}\to K+K^{-}$ and
$D^{0}\to\pi^{+}\pi^{-}$. It is found that when the constraint from
$(\epsilon^{\prime}/\epsilon)_{K}$ is considered, the proposed LR mixing
mechanism could be compatible with the value of $\Delta A_{CP}$ averaged by
the LHCb and CDF new data. To illustrate the influence of the LR mixing
effects, we have adopted two cases for the new flavor mixing matrix
$\bar{V}^{R}$ to explain the large $\Delta A_{CP}$. In Case I, we have found
that $A_{CP}(D^{0}\to K^{+}K^{-})\approx-A_{CP}(D^{0}\to\pi^{+}\pi^{-})$ can
be achieved. In Case II, we have used two special forms for $V^{R}$, resulting
in $A_{CP}(D^{0}\to\pi^{+}\pi^{-})\approx 0$ and $A_{CP}(D^{0}\to
K^{+}K-)\approx 0$, respectively.
Acknowledgments
This work is supported by the National Science Council of R.O.C. under Grant
#s: NSC-100-2112-M-006-014-MY3 (CHC) and NSC-98-2112-M-007-008-MY3 (CQG).
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|
arxiv-papers
| 2012-06-22T14:22:37 |
2024-09-04T02:49:32.067987
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chuan-Hung Chen, Chao-Qiang Geng and Wei Wang",
"submitter": "Chuan Hung Chen",
"url": "https://arxiv.org/abs/1206.5158"
}
|
1206.5160
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-171 LHCb-PAPER-2011-037
Measurement of prompt hadron production ratios in $pp$ collisions at
$\sqrt{s}=$ 0.9 and 7 TeV
The LHCb collaboration111Authors are listed on the following pages.
The charged-particle production ratios $\bar{p}/p$, $K^{-}/K^{+}$,
$\pi^{-}/\pi^{+}$, $(p+\bar{p})/(\pi^{+}+\pi^{-})$,
$(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ and $(p+\bar{p})/(K^{+}+K^{-})$ are measured
with the LHCb detector using $0.3\,{\rm nb^{-1}}$ of $pp$ collisions delivered
by the LHC at $\sqrt{s}=0.9$ TeV and $1.8\,{\rm nb^{-1}}$ at $\sqrt{s}=7$ TeV.
The measurements are performed as a function of transverse momentum $p_{\rm
T}$ and pseudorapidity $\eta$. The production ratios are compared to the
predictions of several Monte Carlo generator settings, none of which are able
to describe adequately all observables. The ratio $\bar{p}/p$ is also
considered as a function of rapidity loss, $\Delta y\equiv y_{\rm beam}-y$,
and is used to constrain models of baryon transport.
Accepted by Eur. Phys. J. C
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, J. Anderson37, R.B. Appleby51, O. Aquines Gutierrez10,
F. Archilli18,35, A. Artamonov 32, M. Artuso53, E. Aslanides6, G.
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Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14,c, G. Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50,
K. Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, R. Currie47, C. D’Ambrosio35, P.
David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De Capua21,k, M. De
Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D. Decamp4, M.
Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D. Derkach14,35,
O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P. Diniz Batista1,
F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil Suárez34, D.
Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A. Dziurda23, A.
Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31, D. van Eijk38,
F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El Rifai5, Ch.
Elsasser37, D. Elsby42, D. Esperante Pereira34, A. Falabella14,e, C. Färber11,
G. Fardell47, C. Farinelli38, S. Farry12, V. Fave36, V. Fernandez Albor34, F.
Ferreira Rodrigues1, M. Ferro-Luzzi35, S. Filippov30, C. Fitzpatrick47, M.
Fontana10, F. Fontanelli19,i, R. Forty35, O. Francisco2, M. Frank35, C.
Frei35, M. Frosini17,f, S. Furcas20, A. Gallas Torreira34, D. Galli14,c, M.
Gandelman2, P. Gandini52, Y. Gao3, J-C. Garnier35, J. Garofoli53, J. Garra
Tico44, L. Garrido33, D. Gascon33, C. Gaspar35, R. Gauld52, N. Gauvin36, E.
Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph. Ghez4, V. Gibson44, V.V.
Gligorov35, C. Göbel54, D. Golubkov28, A. Golutvin50,28,35, A. Gomes2, H.
Gordon52, M. Grabalosa Gándara33, R. Graciani Diaz33, L.A. Granado Cardoso35,
E. Graugés33, G. Graziani17, A. Grecu26, E. Greening52, S. Gregson44, O.
Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32, T. Gys35, C. Hadjivasiliou53,
G. Haefeli36, C. Haen35, S.C. Haines44, T. Hampson43, S. Hansmann-Menzemer11,
N. Harnew52, S.T. Harnew43, J. Harrison51, P.F. Harrison45, T. Hartmann55, J.
He7, V. Heijne38, K. Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van
Herwijnen35, E. Hicks49, D. Hill52, M. Hoballah5, P. Hopchev4, W.
Hulsbergen38, P. Hunt52, T. Huse49, N. Hussain52, R.S. Huston12, D.
Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J. Imong43, R.
Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F. Jansen38, P.
Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R. Jones44, B.
Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9, J.
Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji20,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
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Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, K. Kruzelecki35, M.
Kucharczyk20,23,35,j, V. Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36,
D. Lacarrere35, G. Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E.
Lanciotti35, G. Lanfranchi18,35, C. Langenbruch35, T. Latham45, C.
Lazzeroni42, R. Le Gac6, J. van Leerdam38, J.-P. Lees4, R. Lefèvre5, A.
Leflat29,35, J. Lefrançois7, O. Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li
Gioi5, M. Lieng9, M. Liles49, R. Lindner35, C. Linn11, B. Liu3, G. Liu35, J.
von Loeben20, J.H. Lopes2, E. Lopez Asamar33, N. Lopez-March36, H. Lu3, J.
Luisier36, A. Mac Raighne48, F. Machefert7, I.V. Machikhiliyan4,28, F.
Maciuc10, O. Maev27,35, J. Magnin1, S. Malde52, R.M.D. Mamunur35, G.
Manca15,d, G. Mancinelli6, N. Mangiafave44, U. Marconi14, R. Märki36, J.
Marks11, G. Martellotti22, A. Martens8, L. Martin52, A. Martín Sánchez7, M.
Martinelli38, D. Martinez Santos35, A. Massafferri1, Z. Mathe12, C.
Matteuzzi20, M. Matveev27, E. Maurice6, B. Maynard53, A. Mazurov16,30,35, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran51,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, J. Palacios37, A. Palano13,b, M.
Palutan18, J. Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J.
Parkinson50, G. Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C.
Patrignani19,i, C. Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G.
Penso22,l, M. Pepe Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo34, A. Pérez-Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, A. Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie
Valls33, B. Pietrzyk4, T. Pilař45, D. Pinci22, R. Plackett48, S. Playfer47, M.
Plo Casasus34, F. Polci8, G. Polok23, A. Poluektov45,31, E. Polycarpo2, D.
Popov10, B. Popovici26, C. Potterat33, A. Powell52, J. Prisciandaro36, V.
Pugatch41, A. Puig Navarro33, W. Qian53, J.H. Rademacker43, B.
Rakotomiaramanana36, M.S. Rangel2, I. Raniuk40, G. Raven39, S. Redford52, M.M.
Reid45, A.C. dos Reis1, S. Ricciardi46, A. Richards50, K. Rinnert49, D.A. Roa
Romero5, P. Robbe7, E. Rodrigues48,51, P. Rodriguez Perez34, G.J. Rogers44, S.
Roiser35, V. Romanovsky32, M. Rosello33,n, J. Rouvinet36, T. Ruf35, H. Ruiz33,
G. Sabatino21,k, J.J. Saborido Silva34, N. Sagidova27, P. Sail48, B.
Saitta15,d, C. Salzmann37, B. Sanmartin Sedes34, M. Sannino19,i, R.
Santacesaria22, C. Santamarina Rios34, R. Santinelli35, E. Santovetti21,k, M.
Sapunov6, A. Sarti18,l, C. Satriano22,m, A. Satta21, M. Savrie16,e, D.
Savrina28, P. Schaack50, M. Schiller39, H. Schindler35, S. Schleich9, M.
Schlupp9, M. Schmelling10, B. Schmidt35, O. Schneider36, A. Schopper35, M.-H.
Schune7, R. Schwemmer35, B. Sciascia18, A. Sciubba18,l, M. Seco34, A.
Semennikov28, K. Senderowska24, I. Sepp50, N. Serra37, J. Serrano6, P.
Seyfert11, M. Shapkin32, I. Shapoval40,35, P. Shatalov28, Y. Shcheglov27, T.
Shears49, L. Shekhtman31, O. Shevchenko40, V. Shevchenko28, A. Shires50, R.
Silva Coutinho45, T. Skwarnicki53, N.A. Smith49, E. Smith52,46, M. Smith51, K.
Sobczak5, F.J.P. Soler48, A. Solomin43, F. Soomro18,35, D. Souza43, B. Souza
De Paula2, B. Spaan9, A. Sparkes47, P. Spradlin48, F. Stagni35, S. Stahl11, O.
Steinkamp37, S. Stoica26, S. Stone53, B. Storaci38, M. Straticiuc26, U.
Straumann37, V.K. Subbiah35, S. Swientek9, M. Szczekowski25, P. Szczypka36,35,
T. Szumlak24, S. T’Jampens4, M. Teklishyn7, E. Teodorescu26, F. Teubert35, C.
Thomas52, E. Thomas35, J. van Tilburg11, V. Tisserand4, M. Tobin37, S. Tolk39,
S. Topp-Joergensen52, N. Torr52, E. Tournefier4,50, S. Tourneur36, M.T.
Tran36, A. Tsaregorodtsev6, N. Tuning38, M. Ubeda Garcia35, A. Ukleja25, U.
Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez Gomez33, P. Vazquez Regueiro34,
S. Vecchi16, J.J. Velthuis43, M. Veltri17,g, M. Vesterinen35, B. Viaud7, I.
Videau7, D. Vieira2, X. Vilasis-Cardona33,n, J. Visniakov34, A. Vollhardt37,
D. Volyanskyy10, D. Voong43, A. Vorobyev27, V. Vorobyev31, C. Voß55, H.
Voss10, R. Waldi55, R. Wallace12, S. Wandernoth11, J. Wang53, D.R. Ward44,
N.K. Watson42, A.D. Webber51, D. Websdale50, M. Whitehead45, J. Wicht35, D.
Wiedner11, L. Wiggers38, G. Wilkinson52, M.P. Williams45,46, M. Williams50,
F.F. Wilson46, J. Wishahi9, M. Witek23, W. Witzeling35, S.A. Wotton44, S.
Wright44, S. Wu3, K. Wyllie35, Y. Xie47, F. Xing52, Z. Xing53, Z. Yang3, R.
Young47, X. Yuan3, O. Yushchenko32, M. Zangoli14, M. Zavertyaev10,a, F.
Zhang3, L. Zhang53, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, L. Zhong3, A.
Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
## 1 Introduction
All underlying interactions responsible for $pp$ collisions at the Large
Hadron Collider (LHC) and the subsequent hadronisation process can be
understood within the context of quantum chromodynamics (QCD). In the non-
perturbative regime, however, precise calculations are difficult to perform
and so phenomenological models must be employed. Event generators based on
these models must be optimised, or ‘tuned’, to reproduce experimental
observables. The observables exploited for this purpose include event
variables, such as particle multiplicities, the kinematical distributions of
the inclusive particle sample in each event, and the corresponding
distributions for individual particle species. The generators can then be used
in simulation studies when analysing data to search for physics beyond the
Standard Model.
The relative proportions of each charged quasi-stable hadron, and the ratio of
antiparticles to particles in a given kinematical region, are important inputs
for generator tuning. Of these observables, the ratio of antiprotons to
protons is of particular interest. Baryon number conservation requires that
the disintegration of the beam particles that occurs in high-energy inelastic
non-diffractive $pp$ collisions must be balanced by the creation of protons or
other baryons elsewhere in the event. This topic is known as baryon-number
transport. Several models exist to describe this transport, but it is not
clear which mechanisms are most important in driving the phenomenon[1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Pomeron exchange is expected to play a
significant role, but contributions may exist from other sources, for example
the Odderon, the existence of which has not yet been established[13, 14, 15].
Experimentally, baryon-number transport can be studied by measuring
$\bar{p}/{p}$, the ratio of the number of produced antiprotons to protons, as
a function of suitable kinematical variables.
In this paper results are presented from the LHCb experiment for the following
production ratios: $\bar{p}/{p}$, $K^{-}/K^{+}$, $\pi^{-}/\pi^{+}$,
$(p+\bar{p})/(\pi^{+}+\pi^{-})$, $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ and
$(p+\bar{p})/(K^{+}+K^{-})$. The first three of these observables are termed
the same-particle ratios and the last three the different-particle ratios.
Only prompt particles are considered, where a prompt particle is defined to be
one that originates from the primary interaction, either directly, or through
the subsequent decay of a resonance. The ratios are measured as a function of
transverse momentum $p_{\rm T}$ and pseudorapidity $\eta=-\ln(\tan\theta/2)$,
where $\theta$ is the polar angle with respect to the beam axis.
Measurements have been performed of the $\bar{p}/p$ ratio in $pp$ collisions
both at the LHC[16], and at other facilities[17, 18, 19, 20, 21, 22]. Studies
have also been made of the production characteristics of pions, kaons and
protons at the LHC at $\sqrt{s}=0.9$ TeV at mid-rapidity[23]. The analysis
presented in this paper exploits the unique forward coverage of the LHCb
spectrometer, and the powerful particle separation capabilities of the ring-
imaging Cherenkov (RICH) system, to yield results for the production ratios in
the range $2.5<\eta<4.5$ at both $\sqrt{s}=0.9$ TeV and $\sqrt{s}=7$ TeV. LHCb
has previously published studies of baryon transport and particle ratios with
neutral strange hadrons[24], and results for strange baryon observables at the
LHC are also available in the midrapidity region[25, 26]. New analyses have
also been made public since the submission of this paper [27].
The paper is organised as follows. Section 2 introduces the LHCb detector and
the datasets used. Section 3 describes the selection of the analysis sample,
while Sect. 4 discusses the calibration of the particle identification
performance. The analysis procedure is explained in Sect. 5. The assignment of
the systematic uncertainties is described in Sect. 6 and the results are
presented and discussed in Sect. 7, before concluding in Sect. 8. Full tables
of numerical results may be found in Appendix A. Throughout, unless specified
otherwise, particle types are referred to by their name (e.g. proton) when
both particles and antiparticles are being considered together, and by symbol
(e.g. $p$ or $\bar{p}$) when it is necessary to distinguish between the two.
## 2 Data samples and the LHCb detector
The LHCb experiment is a forward spectrometer at the Large Hadron Collider
with a pseudorapidity acceptance of approximately $2<\eta<5$. The tracking
system begins with a silicon strip Vertex Locator (VELO). The VELO consists of
23 sequential stations of silicon strip detectors which retract from the beam
during injection. A large area silicon tracker (TT) follows upstream of a
dipole magnet, downstream of which there are three tracker stations, each
built with a mixture of straw tube and silicon strip detectors. The dipole
field direction is vertical, and charged tracks reconstructed through the full
spectrometer are deflected by an integrated $B$ field of around 4 Tm. Hadron
identification is provided by the RICH system, which consists of two
detectors, one upstream of the magnet and the other downstream, and is
designed to provide particle identification over a momentum interval of 2–100
GeV/$c$. Also present, but not exploited in the current analysis, are a
calorimeter and muon system. A full description of the LHCb detector may be
found in [28].
The data sample under consideration derives from the early period of the 2010
LHC run. Inelastic interactions were triggered by requiring at least one track
in either the VELO or the tracking stations downstream of the magnet. This
trigger was more than 99% efficient for all offline selected events that
contain at least two tracks reconstructed through the whole system. Collisions
were recorded both at $\sqrt{s}=0.9$ TeV and $7$ TeV. During $0.9$ TeV
running, where the beams were wider and the internal crossing-angle of the
beams within LHCb was larger, detector and machine safety considerations
required that each VELO half was retracted by 10 mm from the nominal closed
position. For 7 TeV operation the VELO was fully closed.
The analysis exploits a data sample of around $0.3\,{\rm nb}^{-1}$ recorded at
$\sqrt{s}=0.9$ TeV and $1.8\,{\rm nb}^{-1}$ at $\sqrt{s}=7$ TeV. In order to
minimise potential detector-related systematic biases, the direction of the
LHCb dipole field was inverted every 1–2 weeks of data taking. At 0.9 TeV the
data divide approximately equally between the two polarities, while at 7 TeV
around two-thirds were collected in one configuration. The analysis is
performed separately for each polarity.
The beams collided with a crossing angle in the horizontal plane which was set
to compensate for the field of the LHCb dipole. This angle was 2.1 mrad in
magnitude at $\sqrt{s}=0.9$ TeV and 270 $\upmu$rad at $\sqrt{s}=7$ TeV.
Throughout this analysis momenta and any derived quantities are computed in
the centre-of-mass frame.
Monte Carlo simulated events are used to calculate efficiencies and estimate
systematic uncertainties. A total of around 140 million events are simulated
at 0.9 TeV and 130 million events at 7 TeV. The $pp$ collisions are generated
by Pythia6.4 [29] and the parameters tuned as described in Ref. [30]. The
decays of emerging particles are implemented with the EvtGen package [31],
with final state radiation described by Photos [32]. The resulting particles
are transported through LHCb by Geant4 [33, 34], which models hits in the
sensitive regions of the detector as well as material interactions as
described in Ref. [35]. The decay of secondary particles produced in these
interactions is controlled by Geant4. Additional Pythia6.4 samples with
different generator tunes were produced in order to provide references with
which to compare the results. These were Perugia 0, which was tuned on
experimental results from SPS, LEP and the Tevatron, and Perugia NOCR, which
includes an extreme model of baryon transport[36].
## 3 Selection of the analysis sample
The measurement is performed using the analysis sample, the selection of which
is described here. Understanding of the particle identification (PID)
performance provided by the RICH sample is obtained from the calibration
sample, which is discussed in Sect. 4.
Events are selected which contain at least one reconstructed primary vertex
(PV) within $20$ cm of the nominal interaction point. The primary vertex
finding algorithm requires at least three reconstructed tracks.222The PV
requirement can be approximated in Monte Carlo simulation by imposing a filter
at generator level which demands at least three charged particles with
lifetime $c\tau>10^{-9}$ m, momentum $p>0.3$ GeV/$c$ and polar angle
$15<\theta<460$ mrad.
Tracks are only considered that have hits both in the VELO detector and in the
tracking stations downstream of the magnet, and for which the track fit yields
an acceptable $\chi^{2}$ per number of degrees of freedom (ndf). In order to
suppress background from decays of long-lived particles, or particles produced
in secondary interactions, an upper bound is placed on the goodness of fit
when using the track’s impact parameter (IP) to test the hypothesis that the
track is associated with the PV ($\chi^{2}_{\rm IP}<49$). To reduce systematic
uncertainties in the calculation of the ratio observables, a momentum cut is
imposed of $p>5$ GeV/$c$, as below this value the cross-section for strong
interaction with the beampipe and detector elements differs significantly
between particle and anti-particle for kaons and protons. If a pair of tracks,
$i$ and $j$, are found to have very similar momenta
($|{\mathbf{p}}_{i}-{\mathbf{p}}_{j}|/|{\mathbf{p}}_{i}+{\mathbf{p}}_{j}|<0.001$),
then one of the two is rejected at random. This requirement is imposed to
suppress ‘clones’, which occur when two tracks are reconstructed from the hit
points left by a single particle, and eliminates ${\cal{O}}(1\%)$ of
candidates.
The analysis is performed in bins of $p_{\rm T}$ and $\eta$. In $p_{\rm T}$
three separate regions are considered: $p_{\rm T}<0.8$ GeV/$c$, $0.8\leq
p_{\rm T}<1.2$ GeV/$c$ and $p_{\rm T}\geq 1.2$ GeV/$c$. In $\eta$ half-integer
bins are chosen over the intervals $3.0<\eta<4.5$ for $p_{\rm T}<0.8$ GeV/$c$,
and $2.5<\eta<4.5$ for higher $p_{\rm T}$ values. The $\eta$ acceptance is not
constant with $p_{\rm T}$ because the limited size of the calibration samples
does not allow for the PID performance to be determined with adequate
precision below $\eta=3$ in the lowest $p_{\rm T}$ bin. The bin size is large
compared to the experimental resolution and hence bin-to-bin migration effects
are negligible in the analysis.
The RICH is used to select the analysis sample at both energy points from
which the ratio observables are determined. A pattern recognition and particle
identification algorithm uses information from the RICH and tracking detectors
to construct a negative log likelihood for each particle hypothesis ($e$,
$\mu$, $\pi$, $K$ or $p$). This negative log likelihood is minimised for the
event as a whole. After minimisation, the change in log likelihood (DLL) is
recorded for each track when the particle type is switched from that of the
preferred assignment to another hypothesis. Using this information the
separation in log likelihood DLL($x-y$) can be calculated for any two particle
hypotheses $x$ and $y$, where a positive value indicates that $x$ is the
favoured option. In the analysis, cuts are placed on DLL($p-K$) versus
DLL($p-\pi$) to select protons and on DLL($K-p$) versus DLL($K-\pi$) to select
kaons. Pions are selected with a simple cut on DLL($\pi-K$). As the RICH
performance varies with momentum and track density, different cuts are applied
in each $(p_{\rm T},\eta)$ bin. The selection cuts are chosen in order to
optimise purity, together with the requirement that the identification
efficiency be at least 10%. Figure 1 shows the background-subtracted two-
dimensional distribution of DLL($p-K$) and DLL($p-\pi$) for protons, kaons and
pions in the calibration sample for one example bin. The approximate number of
positive and negative tracks selected in each PID category is given in Tables
1 and 2. 333The journal version of this paper has incorrect entries in Table
2. A charge asymmetry can be observed in many bins, most noticeably for the
protons.
Figure 1: Two-dimensional distribution of the change in log likelihood DLL($p-K$) and DLL($p-\pi$) for (a) protons, (b) kaons and (c) pions (here shown for negative tracks and one magnet polarity) in the calibration sample with $p_{\rm T}>1.2$ GeV/$c$ and $3.5<\eta\leq 4.0$. The region indicated by the dotted lines in the top right corner of each plot is that which is selected in the analysis to isolate the proton sample. The selection of the calibration sample is discussed in Sect. 4. Table 1: Number of particle candidates in the analysis sample at $\sqrt{s}=0.9$ TeV, separated into positive and negative charge ($Q$). | | $p_{\rm T}<0.8$ GeV/$c$ | $0.8\leq p_{\rm T}<1.2$ GeV/$c$ | $p_{\rm T}\geq 1.2$ GeV/$c$
---|---|---|---|---
| $Q$ | $p$ | $K$ | $\pi$ | $p$ | $K$ | $\pi$ | $p$ | $K$ | $\pi$
$2.5<\eta<3.0$ | $+$ | – | – | – | 16k | 39k | 270k | 19k | 36k | 130k
| $-$ | – | – | – | 13k | 35k | 270k | 13k | 31k | 120k
$3.0\leq\eta<3.5$ | $+$ | 21k | 78k | 1.1M | 30k | 63k | 260k | 34k | 39k | 120k
| $-$ | 17k | 69k | 1.1M | 21k | 55k | 250k | 20k | 31k | 100k
$3.5\leq\eta<4.0$ | $+$ | 55k | 120k | 1.9M | 55k | 60k | 240k | 31k | 33k | 97k
| $-$ | 38k | 100k | 1.9M | 33k | 49k | 230k | 14k | 23k | 85k
$4.0\leq\eta<4.5$ | $+$ | 26k | 90k | 1.2M | 23k | 30k | 100k | 14k | 11k | 39k
| $-$ | 21k | 86k | 1.2M | 11k | 22k | 88k | 4.2k | 6.6k | 30k
Table 2: Number of particle candidates in the analysis sample at $\sqrt{s}=7.0$ TeV, separated into positive and negative charge ($Q$). | | $p_{\rm T}<0.8$ GeV/$c$ | $0.8\leq p_{\rm T}<1.2$ GeV/$c$ | $p_{\rm T}\geq 1.2$ GeV/$c$
---|---|---|---|---
| $Q$ | $p$ | $K$ | $\pi$ | $p$ | $K$ | $\pi$ | $p$ | $K$ | $\pi$
$2.5<\eta<3.0$ | $+$ | – | – | – | 180k | 850k | 6.8M | 500k | 1.3M | 4.6M
| $-$ | – | – | – | 170k | 820k | 6.8M | 450k | 1.2M | 4.7M
$3.0\leq\eta<3.5$ | $+$ | 230k | 1.5M | 22M | 380k | 1.6M | 6.7M | 850k | 1.4M | 4.4M
| $-$ | 220k | 1.4M | 23M | 350k | 1.6M | 6.7M | 760k | 1.4M | 4.4M
$3.5\leq\eta<4.0$ | $+$ | 740k | 2.5M | 38M | 930k | 1.6M | 6.4M | 880k | 1.2M | 3.8M
| $-$ | 690k | 2.4M | 38M | 840k | 1.5M | 6.3M | 760k | 1.2M | 3.7M
$4.0\leq\eta<4.5$ | $+$ | 460k | 3.4M | 44M | 490k | 1.3M | 4.6M | 480k | 650k | 2.6M
| $-$ | 450k | 3.2M | 43M | 420k | 1.3M | 4.4M | 390k | 580k | 2.5M
## 4 Calibration of particle identification
The calibration sample consists of the decays444In this section the inclusion
of the charge conjugate decay $\bar{\Lambda}\rightarrow\bar{p}\pi^{+}$ is
implicit. $K^{0}_{\rm S}\rightarrow\pi^{+}\pi^{-}$, $\Lambda\rightarrow
p\pi^{-}$ and $\phi\rightarrow K^{+}K^{-}$, all selected from the 7 TeV data.
The signal yields in each category are 4.7 million, 1.4 million and 5.5
million, respectively.
The $K^{0}_{\rm S}$ and $\Lambda$ (collectively termed $V^{0}$) decays are
reconstructed through a selection algorithm devoid of RICH PID requirements,
identical to that used in Ref. [24], providing samples of pions and protons
which are unbiased for PID studies. The purity of the samples varies across
the $p_{\rm T}$ and $\eta$ bins, but is found always to be in excess of 83%
and 87%, for $K^{0}_{\rm S}$ and $\Lambda$, respectively. Isolating
$\phi\rightarrow K^{+}K^{-}$ decays with adequate purity is only achievable by
exploiting RICH information. A PID requirement of DLL$(K-\pi)>15$ is placed on
one of the two kaon candidates, chosen at random, so as to leave the other
candidate unbiased for calibration studies. The purity of this selection
ranges from 17% to 68%, over the kinematic range. Examples of the invariant
mass distributions obtained in a typical analysis bin for each of the three
calibration modes are shown in Fig. 2.
Figure 2: Invariant mass distributions reconstructed for one magnet polarity
from the $\sqrt{s}=7\,\mathrm{TeV}$ data in the analysis bin for which the
positive final-state particle has $\mbox{$p_{\rm T}$}\geq 1.2~{}{\rm GeV}$ and
$3.5\leq\eta<4.0$ for (a) $K^{0}_{\rm S}\rightarrow\pi^{+}\pi^{-}$, (b)
$\Lambda\rightarrow p\pi^{-}$ and (c) $\phi\rightarrow K^{+}K^{-}$. The
results of unbinned maximum likelihood fits to the data are superimposed.
In order to study the PID performance on the unbiased $K^{\pm}$ tracks
associated with genuine $\phi$ decays the sPlot[37] technique is employed,
using the invariant mass as the uncorrelated discriminating variable, to
produce distributions of quantities such as the RICH DLL$(K-\pi)$. Although
the background contamination in the $V^{0}$ selections is small in comparison,
the same strategy is employed to extract the true DLL distributions from all
unbiased track samples in each analysis bin. The two $V^{0}$ signal peaks are
parameterised by a double Gaussian function, while the strongly decaying
$\phi$ is described by a Breit-Wigner function convoluted with a Gaussian. The
background is modelled by a first and third order Chebyshev polynomial for the
$V^{0}$ and $\phi$ distributions, respectively.
The resulting distributions cannot be applied directly to the analysis sample
for two reasons. The first is that the PID performance varies with momentum,
and the finite size of the ($p_{\rm T}$, $\eta$) bins means that the momentum
spectrum within each bin is in general different between the calibration and
analysis samples. The second is that the PID performance is also dependent on
multiplicity, and here significant differences exist between the calibration
and analysis samples, most noticeably for the 0.9 TeV data. To obtain rates
applicable to the 0.9 TeV and 7 TeV analysis samples, it is therefore
necessary to reweight the calibration tracks such that their distributions in
momentum and track multiplicity match those of a suitable reference sample. A
single reference sample cannot be adopted for all particle types, as the
unbiased momentum spectrum is in general different particle-to-particle. For
this reason, the analysis samples are used, but with the final selection
replaced by looser PID requirements. This modified selection minimises
distortions to the momentum spectra, while providing sufficient purity for the
differences in distributions between particle species to be still evident. In
each ($p_{\rm T}$, $\eta$) bin the reference and calibration samples are
subdivided into six momentum and four track multiplicity cells, and the
relative proportion of tracks within each cell is used to calculate a weight.
The PID performance as determined from the calibration samples after
reweighting is then applied in the analysis.
Figure 3: Monte Carlo PID efficiency study for protons (a), kaons (b) and
pions (c). Shown is a comparison of measured efficiencies from a Monte Carlo
calibration sample, after background subtraction and reweighting, with the
true values in the Monte Carlo analysis sample. The diagonal line on each plot
is drawn with unit gradient.
The reliability of the calibration can be assessed by comparing the results
for the measured PID efficiencies from a Monte Carlo simulated calibration
sample, after background subtraction and reweighting, to the true values in
the Monte Carlo analysis sample. The results are shown in Fig. 3, where each
entry comes from a separate ($p_{\rm T}$, $\eta$) bin. In general good
agreement is observed over a wide range of working points, with some residual
biases seen at low $p_{\rm T}$. These biases can be attributed to minor
deficiencies in the reweighting procedure, which are expected to be most
prevalent in this region.
## 5 Analysis procedure
The number of particles, $N^{\rm S}_{i}$, selected in each of the three
classes $i=p,\,K$ or $\pi$, is related to the true number of particles before
particle identification, $N^{\rm T}_{i}$, by the relationship
$\left(\begin{array}[]{c}N^{\rm S}_{p}\\\ N^{\rm S}_{K}\\\ N^{\rm
S}_{\pi}\end{array}\right)\,=\,\left(\begin{array}[]{ccc}\epsilon_{p\rightarrow
p}&\epsilon_{K\rightarrow p}&\epsilon_{\pi\rightarrow p}\\\
\epsilon_{p\rightarrow K}&\epsilon_{K\rightarrow K}&\epsilon_{\pi\rightarrow
K}\\\
\epsilon_{p\rightarrow\pi}&\epsilon_{K\rightarrow\pi}&\epsilon_{\pi\rightarrow\pi}\end{array}\right)\left(\begin{array}[]{c}N^{\rm
T}_{p}\\\ N^{\rm T}_{K}\\\ N^{\rm T}_{\pi}\end{array}\right),$ (1)
where the matrix element $\epsilon_{i\rightarrow j}$ is the probability of
identifying particle type $i$ as type $j$. This expression is valid for the
purposes of the measurement since the fraction of other particle types, in
particular electrons and muons, contaminating the selected sample is
negligible. As $N^{\rm S}_{i}$ and $\epsilon_{i\rightarrow j}$ are known, the
expression can be inverted to determine $N^{\rm T}_{i}$. This is done for each
$(\mbox{$p_{\rm T}$},\eta)$ bin, at each energy point and magnet polarity
setting. After this step (and including the low $p_{\rm T}$ scaling factor
correction discussed below) the purities of each sample can be calculated.
Averaged over the analysis bins the purities at 0.9 TeV (7 TeV) are found to
be 0.90 (0.84), 0.89 (0.87) and 0.98 (0.97) for the protons, kaons and pions,
respectively.
In order to relate $N^{\rm T}_{i}$ to the number of particles produced in the
primary interaction it is necessary to correct for the effects of non-prompt
contamination, geometrical acceptance losses and track finding inefficiency.
The non-prompt correction, according to simulation, is typically 1–2%, and is
similar for positive and negative particles. The most important correction
when calculating the particle ratios is that related to the track finding
inefficiency, as different interaction cross-sections and decays in flight
mean that this effect does not in general cancel. All correction factors are
taken from simulation, and are applied bin-by-bin, after which the particle
ratios are determined. The corrections typically lead to a change of less than
a relative 10% on the ratios.
The analysis procedure is validated on simulated events in which the measured
ratios are compared with those expected from generator level. A $\chi^{2}$ is
formed over all the $\eta$ bins at low $p_{\rm T}$, summed over the different-
particle ratios. Good agreement is found for the same-particle ratios over all
$\eta$ and $p_{\rm T}$, and for the different-particle ratios at mid and high
$p_{\rm T}$. Discrepancies are however observed at low $p_{\rm T}$ for the
different-particle ratios, which are attributed to imperfections in the PID
reweighting procedure for this region. The $\chi^{2}$ in the low $p_{\rm T}$
bin is then minimised by applying charge-independent scaling factors of $1.33$
($1.10$) and $0.90$ ($0.86$) for the proton and kaon efficiencies,
respectively, at 0.9 TeV (7 TeV). An uncertainty of $\pm 0.11$ is assigned to
the scaling factors, uncorrelated bin-to-bin, in order to obtain
$\chi^{2}/{\rm ndf}\approx 1$ at both energy points. This uncertainty is fully
correlated between positive and negative tracks. Although no bias is observed
at mid and high $p_{\rm T}$, an additional relative uncertainty of $\pm 0.03$
is assigned to the proton and kaon efficiencies for these bins to yield an
acceptable scatter (i.e. $\chi^{2}/{\rm ndf}\approx 1$). This uncertainty is
also taken to be uncorrelated bin-to-bin, but fully correlated between
positive and negative tracks. The scaling factors and uncertainties from these
studies are adopted for the analysis of the data.
## 6 Systematic uncertainties
The contribution to the systematic uncertainty of all effects considered is
summarised in Tables 3 and 4 for the same-particle ratios, and in Tables 5 and
6 for the different-particle ratios.
The dominant uncertainty is associated with the understanding of the PID
performance. Each element in the identification matrix (Eq. 1), is smeared by
a Gaussian of width corresponding to the uncertainty in the identification (or
misidentification) efficiency of that element, and the full set of particle
ratios is recalculated. This uncertainty is the sum in quadrature of the
statistical error from the calibration sample after reweighting, as discussed
in Sect. 4, and the additional uncertainty assigned after the analysis
validation, described in Sect. 5. The procedure is repeated many times and the
width of the resulting distributions is assigned as the systematic
uncertainty. As can be seen in Tables 3–6 there is a large range in the
magnitude of this contribution. The uncertainty is smallest at high $p_{\rm
T}$ and $\eta$, on account of the distribution of the events in the
calibration sample. For each observable the largest value is found in the
lowest $\eta$ bin at mid-$p_{\rm T}$. If this bin and the lowest $\eta$ bin at
low $p_{\rm T}$ are discounted, the variation in uncertainty of the remainder
of the acceptance is much smaller, being typically a factor of two or three.
Knowledge of the interaction cross-sections and the amount of material
encountered by particles in traversing the spectrometer is necessary to
determine the fraction of particles that cannot be reconstructed due to having
undergone a strong interaction. The interaction cross-sections as implemented
in the LHCb simulation agree with measurements [38] over the momentum range of
interest to a precision of around 20% for protons and kaons, and 10% for
pions. The material description up to and including the tracking detectors is
correct within a tolerance of 10%. The effect of these uncertainties is
propagated through in the calculation of the track loss for each particle type
from strong interaction effects.
The detection efficiency of positive and negative tracks need not be identical
due to the fact that each category is swept by the dipole field, on average,
to different regions of the spectrometer. Studies using
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decays in which one track is selected by
muon chamber information alone constrain any charge asymmetry in the track
reconstruction efficiency to be less than 1.0 (0.5)% for the 0.9 (7) TeV data.
These values are used to assign systematic uncertainties on the particle
ratios. The identification efficiencies in the RICH system are measured
separately for each charge, and so this effect is accounted for in the inputs
to the analysis. A cross-check that there are no significant reconstruction
asymmetries left unaccounted for is provided by a comparison of the results
obtained with the two polarity settings of the dipole magnet. Consistent
results are found for all observables.
A possible source of bias arises from the contribution of ‘ghost’ tracks;
these are tracks which have no correspondence with the trajectory of any
charged particle in the event, but are reconstructed from the incorrect
association of hit points in the tracking detectors. Systematic uncertainties
are therefore assigned in each $(p_{T},\eta)$ bin for each category of ratio
by subtracting the estimated contribution of ghost tracks for each particle
assignment, and determining the resulting shifts in the calculated ratios. A
sample enriched in ghost tracks can be obtained by selecting tracks where the
number of hits associated with the track in the TT detector is significantly
less than that expected for a particle with that trajectory. Comparison of the
fraction of tracks of this nature in data and simulation is used to determine
the ghost-track rate in data by scaling the known rate in simulation. This
exercise is performed independently for identified tracks which are above and
below the Cherenkov threshold in the RICH system. The contamination from ghost
tracks is lower in the above-threshold category since the presence of
photodetector hits is indicative of a genuine track. The total ghost-track
fraction for pions and kaons is found to be typically below 1%, rising to
around 2% in certain bins. The ghost-track fraction for protons rises to 5% in
some bins, on account of the larger fraction of this particle type lying below
the RICH threshold. The charge asymmetry for this background is found to be
small and the assigned systematic uncertainty is in general around 0.1%. To
provide further confirmation that ghost tracks are not a significant source of
bias the analysis is repeated with different cut values on the track-fit
$\chi^{2}/{\rm ndf}$ and stable results are found.
Clones are suppressed by the requirement that only one track is retained from
pairs of tracks that have very similar momentum. The analysis is repeated with
the requirement removed, and negligible changes are seen for all observables.
Contamination from non-prompt particles induces a small uncertainty in the
measurement, as this source of background is at a low level and cancels to
first order in the ratios. The error is assigned by repeating the analysis and
doubling the assumed charge asymmetry of these tracks compared with the value
found from the simulation. No significant variations are observed when the
analysis is repeated with different cut values on the prompt-track selection
variable $\chi^{2}_{\rm IP}$.
The total systematic uncertainty for each observable is obtained by summing in
quadrature the individual contributions in each ($p_{\rm T}$, $\eta$) bin. In
general, the systematic uncertainty is significantly larger than the
statistical uncertainty, with the largest contribution coming from the
knowledge of the PID performance, which is limited by the size of the
calibration sample.
Table 3: Range of systematic uncertainties, in percent, for same-particle ratios at $\sqrt{s}=0.9$ TeV. | $\bar{p}/{p}$ | $K^{-}/K^{+}$ | $\pi^{-}/\pi^{+}$
---|---|---|---
PID | 7.5 $-$ 46.7 | 4.9 $-$ 42.4 | 0.8 $-$ 6.0
Cross-sections | 0.2 $-$ 1.6 | 0.1 $-$ 1.5 | $<$0.1 $-$ 0.8
Detector material | 0.1 $-$ 0.8 | 0.1 $-$ 0.7 | $<$0.1 $-$ 0.8
Ghosts | $<$0.1 $-$ 0.1 | $<$0.1 $-$ 0.1 | $<$0.1 $-$ 0.1
Tracking asymmetry | 1.0 | 1.0 | 1.0
Non-prompt | $<$0.1 $-$ 0.2 | $<$0.1 $-$ 0.1 | $<$0.1 $-$ 0.1
Total | 7.7 $-$ 46.7 | 5.0 $-$ 42.4 | 1.3 $-$ 6.0
Table 4: Range of systematic uncertainties, in percent, for same-particle ratios at $\sqrt{s}=7$ TeV. | $\bar{p}/{p}$ | $K^{-}/K^{+}$ | $\pi^{-}/\pi^{+}$
---|---|---|---
PID | 3.4 $-$ 26.4 | 2.0 $-$ 15.8 | 0.6 $-$ 2.7
Cross-sections | 0.3 $-$ 1.8 | 0.3 $-$ 0.7 | $<$0.1 $-$ 0.2
Detector material | 0.2 $-$ 0.9 | 0.1 $-$ 0.4 | $<$0.1 $-$ 0.2
Ghosts | $<$0.1 $-$ 0.4 | $<$0.1 $-$ 0.1 | $<$0.1
Tracking asymmetry | 0.5 | 0.5 | 0.5
Non-prompt | $<$0.1 $-$ 0.2 | $<$0.1 $-$ 0.1 | $<$0.1 $-$ 0.1
Total | 3.5 $-$ 26.5 | 2.1 $-$ 15.8 | 0.8 $-$ 2.8
Table 5: Range of systematic uncertainties, in percent, for different-particle ratios at $\sqrt{s}=0.9$ TeV. | $(p+\bar{p})/(\pi^{+}+\pi^{-})$ | $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ | $(p+\bar{p})/(K^{+}+K^{-})$
---|---|---|---
PID | 10.2 $-$ 63.7 | 8.1 $-$ 46.8 | 5.9 $-$ 42.6
Cross-sections | 0.1 $-$ 1.6 | 0.4 $-$ 1.3 | 0.2 $-$ 2.4
Detector material | $<$0.1 $-$ 0.8 | 0.2 $-$ 0.7 | 0.1 $-$ 1.2
Ghosts | $<$0.1 $-$ 0.1 | $<$0.1 $-$ 0.1 | $<$0.1 $-$ 0.1
Tracking asymmetry | $<$0.1 | $<$0.1 | $<$0.1
Non-prompt | $<$0.1 $-$ 0.2 | 0.1 | $<$0.1 $-$ 0.1
Total | 10.2 $-$ 63.7 | 8.6 $-$ 46.8 | 6.0 $-$ 42.6
Table 6: Range of systematic uncertainties, in percent, for different-particle ratios at $\sqrt{s}=7$ TeV. | $(p+\bar{p})/(\pi^{+}+\pi^{-})$ | $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ | $(p+\bar{p})/(K^{+}+K^{-})$
---|---|---|---
PID | 5.9 $-$ 31.1 | 4.6 $-$ 26.6 | 3.7 $-$ 16.1
Cross-sections | 0.3 $-$ 2.2 | 1.2 $-$ 1.5 | 0.2 $-$ 2.1
Detector material | 0.2 $-$ 1.1 | 0.6 $-$ 0.8 | 0.1 $-$ 1.0
Ghosts | $<$0.1 $-$ 0.3 | $<$0.1 $-$ 0.3 | $<$0.1 $-$ 0.2
Tracking asymmetry | $<$0.1 | $<$0.1 | $<$0.1
Non-prompt | $<$0.1 $-$ 0.3 | 0.1 $-$ 0.2 | $<$0.1 $-$ 0.2
Total | 6.0 $-$ 31.1 | 4.8 $-$ 26.7 | 3.7 $-$ 16.2
## 7 Results
The measurements of the same-particle ratios are plotted in Figs. 4, 5 and 6,
and those of the different-particle ratios in Figs. 7, 8 and 9. The numerical
values can be found in Appendix A. Also shown are the predictions of several
Pythia6.4 generator settings, or ‘tunes’: LHCb MC[30], Perugia 0 and Perugia
NOCR[36]. At 0.9 TeV the $\bar{p}/p$ ratio falls from around 0.8 at low $\eta$
to around 0.4 in the highest $p_{\rm T}$ and $\eta$ bin. At this energy point
there is a significant spread between models for the Monte Carlo predictions,
with the data lying significantly below the LHCb MC and Perugia 0
expectations, but close to those of Perugia NOCR. At higher energy the
$\bar{p}/p$ ratio is higher and varies more slowly, in good agreement with
LHCb MC and Perugia 0 and less so with Perugia NOCR. The $K^{-}/K^{+}$ and
$\pi^{-}/\pi^{+}$ ratios also differ from unity, most noticeably at high
$p_{\rm T}$ and high $\eta$. This behaviour is in general well modelled by all
the generator tunes, which give similar predictions for these observables.
Small discrepancies are observed at 7 TeV for $K^{-}/K^{+}$ at low $p_{\rm
T}$, and $\pi^{-}/\pi^{+}$ at high $p_{\rm T}$. When comparing the
measurements and predictions for the different-particle ratios the most
striking differences occur for $(p+\bar{p})/(\pi^{+}+\pi^{-})$ and
$(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$, where there is a tendency for the data to
lie significantly higher than the Perugia 0 and NOCR expectations. The
agreement with the LHCb MC for these observables is generally good.
It is instructive to consider the $\bar{p}/p$ results as a function of
rapidity loss, $\Delta y\equiv y_{\rm beam}-y$, where $y_{\rm beam}$ is the
rapidity of the protons in the LHC beam which travels forward in the
spectrometer ($y_{\rm beam}=6.87$ at 0.9 TeV and 8.92 at 7 TeV). For the same-
particle ratios it is possible to determine the rapidity value to which the
measurement in each $\eta$ bin corresponds. In each bin the mean and RMS
spread of the rapidity of the tracks in the analysis sample is determined.
Correlations are accounted for, but these are in general negligible as the
uncertainties are dominated by the PID errors, which for these observables are
statistical in nature. A small correction is applied to this mean, obtained
from Monte Carlo, to account for the distortion to the unbiased spectrum that
is induced by the reconstruction and PID requirements. The values of the mean
and RMS spread of the rapidities for $\bar{p}/p$ can be found in Appendix A,
together with those of $K^{-}/K^{+}$ and $\pi^{-}/\pi^{+}$. As no evidence is
seen of any $p_{\rm T}$ dependence in the distribution of the $\bar{p}/p$
results against $\Delta y$ the measurements in each $\eta$ bin at each energy
point are integrated over $p_{\rm T}$, with the uncertainties on the
individual values of the ratios used to determine the weights of each input
entering into the mean. The mean $\bar{p}/p$ ratios are given as a function of
$\Delta y$ in Table 7 and plotted in Fig. 10, with the results from other
experiments [16, 17, 18, 19, 20, 21] superimposed. The LHCb results cover a
wider range of $\Delta y$ than any other single experiment and significantly
improve the precision of the measurements in the region $\Delta y<6.5$.
Table 7: Results for $\bar{p}/p$ ratio integrated over $p_{\rm T}$ in $\eta$ bins as a function of the rapidity loss $\Delta y$. $\sqrt{\mathrm{s}}$ | $\eta$ range | $\Delta y$ | Ratio
---|---|---|---
$0.9\,\mathrm{TeV}$ | $4.0-4.5$ | $3.1\pm 0.2$ | $0.48\pm 0.03$
| $3.5-4.0$ | $3.5\pm 0.2$ | $0.57\pm 0.02$
| $3.0-3.5$ | $3.9\pm 0.2$ | $0.65\pm 0.03$
| $2.5-3.0$ | $4.3\pm 0.1$ | $0.81\pm 0.09$
$7\,\mathrm{TeV}$ | $4.0-4.5$ | $5.1\pm 0.2$ | $0.90\pm 0.03$
| $3.5-4.0$ | $5.5\pm 0.2$ | $0.92\pm 0.02$
| $3.0-3.5$ | $5.9\pm 0.2$ | $0.91\pm 0.02$
| $2.5-3.0$ | $6.3\pm 0.1$ | $0.89\pm 0.04$
Within the Regge model, baryon production at high energy is driven by Pomeron
exchange and baryon transport by string-junction exchange [9]. Assuming this
picture the $\Delta y$ dependence of the $\bar{p}/p$ ratio approximately
follows the form $1/\left(1+C\exp[(\alpha_{J}-\alpha_{P})\Delta y]\right)$,
where $C$ determines the relative contributions of the two mechanisms, and
$\alpha_{J}$ ($\alpha_{P}$) is the intercept of the string junction (Pomeron)
Regge trajectory. Figure 10 shows the results of fitting this expression to
both the LHCb and, in order to constrain the high $\Delta y$ region, the ALICE
data. Both $C$ and $(\alpha_{J}-\alpha_{P})$ are free parameters of the fit
and are determined to be $22.5\pm 6.0$ and $-0.98\pm 0.07$ respectively with a
$\chi^{2}$/ndf of $8.7/8$. Taking $\alpha_{P}=1.2$ [39] suggests a low value
of $\alpha_{J}$, significantly below the $\alpha_{J}\approx 0.5$ expected if
the string-junction intercept is associated with that of the standard Reggeon
(or meson). The value of $\alpha_{J}\approx 0.9$ which would be expected if
the string junction is associated with the Odderon[13] is excluded using this
fit model. The same conclusion applies if the LHCb and ALICE $\bar{p}/p$ ratio
values are fitted with an alternative parameterisation [11]
$C^{\prime}\cdot(s[{\rm
GeV}^{2}])^{(\alpha_{J}-\alpha_{P})/2}\cdot\cosh[y(\alpha_{J}-\alpha_{P})]$,
which yields the results $C^{\prime}=10.2\pm 1.8$,
$(\alpha_{J}-\alpha_{P})=-0.86\pm 0.05$ with a $\chi^{2}$/ndf of $10.2/8$.
Figure 4: Results for the $\bar{p}/p$ ratio at 0.9 TeV (a) and 7 TeV (b).
Figure 5: Results for the $K^{-}/K^{+}$ ratio at 0.9 TeV (a) and 7 TeV (b).
Figure 6: Results for the $\pi^{-}/\pi^{+}$ ratio at 0.9 TeV (a) and 7 TeV
(b).
Figure 7: Results for the $(p+\bar{p})/(\pi^{+}+\pi^{-})$ ratio at 0.9 TeV (a)
and 7 TeV (b).
Figure 8: Results for the $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ ratio at 0.9 TeV
(a) and 7 TeV (b).
Figure 9: Results for the $(p+\bar{p})/(K^{+}+K^{-})$ ratio at 0.9 TeV (a) and
7 TeV (b). Figure 10: Results for the $\bar{p}/p$ ratio against the rapidity
loss $\Delta y$ from LHCb. Results from other experiments are also shown [16,
17, 18, 19, 20, 21]. Superimposed is a fit to the LHCb and ALICE [16]
measurements that is described in the text.
## 8 Conclusions
Measurements have been presented of the charged-particle production ratios
$\bar{p}/{p}$, $K^{-}/K^{+}$, $\pi^{-}/\pi^{+}$, $(p+\bar{p})/(K^{+}+K^{-})$,
$(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ and $(p+\bar{p})/(\pi^{+}+\pi^{-})$ at both
$\sqrt{s}=0.9$ TeV and $\sqrt{s}=7$ TeV. The results at $7$ TeV are the first
studies of pion, kaon and proton production to be performed at this energy.
Comparisons have been made with several generator tunes (LHCb MC, Perugia 0
and Perugia NOCR). No single tune is able to describe well all observables.
The most significant discrepancies occur for the
$(p+\bar{p})/(\pi^{+}+\pi^{-})$ and $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ ratios,
where the measurements are much higher than the Perugia 0 and Perugia NOCR
predictions, but lie reasonably close to the LHCb MC expectation.
The $\bar{p}/p$ ratio has been studied as a function of rapidity loss, $\Delta
y$. The results span the $\Delta y$ interval $3.1$ to $6.3$, and are more
precise than previous measurements in this region. Fitting a simple Regge
theory inspired model to the LHCb measurements, and those from the midrapidity
region obtained by ALICE [16], yields a result with a string-junction
contribution with low intercept value.
These results, together with those for related observables obtained by LHCb
[24], will help in understanding the phenomenon of baryon-number transport,
and the development of hadronisation models to improve the description of
Standard Model processes in the forward region at the LHC.
## Acknowledgements
We thank Yuli Shabelski for several useful discussions. We express our
gratitude to our colleagues in the CERN accelerator departments for the
excellent performance of the LHC. We thank the technical and administrative
staff at CERN and at the LHCb institutes, and acknowledge support from the
National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); CERN; NSFC (China);
CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN
(Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS (Romania); MinES
of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT (Spain); SNSF and
SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7 and the Region
Auvergne.
Appendix
## Appendix A Tables of results
The results for the same-particle ratios, including the rapidity to which the
events in each pseudorapidity bin correspond, are given in Tables 8, 9 and 10.
The results for the different-particle ratios can be found in Tables 11, 12
and 13.
Table 8: Results for the $\bar{p}/p$ ratio with statistical and systematic uncertainties, as a function of $p_{\rm T}$ and $\eta$. Also shown is the mean rapidity, $y$, and RMS spread for the sample in each $\eta$ bin. | $\mbox{$p_{\rm T}$}<0.8$ GeV/$c$ | $0.8\leq\mbox{$p_{\rm T}$}<1.2$ GeV/$c$ | $\mbox{$p_{\rm T}$}\geq 1.2$ GeV/$c$
---|---|---|---
| $y$ (RMS) | Ratio | $y$ (RMS) | Ratio | $y$ (RMS) | Ratio
$\sqrt{s}=0.9$ TeV | | | | | |
$2.5<\eta<3.0$ | – | – | $2.42$ $(0.24)$ | ${1.107\pm 0.020\pm 0.349}$ | $2.63$ $(0.16)$ | ${0.794\pm 0.015\pm 0.089}$
$3.0\leq\eta<3.5$ | $2.58$ $(0.27)$ | ${0.751\pm 0.011\pm 0.163}$ | $2.96$ $(0.25)$ | ${0.684\pm 0.010\pm 0.049}$ | $3.08$ $(0.23)$ | ${0.614\pm 0.010\pm 0.047}$
$3.5\leq\eta<4.0$ | $2.96$ $(0.11)$ | ${0.729\pm 0.007\pm 0.040}$ | $3.40$ $(0.22)$ | ${0.576\pm 0.007\pm 0.032}$ | $3.56$ $(0.24)$ | ${0.456\pm 0.009\pm 0.033}$
$4.0\leq\eta<4.5$ | $3.34$ $(0.24)$ | ${0.660\pm 0.009\pm 0.046}$ | $3.87$ $(0.14)$ | ${0.451\pm 0.009\pm 0.038}$ | $4.02$ $(0.25)$ | ${0.328\pm 0.010\pm 0.049}$
$\sqrt{s}=7$ TeV | | | | | |
$2.5<\eta<3.0$ | – | – | $2.41$ $(0.25)$ | ${1.181\pm 0.020\pm 0.195}$ | $2.63$ $(0.16)$ | ${0.880\pm 0.009\pm 0.039}$
$3.0\leq\eta<3.5$ | $2.55$ $(0.27)$ | ${0.734\pm 0.011\pm 0.124}$ | $2.98$ $(0.25)$ | ${0.942\pm 0.011\pm 0.036}$ | $3.12$ $(0.22)$ | ${0.905\pm 0.008\pm 0.026}$
$3.5\leq\eta<4.0$ | $2.96$ $(0.09)$ | ${1.015\pm 0.009\pm 0.037}$ | $3.40$ $(0.23)$ | ${0.916\pm 0.007\pm 0.022}$ | $3.59$ $(0.24)$ | ${0.903\pm 0.008\pm 0.023}$
$4.0\leq\eta<4.5$ | $3.34$ $(0.21)$ | ${0.957\pm 0.010\pm 0.051}$ | $3.86$ $(0.19)$ | ${0.906\pm 0.010\pm 0.039}$ | $4.06$ $(0.25)$ | ${0.831\pm 0.010\pm 0.050}$
Table 9: Results for the $K^{-}/K^{+}$ ratio with statistical and systematic uncertainties, as a function of $p_{\rm T}$ and $\eta$. Also shown is the mean rapidity, $y$, and RMS spread for the sample in each $\eta$ bin. | $\mbox{$p_{\rm T}$}<0.8$ GeV/$c$ | $0.8\leq\mbox{$p_{\rm T}$}<1.2$ GeV/$c$ | $\mbox{$p_{\rm T}$}\geq 1.2$ GeV/$c$
---|---|---|---
| $y$ (RMS) | Ratio | $y$ (RMS) | Ratio | $y$ (RMS) | Ratio
$\sqrt{s}=0.9$ TeV | | | | | |
$2.5<\eta<3.0$ | – | – | $2.65$ $(0.19)$ | ${0.870\pm 0.010\pm 0.267}$ | $2.69$ $(0.14)$ | ${0.936\pm 0.013\pm 0.069}$
$3.0\leq\eta<3.5$ | $2.99$ $(0.25)$ | ${0.834\pm 0.007\pm 0.069}$ | $3.12$ $(0.21)$ | ${0.847\pm 0.009\pm 0.040}$ | $3.18$ $(0.15)$ | ${0.783\pm 0.011\pm 0.037}$
$3.5\leq\eta<4.0$ | $3.32$ $(0.25)$ | ${1.001\pm 0.007\pm 0.064}$ | $3.62$ $(0.22)$ | ${0.792\pm 0.009\pm 0.028}$ | $3.70$ $(0.17)$ | ${0.723\pm 0.012\pm 0.031}$
$4.0\leq\eta<4.5$ | $3.67$ $(0.18)$ | ${1.002\pm 0.007\pm 0.093}$ | $4.11$ $(0.25)$ | ${0.680\pm 0.010\pm 0.041}$ | $4.20$ $(0.21)$ | ${0.506\pm 0.014\pm 0.050}$
$\sqrt{s}=7$ TeV | | | | | |
$2.5<\eta<3.0$ | – | – | $2.65$ $(0.19)$ | ${0.995\pm 0.008\pm 0.101}$ | $2.70$ $(0.13)$ | ${0.991\pm 0.007\pm 0.021}$
$3.0\leq\eta<3.5$ | $3.02$ $(0.25)$ | ${0.992\pm 0.006\pm 0.063}$ | $3.12$ $(0.21)$ | ${0.966\pm 0.006\pm 0.019}$ | $3.20$ $(0.14)$ | ${0.999\pm 0.006\pm 0.016}$
$3.5\leq\eta<4.0$ | $3.34$ $(0.25)$ | ${1.062\pm 0.005\pm 0.040}$ | $3.62$ $(0.21)$ | ${0.948\pm 0.006\pm 0.014}$ | $3.70$ $(0.15)$ | ${0.930\pm 0.006\pm 0.017}$
$4.0\leq\eta<4.5$ | $3.72$ $(0.22)$ | ${1.161\pm 0.005\pm 0.055}$ | $4.11$ $(0.23)$ | ${0.898\pm 0.006\pm 0.025}$ | $4.21$ $(0.18)$ | ${0.958\pm 0.009\pm 0.049}$
Table 10: Results for the $\pi^{-}/\pi^{+}$ ratio with statistical and systematic uncertainties, as a function of $p_{\rm T}$ and $\eta$. Also shown is the mean rapidity, $y$, and RMS spread for the sample in each $\eta$ bin. | $\mbox{$p_{\rm T}$}<0.8$ GeV/$c$ | $0.8\leq\mbox{$p_{\rm T}$}<1.2$ GeV/$c$ | $\mbox{$p_{\rm T}$}\geq 1.2$ GeV/$c$
---|---|---|---
| $y$ (RMS) | Ratio | $y$ (RMS) | Ratio | $y$ (RMS) | Ratio
$\sqrt{s}=0.9$ TeV | | | | | |
$2.5<\eta<3.0$ | – | – | $2.74$ $(0.07)$ | ${0.987\pm 0.010\pm 0.013}$ | $2.75$ $(0.05)$ | ${0.970\pm 0.016\pm 0.014}$
$3.0\leq\eta<3.5$ | $3.23$ $(0.09)$ | ${0.979\pm 0.005\pm 0.010}$ | $3.23$ $(0.07)$ | ${0.971\pm 0.011\pm 0.010}$ | $3.24$ $(0.05)$ | ${0.926\pm 0.017\pm 0.014}$
$3.5\leq\eta<4.0$ | $3.71$ $(0.15)$ | ${0.968\pm 0.004\pm 0.011}$ | $3.75$ $(0.08)$ | ${0.951\pm 0.012\pm 0.010}$ | $3.75$ $(0.05)$ | ${0.871\pm 0.019\pm 0.012}$
$4.0\leq\eta<4.5$ | $4.15$ $(0.24)$ | ${0.929\pm 0.004\pm 0.017}$ | $4.30$ $(0.10)$ | ${0.971\pm 0.016\pm 0.019}$ | $4.30$ $(0.07)$ | ${0.816\pm 0.025\pm 0.029}$
$\sqrt{s}=7$ TeV | | | | | |
$2.5<\eta<3.0$ | – | – | $2.74$ $(0.07)$ | ${1.002\pm 0.007\pm 0.006}$ | $2.74$ $(0.04)$ | ${1.015\pm 0.010\pm 0.005}$
$3.0\leq\eta<3.5$ | $3.23$ $(0.09)$ | ${1.011\pm 0.004\pm 0.006}$ | $3.24$ $(0.07)$ | ${0.998\pm 0.007\pm 0.004}$ | $3.24$ $(0.04)$ | ${0.998\pm 0.010\pm 0.004}$
$3.5\leq\eta<4.0$ | $3.70$ $(0.14)$ | ${1.002\pm 0.003\pm 0.006}$ | $3.74$ $(0.07)$ | ${1.003\pm 0.008\pm 0.004}$ | $3.75$ $(0.05)$ | ${1.000\pm 0.011\pm 0.005}$
$4.0\leq\eta<4.5$ | $4.14$ $(0.22)$ | ${0.976\pm 0.003\pm 0.006}$ | $4.26$ $(0.08)$ | ${0.998\pm 0.009\pm 0.008}$ | $4.26$ $(0.05)$ | ${0.974\pm 0.012\pm 0.017}$
Table 11: Results for the $(p+\bar{p})/(\pi^{+}+\pi^{-})$ ratio with statistical and systematic uncertainties, as a function of $p_{\rm T}$ and $\eta$. | $\mbox{$p_{\rm T}$}<0.8$ GeV/$c$ | $0.8\leq\mbox{$p_{\rm T}$}<1.2$ GeV/$c$ | $\mbox{$p_{\rm T}$}\geq 1.2$ GeV/$c$
---|---|---|---
$\sqrt{s}=0.9$ TeV | | |
$2.5<\eta<3.0$ | – | ${0.328\pm 0.007\pm 0.104}$ | ${0.300\pm 0.008\pm 0.034}$
$3.0\leq\eta<3.5$ | ${0.086\pm 0.001\pm 0.021}$ | ${0.208\pm 0.004\pm 0.016}$ | ${0.272\pm 0.007\pm 0.023}$
$3.5\leq\eta<4.0$ | ${0.062\pm 0.001\pm 0.008}$ | ${0.175\pm 0.003\pm 0.011}$ | ${0.252\pm 0.007\pm 0.020}$
$4.0\leq\eta<4.5$ | ${0.076\pm 0.001\pm 0.010}$ | ${0.233\pm 0.006\pm 0.022}$ | ${0.301\pm 0.013\pm 0.047}$
$\sqrt{s}=7$ TeV | | |
$2.5<\eta<3.0$ | – | ${0.235\pm 0.004\pm 0.039}$ | ${0.262\pm 0.004\pm 0.014}$
$3.0\leq\eta<3.5$ | ${0.085\pm 0.001\pm 0.017}$ | ${0.174\pm 0.002\pm 0.009}$ | ${0.245\pm 0.003\pm 0.011}$
$3.5\leq\eta<4.0$ | ${0.069\pm 0.001\pm 0.008}$ | ${0.156\pm 0.002\pm 0.006}$ | ${0.242\pm 0.003\pm 0.010}$
$4.0\leq\eta<4.5$ | ${0.051\pm 0.001\pm 0.007}$ | ${0.184\pm 0.003\pm 0.010}$ | ${0.244\pm 0.004\pm 0.017}$
Table 12: Results for the $(K^{+}+K^{-})/(\pi^{+}+\pi^{-})$ ratio with statistical and systematic uncertainties, as a function of $p_{\rm T}$ and $\eta$. | $\mbox{$p_{\rm T}$}<0.8$ GeV/$c$ | $0.8\leq\mbox{$p_{\rm T}$}<1.2$ GeV/$c$ | $\mbox{$p_{\rm T}$}\geq 1.2$ GeV/$c$
---|---|---|---
$\sqrt{s}=0.9$ TeV | | |
$2.5<\eta<3.0$ | – | ${0.184\pm 0.003\pm 0.056}$ | ${0.351\pm 0.008\pm 0.028}$
$3.0\leq\eta<3.5$ | ${0.180\pm 0.002\pm 0.026}$ | ${0.267\pm 0.004\pm 0.015}$ | ${0.319\pm 0.008\pm 0.018}$
$3.5\leq\eta<4.0$ | ${0.171\pm 0.001\pm 0.023}$ | ${0.247\pm 0.004\pm 0.011}$ | ${0.314\pm 0.009\pm 0.017}$
$4.0\leq\eta<4.5$ | ${0.173\pm 0.001\pm 0.025}$ | ${0.268\pm 0.006\pm 0.018}$ | ${0.281\pm 0.012\pm 0.031}$
$\sqrt{s}=7$ TeV | | |
$2.5<\eta<3.0$ | – | ${0.224\pm 0.002\pm 0.024}$ | ${0.371\pm 0.004\pm 0.014}$
$3.0\leq\eta<3.5$ | ${0.181\pm 0.001\pm 0.024}$ | ${0.263\pm 0.003\pm 0.010}$ | ${0.357\pm 0.004\pm 0.012}$
$3.5\leq\eta<4.0$ | ${0.173\pm 0.001\pm 0.021}$ | ${0.262\pm 0.003\pm 0.009}$ | ${0.367\pm 0.005\pm 0.013}$
$4.0\leq\eta<4.5$ | ${0.131\pm 0.001\pm 0.016}$ | ${0.275\pm 0.003\pm 0.011}$ | ${0.328\pm 0.005\pm 0.020}$
Table 13: Results for the $(p+\bar{p})/(K^{+}+K^{-})$ ratio with statistical and systematic uncertainties, as a function of $p_{\rm T}$ and $\eta$. | $\mbox{$p_{\rm T}$}<0.8$ GeV/$c$ | $0.8\leq\mbox{$p_{\rm T}$}<1.2$ GeV/$c$ | $\mbox{$p_{\rm T}$}\geq 1.2$ GeV/$c$
---|---|---|---
$\sqrt{s}=0.9$ TeV | | |
$2.5<\eta<3.0$ | – | ${1.831\pm 0.039\pm 0.822}$ | ${0.855\pm 0.020\pm 0.119}$
$3.0\leq\eta<3.5$ | ${0.481\pm 0.008\pm 0.139}$ | ${0.779\pm 0.014\pm 0.073}$ | ${0.851\pm 0.019\pm 0.084}$
$3.5\leq\eta<4.0$ | ${0.363\pm 0.004\pm 0.066}$ | ${0.709\pm 0.012\pm 0.055}$ | ${0.799\pm 0.021\pm 0.076}$
$4.0\leq\eta<4.5$ | ${0.433\pm 0.007\pm 0.086}$ | ${0.865\pm 0.021\pm 0.097}$ | ${1.067\pm 0.045\pm 0.200}$
$\sqrt{s}=7$ TeV | | |
$2.5<\eta<3.0$ | – | ${1.051\pm 0.020\pm 0.204}$ | ${0.705\pm 0.009\pm 0.046}$
$3.0\leq\eta<3.5$ | ${0.465\pm 0.008\pm 0.111}$ | ${0.660\pm 0.009\pm 0.039}$ | ${0.682\pm 0.007\pm 0.038}$
$3.5\leq\eta<4.0$ | ${0.398\pm 0.004\pm 0.067}$ | ${0.593\pm 0.006\pm 0.031}$ | ${0.659\pm 0.007\pm 0.037}$
$4.0\leq\eta<4.5$ | ${0.379\pm 0.004\pm 0.068}$ | ${0.671\pm 0.009\pm 0.046}$ | ${0.744\pm 0.011\pm 0.069}$
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|
arxiv-papers
| 2012-06-22T14:33:19 |
2024-09-04T02:49:32.074945
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, J. Anderson, R. B. Appleby, O. Aquines Gutierrez, F.\n Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma, S. Bachmann, J.\n J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, R. Currie,\n C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn, S. De\n Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D. Decamp, M.\n Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach, O.\n Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F. Domingo\n Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R. Ekelhof, L.\n Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante Pereira, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, C.\n Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C.\n Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman, P.\n Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, R. Gauld, N. Gauvin, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.\n A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S. C. Haines, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata,\n E. van Herwijnen, E. Hicks, D. Hill, M. Hoballah, P. Hopchev, W. Hulsbergen,\n P. Hunt, T. Huse, N. Hussain, R. S. Huston, D. Hutchcroft, D. Hynds, V.\n Iakovenko, P. Ilten, J. Imong, R. Jacobsson, A. Jaeger, M. Jahjah Hussein, E.\n Jans, F. Jansen, P. Jaton, B. Jean-Marie, F. Jing, M. John, D. Johnson, C. R.\n Jones, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T. M. Karbach, J.\n Keaveney, I. R. Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim,\n M. Knecht, O. Kochebina, I. Komarov, R. F. Koopman, P. Koppenburg, M.\n Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P.\n Krokovny, F. Kruse, K. Kruzelecki, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V. N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R. W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y. Li, L. Li Gioi, M. Lieng, M.\n Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J. von Loeben, J. H. Lopes, E.\n Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier, A. Mac Raighne, F.\n Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J. Magnin, S. Malde, R.\n M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave, U. Marconi, R.\n M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z. Mathe, C.\n Matteuzzi, M. Matveev, E. Maurice, B. Maynard, A. Mazurov, J. McCarthy, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, J. Palacios, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, R. Plackett, S. Playfer, M.\n Plo Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B.\n Popovici, C. Potterat, A. Powell, J. Prisciandaro, V. 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Websdale, M. Whitehead, J. Wicht, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Guy Wilkinson",
"url": "https://arxiv.org/abs/1206.5160"
}
|
1206.5299
|
# A note on the ($h,q$)-Zeta type function with weight $\alpha$
Elif Cetin Uludag University, Faculty of Arts and Science, Department of
Mathematics, Bursa, Turkey elifc2@hotmail.com , Mehmet Acikgoz University of
Gaziantep, Faculty of Science and Arts, Department of Mathematics, 27310
Gaziantep, TURKEY acikgoz@gantep.edu.tr , Ismail Naci Cangul Uludag
University, Faculty of Arts and Science, Department of Mathematics, Bursa,
Turkey ncangul@gmail.com and Serkan Araci University of Gaziantep, Faculty of
Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY
mtsrkn@hotmail.com
###### Abstract.
The objective of this paper is to derive symmetric property of ($h,q$)-Zeta
function with weight $\alpha$. By using this property, we give some
interesting identities for ($h,q$)-Genocchi polynomials with weight $\alpha$.
As a result, our applications possess a number of interesting property which
we state in this paper.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. ($h,q$)-Genocchi numbers and polynomials with weight
$\alpha$, ($h,q$)-Zeta function with weight $\alpha$, $p$-adic $q$-integral on
$\mathbb{Z}_{p}$.
## 1\. INTRODUCTION
Recently, T. Kim has developed a new method by using $q$-Volkenborn integral
(or $p$-adic $q$-integral on $\mathbb{Z}_{p}$) which has added a weight to
$q$-Bernoulli polynomials and investigated their properties (see [8]). He also
showed that this polynomials are closely related to weighted $q$-Bernstein
polynomials and derived novel properties of $q$-Bernoulli numbers with weight
$\alpha$ by using symmetric property of weighted $q$-Bernstein polynomials on
the $q$-Volkenborn integral (for more details, see [10]). After, Araci $et$
$al$. have introduced weighted ($h,q$)-Genocchi polynomials and so defined
($h,q$)-Zeta type function with weight by applying Mellin transformation to
generating function of ($h,q$)-Genocchi polynomials with weight $\alpha$ which
interpolates for ($h,q$)-Genocchi polynomials with weight $\alpha$ at negative
integers (for details, see [20]). In this paper, we also consider
($h$,$q$)-Zeta type function with weight and derive some interesting
properties.
We firstly list some notations as follows:
Imagine that $p$ be a fixed odd prime. Throughout this work $\mathbb{Z},$
$\mathbb{Z}_{p},$ $\mathbb{Q}_{p}$ and $\mathbb{C}_{p}$ will denote by the
ring of integers, the field of $p$-adic rational numbers and the completion of
the algebraic closure of $\mathbb{Q}_{p},$ respectively. Also we denote
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$ and
$\exp\left(x\right)=e^{x}.$ Let
$v_{p}:\mathbb{C}_{p}\rightarrow\mathbb{Q}\cup\left\\{\infty\right\\}$
($\mathbb{Q}$ is the field of rational numbers) denote the $p$-adic valuation
of $\mathbb{C}_{p}$ normalized so that $v_{p}\left(p\right)=1$. The absolute
value on $\mathbb{C}_{p}$ will be denoted as $\left|\text{ }.\right|$, and
$\left|x\right|_{p}=p^{-v_{p}\left(x\right)}$ for $x\in\mathbb{C}_{p}.$ When
one speaks of $q$-extensions, $q$ is considered in many ways, e.g. as an
indeterminate, a complex number $q\in\mathbb{C},$ or a $p$-adic number
$q\in\mathbb{C}_{p},$ If $q\in\mathbb{C}$ we assume that $\left|q\right|<1.$
If $q\in\mathbb{C}_{p},$ we assume $\left|1-q\right|_{p}<p^{-\frac{1}{p-1}},$
so that $q^{x}=\exp\left(x\log q\right)$ for $\left|x\right|_{p}\leq 1.$ We
use the following notation
(1) $\left[x\right]_{q}=\frac{1-q^{x}}{1-q},\text{ \
}\left[x\right]_{-q}=\frac{1-\left(-q\right)^{x}}{1+q}$
where we want to note that $\lim_{q\rightarrow 1}\left[x\right]_{q}=x;$ cf.
[1-21].
For a fixed positive integer $d$, set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overleftarrow{n}}\mathbb{Z}/dp^{n}\mathbb{Z},$ $\displaystyle
X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$
and
$a+dp^{n}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{n}\right)\right\\},$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{n}$ (see [1-21]).
The following $q$-Haar distribution is defined by T. Kim
$\mu_{q}\left(x+p^{n}\mathbb{Z}_{p}\right)=\frac{q^{x}}{\left[p^{n}\right]_{q}}$
for any positive $n$ (see [11], [12]).
Let $UD\left(\mathbb{Z}_{p}\right)$ be the set of uniformly differentiable
function on $\mathbb{Z}_{p}$. We say that $f$ is a uniformly differentiable
function at a point $a\in\mathbb{Z}_{p},$ if the difference quotient
$F_{f}\left(x,y\right)=\frac{f\left(x\right)-f\left(y\right)}{x-y}$
has a limit $f^{{\acute{}}}\left(a\right)$ as
$\left(x,y\right)\rightarrow\left(a,a\right)$ and denote this by $f\in
UD\left(\mathbb{Z}_{p}\right).$ In [11] and [12], the $p$-adic $q$-integral of
the function $f\in UD\left(\mathbb{Z}_{p}\right)$ is defined by Kim
(2)
$I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{q}\left(\xi\right)=\lim_{n\rightarrow\infty}\sum_{\xi=0}^{p^{n}-1}f\left(\xi\right)\mu_{q}\left(\xi+p^{n}\mathbb{Z}_{p}\right)$
The bosonic integral is considered as the bosonic limit $q\rightarrow 1,$
$I_{1}\left(f\right)=\lim_{q\rightarrow 1}I_{q}\left(f\right)$. Similarly, the
$p$-adic fermionic integration on $\mathbb{Z}_{p}$ is defined by Kim [5] as
follows:
(3)
$I_{-q}\left(f\right)=\lim_{q\rightarrow-q}I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)$
By using fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$, ($h,q$)-Genocchi
polynomials are defined by [20]
$\displaystyle\frac{\widetilde{G}_{n+1,q}^{\left(\alpha,h\right)}\left(x\right)}{n+1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{-q}}\sum_{\xi=0}^{p^{n}-1}\left(-1\right)^{\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}q^{h\xi}\text{.}$
For $x=0$ in (1), we have
$\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(0\right):=\widetilde{G}_{n,q}^{\left(\alpha,h\right)}$
are called ($h,q$)-Genocchi numbers with weight $\alpha$ which is defined by
$\widetilde{G}_{0,q}^{\left(\alpha,h\right)}=0\text{ and
}q^{h}\frac{\widetilde{G}_{m+1}^{\left(\alpha,h\right)}\left(1\right)}{m+1}+\frac{\widetilde{G}_{m+1}^{\left(\alpha,h\right)}}{m+1}=\left\\{\begin{array}[]{cc}\left[2\right]_{q}\text{,
}&\text{if }m=0,\\\ 0\text{,}&\text{if }m\neq 0\text{.}\end{array}\right.$
By (1), we have distribution formula for ($h,q$)-Genocchi polynomials, which
is shown by [20]
$\widetilde{G}_{n+1,q}^{\left(\alpha,h\right)}\left(x\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{a}}}\left[a\right]_{q^{\alpha}}^{n}\sum_{j=0}^{a-1}\left(-1\right)^{j}q^{jh}\widetilde{G}_{n+1,q^{a}}^{\left(\alpha,h\right)}\left(\frac{x+j}{a}\right)\text{.}$
By applying some elementary methods, we shall give symmetric properties of
weighted ($h,q$)-Genocchi polynomials and weighted ($h,q$)-Zeta type function.
Consequently, our applications seem to be interesting and worthwhile for
studying in Theory of Analytic Numbers.
## 2\. ON THE ($h,q$)-ZETA-TYPE FUNCTION
In this part, we firstly recall the ($h,q$)-Zeta type function with weight
$\alpha$ which is derived in [20] as follows:
(5)
$\widetilde{\zeta}_{q}^{\left(\alpha,h\right)}\left(s,x\right)=\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{mh}}{\left[m+x\right]_{q^{\alpha}}^{s}}$
where $q\in\mathbb{C}$, $h\in\mathbb{N}$ and $\Re\left(s\right)>1$. It is
clear that the special case $h=0$ and $q\rightarrow 1$ in (5), it reduces to
the ordinary Hurwitz-Euler zeta function. Now, we consider (5) in this form
$\widetilde{\zeta}_{q^{a}}^{\left(\alpha,h\right)}\left(s,bx+\frac{bj}{a}\right)=\left[2\right]_{q^{a}}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{mah}}{\left[m+bx+\frac{bj}{a}\right]_{q^{a\alpha}}^{s}}$
By applying some basic operations to the above identity, that is, for any
positive integers $m$ and $b$, there exist unique non-negative integers $k$
and $i$ such that $m=bk+i$ with $0\leq i\leq b-1$. For $a\equiv
1(\mathop{\mathrm{m}od}2)$ and $b\equiv 1(\mathop{\mathrm{m}od}2)$. Thus, we
can compute as follows:
(6)
$\displaystyle\widetilde{\zeta}_{q^{a}}^{\left(\alpha,h\right)}\left(s,bx+\frac{bj}{a}\right)$
$\displaystyle=\left[a\right]_{q^{\alpha}}^{s}\left[2\right]_{q^{a}}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{mah}}{\left[ma+abx+bj\right]_{q^{a\alpha}}^{s}}$
$\displaystyle=\left[a\right]_{q^{\alpha}}^{s}\left[2\right]_{q^{a}}\sum_{m=0}^{\infty}\sum_{i=0}^{b-1}\frac{\left(-1\right)^{i+mb}q^{\left(i+mb\right)ah}}{\left[\left(i+mb\right)a+abx+bj\right]_{q^{a\alpha}}^{s}}$
$\displaystyle=\left[a\right]_{q^{\alpha}}^{s}\left[2\right]_{q^{a}}\sum_{i=0}^{b-1}\left(-1\right)^{i}q^{iah}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{mbah}}{\left[ab\left(m+x\right)+ai+bj\right]_{q^{\alpha}}^{s}}$
From this, we can easily discover the following
(7)
$\displaystyle\sum_{j=0}^{a-1}\left(-1\right)^{j}q^{jbh}\widetilde{\zeta}_{q^{a}}^{\left(\alpha,h\right)}\left(s,bx+\frac{bj}{a}\right)=$
$\displaystyle\left[a\right]_{q^{\alpha}}^{s}\left[2\right]_{q^{a}}\sum_{j=0}^{a-1}\left(-1\right)^{j}q^{jbh}\sum_{i=0}^{b-1}\left(-1\right)^{i}q^{iah}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{mbah}}{\left[ab\left(m+x\right)+ai+bj\right]_{q^{\alpha}}^{s}}$
Replacing $a$ by $b$ and $j$ by $i$ in (6) and so we have the following
$\widetilde{\zeta}_{q^{b}}^{\left(\alpha,h\right)}\left(s,ax+\frac{ai}{b}\right)=\left[b\right]_{q^{\alpha}}^{s}\left[2\right]_{q^{b}}\sum_{j=0}^{a-1}\left(-1\right)^{j}q^{jbh}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{mbah}}{\left[ab\left(m+x\right)+ai+bj\right]_{q^{\alpha}}^{s}}$
By considering the above identity in (7), we can easily state the following
theorem.
###### Theorem 2.1.
The following
$\frac{\left[2\right]_{q^{b}}}{\left[a\right]_{q^{\alpha}}^{s}}\sum_{i=0}^{a-1}\left(-1\right)^{i}q^{ibh}\widetilde{\zeta}_{q^{a}}^{\left(\alpha,h\right)}\left(s,bx+\frac{bi}{a}\right)=\frac{\left[2\right]_{q^{a}}}{\left[b\right]_{q^{\alpha}}^{s}}\sum_{i=0}^{b-1}\left(-1\right)^{i}q^{iah}\widetilde{\zeta}_{q^{b}}^{\left(\alpha,h\right)}\left(s,ax+\frac{ai}{b}\right)$
is true.
Now, setting $b=1$ in Theorem 2.1, we have the following distribution formula
(8)
$\widetilde{\zeta}_{q}^{\left(\alpha,h\right)}\left(s,ax\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{a}}\left[a\right]_{q^{\alpha}}^{s}}\sum_{i=0}^{a-1}\left(-1\right)^{i}q^{ih}\widetilde{\zeta}_{q^{a}}^{\left(\alpha,h\right)}\left(s,x+\frac{i}{a}\right)\text{.}$
If putting $a=2$ in (8) leads to the following corollary.
###### Corollary 2.2.
The following identity holds true:
$\widetilde{\zeta}_{q}^{\left(\alpha,h\right)}\left(s,2x\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{2}}\left[2\right]_{q^{\alpha}}^{s}}\left(\widetilde{\zeta}_{q^{2}}^{\left(\alpha,h\right)}\left(s,x\right)-q^{h}\widetilde{\zeta}_{q^{2}}^{\left(\alpha,h\right)}\left(s,x+\frac{1}{2}\right)\right)\text{.}$
Taking $s=-m$ into Theorem 2.1, we have the symmetric property of
($h,q$)-Genocchi polynomials by the following theorem.
###### Theorem 2.3.
The following identity
$\left[2\right]_{q^{b}}\left[a\right]_{q^{\alpha}}^{m-1}\sum_{j=0}^{a-1}\left(-1\right)^{i}q^{ibh}\widetilde{G}_{m,q^{a}}^{\left(\alpha,h\right)}\left(bx+\frac{bi}{a}\right)=\left[2\right]_{q^{a}}\left[b\right]_{q^{\alpha}}^{m-1}\sum_{i=0}^{b-1}\left(-1\right)^{i}q^{iah}\widetilde{G}_{m,q^{b}}^{\left(\alpha,h\right)}\left(ax+\frac{ai}{b}\right)$
is true.
Now also, setting $b=1$ and replacing $x$ by $\frac{x}{a}$ on the above
theorem, we can rewrite the following ($h,q$)-Genocchi polynomials with weight
$\alpha$.
$\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(x\right)=\frac{\left[2\right]_{q}}{\left[2\right]_{q^{a}}}\left[a\right]_{q^{\alpha}}^{n-1}\sum_{i=0}^{a-1}\left(-1\right)^{i}q^{ih}\widetilde{G}_{n,q^{a}}^{\left(\alpha,h\right)}\left(\frac{x+i}{a}\right)\text{
}\left(2\nmid a\right)\text{.}$
Due to Araci $et$ $al$. [20], we develop as follows
$\displaystyle\sum_{n=0}^{\infty}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(x+y\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}t\sum_{m=0}^{\infty}\left(-1\right)^{m}q^{mh}e^{t\left[x+y+m\right]_{q^{\alpha}}}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}t\sum_{m=0}^{\infty}\left(-1\right)^{m}q^{mh}e^{t\left[y\right]_{q^{\alpha}}}e^{\left(q^{\alpha
y}t\right)\left[x+m\right]_{q^{\alpha}}}$ $\displaystyle=$
$\displaystyle\left(\sum_{n=0}^{\infty}\left[y\right]_{q^{\alpha}}^{n}\frac{t^{n}}{n!}\right)\left(\sum_{n=0}^{\infty}q^{\alpha\left(n-1\right)y}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(x\right)\frac{t^{n}}{n!}\right)$
by using Cauchy product, we see that
$\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\binom{n}{j}q^{\alpha\left(j-1\right)y}\widetilde{G}_{j,q}^{\left(\alpha,h\right)}\left(x\right)\left[y\right]_{q^{\alpha}}^{n-j}\right)\frac{t^{n}}{n!}\text{.}$
Thus, by comparing the coefficients of $\frac{t^{n}}{n!}$, we state the
following corollary.
###### Corollary 2.4.
The following equality holds true:
(9)
$\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(x+y\right)=\sum_{j=0}^{n}\binom{n}{j}q^{\alpha\left(j-1\right)y}\widetilde{G}_{j,q}^{\left(\alpha,h\right)}\left(x\right)\left[y\right]_{q^{\alpha}}^{n-j}\text{.}$
By using Theorem 2.3 and (9), we readily derive the following symmetric
relation after some applications.
###### Theorem 2.5.
The following equality holds true:
$\displaystyle\left[2\right]_{q^{b}}\sum_{i=0}^{m}\binom{m}{i}\left[a\right]_{q^{\alpha}}^{i-1}\left[b\right]_{q^{\alpha}}^{m-i}\widetilde{G}_{i,q^{a}}^{\left(\alpha,h\right)}\left(bx\right)\widetilde{S}_{m-i:q^{b},h+i-1}^{\left(\alpha\right)}\left(a\right)$
$\displaystyle=\left[2\right]_{q^{a}}\sum_{i=0}^{m}\binom{m}{i}\left[b\right]_{q^{\alpha}}^{i-1}\left[a\right]_{q^{\alpha}}^{m-i}\widetilde{G}_{i,q^{b}}^{\left(\alpha,h\right)}\left(ax\right)\widetilde{S}_{m-i:q^{a},h+i-1}^{\left(\alpha\right)}\left(b\right)$
where
$\widetilde{S}_{m:q,i}^{\left(\alpha\right)}\left(a\right)=\sum_{j=0}^{a-1}\left(-1\right)^{j}q^{ji}\left[j\right]_{q^{\alpha}}^{m}$.
When $q\rightarrow 1$ into Theorem 2.5, it leads to the following corollary.
###### Corollary 2.6.
The following identity holds true:
$\displaystyle\sum_{i=0}^{m}\binom{m}{i}a^{i-1}b^{m-i}G_{i}\left(bx\right)S_{m-i}\left(a\right)$
$\displaystyle=\sum_{i=0}^{m}\binom{m}{i}b^{i-1}a^{m-i}G_{i}\left(ax\right)S_{m-i}\left(b\right)$
where $S_{m}\left(a\right)=\sum_{j=0}^{a-1}\left(-1\right)^{j}j^{m}$ and
$G_{n}\left(x\right)$ are called the ordinary Genocchi polynomials which is
defined via the following generating function
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{2t}{e^{t}+1}e^{xt}\text{.}$
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* [14] Lee Chae Jang, The $q$-analogue of twisted Lerch type Euler Zeta functions, Bull. Korean Math. Soc. 47 (2010), No. 6, pp. 1181-1188.
* [15] H. Ozden, I. N. Cangul, Y. Simsek, Multivariate interpolation functions of higher order $q$-Euler numbers and their applications, Abstract and Applied Analysis 2008 (2008), Article ID 390857, 16 pages.
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|
arxiv-papers
| 2012-06-22T17:19:40 |
2024-09-04T02:49:32.091357
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Cetin, M. Acikgoz, I. N. Cangul and S. Araci",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1206.5299"
}
|
1206.5328
|
# The configurational space of colloidal patchy polymers with heterogeneous
sequences
Ivan Coluzza Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090
Vienna, Austria Christoph Dellago Faculty of Physics, University of Vienna,
Boltzmanngasse 5, 1090 Vienna, Austria
(August 27, 2024)
###### Abstract
In this work we characterize the configurational space of a short chain of
colloidal particles as function of the range of directional and heterogeneous
isotropic interactions. The individual particles forming the chain are
colloids decorated with patches that act as interaction sites between them. We
show, using computer simulations, that it is possible to sample the relative
probability of occurrence of a structure with a sequence in the space of all
possible realizations of the chain. The results presented here represent a
first attempt to map the space of possible configurations that a chain of
colloidal particles may adopt. Knowledge of such a space is crucial for a
possible application of colloidal chains as models for designable self-
assembling systems.
###### pacs:
64.70.pv, 64.75.Yz,64.60.De
## I Introduction
Self-assembly is the process by which a substance exclusively driven by non-
covalent interactions spontaneously develops into a specific long-lived
conformation with a well defined structure Lawrence:10.1021/cr00038a018 .
Self-assembling is common in natural bio-polymers, such as DNA, RNA and, in
particular, proteins, that exhibit exceptional self-assembling properties.
Proteins, the fundamental building blocks of every living organisms, have the
remarkable property that their structure and, therefore, their function is
encoded in the one-dimensional sequence of the structural elements that
compose them. Although proteins vary strongly in size, structure and function,
they are all composed of the same 20 fundamental chemical units called amino
acids. This protein alphabet insures an enormous variety of combinations,
however only few sequences will have a well defined stable ground state, each
of which consists of complex arrangements of predominantly three types of
secondary structures: alpha helices, beta sheets and random coils
ANFINSEN:1973p2310 ; Dobson:1998p2346 ; KeithDunker:2002p2365 . Hence, bio-
polymers are a clear reference point for the development of artificial self-
assembling systems based on modular subunits.
Recently, it has been shown Coluzza:2011p0020853 that the minimal set of
constraints imposed on configurational space by the hydrogen bonds along the
protein backbone and the self-avoidance of the residues are sufficient to
enable the successful folding of real protein structures. More generally, it
is known that the specific geometry of the backbone and the directionality of
the hydrogen bonds “pre-sculpt” the configurational space of proteins to the
sub-space of currently known proteins structures Maritan00 ; Hoang:2004p1364 ;
Magee06 ; Banavar:2009p6900 , and allow for the design of sequences capable of
folding back to their target structures Coluzza:2011p0020853 . We believe that
it is possible to extend the concept of configurational space pre-sculpting to
generalized chains of particles capable of folding into geometries very
different from those observed for natural proteins. However, for the future
design of complex structures it is crucial to characterize the variety of
configurations that a given set of interactions can generate. This is
equivalent to the still ongoing process of mapping the “protein universe”,
where huge efforts are devoted to finding all protein structures corresponding
to all sequences expressed in living organisms PDB .
In this work, we explore the total phase space of colloidal patchy polymers.
Such phase space is the ensemble of all chain configurations defined by a
structure and a sequence along the chain of particles. Each particle is
decorated by spots, or patches, with interactions that mimic the
directionality of the hydrogen bonds. Such directional interactions confine
the configurational space of the synthetic protein analogue to a sub-space of
compatible structures. We map the phase space for a range of values of the
interaction potential and for two different forms of the interaction
potential. We have chosen as subunits patchy particles, because in addition to
mimicking the properties of hydrogen bonds, they have a rich ensemble of self-
assembling properties 0953-8984-19-32-323101 ; Bianchi2011:10.1039/c0cp02296a
; PhysRevLett.106.255501 . Experimentally, patchy colloidal particles are
spherical particles with a hard core and a radius that varies from the
nanometer to the micrometer scale and several methods have been developed for
the experimental realization of such particles Cho:10.1021/ja0550632 ;
Zhang:2005p143 ; Mirkin:1996p607 . A beautiful review on the experimental
methods to produce patchy particles can be found in the paper of Bianchi et
al. Bianchi2011:10.1039/c0cp02296a .
The results presented in this paper represent a first attempt to link the
configurational space of a generic chain of colloidal particles to simple
geometrical parameters of the interaction potential between the particles.
Such a space represents a “phase diagram“ of the designability of the system,
because the occurrence probability reflects the propensity of a generic
sequence to adopt a given structure. Hence, the larger the space the more
versatile is the corresponding model. Below, we will first introduce the
details of the model and explain the computational methods used to sample the
complex configurational space. Then, we will study the properties of chains of
particles with three patches each, where each patch has different interaction
ranges relative to the particle-particle interaction, as well as two different
functional forms.
## II Methods
### II.1 Model
Figure 1: Real-space representation of the backbone of the patchy polymer. The
patches are indicated by small white spheres, while the large turquoise
spheres represent the hard core of the colloidal particles. If a comparison
between this chain and the caterpillar backbone is made Coluzza:2011p0020853 ,
it is easy to see how the patches are reminiscent of the O and H atoms of the
protein backbone. The insets show a schematic representation of the distance
$R$ between the patches and the alignment angles $\theta_{1}$ and
$\theta_{2}$, which, according to Eq. 3 and Eq. 4 , determine the interactions
between patches not involved in the polymer backbone.
The system we study here consists of a single chain of particles, each of
which can occur in different types mimicking the various residues of a real
protein. Each particle is decorated with 3 patches, two of which are used to
link the particle with neighboring particles and form the chain as illustrated
in Fig. 1. These connecting patches interact via a simple harmonic spring
potential,
$E_{\text{link}}(r)=\frac{\kappa}{2}\,r^{2}.$ (1)
Here, $r$ is the distance between the two anchoring patches and
$\kappa=5\>k_{\rm B}T\,R_{HC}^{-2}$, where $R_{HC}$ is the “hard core” radius
of the colloids. In what follows all the energies are given in units of
$k_{B}T_{\text{ref}}$, where $T_{\text{ref}}$ is a reference temperature that
sets the scale of all interactions, hence in what follows the temperatures do
not have units.
The model colloidal polymer studied here is based on the caterpillar model
Coluzza:2011p0020853 . Accordingly, the colloidal particles interact pairwise
via a smoothed square well pair potential
$E_{AB}\left(r\right)=\left\\{\begin{array}[]{ll}\epsilon_{AB}\left[1-\frac{1}{1+e^{\left(r_{\textrm{max}}-r\right)/K}}\right]&\mbox{
if $r>R_{HC},$}\\\ \infty&\mbox{ if $r\leq R_{HC},$}\end{array}\right.$ (2)
where the subscripts $A$ and $B$ refer to the type of two interacting
particles and $r$ is the distance between their centers. The parameter
$K=0.2R_{HC}$ determines the width of the range over which the potential
changes from $\simeq 0$ to $\simeq\epsilon_{AB}$, and $r_{\textrm{max}}$ is
the distance at which $E_{AB}(r_{\rm rmax})=\epsilon_{AB}/2$. We have varied
$r_{\textrm{max}}$ in the range
$r_{\textrm{max}}={2.2,2.5,3.0,4.0,6.0}\;[R_{HC}]$. The parameter
$\epsilon_{AB}$, which depends on the types $A$ and $B$ of the interacting
particles, determines the depth of the well, i.e., the strength of the
interactions. Depending on the particular type of the interacting particles,
$\epsilon_{AB}$ can be positive or negative, leading to repulsive or
attractive interactions, respectively. Each particle is then assigned a
particular identity that will define the interactions with the other particles
along the chain. There is no unique way to choose the values of the
$\epsilon_{AB}$, and we will show that with just an alphabet of 2 letters it
is possible to produce a wide range of structures. The 2 letters will only
distinguish between hydrophilic (P) and hydrophobic particles (H) with
$\epsilon_{PP}=0$, $\epsilon_{HH}=-\alpha\,k_{B}T_{\text{ref}}$,
$\epsilon_{HP}=-\alpha/2\,k_{B}T_{\text{ref}}$. Here $\alpha$ is an adjustable
parameter that we will fix to keep the second virial coefficient
$B^{AB}_{2}\sim 1$ constant for the various ranges $r_{\textrm{max}}$ of the
potential. The value of $B^{AB}_{2}$ corresponds to the contribution of
$E_{AB}$ in Eq. 2 to the total energy and was calculated with numerical
integration for each value of $r_{\textrm{max}}$.
As patch-patch interaction, acting only among the free patches not involved in
the links along the chain, we use two sets of directional interactions. The
first one is inspired by the hydrogen bond interaction from the caterpillar
model, which is represented by a 10-12 Lennard-Jones type potential with an
additional angular dependence that takes into account the directionality of
the patch bonds Irback:2000p1573 ,
$E_{P1}=-\epsilon_{P1}\left(\cos{\theta_{1}}\cos{\theta_{2}}\right)^{\nu}\left[5\left(\frac{\sigma_{P1}}{r}\right)^{12}-6\left(\frac{\sigma_{P1}}{r}\right)^{10}\right].$
(3)
Here, $\nu=2$ as given in Irback:2000p1573 , $r$ is the distance between the
patches, and $\theta_{1}$ and $\theta_{2}$ are angles between the patches and
the centers of their corresponding spheres, as indicated in the left inset of
Fig. 1. $\sigma_{P1}$ represents the position of the minimum of the potential;
once a value for $\sigma_{P1}$ is fixed then also the range of the potential
is determined. The smaller the value of $\sigma_{P1}$ the shorter the range of
the potential will be. For each of the values of $r_{\textrm{max}}$ in Eq. 2
we considered short and long range directional interactions summarized below.
As for $\alpha$ that defines the $\epsilon_{AB}$ scale factor in Eq. 2, we
determined the parameter $\epsilon_{P1}$ by imposing that the second virial
coefficient $B^{P1}_{2}$ of the directional potential in Eq. 3 is equal to the
second virial coefficient $B^{AB}_{2}$ for each pair
$(r_{\textrm{max}},\sigma_{P1})$. The hard core of the spheres ensures that
only the maximum of angular term close to $\pi$ is accessible, i.e. $-\pi$
corresponds to configurations that are sterically inaccessible.
The second type of directional potential is represented by the same square-
well like potential defined in Eq. 2 modulated by the same angular dependence
used for $E_{P1}$ in Eq. 3, but a different definition of the angles
$\theta_{1}$ and $\theta_{2}$,
$E_{P2}=\left\\{\begin{array}[]{ll}-\epsilon_{P2}\left(\cos{\theta_{1}}\cos{\theta_{2}}\right)^{\nu}\left[1-\frac{1}{1+e^{\left(\sigma_{P2}-r\right)/K}}\right]&\mbox{
if $r>R_{HC}$,}\\\ \infty&\mbox{ if $r\leq R_{HC}$,}\end{array}\right.$ (4)
Here, $\nu=2$, $K=0.2R_{HC}$, $r$ is now the distance between the centers of
the spheres, $\sigma_{P2}$ defines the range of the potential, and
$\theta_{1}$ and $\theta_{2}$ are now the angles of the projection of each
patches on the vector joining the centers of the interacting spheres, as
indicated in the right inset of Fig. 1. We choose a different definition for
the distance $r$ in the potential $E_{P2}$, compared to $E_{P1}$, because we
compute the same distance for the potential $E_{AB}$ in Eq. 2. As the
interaction P2 in Eq. 4 is defined between the centers of the spheres, the
range $\sigma_{P2}$ will be longer than $\sigma_{P1}$ from Eq. 3 by exactly 2
colloids radius $R_{HC}$. Again, for each of the values of $r_{\textrm{max}}$
in Eq. 2 we considered short and long range directional interactions
summarized in Table 1. The parameter $\epsilon_{P2}$ is determined by imposing
that the second virial coefficient $B^{P2}_{2}$ of the directional potential
in Eq. 4 is equal to the second virial coefficient $B^{AB}_{2}$ for each pair
$(r_{\textrm{max}},\sigma_{P2})$. The directionality of the patch bonds,
encoded in the angular term in Eqs. 3 and 4 , are each essential to pre-sculpt
the conformational space in analogy to the role of hydrogen bonds that
characterize the secondary structure elements typical of proteins.
We considered the directional interactions in Eqs. 3 and 4 as a way to
estimate the dependence of configurational space on the different ways of
imposing the directionality. Since the directional contribution as well as the
window of interaction ranges considered are the same, the only difference
between the two interactions is in the shape of the potential. While the
interaction in Eq. 3 has a sharp minimum, the square-well profile of the
potential in Eq. 4, introduces less frustration when combined with the
isotropic interaction of Eq. 2, which has the same functional form. Moreover,
there are also experimental reasons to consider the above mentioned
interactions. The first potential can represent a short-range directional
attraction on the surface (e.g. Hydrogen Bonds, short DNA), while the second
is representative of a class of particles, where the patch is produced by
creating a dent in the isotropic interaction.
Parameter | Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | Set 6 | Set 7 | Set 8
---|---|---|---|---|---|---|---|---
$r_{\textrm{max}}$ $[R_{HC}]$ | 2.2 | 2.5 | 3.0 | 4.0 | 4.0 | 6.0 | 6.0 | 6.0
$\alpha$ $[k_{B}T_{\text{ref}}]$ | 2.6 | 1.7 | 1.0 | 0.441 | 0.441 | 0.14 | 0.14 | 0.14
$\sigma_{P1}$ $[R_{HC}]$ | 0.2 | 0.2 | 0.2 | 0.2 | 1.0 | 0.2 | 1.0 | 2.0
$\epsilon_{P1}$ $[k_{B}T_{\text{ref}}]$ | 5.95 | 5.95 | 5.95 | 5.95 | 4.0 | 5.95 | 4.0 | 3.1
$\sigma_{P2}$ $[R_{HC}]$ | 2.2 | 2.2 | 2.2 | 2.2 | 3.0 | 2.2 | - | 4.0
$\epsilon_{P2}$ $[k_{B}T_{\text{ref}}]$ | 5.0 | 5.0 | 5.0 | 5.0 | 2.7 | 5.0 | - | 1.38
Table 1: Values of the potential parameters used in our simulations. It is
important to remember that as the range $\sigma_{P2}$ is calculated between
the centers of the colloids it will be longer than $\sigma_{P1}$ by exactly 2
colloids radius $R_{HC}$
### II.2 Simulation methodology
The first step in the characterization of the patchy polymer model is the
classification of the different structures according to how easy it is to
design them. In order to do so, we will compare the designability of many
chain configurations by estimating the number of sequences that fold into each
structure. In order to explore this vast space we combine a particle identity
mutation move (see below) with the standard set of pivot crankshaft and single
particle moves used to explore the configurations of chains of particles
FrenkelSmit . For relatively short chains it is possible to identify the most
recurrent structures and compare their designability using the algorithm
described below. We considered first the simple case of a chain of 20
colloidal particles chosen from an alphabet of size 2 as described in the
section II.A.
As we aim to imitate the designability property of natural proteins, we based
the identity mutations move on the design scheme successfully used in protein
studies Coluzza:2011p0020853 . As in the conventional Metropolis scheme, the
acceptance of such trial moves depends on the ratio of the Boltzmann weights
of the new and old states for each temperature $T$ FrenkelSmit . However, if
this were the only criterion, there would be a tendency to generate
homopolymer chains with a low energy, rather than chains that fold selectively
into a specific target structure. To ensure a type composition far from the
homopolymer region of sequence space, we sample the equilibrium sequences for
the generalized energy function
$W=E-\epsilon_{p}\ln N_{P}$ (5)
where $E$ is the energy of the patchy polymer defined as the sum of all the
contributions calculated according to Eq. 2 (as the structure does not change
the spring and directional terms are constant), $\epsilon_{p}$ is a scale
factor adjusting the relative importance of the two terms in the equation, and
$N_{P}$ is the number of permutations that are possible for a given set of
particle of given types,
$N_{P}=\frac{N!}{n_{1}!n_{2}!n_{3}!\cdots n_{M}!}.$ (6)
Here, $N$ is the total number of particles in the chain, $M$ is the number of
different particles types, and $n_{1},n_{2},\cdots,n_{M}$, etc. are the
numbers of particles of type $1,2,\cdots,M$, respectively. In the present
study we have considered an alphabet of only two particles types, namely $H$
and $P$. Hence, it is simply the ratio:
$N_{P}=\frac{N!}{n_{H}!n_{P}!}.$ (7)
While sampling sequence space with the Monte Carlo scheme, we set
$\epsilon_{p}$ to a sufficiently high value, $\epsilon_{p}=5\>k_{\rm B}T$, to
generate sequences with a heterogeneous composition. Note that the additional
term to the generalized energy of Eq. (5) is a phenomenological bias that
drives the simulation towards the region of heterogeneous sequences. Although
due to this particular bias we may miss some low energy configurations, we
believe that an exhaustive sampling of the entire sequence space of each
structure is not necessary to compare the designability.
During each simulation we projected the configurational space on the
collective variable $Q_{S}$, which counts the number of contacts between the
spheres. We then compute the free energy $F(Q_{S})$ as a function of $Q_{S}$,
$F\left(Q_{S}\right)=-kT\ln\left[P\left(Q_{S}\right)\right].$ (8)
Here, $P(Q_{S})$ denotes the normalized histogram of the number of sampled
conformations with order parameter $Q_{S}$. In practice, a direct calculation
of this histogram is not efficient, since even such short chains tends to be
trapped in local minima, especially at low temperatures. To induce escape from
these local minima, we made use of the Virtual Move Parallel Tempering Monte
Carlo sampling scheme proposed by Coluzza and Frenkel Coluzza:2005p596 , based
on the Waste Recycling approach Frenkel:2004p329 . This scheme is very
efficient in sampling both high and low free energy states.
Figure 2: Free energy $F(Q_{S})/k_{\rm B}T$ as a function of the number
$Q_{S}$ of contacts among the spheres obtained for a patchy polymer of length
$N=20$ with 3 patches per particle interacting with the directional
interaction defined in Eq. 3. The simulations were performed by altering the
structure of the chain as well as by mutating the identities of the spheres
that intervene in $E_{AB}$ interaction in Eq. 2, at the temperature $T=0.2$.
Each curve represents a simulation with a different pair of values
$(r_{\textrm{max}},\sigma_{P1})$ as indicated in Table 1. To each pair
corresponds a different symbol, however we have grouped the curves with the
same range of the directional potential $E_{P1}$ defined in Eq. 3.
## III Results
We first carried out a simulation in which we varied both the configuration as
well as the sequence of the chain for the patch-patch interaction P1 from Eq.
3. We considered the combination of parameters in Table 1 and we projected the
structure space on the collective variable $Q_{S}$, which is a count of the
number of contacts between the spheres. In Fig. 2, we show the free energy
profiles $F(Q_{S})/k_{\rm B}T$ as a function of $Q_{S}$ at various
combinations of the potentials ranges $r_{\text{max}}$ and $\sigma_{P1}$. From
the plot we can see that for $r_{\text{max}}=2.2,2.5$ and $\sigma_{P1}=0.2$
the landscapes present a single sharp minimum, indicating that there is a
preferred number of contacts for the structures that have high occurrence. As
$r_{\text{max}}$ is increased, the position of the minimum, not surprisingly,
shifts towards higher values of $Q_{S}$ and it also widens suggesting a richer
structural space. We further characterized the ensemble of configurations by
looking at the structures that correspond to the global minimum of each free
energy profile. In Figs. 3 and 4 we show a real space representation of such
structures. The first aspect that becomes apparent is that there is a wide
range of possible structures (it is important to remember that the structures
in the minimum are only the most common and not the only possible ones). In
addition, we observed the appearance of double helices at the values
$r_{\text{max}}$ and $\sigma_{P1}$ for which the interacting patches sit very
close to the half distance $r_{\text{max}}$ between the centers of the spheres
( the case $r_{\text{max}}=2.2,\sigma_{P1}=0.2$ is a bit of an exception as it
shows only a partial helical configuration). It is important to notice that
the pitch of the double helical structures in Fig. 4 shrinks with increasing
values of $r_{\text{max}}$. One could imagine a system where the range of the
isotropic interaction is controllable by external parameters (e.g. PH, DNA
linkers, depletion), which in turns will allow for the control over the
extension of the helix.
Figure 3: Typical configurations obtained from the simulations of the
interaction potential P1 (Eq. 3). For clarity we have not represented each
sphere along the chain, but instead we draw only the backbone as a thick line
joining the centers of the spheres. The patches are represented as white stick
anchored on the center of the colloid and of length equal to the radius of the
colloid, so that the end of the stick corresponds to the position in space of
the patch. The structures have been taken from the ensemble of structures
corresponding to the minimum of the free energy $F(Q_{S})/k_{\rm B}T$ for the
combinations of parameters: a) $r_{\text{max}}=2.2\;\sigma_{P1}=0.2$, b)
$r_{\text{max}}=3.0\;\sigma_{P1}=0.2$, c)
$r_{\text{max}}=4.0\;\sigma_{P1}=0.2$, d)
$r_{\text{max}}=6.0\;\sigma_{P1}=0.2$, e)
$r_{\text{max}}=6.0\;\sigma_{P1}=1.0$. The configurations have a loop
structure that folds on itself very similar to what in proteins is called a
beta-sheet. We colored structure (a) in red to highlight the loop-helix nature
of the backbone which resembles the helices in Fig. 4 corresponding to the
other set of parameters. Figure 4: As for the calculations shown in Fig. 3 we
have isolated the characteristic structures of the minimum free energy
$F(Q_{S})/k_{\rm B}T$ but now for the combinations of parameters: a)
$r_{\text{max}}=2.5\;\sigma_{P1}=0.2$, b)
$r_{\text{max}}=4.0\;\sigma_{P1}=1.0$, c)
$r_{\text{max}}=6.0\;\sigma_{P1}=2.0$. All the systems exhibit a strong
preference for helical configurations of the backbone. To be noted is the
shrinking effect that helices show upon increase of the range $r_{\text{max}}$
of the isotropic interaction.
We now consider the second interaction potential P2 from Eq. 4. We repeated
the same procedure described above for the P1 interaction potential. In Fig.
5, we plot the free energies $F(Q_{S})/k_{\rm B}T$ for the values of the
parameters $r_{\textrm{max}},\sigma_{P2}$ in Table 1, as a function of the
same collective variable $Q_{S}$. As before we observe a clear trend that the
value of $Q_{S}$ at the minimum increases for increasing values of
$r_{\textrm{max}}$. However, if we look at the actual structures we noticed
that they are all very close to each other (see Fig. 6) indicating that the
displacement of the minimum is just an effect of counting more neighbors per
particle. This is true also for the longer ranges of the directional
potential, that show for $\sigma_{P2}=3.0$ a double minimum and for
$\sigma_{P2}=4.0$ a very wide basin. The latter is the result of large
fluctuations that the long range potentials can accommodate. It is striking to
obtain such a dramatic preference for a specific configuration just by
slightly altering the form of the interaction potential. By using the same
form of the potentials for both isotropic and directional contributions we
have reduced the frustration between the two potentials, allowing the system
to find an optimal structure for all sets of parameters. We stress that we
have chosen the values of the parameters in order to fix the second viral
coefficient of all the terms of the potential constant across all simulations.
Slightly different configurations occur when the range of the directional
interaction $\sigma_{P2}$ is half of range $r_{\textrm{max}}$ the isotropic
interaction (See (e) and (g) of Fig. 6). The main difference between the
configurations (e) and (g) and the others is in the central loop of the
backbone, that for larger values of $\sigma_{P2}$ expands into an extended
arch. In particular the arch allows for large compressions of the molecule
that results in the wide minima for $\sigma_{P2}=3.0$ in Fig. 5.
Figure 5: Free energy $F(Q_{S})/k_{\rm B}T$ as a function of $Q_{S}$ for the
same simulation conditions and parameters as in Fig. 2, but with the patch-
patch interaction from Eq. 4. Each curve represents a simulation with a
different pair of values $(r_{\textrm{max}},\sigma_{P2})$ as indicated in
Table 1. To each pair corresponds a different symbol (see inset), however we
have grouped the curves with the same range of the directional potential
$E_{P2}$ defined in Eq. 4. The wide free energy minima observed for the
$r_{\textrm{max}}=6.0,\sigma_{P2}=4.0$ case is due to the large fluctuations
that the long range potentials can tolerate. Figure 6: Minimum free energy
configurations (see Fig. 5), obtained from the simulations of the P2 (Eq. 4)
interaction potential. Each structure is representative of different
combinations of the potentials range: a)
$r_{\text{max}}=2.2\;\sigma_{P2}=2.2$, b)
$r_{\text{max}}=2.5\;\sigma_{P2}=2.2$, c)
$r_{\text{max}}=3.0\;\sigma_{P2}=2.2$, d)
$r_{\text{max}}=4.0\;\sigma_{P2}=2.2$, e)
$r_{\text{max}}=4.0\;\sigma_{P2}=2.0$, f)
$r_{\text{max}}=6.0\;\sigma_{P2}=2.2$, g)
$r_{\text{max}}=6.0\;\sigma_{P2}=3.0$. We colored structures (e) and (g) in
blue because they correspond to the same set of parameters that for the P1
interaction potential gave rise to the structures in Fig. 4. Unexpectedly all
the structures are almost identical (with an average root mean square distance
between the centers of the spheres of 1 $R_{HC}$). The common backbone
arrangement appears to be a distorted helix where the patches interact via the
tails of the potential in Eq. 4.
## IV Summary and Discussion
In this work, we present a methodology to explore the configurational space of
a chain of colloidal particles decorated with a single patch. The particles
interact through an isotropic potential that depends on the identity of the
particles as well as through a direction dependent patch-patch interaction. We
developed a methodology to sample the probability of observing a structure
with many different sequences therefore allowing for a comparison of the
relative probability of observing each configuration. The complexity of such a
space gives an idea of how a particular model is suitable for the design of
self-assembling structures. We then defined two different forms for the
interaction potential and for each of the potentials we considered a wide
range of values for the parameters of the potential. From the analysis of the
projection of such a space on two collective variables, we identified a wide
range of structures with high probability of being observed.
Our analysis demonstrates that the combination of interactions, with which the
chain of colloids is decorated, is capable of shaping the configurational
space to a reduced ensemble of configurations. Furthermore, we have shown that
the reduced ensemble can be dramatically altered by changing few parameters of
the interaction potential (e.g., the interaction range). Such control over the
size and the structure heterogeneity of the configurational space makes the
chain of patchy-colloids an ideal candidate for the experimental realization
of new self-assembling systems. Moreover, we have provided initial but strong
indications that, by inducing frustration between the directional (Eq. 3) and
the isotropic potential the chain is forced to adopt a wider range of
structures, compared to the scenario where both potentials shared the same
functional form (Eq. 4). In particular, for specific parameters the backbone
of the chain has a strong preference for ordered double helices with a pitch
controllable by the range of the isotropic interaction.
Our results provide an important starting point for the realization of
designable chains of patchy colloids. The methodology used here can be easily
extended to an arbitrary set of interactions and provide, as a result, a set
of target structure for the design of chains of patchy colloids.
###### Acknowledgements.
IC would like to thank Prof. Erik Reimhult, Dr. Peter van Oostrum and Dr.
Ronald Zirbs for the enlightening scientific discussions. We acknowledge
support from the Austrian Science Fund (FWF) within the SFB ViCoM (F 41). All
simulations presented in this paper were carried out on the Vienna Scientific
Cluster (VSC).
## References
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* (5) I. Coluzza, Plos ONE 6, e20853 (2011).
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* (7) T. Hoang et al., Proc. Natl. Acad. Sci. USA 101, 7960 (2004).
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* (10) H. Berman et al., Nuc. Acid. Res. 28, 235 (2000).
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* (12) E. Bianchi, R. Blaak, and C. N. Likos, Phys. Chem. Chem. Phys. 13, 6397 (2011).
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* (14) Y.-S. Cho et al., J. Am. Che. Soc. 127, 15968 (2005).
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* (16) C. A. Mirkin, R. L. Letsinger, R. C. Mucic, and J. J. Storhoff, Nature 382, 607 (1996).
* (17) A. Irbäck, F. Sjunnesson, and S. Wallin, Proc. Natl. Acad. Sci. USA 97, 13614 (2000).
* (18) D. Frenkel and B. Smit, Understanding Molecular Simulations (Accademic Press, ADDRESS, 2002).
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* (20) D. Frenkel, Proc. Natl. Acad. Sci. USA 101, 17571 (2004).
|
arxiv-papers
| 2012-06-22T21:59:38 |
2024-09-04T02:49:32.097777
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ivan Coluzza and Christoph Dellago",
"submitter": "Ivan Coluzza",
"url": "https://arxiv.org/abs/1206.5328"
}
|
1206.5333
|
# TempEval-3: Evaluating Events, Time Expressions, and Temporal Relations
Naushad UzZaman∗, Hector Llorens†, James Allen∗,
Leon Derczynski‡, Marc Verhagen⋄ and James Pustejovsky⋄
$\ast$: {naushad,james}@cs.rochester.edu University of Rochester, NY, USA
$\dagger$: hllorens@dlsi.ua.es University of Alicante, Spain
$\ddagger$: leon@dcs.shef.ac.uk University of Sheffield, UK
$\diamond$: {jamesp,marc}@cs.brandeis.edu Brandeis University, MA, USA
###### Abstract
In this proposal, we describe the forthcoming TempEval-3 task which is
currently in preparation for the SemEval-2013 evaluation exercise. The aim of
TempEval is to advance research on temporal information processing. TempEval-3
follows on from previous TempEval events, incorporating: a three-part task
structure covering event, temporal expression and temporal relation
extraction; a larger dataset; and single overall task quality scores.
## 1 Introduction
The TempEval task was added as a new task in SemEval-2007 (Verhagen et al.,
2009), focusing on the identification of temporal relations. The automatic
identification of all temporal referring expressions, events, and temporal
relations within a text is the ultimate aim of research in this area. The area
is too broad to address completely in a first evaluation challenge, and a
staged approach was taken instead. TempEval (henceforth TempEval-1) was an
initial evaluation exercise based on three fixed-scope tasks (identifying
links between: events and timexes in the same sentence; events and document
creation time DCT; main events in successive sentences) that were considered
realistic both from the perspective of assembling resources for development
and testing and from the perspective of developing systems capable of
addressing the tasks.
TempEval-2 (Verhagen et al., 2010) extended TempEval-1, growing into a
multilingual task, and consisting of six subtasks rather than three. This
included event and timex extraction, as well as the three relation tasks from
TempEval-1, with the addition of a relation task where one event subordinates
another.
Temporal annotation is a time-consuming task for humans, which has limited the
size of annotated data in previous TempEval exercises. Current systems,
however, are performing close to the inter-annotator reliability for entity
recognition. This suggests that larger corpora could be built from
automatically annotated data with minor human reviews. As part of TempEval-3,
we explore whether there is value in adding a large automatically created
silver standard to a hand-crafted gold standard.
Automatic performance on temporal relation annotation is still limited; in
TempEval-2, systems achieved an above-baseline error reduction of less than
20% for most tasks. This suggests that the temporal relation problem is still
open and remains a topic of intensive contemporary research.
With these points in mind, this paper describes the next upcoming temporal
evaluation shared task – TempEval-3 – to be held with SemEval-2013.
## 2 TempEval changes
As proposed, TempEval-3 is a follow-up to TempEval-1 and 2. TempEval-3 differs
from its ancestors in the following respects:
1. (i)
size of the corpus: the dataset used comprises about 500K tokens of silver
standard data and about 100K tokens of gold standard data for training,
compared to the corpus of roughly 50K tokens corpus used in TempEval 1 and 2;
2. (ii)
temporal relation task: the temporal relation classification tasks are to be
performed from raw text, i.e. participants need to extract events and temporal
expressions first, determine which ones to link and then obtain the relation
types;
3. (iii)
tasks not independent: participants must annotate temporal expressions and
events in order to do the relation task;
4. (iv)
temporal relation types: the full set of temporal interval relations in TimeML
(Pustejovsky et al., 2005) is used, rather than the reduced set used in
earlier TempEvals;
5. (v)
annotation: most of the corpus was automatically annotated by the state-of-
the-art systems from TempEval-2, a portion of the corpus, including the test
dataset, that is human reviewed;
6. (vi)
evaluation: we will report a temporal awareness score for evaluating temporal
relations, to help to rank systems with a single score.
## 3 Tasks
The tasks proposed for TempEval-3 are:
### 3.1 Task A: Temporal expression extraction and normalization
Determine the extent of the time expressions in a text as defined by the
TimeML TIMEX3 tag. In addition, determine the value of the features TYPE and
VAL. The possible values of TYPE are time, date, duration, and set; the value
of VAL is a normalized value as defined by the TIMEX3 standard. The main
attribute to annotate is VAL.
### 3.2 Task B: Event extraction
As in TempEval-2, participants will determine the extent of the events in a
text as defined by the TimeML EVENT tag. In addition, systems may determine
the value of the features CLASS, TENSE, ASPECT, POLARITY, MODALITY and also
identify if the event is a main event or not. The main attribute to annotate
is CLASS.
### 3.3 Task C: Annotating temporal relations
Identify the pairs of temporal entities (events or temporal expressions) that
have a temporal link and classify the temporal relation between them as a
TLINK. Possible pairs of entities that can have a temporal link are: (i) event
and temporal expressions in the same sentence, (ii) event and document
creation time, (iii) main events of consecutive sentences and (iv) pairs of
events in the same sentence. For this task, we now require that the
participating systems determine which entities need to be linked.
The relation labels will be same as in TimeML, i.e.: before, after, includes,
is-included, during, simultaneous, immediately after, immediately before,
identity, begins, ends, begun-by and ended-by.
### 3.4 Task selection
Participants may choose to do task A, B, or C. Choosing task C (relation
annotation) entails doing tasks A and B (interval annotation). However, a
participant may perform only task C by applying existing tools to carry out
tasks A and B.
## 4 Dataset creation
In TempEval-3, we release new data, as well as significantly reviewing and
modifying existing corpora.
### 4.1 Reviewing Existing Corpora
We considered the existing TimeBank (Pustejovsky et al., 2003), TempEval-1,
TempEval-2 and AQUAINT111http://timeml.org/site/timebank/timebank.html data
for review in TempEval-3. TimeBank v1.2, TempEval-1 and TempEval-2 had the
same documents but different relation types and sometimes different sets of
events. We will refer to this body of temporally-annotated newswire documents
as TimeBank.
For both TimeBank and AQUAINT, we cleaned up the formatting for all files
making it easy to review and read, made all files XML and TimeML schema
compatible and added some missing events and temporal expressions. In AQUAINT,
we added the temporal relations between event and DCT (document creation
time), which was missing for many documents in that corpus. In particular, for
the TimeBank documents, we borrowed the events from the TempEval-2 corpus and
the temporal relations from the TimeBank corpus, which contains a full set of
temporal relations (TempEval-2 used a simpler, coarse-grained set of temporal
relations).
A standard datafile format has been adopted, which is a subset of valid ISO-
TimeML. It begins with an outer TimeML element as normal. The document name is
contained in a child DOCID element, any newswire preamble in an
optional EXTRAINFO element, headline in an optional TITLE element, the
document timestamp in a DCT element (usually with an ID of t0) and the main
body of the text to be annotated in a TEXT element. For example:
<?xml version="1.0" ?>
<TimeML xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="http://timeml.org/timeMLdocs/TimeML_1.2.1.xsd">
<DOCID>XIN_ENG_20061119.0021</DOCID>
<DCT>HANOI, <TIMEX3 functionInDocument="CREATION_TIME"
temporalFunction="false" tid="t0" type="TIME" value="2006-11-19">Nov. 19 ,
2006</TIMEX3> (Xinhua)</DCT>
<TITLE>URGENT: Russia, US sign agreement on WTO deal in Vietnam</TITLE>
<TEXT>
Russia and the United States Sunday <EVENT aspect="NONE" class="OCCURRENCE"
eid="e1" eiid="ei1" polarity="POS" pos="VERB" tense="PAST">signed</EVENT> a
bilateral <EVENT aspect="NONE" class="OCCURRENCE" eid="e2" eiid="ei2"
polarity="POS" pos="NOUN" tense="PAST">agreement</EVENT> on Russia’s accession
to
the World Trade Organization (WTO) on the sidelines of the ongoing Asia-
Pacific
Economic Cooperaiton Economic Leaders’ Meeting in Hanoi.
</TEXT>
<TLINK eventInstanceID="ei1" lid="l1" relType="NONE" relatedToTime="t0"/>
<TLINK eventInstanceID="ei2" lid="l2" relType="NONE"
relatedToEventInstance="ei1"/>
</TimeML>
### 4.2 Automatically Creating New Large Corpora
A large portion of the TempEval-3 data is automatically generated, using a
temporal merging system. We collected the half-million token text corpus from
English
Gigaword222http://www.ldc.upenn.edu/Catalog/catalogEntry.jsp?catalogId=LDC2011T07.
We automatically annotated this corpus using TIPSem, TIPSem-B (Llorens et al.,
2010) and TRIOS (UzZaman and Allen, 2010). These systems were re-trained on
the TimeBank and AQUAINT corpus, using the TimeML temporal relation set. We
then merged these three state-of-the-art system outputs using our merging
algorithm (UzZaman et al., 2012). In our merging configuration, all entities
and relations suggested by the best system (TIPSem) are added to the merge
output. Suggestions from two other systems (TRIOS and TIPSem-B) are added to
the merge output if they are supported by at least 2 of the 3 systems overall.
The weights used in our configuration are: TIPSem 0.36, TIPSemB 0.32, TRIOS
0.32.
This automatically created corpus is referred as silver data. A portion of the
silver data is in the process of human reviewing for release as additional
gold training data, in addition to reviewed and re-curated versions of
TimeBank and AQUAINT. The parts described in Table 1 comprise our released
dataset.
Table 1: Available corpus released for TempEval-3. (*: reviewing in progress) Corpus | Number of tokens | Purpose | Standard
---|---|---|---
TimeBank | 61 418 | Training | Gold
AQUAINT | 33 973 | Training | Gold
TempEval-3 Silver | 666 309 | Training | Silver
TempEval-3 Gold | 20 000* | Training | Gold
TempEval-3 Evaluation | 20 000* | Evaluation | Gold
The exploration of the benefits of both very large automatically temporally
annotated corpora (silver data) and of smaller human annotated/reviewed
temporal annotated corpora (gold data) with our TempEval-3 release is left to
task participants and to future research.
## 5 Evaluation
Evaluation on tasks A and B will be a standard F-score (incorporating
Precision and Recall metrics) on extents and F-score/Kappa on attributes on
the response extents that overlap with the key extents. Evaluation on task C
will be incorporated from our proposed graph-based evaluation metric (see
UzZaman and Allen (2011) for details). This metric uses temporal closure to
reward relation annotations that are equivalent but distinct and then finds
precision and recall. Our temporal awareness score is a combined measure of a
system’s performance (i.e. it evaluates how a system extracts events, temporal
expressions and also identifies all temporal relations).
## 6 Conclusion
We have described the task, dataset and evaluation style for TempEval-3. The
event will be part of SemEval-2013. Training will begin in autumn 2012, and
the evaluation period ends January 2013. Further information can be found on
the task website333http://www.cs.york.ac.uk/semeval-2013/ and via the TempEval
group list444http://groups.google.com/group/tempeval.
## References
* Llorens et al. (2010) Llorens, H., E. Saquete, and B. Navarro (2010), “TIPSem (English and Spanish): Evaluating CRFs and Semantic Roles in TempEval-2.” In _Proceedings of the 5th International Workshop on Semantic Evaluation_ , 284–291, Association for Computational Linguistics.
* Pustejovsky et al. (2003) Pustejovsky, J., P. Hanks, R. Sauri, A. See, R. Gaizauskas, A. Setzer, D. Radev, B. Sundheim, D. Day, L. Ferro, et al. (2003), “The TimeBank corpus.” In _Corpus Linguistics_ , volume 2003, 40.
* Pustejovsky et al. (2005) Pustejovsky, J., B. Ingria, R. Sauri, J. Castano, J. Littman, R. Gaizauskas, A. Setzer, G. Katz, and I. Mani (2005), “The specification language TimeML.” _The Language of Time: A reader_ , 545–557.
* UzZaman and Allen (2010) UzZaman, N. and J.F. Allen (2010), “TRIPS and TRIOS system for TempEval-2: Extracting temporal information from text.” In _Proceedings of the 5th International Workshop on Semantic Evaluation_ , 276–283, Association for Computational Linguistics.
* UzZaman and Allen (2011) UzZaman, N. and J.F. Allen (2011), “Temporal Evaluation.” In _Proceedings of The 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies (Short Paper), Portland, Oregon, USA_.
* UzZaman et al. (2012) UzZaman, N., H. Llorens, and J.F. Allen (2012), “Merging Temporal Annotations.” In _Proceedings of the TIME Conference_.
* Verhagen et al. (2009) Verhagen, M., R. Gaizauskas, F. Schilder, M. Hepple, J. Moszkowicz, and J. Pustejovsky (2009), “The TempEval challenge: identifying temporal relations in text.” _Language Resources and Evaluation_ , 43, 161–179.
* Verhagen et al. (2010) Verhagen, M., R. Sauri, T. Caselli, and J. Pustejovsky (2010), “SemEval-2010 task 13: TempEval-2.” In _Proceedings of the 5th International Workshop on Semantic Evaluation_ , 57–62, Association for Computational Linguistics.
|
arxiv-papers
| 2012-06-22T22:30:44 |
2024-09-04T02:49:32.104723
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Naushad UzZaman, Hector Llorens, James Allen, Leon Derczynski, Marc\n Verhagen and James Pustejovsky",
"submitter": "Leon Derczynski",
"url": "https://arxiv.org/abs/1206.5333"
}
|
1206.5414
|
# Eta-meson production in the resonance energy region. 111Supported by
Transregio SFB/TR16, project B.7
V. Shklyar shklyar@theo.physik.uni-giessen.de H. Lenske U. Mosel Institut
für Theoretische Physik, Universität Giessen, D-35392 Giessen, Germany
###### Abstract
We perform an updated coupled-channel analysis of eta-meson production
including all recent photoproduction data on the proton. The dip observed in
the differential cross sections at c.m. energies W=1.68 GeV is explained by
destructive interference between the $S_{11}(1535)$ and $S_{11}(1560)$ states.
The effect from $P_{11}(1710)$ is found to be small but still important to
reproduce the correct shape of the differential cross section. For the
$\pi^{-}N\to\eta N$ scattering we suggest a reaction mechanism in terms of the
$S_{11}(1535)$, $S_{11}(1560)$, and $P_{11}(1710)$ states. Our conclusion on
the importance of the $S_{11}(1535)$, $S_{11}(1560)$, and $P_{11}(1710)$
resonances in the eta-production reactions is in line with our previous
results. No strong indication for a narrow state with a width of 15 MeV and
the mass of 1680 MeV is found in the analysis. $\eta N$ scattering length is
extracted and discussed.
###### pacs:
11.80.-m,13.75.Gx,14.20.Gk,13.30.Gk
## I Introduction
The discovery of nucleon resonances in the first pion-nucleon scattering
experiments provided first indications for a complicated intrinsic structure
of the nucleon. With establishing the quark picture of hadrons and
developments of the constituent quark models the interest in the study of the
nucleon excitation spectra was renewed. The major question was the number of
the excited states and their properties. This problem was attacked both
experimentally and theoretically. On the theory side constituent quark (CQM)
models, lattice QCD and Dyson-Schwinger approaches have been developed to
describe and predict the nucleon resonance spectra (see e.g. Aznauryan:2009da
for a review). The main problem remains, however, a serious disagreement
between the theoretical calculations and the experimentally observed baryon
spectra. This concerns both the number and the properties of excited states.
On the experimental side pion-induced reactions have been studied to establish
resonance spectra. However, due to difficulties in detecting neutral particles
most experiments were limited to pion-nucleon elastic scattering with charged
particles in the final state. Being the lightest non-strange particle next to
the pion the $\eta$-meson also becomes an interesting probe to study nucleon
excitations. A few experiments have been made in the past to investigate
$\eta$-production. The first near-threshold measurements Jones:1966zza ;
Richards:1970cy ; Bulos:1970zk demonstrated that the reaction proceeds
through a strong $S$-wave resonance excitation which was later identified with
$S_{11}(1535)$. An extensive study of the $\pi^{-}p\to\eta n$ reaction above
W$>$1.7 GeV has been made in Brown:1979ii ; Baker:1979aw . Both differential
cross section and asymmetry data have been obtained. However, due to possible
problems with the energy-momentum calibration Clajus:1992dh the use of these
data might lead to wrong conclusions on the reaction mechanism. Note that
these problems are present not only in the $\eta$-measurements Brown:1979ii ;
Baker:1979aw but also in the charge-exchange data obtained in the same
experiment.
Presently the development of the high-duty electron facilities (ELSA, JLAB,
MAMI, SPring) offers new possibilities to study the $\eta$-photoproduction
both on the proton ($\eta p$) and on the neutron ($\eta n$). The first
measurement of the $\eta$-photoproduction on the neutron reported an
indication for a resonance-like structure in the reaction cross section at
W=1.68 GeV Kuznetsov:2006 ; Kuznetsov:2006kt . Independent experimental
studies Jaegle:2008ux ; Jaegle:2011sw confirmed the existence of this effect
in the $\gamma n$ reaction. This phenomenon was predicted in Azimov:2005jj as
a signal from a narrow state - a possible non-strange partner of the
pentaquark Diakonov:1997mm . Another explanation has been suggested in
Shklyar:2006xw where the observed effect was described by the contributions
from the $S_{11}(1650)$ or $P_{11}(1710)$ states. Due to the lack of knowledge
of the $S_{11}(1650)$ and $P_{11}(1710)$ resonance couplings to $\gamma n$ a
clean separation of the relative contributions from these states is difficult.
The general conclusion made in Shklyar:2006xw is that both states might be
good candidates to explain the observed structure.
By fitting to the $\eta n$ cross sections and beam asymmetry the Bonn-Gatchina
group provided an explanation Anisovich:2008wd for the second peak in terms
of the $S_{11}(1650)$ state. Another contribution to the field has been made
by the authors of Doring:2009qr . There the peak in the
$\sigma_{p}/\sigma_{n}$ cross section ratio was explained by a cusp effect
from the $K\Sigma$ and $K\Lambda$ rescattering channel. All these studies have
been done assuming scattering on a quasi-free nucleon. At the same time a
realistic analysis of meson photoproduction on the quasi-free neutron should
include the nucleon-nucleon and meson-nucleon correlations (FSI-effect) which
were shown to be very important Tarasov:2011ec and take into account
corresponding experimental cuts applied by the extraction of the quasi-free
neutron data from $\gamma D$-scattering. The later issue might be crucial for
the unambiguous identification of the narrow resonance contribution as
discussed in MartinezTorres:2010zzb .
If it is granted that the signal observed in the $\gamma n$ scattering
Kuznetsov:2006 ; Kuznetsov:2006kt ; Jaegle:2008ux ; Jaegle:2011sw is due to
the narrow (exotic) state one may expect to observe a similar effect in other
eta-production reactions at the same energies, e.g in gamma-proton scattering.
The experimental investigations of the $\eta$-production on the proton made by
the CLAS, GRAAL, and CB-ELSA/TAPS collaborations Dugger:2002ft ; Crede:2009zzb
; Bartholomy:2007zz ; Bartalini:2007fg have found an indication of the dip
structure around W=1.68 GeV in the differential cross section but not a
resonance-like structure. This effect was also accompanied by the change in
the angular distribution of the differential cross section. However, despite
of extensive theoretical studies of the $\eta$ -production the reaction
mechanism is still under discussion PhysRevC.84.045207 ; Chiang:2002vq ;
Bartholomy:2007zz ; Feuster:1998b ; An:2011sb ; Anisovich:2011ka ;
Shyam:2008fr ; Nakayama:2008tg ; Choi:2007gy ; Zhong:2007fx ; Fix:2007st ;
Gasparyan:2003fp ; Ruic:2011wf .
Recently the $\eta$-photoproduction on the proton has been measured with high-
precision by the Crystal Ball collaboration at MAMI McNicoll:2010qk . These
high-resolution data provides a new step forward in understanding the reaction
dynamics and in the search for a signal from the ’weak’ resonance states. The
main result reported in McNicoll:2010qk is a very clean signal of a dip
structure around W=1.68 GeV. It is interesting to note that the old
measurements of the $\pi N\to\eta N$ reaction Richards:1970cy also give an
indication for the second structure in the differential cross section at W=1.7
GeV. This raises a question whether the dip reported in the $\eta p$ reaction,
the resonance-like signal observed in $\eta n$ and the possible structure in
the $\pi N\to\eta N$ cross section are originating from the same degrees of
freedom or not. The second question is whether one of these phenomena can be
attributed to the signal from a narrow (exotic) resonance state as discussed
in Azimov2 ; Azimov:2005jj ; Azimov:2005hv .
In our previous coupled-channel PWA study Shklyar:2006xw we proposed an
explanation of the possible dip in the $\eta$-proton cross section in terms of
the destructive interference of the $S_{11}(1535)$ and $S_{11}(1650)$ states.
The result was based on the $\eta p$ photoproduction data taken before 2006
Dugger:2002ft ; Crede:2003ax . The aim of the present study is to extend our
previous coupled-channel analysis of the $\gamma p\to\eta p$ reaction by
including the data from the high-precision measurements McNicoll:2010qk . The
main question is whether the $\eta p$ reaction dynamics can be understood in
terms of the established resonance states. We emphasize that for reliable
identification of the resonance contributions the calculations should maintain
unitarity. Another complication comes from the fact that the most
contributions to the resonance self-energy (total decay width) is driven by
its hadronic couplings. Therefore the analysis of the photoproduction data
requires the knowledge of the hadronic transition amplitudes. Hence the
simultaneous analysis of all open channels (both hadronic and electromagnetic
) is inevitable for the identification of the resonances and extraction of
their properties. In the present study we concentrate on the combined
description of the $(\gamma/\pi)p\to\eta p$ scattering taking also the
$(\gamma/\pi)\to\pi N$, $2\pi N$, $\omega N$, $K\Lambda$ channels into
account. The results on the $\eta n$ reaction will be reported elsewhere.
First, we corroborate our previous findings Penner:2002a ; Penner:2002b ;
Shklyar:2006xw where the important contributions from the $S_{11}(1535)$,
$S_{11}(1650)$ and $P_{11}(1710)$ resonances to the $\pi N\to\eta N$ reaction
have been found. The major effect comes from the $S_{11}$ and $P_{11}$ partial
waves. The interference between the $S_{11}(1535)$ and $S_{11}(1650)$ states
produces a dip in the $S_{11}$ amplitude. The $P_{11}$ amplitude is influenced
by the contributions from the $P_{11}(1710)$ state. The interference between
the $S_{11}$ and $P_{11}$ partial waves leads to the forward peak in the
differential cross section around W=1.7 GeV. We stress that the interference
between two nearby states also includes rescattering and coupled-channel
effects which are hard to simulate by the simple sum of two Breit-Wigner
forms.
We also confirm our previous finding that the interference between
$S_{11}(1535)$ and $S_{11}(1650)$ is responsible for the dip seen in the $\eta
p$ data. The effect from the $\omega N$ threshold is found to be relatively
small which is also in line with the conclusion of Shklyar:2006xw . Opposite
to Anisovich:2011ka we do not find any strong indications for a narrow state
in the Crystal Ball/Taps data around W=1.68 GeV. We have also checked our
results for the $\eta p$ reaction above W=2 GeV where a number of new
experimental data are available. Note that we do not use Reggezied $t-$channel
exchange but include all $t$-channel contributions consistently into our
unitarization procedure. Because of the normalization problem Sibirtsev:2010yj
; Dey:2011rh between the CLAS Williams:2009yj and the CB-ELSA Crede:2009zzb
datasets the simultaneous description of these data is not possible. Above W=2
GeV our calculations are found to be in closer agreement with the CLAS
measurements Williams:2009yj . The CB-ELSA data Crede:2009zzb demonstrates a
step rise around W=1.925 GeV for the scattering angles
$\cos\theta=0.85...0.95$. It is not clear whether this phenomenon could be
related to a threshold effect (e.g. $\phi N$, $a_{0}(980)N$, $f_{0}(980)$, or
$\eta^{\prime}N$) or attributed to other reaction mechanisms.
We conclude that further progress in understanding of the $\eta$-meson
production dynamics would be hardly possible without new measurements of the
$\pi N\to\eta N$ reaction.
## II Database
Here we present a short overview of the experimental database relevant for the
present calculations. The details on the $K\Lambda$, $K\Sigma$, $\omega N$
channels will be given elsewhere.
$\pi N\to\eta N$: The thorough overview of the $\pi N\to\eta N$ experimental
data (except the recently published Crystal Ball measurements Prakhov ), is
given in Clajus:1992dh . As already mentioned in Introduction only few
measurements of the $\eta$-production have been made with pion beams: except
for Danburg:1971 where the eta-meson was produced in $\pi^{+}D$ collisions,
all the data have been taken from the $\pi^{-}p$ scattering Prakhov ;
Baker:1979aw ; Brown:1979ii ; Bulos:1970zk ; Richards:1970cy ; Jones:1966zza ;
Debenham ; Deinet:1969cd . Unfortunately due to numerous problems with the
experimental data from Baker:1979aw ; Brown:1979ii (see discussion in
Clajus:1992dh and references therein) the use of these measurements in the
analysis might lead to wrong conclusions for the reaction mechanism.
Therefore, opposite to Shrestha:2012va we do not include these data in the
analysis. Another measurement available above W=1.65 GeV is the data from
Richards et al Richards:1970cy . In the first resonance energy region this
cross section tends to be lower than results from other experiments. Since the
old measurements quote only statistical uncertainties the reason for these
differences is unclear. In their study the authors of Batinic:1995 added
systematical errors to all differential cross sections. We do not follow this
procedure and include only quoted uncertainties in the analysis.
$\gamma p\to\eta p$: a number of experimental studies have been performed in
the resonance energy region McNicoll:2010qk ; Williams:2009yj ; GRAAL:2002 ;
Dugger:2002ft ; Krusche:1995nv ; Bartalini:2007fg ; Nakabayashi:2006ut ;
Crede:2009zzb ; Crede:2003ax ; Bartholomy:2007zz ; Elsner:2007hm ; Bock:1998rk
; Ajaka:1998zi . Most of these measurement are differential cross sections.
The target asymmetry has been studied in Bock:1998rk . It has been observed
that close to the $\eta N$ production threshold the asymmetry changes the sign
at moderate scattering angles. The previous calculations of the Giessen Model
Penner:2002a ; Shklyar:2006xw and the Mainz group PhysRevC.60.035210 could
not explain this feature. The description of this data would require an
unexpected phase shift between the $S_{11}$ and $D_{13}$ resonances as noted
in PhysRevC.60.035210 . One may hope that the upcoming new measurements of the
target asymmetry at the ELSA facility will solve this puzzle Hartmann:2011uv .
For the beam asymmetry we use the recent data from the GRAAL Bartalini:2007fg
and CB-ELSA/TAPS Elsner:2007hm collaborations which cover the energy region
up to W=1.91 GeV. For the differential cross section we use the recent high-
quality Crystal Ball data McNicoll:2010qk . Above W=1.89 GeV our calculations
are constrained by the amalgamated data set from experiments Williams:2009yj ;
Crede:2009zzb ; Crede:2003ax ; Bartholomy:2007zz . Since the experimental
uncertainties of the data Williams:2009yj ; Crede:2009zzb ; Crede:2003ax ;
Bartholomy:2007zz are much larger than those in McNicoll:2010qk we reduce
them by factor of 2.
In the $(\pi/\gamma)N\to\pi N$ channels our calculations are constrained by
the single-energy solutions from the GWU (former SAID) analysis Workman:2011vb
; Arndt:2006bf ; Arndt:2008zz . For the $\pi N\to 2\pi N$ transitions we
follow the procedure described in Penner:2002a ; Penner:2002b ; Feuster:1998a
; Feuster:1998b . We continue to parameterize the $2\pi N$ channel in terms of
the effective $\zeta N$ state, where $\zeta$ is an isovector scalar meson of
two pion mass: $m_{\zeta}=2m_{\pi}$. The final $\zeta N$ state is only allowed
to couple to nucleon resonances. Therefore the decay $N^{*}\to\zeta N$ stands
for the sum of transitions $N^{*}\to\Delta\pi$, $\sigma N$, $\rho N$ etc. This
procedure allows for the good description of the $\pi N\to 2\pi N$ partial
wave cross sections extracted in Manley:1984 . However of case of the $\gamma
p\to 2\pi N$ the same agreement cannot be expected. This is because of the
enhanced role of the background contributions (due to e.g. the contact
$\gamma\rho NN$ interaction in the $\gamma N\to\rho N$ transitions). After
fixing the database a $\chi^{2}$ minimization is performed to fix the model
parameters.
## III Giessen Model
Here we briefly outline the main ingredients of the model. More details can be
found in Feuster:1998a ; Feuster:1998b ; Penner:2002a ; Penner:2002b ;
shklyar:2004a ; shklyar:2005c . The Bethe-Salpeter equation is solved in the
$K$-matrix approximation to obtain multi-channel scattering $T$-matrix:
$\displaystyle
T(\sqrt{s},p,p^{\prime})=K(\sqrt{s},p,p^{\prime})+\int\frac{d^{4}q}{(2\pi)^{4}}K(\sqrt{s},p,q)G_{BS}(\sqrt{s},q)T(\sqrt{s},q,p^{\prime}),$
(1)
where $p$ ($k$) and $p^{\prime}$ ($k^{\prime}$) are the incoming and outgoing
baryon (meson) four-momenta, $T(\sqrt{s},p,p^{\prime})$ is a coupled-channel
scattering amplitude, $G_{BS}$ is a meson-nucleon propagator and
$K(\sqrt{s},p,p^{\prime})$ is an interaction kernel. The quantities
$T(\sqrt{s},p,p^{\prime})$, $G_{BS}$, and $K(\sqrt{s},p,p^{\prime})$ are in
fact multidimensional matrices where the elements of the matrix stand for the
different scattering reactions.
To solve the coupled-channel scattering problem with a large number of
inelastic channels, we apply the so-called K-matrix approximation by
neglecting the real part of the BSE propagator $G_{BS}$. After the integration
over the relative energy, Eq. (1) reduces to
$\displaystyle
T^{\lambda_{f}\lambda_{i}}_{fi}=K^{\lambda_{f}\lambda_{i}}_{fi}+i\int
d\Omega_{n}\sum_{n}\sum_{\lambda_{n}}T^{\lambda_{f}\lambda_{n}}_{fn}K^{\lambda_{n}\lambda_{i}}_{ni},$
(2)
where $T_{fi}$ is a scattering matrix and $\lambda_{i}$($\lambda_{f}$) stands
for the quantum numbers of initial(final) states $f,i,n=$ $\gamma N$, $\pi N$,
$2\pi N$, $\eta N$, $\omega N$, $K\Lambda$, $K\Sigma$. Using the partial-wave
decomposition of $T$, $K$ in terms of Wigner d-functions the angular
integration can be easily carried out and the equation is further simplified
to the algebraic form
$\displaystyle
T^{J\pm,I}_{fi}=\left[\frac{K^{J\pm,I}}{1-iK^{J\pm,I}}\right]_{fi}.$ (3)
The validity of this approximation was demonstrated by Pearce and Jennings in
Pearce:1990uj by studying different approximations to the BSE for $\pi N$
scattering. Considering different BSE propagators they concluded that an
important feature of the reduced intermediate two particle propagator is the
on-shell part of $G_{BS}$. It has been argued that there is no much difference
between physical parameters obtained using the $K$-matrix approximation and
other schemes. It has also been shown in Goudsmit:1993cp ; Oset:1997it that
for $\pi N$ and $\bar{K}N$ scattering the main effect from the off-shell part
is a renormalization of the couplings and the masses.
Due to the smallness of the electromagnetic coupling the dominant
contributions to the self energy stem from the hadronic part. Therefore we
treat the photoproduction reactions perturbatively. This is equivalent to
neglecting $\gamma N$ in the sum over intermediate states $n$ in Eq. (2).
Thus, for a photoproduction process the equation (3) can be rewritten as
follows Penner:2002b ; Feuster:1998b
$\displaystyle
T^{J\pm,I}_{f\gamma}=K^{J\pm,I}_{f\gamma}+i\sum_{n}T^{J\pm,I}_{fn}K^{J\pm,I}_{n\gamma},$
(4)
where the summation in Eq.(4) is done over all hadronic intermediate states.
Here the matrix $T^{J\pm,I}_{fn}$ stems only from the hadronic transitions:
indices $f$ and $n$ run over $\pi N$, $2\pi N$, $\eta N$, $K\Lambda$,
$K\Sigma$, $\omega N$ channels. The sum in Eq. (4) reflects the importance of
the hadronic part of the transition amplitude in the description of
photoproduction reactions. In other words, the amplitudes for the $\pi N\to\pi
N$, $\eta N$, $\omega N$ etc. transitions should always be included in the
calculation of the photoproduction amplitudes.
### III.1 Interaction kernel and resonance parameters
Here we present the main ingredients of the interaction kernel to the BSE
Eq.(1) relevant for $\eta$-production. More details on other reactions can be
found in shklyar:2004a ; Penner:2002a ; Penner:2002b ; Feuster:1998a ;
Feuster:1998b ; shklyar:2004b . The interaction potential ($K$-matrix) of the
BSE is built up as a sum of $s$-, $u$-, and $t$-channel contributions
corresponding to the tree level Feynman diagrams shown in Fig. (1).
Figure 1: $s$-,$u$-, and $t$\- channel contributions to the interaction
potential. $i$ and $f$ stand for the initial and final $\gamma N$, $\pi N$,
$2\pi N$, $\eta N$, $\omega N$, $K\Lambda$, $K\Sigma$ states. $m$ denotes
intermediate $t$-channel meson.
In the isospin $I=\frac{1}{2}$ channel we checked for the contributions from
the $S_{11}(1535)$, $S_{11}(1650)$, $P_{11}(1440)$, $P_{11}(1710)$,
$P_{13}(1720)$, $P_{13}(1900)$, $D_{13}(1520)$ $D_{13}(1900)$, $D_{15}(1675)$,
$F_{15}(1680)$ , $F_{15}(2000)$ resonances. The resonance and background
contributions are consistently generated from the same effective interaction.
The Lagrangian densities are given in shklyar:2004a ; Penner:2002a ;
Penner:2002b ; Feuster:1998a ; Feuster:1998b ; shklyar:2004b and respect the
chiral symmetry in low-energy regime. The properties of the $t$-channel mesons
important for $\eta$ production are given in Table 1.
| mass [GeV] | $J^{P}$ | $I$ | final state
---|---|---|---|---
$\omega$ | 0.783 | $1^{-}$ | $0$ | $(\gamma,\eta)$
$\rho$ | 0.769 | $1^{-}$ | $1$ | $(\pi,\eta)(\gamma,\eta)$
$a_{0}$ | 0.983 | $0^{+}$ | $1$ | $(\pi,\eta)$
$\phi$ | 1.02 | $1^{-}$ | $0$ | $(\gamma,\eta)$
Table 1: Properties of mesons which give contributions to the $\eta N$ final
state via the $t$-channel exchange. The notation $(\gamma,\eta)$ means $\gamma
N\to\eta N$ etc.
Using the interaction Lagrangians and values of the corresponding meson decay
widths taken from the PDG pdg the following coupling constants are obtained:
$\displaystyle\begin{array}[]{lcrclcr}g_{a_{0}\eta\pi}&=&-2.100\;,&&g_{\omega\eta\gamma}&=&-0.27\;,\\\
g_{\rho\eta\gamma}&=&-0.64\;,&&g_{\phi\eta\gamma}&=&-0.385\;.\\\ \end{array}$
(7)
All other coupling constants were allowed to be varied during the fit. The
obtained values are given in Table 2. For the $\eta NN$ interaction we use
pseudoscalar coupling , which has been also utilized in our previous studies
Feuster:1998b ; Feuster:1998a ; Penner:2002a ; Penner:2002b ; Shklyar:2006xw .
The derived $g_{\eta NN}$ constant is found to be small which is in line with
our previous results Shklyar:2006xw ; Penner:2002b . To check the dependence
of our results on the choice of the $\eta NN$ interaction we have also
performed calculations with the pseudovector coupling. However also in the
latter case only a small $g_{\eta NN}$ coupling constant has been found.
Since the PDG gives only the upper limit for the decay branching ratio
R$(\rho\to\pi\eta)<6\times 10^{-3}$ we allowed this constant to be varied
during fit. However due to lack of experimental constraints this coupling
cannot be fully fixed in the present calculation. We find a small overall
contribution from the $t$-channel $\rho$-meson exchange to the
$\pi^{-}p\to\eta n$ reaction. The $g_{\phi NN}$ coupling is calculated from
$g_{\omega NN}$ using the relation
$\frac{g_{\phi NN}}{g_{\omega NN}}=-\tan\Delta\theta_{\phi/\omega},$
where $\Delta\theta_{\phi/\omega}$ is a deviation from the ideal
$\phi$-$\omega$ mixing angle. Taking $\Delta\theta_{\phi/\omega}=3.7^{0}$ from
pdg one gets for the ratio $g_{\phi NN}/g_{\omega NN}\approx-1/15$. Using
this value a very small contribution from the $t$\- channel $\phi$-meson
exchange to the $\eta$-photoproduction has been found.
$g_{\pi NN}$ | 12.85 | $g_{\rho\eta\pi}$ | 0.133 | $g_{\rho NN}$ | 4.98 | $\kappa_{\rho}$ | 2.18
---|---|---|---|---|---|---|---
$g_{\eta NN}$ | 0.31 | $g_{a_{0}NN}$ | -44.37 | $g_{\omega NN}$ | 7.23 | $\kappa_{\omega}$ | -1.50
Table 2: Nucleon and $t$-channel couplings obtained in the present study.
To take into account the finite size of mesons and baryons each vertex is
dressed by a corresponding form factor:
$\displaystyle F_{p}(q^{2},m^{2})$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{4}}{\Lambda^{4}+(q^{2}-m^{2})^{2}},$ (8)
where $q$ is a c.m. four-momentum of an intermediate particle and $\Lambda$ is
a cutoff parameter. The cutoffs $\Lambda$ in Eq. (8) are treated as free
parameters being varied during the calculation. However, we keep the same
cutoffs in all channels for a given resonance spin $J$ : $\Lambda^{J}_{\pi
N}=\Lambda^{J}_{\pi\pi N}=\Lambda^{J}_{\eta N}=...$ etc.,
($J=\frac{1}{2},~{}\frac{3}{2},~{}\frac{5}{2}$). This significantly reduces
the number of free parameters; i.e. for all spin-$\frac{5}{2}$ resonances
there is only one cutoff $\Lambda=\Lambda_{\frac{5}{2}}$ for all decay
channels. However for the photoproduction reactions we use different cutoffs
at the $s$\- and $u$-channel electromagnetic vertices. All values are given in
Table 3. Except for the spin-$\frac{3}{2}$ states, the $s$\- and $u$-channel
cutoffs almost coincide.
$\Lambda_{N}$ [GeV] | $\Lambda_{\frac{1}{2}}^{h}$ [GeV] | $\Lambda_{\frac{3}{2}}^{h}$ [GeV] | $\Lambda_{\frac{5}{2}}^{h}$ [GeV] | $\Lambda_{\frac{1}{2}}^{\gamma}$ [GeV] | $\Lambda_{\frac{3}{2}}^{\gamma}$ [GeV] | $\Lambda_{\frac{5}{2}}^{\gamma}$ [GeV] | $\Lambda_{t}^{h,\gamma}$[GeV]
---|---|---|---|---|---|---|---
0.952 | 3.0 | 0.97 | 1.13 | 1.69 (1.69) | 4.20 (2.9) | 1.17 (1.25) | 0.7
Table 3: Cutoff values for the form factors. The lower index denotes an
intermediate particle, i.e. $N$: nucleon, $\frac{1}{2}$: spin-$\frac{1}{2}$
resonance, $\frac{3}{2}$: spin-$\frac{3}{2}$, $\frac{5}{2}$:
spin-$\frac{5}{2}$ resonance, $t$: $t$-channel meson. The upper index
$h$($\gamma$) denotes whether the value is applied to a hadronic or
electromagnetic vertex. The cutoff values used at electromagnetic $u$-channel
vertices are given in brackets.
The use of vertex form factors requires special care for maintaining the
current conservation when the Born contributions to photoproduction reactions
are considered. Since the resonance and intermediate meson vertices are
constructed from gauge invariant Lagrangians they can be independently
multiplied by the corresponding form factors. For the nucleon contributions to
meson photoproduction we apply the suggestion of Davidson and Workman
Davidson:2001rk and use the crossing symmetric common form factor:
$\displaystyle\tilde{F}(s,u,t)=F(s)+F(u)+F(t)-F(s)F(u)-F(s)F(t)-F(u)F(t)+F(s)F(u)F(t).$
(9)
The extracted resonance parameters given in Table 4 are very close to the
values deduced in our previous calculations shklyar:2004b ; Shklyar:2006xw
which indicates the stability of the obtained solution. However some values
changed upon inclusion of the new MAMI data McNicoll:2010qk . The total width
of $S_{11}(1650)$ tends to be larger than that deduced in our previous
calculations shklyar:2004b . The helicity amplitude is also modified but still
is in good agreement with the parameter range provided by PDG pdg . The
opposite effect is found for the $P_{11}(1710)$ state where the total width is
reduced once the data of McNicoll:2010qk are included. The remaining
resonance parameters are only slightly modified as compared to our previous
results.
The mass and width of the Roper resonance is found to be larger than deduced
in other analyses pdg . However the authors of Vrana:2000 give $490\pm 120$
MeV for the total width. The large decay width $545\pm 170$ MeV has also been
deduced by Cutkosky and Wang Cutkosky:1990 . Note that properties of this
state are strongly influenced by its decay into the $2\pi N$ final state.
Arndt et al Arndt:1990bp found a second pole structure for the Roper
resonance which might be attributed to the coupling to the $\pi\Delta$
subchannel. Since we use a simplified prescription for the $2\pi N$ reaction
this effect cannot be properly described in the present calculations.
The recent GWU(SAID) study of the $\pi N$ data shows no evidence for the
$P_{11}(1710)$ resonance. An indirect indication for the existence of this
state can be concluded from the analysis of the $\pi N$ inelasticity and $2\pi
N$ cross section in the $P_{11}$ partial wave, see discussion in Section IV.
We find a small coupling of this resonance to the $\pi N$ final state. Since a
clear signal from this state is not seen in the recent GWU solution, the
determination of the total width turns out to be difficult. In our
calculations we assume that this resonance has a large decay branching ratio
to the $\eta N$. However the quality of the $\pi^{-}p\to\eta n$ data does not
allow for an unambiguous determination of the properties of this state.
The mass and width of the $D_{13}(1520)$ is more close to the values obtained
by Arndt et al Arndt:1995ak : $1516\pm 10$ MeV and $106\pm 4$ MeV
respectively. It is interesting to note that the mass of this resonance
deduced from the pion photoproduction tends to be 10 MeV lower that the values
derived from the pion-induced reactions pdg . The second $D_{13}(1900)$ has a
very large decay width. We associate this state with $D_{13}(2080)$ as
suggested in PDG. This resonance is rated with two stars and its existence is
still under discussion. In our updated coupled-channel calculation of the
$\omega$-production shklyar:2004b a large $\omega N$ and $2\pi N$ decay
branching ratios have been obtained.
The properties of other resonances are very close to the values given in PDG.
Except for $S_{1}(1535)$ and $P_{11}(1710)$ we find only small resonance
couplings to $\eta N$ which is in accordance with our previous conclusions.
One needs to stress that the smallness of the resonance coupling does not
necessarily mean that the contribution from the state is negligible. The
$S_{11}(1650)$ state produces for example a sizable effect in the eta-
production due to overlapping with $S_{11}(1535)$. Another example is the
effect from the $D_{13}(1520)$ state in $\eta$-photoproduction on the proton.
Here the smallness of the $\eta N$ branching ratio is compensated by the
strong electromagnetic coupling of this resonance. Therefore the effect from
this state could be seen in the $E_{2-}$ and $M_{2-}$ multipoles, see Section
IV.5. However in most cases the resonance contributions with small branching
ratios to the eta are hard to resolve unambiguously.
### III.2 Pole parameters
It is interesting to compare the poles positions and elastic residues with the
results from other studies, see Table 5. The calculated pole masses are very
close to the values obtained in other analyses, see pdg . The agreement
between imaginary parts and elastics residues is also good, though some
differences exist between the present values and the results from other
groups.
For the $S_{11}(1535)$ state we obtain a smaller elastic residue (for
definition of $|R|$ see pdg ) $|R|=15$ MeV which is almost identical to the
result of the GWU group $|R|$=16 MeV Arndt:2006bf . Both values seem to be out
of the range given in PDG pdg 50$\pm$20 MeV. It is interesting to note that
the elastic residue from Arndt:2006bf is included into the estimation made in
pdg but still does not fit to the provided range. The value $\Gamma_{\rm
pole}=89$ MeV for the $S_{11}(1650)$ state is also comparable with the result
from Arndt:2006bf : $\Gamma_{\rm pole}=80$ MeV which are again less than the
lower bound given in pdg .
Though the derived pole mass of $P_{11}(1440)$ is very close to the values
deduced in other calculations we obtain a significantly larger pole width. As
a result the elastic pole residue turns out to be also large $|R|$=126 MeV. We
note, that the extraction of the properties of $P_{11}(1440)$ in the complex
energy plane might require a proper treatment of the
$P_{11}(1440)\to\pi\Delta(1232)\to 2\pi N$ isobar decay channel where the
overlap of the self-energies of the $P_{11}(1440)$ and $\Delta(1232)$ states
might be important for the determination of the properties of $P_{11}(1440)$.
This question will be addressed in shklyar:2012 .
As we already mentioned the results for $P_{11}(1710)$ are controversial. We
find 159 MeV for the pole width. Somewhat greater value of 189 MeV has been
obtained in Tiator:2010rp ; Batinic:2010zz . The recent issue of PDG pdg
summarizes results for the pole parameters taken from four different analyses.
Whereas the calculations Anisovich:2011fc ; Hoehler:1993 give 200 MeV for the
pole width, Cutkosky obtains a significantly lower value $\Gamma$=80 MeV
Cutkosky:1990zh ; Cutkosky:1979fy . This results in a large spread of the
resonance width given by PDG, see Table 5. The elastic residue is found to be
small which is in accordance with the small decay branching ratio to $\pi N$.
The similar conclusion has also been drawn in Tiator:2010rp .
Investigation of the $P_{13}$\- wave inelasticity Arndt:2006bf shows that the
$P_{13}(1720)$ state could have a strong decay flux into the $3\pi N$ channel
Manley:1984 . Therefore the calculation of its pole width might be affected by
deficiencies in description of this channel. PDG estimations are based on
several studies where $\Gamma_{\rm pole}=120\pm 40$ by Cutkosky
Cutkosky:1979fy is the lower limit. The upper bound $\Gamma_{\rm pole}=450\pm
100$ MeV is given by the recent Bonn-Gatchina analysis Anisovich:2011fc .
Neither of these calculations includes the $3\pi N$ channel explicitely.
The situation with the second $P_{13}(1900)$ state is even more complicated.
This resonance is rated by two stars in PDG and supposed to be rather broad.
The latest GWU analysis Arndt:2006bf does not find any indication for this
state. The present information about the pole parameters in PDG is based
solely on the result of the Bonn-Gatchina calculations Anisovich:2011fc which
deduce the pole mass $1900\pm 30$ MeV and the pole width $200^{+100}_{-60}$
MeV. These values are very close to those derived in the present work.
The pole width of the $D_{13}(1520)$ state ( 94 MeV) turns out to be 10 MeV
less than the lower limit given in PDGpdg . The similar value of $\Gamma_{\rm
pole}=$ 95 MeV has also been obtained in the Jülich model Doring:2009yv . Some
analyses find additional poles associated with the $D_{13}(1700)$ and
$D_{13}(1875)$ states pdg . We do not find any indication for $D_{13}(1700)$.
The pole position for the second resonance is close to the results of other
calculation pdg .
Though the elastic residues for the $D_{15}(1675)$ and $F_{15}(1680)$ states
are comparable with the values given in PDG their pole widths are somewhat
lower than those obtained in other studies pdg . We also find an indication
for the second state N(2000) with the pole mass of 1900 MeV and the width of
$123$ MeV, see Table 5. This resonance has a small coupling to the $\pi N$
final state what is in agreement with results from other calculations.
$N^{*}$ | mass (MeV) | $\Gamma_{tot}(MeV)$ | RπN | R2πN | RηN | RωN | $A^{p}_{\frac{1}{2}}$ | $A^{p}_{\frac{3}{2}}$
---|---|---|---|---|---|---|---|---
$S_{11}$(1535) | 1526(2) | 131(12) | 35(3) | $8(2)$ | $58(4)$ | — | 91(4) | —
| 1526 | 136 | 34.4 | $9.5$ | $56.1$ | — | 92 | —
| 1536(10) | 150(25) | 45(10) | $5(5)$ | $42(10)$ | — | 90(30) | —
$S_{11}$(1650) | 1665(2) | 147(14) | 74(3) | $23(2)$ | $1(2)$ | — | 63(6) | —
| 1664 | 131 | 72.4 | $23.1$ | $1.4$ | — | 57 | —
| 1657(13) | 150(30) | 70(20) | $15(5)$ | $10(5)$ | — | 53(16) | —
$P_{11}$(1440) | 1515(15) | 605(90) | 56(2) | $44(2)$ | — | — | -85(3) | —
| 1517 | 608 | 56.0 | $44.0$ | — | — | -84 | —
| 1445(25) | 300(150) | 65(10) | $35(5)$ | — | — | -60(4) | —
$P_{11}$(1710) | 1737(17) | 368(120) | 2(2) | $49(3)$ | $45(4)$ | 3(2) | -50(1) | —
| 1723 | 408 | 1.7 | $49.8$ | $43.0$ | 0.2 | -50 | —
| 1710(30) | 150(100) | 13(7) | $65(25)$ | $20(10)$ | 13(2) | 24(10) | —
$P_{13}$(1720) | 1700(10) | 152(2) | 17(2) | $79(2)$ | $0(1)$ | — | -65(2) | 35(2)
| 1700 | 152 | 17.1 | $78.7$ | $0.2$ | — | -65 | 35
| 1725(24) | 225(125) | 11(3) | $>70$ | $4(1)$ | — | 50(60) | -19(20)
$P_{13}$(1900) | 1998(3) | 359(10) | 25(1) | $61(2)$ | $2(2)$ | 10(3) | -8(1) | 0(1)
| 1998 | 404 | 22.2 | $59.4$ | $2.5$ | 14.9 | -8 | 0
| 1900(-) | 250(-) | 10(-) | — | $12(-)$ | 39(-) | 26(15) | -65(30)
$D_{13}$(1520) | 1505(4) | 100(2) | 57(2) | $44(2)$ | $0(1)$ | — | -15(1) | 146(1)
| 1505 | 100 | 56.6 | $43.4$ | 1.2 | — | -13 | 145
| 1520(5) | 112(12) | 60(5) | $25(5)$ | 2.3$\pm 10^{-3}$ | — | -24(8) | 150(15)
$D_{13}$(1875) | 1934(10) | 857(100) | 11(1) | $69(2)$ | $0(1)$ | 20(5) | 11(1) | 26(1)
| 1934 | 859 | 10.5 | $68.7$ | $0.5$ | 20.1 | 11 | 26
| 1875(45) | 220(100) | 12(10) | $70(20)$ | $3.5(3.5)$ | 21(7) | 18(10) | -9(5)
$D_{15}$(1675) | 1666(2) | 148(1) | 41(1) | $58(1)$ | $0(1)$ | — | 9(1) | 21(1)
| 1666 | 148 | 41.1 | $58.5$ | $0.3$ | — | 9 | 20
| 1675(5) | 150(15) | 40(5) | $55(5)$ | $0(1)$ | — | 19(8) | 15(9)
$F_{15}$(1680) | 1676(2) | 115(1) | 68(1) | $32(1)$ | $0(1)$ | — | 3(1) | 116(1)
| 1676 | 115 | 68.3 | $31.6$ | $0.0$ | — | 3 | 115
| 1685(5) | 130(10) | 67(3) | $35(5)$ | $0(1)$ | — | -15(6) | 132(13)
$F_{15}$(2000) | 1946(4) | 198(2) | 10(1) | $87(1)$ | $2(2)$ | 1(1) | 11(1) | 25(1)
| 1946 | 198 | 9.9 | $87.2$ | $2.0$ | 0.4 | 10 | 25
| 2050(100) | 350(200) | 15(7) | — | — | — | 35(15) | 50(14)
Table 4: Resonance parameters extracted in the present study. The uncertainties are given in brackets. Helicity decay amplitudes are given in $10^{-3}$GeV${}^{-\frac{1}{2}}$. 1st line: present study; 2nd line: shklyar:2004b , 3th line: pdg . (-): the validity range is not given. | Re $z_{0}$(GeV) | -2Im $z_{0}$(MeV) | $|$R$|$(MeV) | $\theta^{0}$
---|---|---|---|---
$S_{11}(1535)$ | 1.49 | 100 | 15 | -51
| 1.49-1.53 | 90-250 | 30…70 | -1…-30
$S_{11}(1650)$ | 1.65 | 89 | 19 | -46
| 1.64-1.67 | 100-170 | 20-50 | -50…-80
$P_{11}(1440)$ | 1.386 | 277 | 126 | -60
| 1.35-1.38 | 160-220 | 40-52 | -75…-100
$P_{11}(1710)$ | 1.67 | 159 | 11 | 9
| 1.67-1.77 | 80-380 | 2-15 | -160…+190
$P_{13}(1720)$ | 1.67 | 118 | 12 | -45
| 1.66-1.69 | 150-400 | 7-23 | -90…-160
$P_{13}(1900)$ | 1.91 | 173 | 10 | -64
| 1.870-1.93 | 140-300 | 1-5 | 45…-25
$D_{13}(1520)$ | 1.492 | 94 | 27 | -35
| 1.505-1.515 | 105-120 | 32-38 | -5…-15
$D_{13}(1875)$ | 1.81 | 98 | 3 | -76
| 1.8-1.95 | 150-250 | 2-10 | 20…180
$D_{15}(1675)$ | 1.64 | 108 | 20 | -49
| 1.655-1.665 | 125-150 | 22-32 | -21…40
$D_{15}(1680)$ | 1.66 | 98 | 33 | -32
| 1.665-1.68 | 110-135 | 35-45 | 0…-30
$F_{15}(2000)$ | 1.90 | 123 | 11 | -6
| 1.92-2.15 | 380-580 | 20-115 | -60…-140
Table 5: Pole positions and elastic pole residues. First line: present study,
second line: values from PDG pdg .
## IV Results and discussion
The lack of the experimental data for the pion-induced reactions does not
provide enough constraints on the resonance parameters. Also the discrepancy
among various measurements (see Section (II)) does not allow for a consistent
description of the data in a full kinematical region. While the contribution
from the $S_{11}(1535)$ state is well established the reaction dynamics above
W=1.6 GeV is still under discussion. One of the early Giessen coupled-channel
calculations Penner:2002a ; Penner:2002b ; Shklyar:2006xw found a destructive
interference between $S_{11}(1535)$ and $S_{11}(1650)$ states. The second
suggestion is a strong contribution from the $P_{11}(1710)$-resonance
excitation above W=1.68 GeV. This resonance was established in the early
single-channel Karlsruhe-Helsinki and Carnegie Mellon-Berkeley analyses (see
PDG pdg and references therein). The independent study of the $\pi
N\to(\pi/\eta)N$ reactions by the Zagreb group Ceci:2006ra provides an
additional evidence for the existence of $P_{11}(1710)$. The result of
Ceci:2006ra confirm the assumption made in Penner:2002a ; Penner:2002b on
the important contribution from this state to the $\eta$-production. However
the recent analysis from the GWU group Arndt:2006bf finds no evidence for
this state. The absence of a clear signal in the $P_{11}$ partial wave of the
elastic $\pi N$ scattering does not necessarily mean that this state does not
exist. If the coupling to the final $\pi N$ state is small, the effect from
this state might not be seen in $\pi N$ scattering. The evidence for the
signal from the $P_{11}(1710)$ resonance has also been reported from the study
of the $\pi N\to K\Lambda$ reaction Ceci:2005vf . On the other hand the result
of the Bayestian analysis performed by the Gent group DeCruz:2012bv
demonstrates that $P_{11}(1710)$ is not needed to describe the $K\Lambda$
photoproduction. An opposite conclusion was drawn by the Bonn-Gatchina group
which finds decay branching ratio of $23\pm 7$% of this state to $K\Lambda$
Anisovich:2011fc .
Figure 2: (Color online) Calculated $\pi N$ inelasticity and $\pi N\to 2\pi N$
cross section in the $P_{11}$ partial wave in comparison with the results from
Arndt:2006bf (GWU 2006) and Manley:1984 (Manley 1984).
Another indication for this state comes from the analysis of an inelastic flux
in the $P_{11}$ partial wave. In Fig. (2) the total inelasticity from the GWU
analysis vs. the total $2\pi$ cross section extracted in Manley:1984 is
compared. The difference between the total $\pi N$ inelasticity and the total
$2\pi N$ cross section at W=1.7 GeV in the $P_{11}$-wave can be attributed to
the sum of inelastic channels like $3\pi N$, $\eta N$, $\eta\pi N$ etc. We
assume here that the observed difference is due to the $\eta N$ production
channel dominated by the $P_{11}(1710)$ state. As $g_{\pi NN^{*}(1710)}$ is
assumed to be small this raises the question about the magnitude of the
$P_{11}(1710)$ contribution in the $\pi N\to\eta N$ reaction. However the
situation in $\eta$-production is different from the $\pi N$ elastic
scattering. Here the contribution from $P_{11}(1710)$ is proportional to the
product $g_{\pi NN^{*}(1710)}g_{\eta NN^{*}(1710)}$, where $g_{\pi
NN^{*}(1710)}$ is the coupling constant at the $N(1710)\to\eta N$ transition
vertex. It follows that the contribution from the $P_{11}(1710)$ can be
significant provided that $g_{\eta NN^{*}(1710)}$ is large enough. The
interplay with background and coupled-channel rescattering would further
increase this effect.
### IV.1 $\pi N\to\eta N$
The results of our calculations are presented in Fig. (3) in comparison with
the world data. The first peak at W=1.54 GeV is related to the well
established $S_{11}(1535)$ resonance contribution.
Figure 3: (Color online) Calculated differential $\pi^{-}p\to\eta n$ cross
section in comparison with the experimental data from: Prakhov 2005:Prakhov ,
Deinet 1969:Deinet:1969cd , Richards 1970:Richards:1970cy , Morrison
2000:Morrison:2000kx .
Though the effect from the $S_{11}(1650)$ state is hardly visible in the
differential cross section this state plays an important role leading to the
destructive interference between $S_{11}(1535)$ and $S_{11}(1650)$ as it has
been pointed out in our previous calculations Penner:2002a ; Penner:2002b .
The second rise is due to the $P_{11}(1710)$ resonance. This state has a small
branching ratio to the $\pi N$ system but due to the large $\eta$-coupling
this resonance affects the production cross section at W=1.7 GeV. The coupled-
channel effects and interference with other partial waves further enlarge the
overall contribution from this state.
The total partial wave cross sections are shown in Fig. 4. The destructive
interference between the $S_{11}(1535)$ and $S_{11}(1650)$ leads to the dip in
the total $S_{11}$-partial wave cross section around W=1.64 GeV (dotted line).
The effect from the $P_{11}(1710)$ state is shown by the dashed line, Fig. 4.
The contributions from other partial waves are found to be small. We also
corroborate our previous results shklyar:2004a where only minor contributions
from spin $J\geq\frac{3}{2}$ resonance states were obtained. Both t-channel
$a_{0}$ and $\rho$ meson exchange and $u-$channel graphs give small effects.
The inclusion of the higher spin state $D_{13}(1520)$ into the calculations is
still important to reproduce the correct shape of the cross section. This
feature is also found in many other calculations, e.g.Penner:2002a ;
Batinic:1995 . It is interesting to note that importance of the $P_{11}(1710)$
resonance contribution has recently been found in Shrestha:2012va which is in
line with our previous results Shklyar:2006xw ; Penner:2002b .
Since the main contributions in our calculations come mainly from the $S_{11}$
and $P_{11}$ partial waves it is interesting to trace back the interference
effect between them. Neglecting the higher partial waves the differential
cross section can be written in the form
$\displaystyle\frac{d\sigma}{d\cos\,(\theta)}~{}\sim~{}1+\alpha\,\sin^{2}\left(\frac{\theta}{2}\right),$
(10)
where $\theta$ is a scattering angle and
$\alpha=\left(\frac{|S_{11}-P_{11}|^{2}}{|S_{11}+P_{11}|^{2}}-1\right)$ only
depends on the c.m. energy. Then the angular distribution should have a
maximum (minimum) at forward angles depending on the relative phase between
the nonvanishing $S_{11}$ and $P_{11}$ amplitudes. In our calculation the
interference between $S_{11}$ and $P_{11}$ partial waves produces a peak at
forward scattering angles and energies above W=1.67 GeV, see Fig. (3). As a
result the signal from the $P_{11}(1710)$ resonance becomes more transparent
for forward scattering. This is in line with the data of Richards et al
Richards:1970cy confirming our guess about the production mechanism. The
inclusion of higher partial waves would modify Eq. (10). However these
contributions are relatively small (see Fig.(4)) thus producing only minor
deviations from the distribution Eq. (10).
Note, that due to numerous problems with the experimental data our
calculations above W=1.6 GeV are only partly constrained by experiment.
Indeed, once the data Baker:1979aw ; Brown:1979ii are neglected there are
only 30 datapoints from experiment Richards:1970cy . This data has relatively
large error bars and seems not to be fully consistent with other measurements
Clajus:1992dh . Therefore, the results for the differential cross section
might be regarded as a prediction rather than an outcome of the fit. This
demonstrates an urgent need for new measurements of the $\pi^{-}N\to\eta N$
reactions above W=1.6 GeV. This would be a challenge for the the upcoming
pion-beams experiment carried out by the HADES collaboration at GSI.
Figure 4: (Color online) Total partial wave cross section $\pi^{-}p\to\eta n$
vs. experimental data.
### IV.2 $\eta N\to\eta N$ amplitude and $\eta N$ scattering lengths
The result for the $\eta N\to\eta N$ transition amplitude in the $S_{11}$
partial wave is presented in Fig. 5. Close to threshold the elastic $\eta N$
scattering is completely determined by the contribution of the $S_{11}(1535)$
resonance. At higher energies the excitation of $S_{11}(1650)$ also becomes
important. The interference between those two $S_{11}$-states produces an
excess structure in the imaginary part of the amplitude at W=1.65 GeV.
The rapid variation of the $S_{11}$-amplitude close to threshold indicates
that this energy dependence should be taken into account when the $\eta N$
scattering length is calculated. Here we use the definition for the effective
range expansion from PhysRevC.55.R2167 :
$\displaystyle\frac{q_{\rm c.m.}}{S_{11}^{\,\eta N}}+iq_{\rm
c.m.}=\frac{1}{a_{\eta N}}+\frac{r_{0}}{2}q_{\rm c.m.}^{2}+s\,q^{4}_{\rm
c.m.},$ (11)
where $S_{11}^{\,\eta N}$ is an elastic partial S-wave amplitude, and $a_{\eta
N}$, $r_{0}$ and $s$ are scattering length, effective range, and effective
volume respectively. The results are shown in Table 6 in comparison with
values deduced from other coupled-channel calculations ( results published
before 1997 are discussed in PhysRevC.55.R2167 ). The obtained value of
$a_{\eta N}$ is very close to our previous results Penner:2002a . The values
for the real part deduced in Batinic:1996me and PhysRevC.55.R2167 are lower
than in this work. The study Batinic:1996me gives 1.550 GeV for the mass and
204 MeV for the width of the $S_{11}(1535)$ state which are somewhat greater
than in the present calculation. This could be one of the reasons for the
differences in $Re\,a_{\eta N}$.
In PhysRevC.55.R2167 only the $S_{11}(1535)$ state is taken into account to
calculate transition amplitudes to the $\eta N$ channel. Since the parameters
of $S_{11}(1535)$ in PhysRevC.55.R2167 are close to the values obtained in
the present study the observed difference in ${\rm Re}\,a_{\eta N}$ might be
attributed to the different treatment of background contributions which have
been assumed in PhysRevC.55.R2167 to be energy-independent. The second piece
of uncertainty is related to the quality of the world data of $\pi N\to\eta N$
scattering. Hence, precise measurements of this reaction would provide an
additional constraint on $\eta N$ scattering length.
The non-vanishing imaginary part of $a_{\eta N}$ is mostly driven by
rescattering in the $\pi N$ channel. Since the largest contributions to the
scattering length are produced by the $S_{11}(1535)$ state the imaginary part
of $a_{\eta N}$ is strongly influenced by the decay branching ratio of this
resonance to $\pi N$. Only a minor effect is found from the rescattering
induced by background contributions and inelastic flux to the $2\pi N$
channel. Since the $\pi NN^{*}(1535)$ coupling is well fixed an agreement in
${\rm Im}(a_{\eta N})$ between various model calculations can be expected
provided that unitarity is maintained.
The obtained value of the scattering length should be taken with care when in-
medium properties of the $\eta$-meson are considered. As it has already been
pointed out in PhysRevC.55.R2167 the $S_{11}$ amplitude has a strong energy
dependence - a feature which might affect the $\eta$-potential. The second
reason is that properties of the $S_{11}(1535)$ resonance might also be
subjected to in-medium modifications Lehr:2003km . Both effects should be
taken into account when $\eta$-meson properties in nuclei are studied.
Figure 5: Calculated $S_{11}$ partial wave amplitude of the elastic $\eta N$ scattering. Reference | $a_{\eta N}$(fm) | $r_{0}$(fm)
---|---|---
present work | 0.99$\pm$0.08 + i0.25$\pm$0.06 | -1.98$\pm$0.1 - i0.43$\pm$0.15
Penner:2002a | 0.99 + i0.34 | -2.08 - i0.81
Batinic:1996me | 0.734$\pm$0.026 + i0.269$\pm 0.019$ |
PhysRevC.55.R2167 | 0.75$\pm 0.04$ \+ i0.27$\pm 0.03$ | -1.5$\pm$0.13 - i0.24$\pm$0.04
Lutz:2001 | 0.43+ i0.21 |
Table 6: Calculated scattering length and effective range in comparison with
results from other works.
### IV.3 $\gamma N\to\eta N$ below 1.89 GeV
The results of our calculation of the differential cross section in comparison
with the recent Crystal Ball/MAMI measurements are shown in Fig. (6). Our
calculations demonstrate a nice agreement with the experimental data in the
whole kinematical region. The first peak is related to the $S_{11}(1535)$
resonance contribution. Similar to the $\pi^{-}p\to\eta n$ reaction the
$S_{11}(1650)$ and $S_{11}(1650)$ states interfere destructively producing a
dip around W=1.68 GeV. Though the effect from the $P_{11}(1710)$ state is only
minor, the contribution from this resonance produces a rapid change in the
$M_{1-}$ photoproduction multipole, see Section IV.5. The coherent sum of all
partial waves leads to the more pronounced effect from the dip at forward
angles. Note that the resonance contribution to the photoproduction reaction
stems from two sources: the first is related to the direct electromagnetic
excitation of the nucleon resonance and the second comes from rescattering
e.g. $\gamma p\to\pi N\to\eta N$, Eq. (4). At this stage the hadronic
transition amplitudes e.g. $T_{\pi N\to\eta N}$ become an important part of
the production mechanism. The sum of these contributions in the $P_{11}$ wave
turns out to be destructive which reduces the overall contribution from the
$P_{11}(1710)$ state. We also corroborate our previous findings Shklyar:2006xw
where a small effect from the $\omega N$ threshold was found.
Figure 6: Differential $\eta p$ cross section vs. recent MAMI data
McNicoll:2010qk .
We also do not find any strong indication for contributions from a hypothetic
narrow $P_{11}$ state with a width of 15-20 MeV around W=1.68 GeV. It is
natural to assume that the contribution from this state would induce a strong
modification of the beam asymmetry for energies close to the mass of this
state. This is because the beam asymmetry is less sensitive to the absolute
magnitude of the various partial wave contributions but strongly affected by
the relative phases between different partial waves. Thus even a small
admixture of a contribution from a narrow state might result into a strong
modification of the beam asymmetry in the energy region of W=1.68 GeV.
Figure 7: (Color online) $\gamma p\to\eta p$ partial wave cross sections vs.
measurements Crede:2003ax ; Crede:2009zzb ; Bartholomy:2007zz .
In Fig. (8) we show the calculation of the photon-beam asymmetry in comparison
with the GRAAL measurements Bartalini:2007fg . One can see that even close to
the $\eta N$ threshold where our calculations exhibit a dominant $S_{11}$
production mechanism (see Fig. (7) ) the beam asymmetry is nonvanishing for
angles $\cos(\theta)\geq-0.2$. This shows that this observable is very
sensitive to very small contributions from higher partial waves. At W=1.68 GeV
and forward angles the GRAAL measurements show a rapid change of the asymmetry
behavior. We explain this effect by a destructive interference between the
$S_{11}(1535)$ and $S_{11}(1650)$ resonances which induces the dip at W=1.68
GeV in the $S_{11}$ partial wave. The strong drop in the $S_{11}$ partial wave
modifies the interference between $S_{11}$ and other partial waves and changes
the asymmetry behavior. Note that the interference between $S_{11}(1535)$ and
$S_{11}(1650)$ and the interference between different partial waves are of
different nature. The overlapping of the $S_{11}(1535)$ and $S_{11}(1650)$
resonances does not simply mean a coherent sum of two independent
contributions, but also includes rescattering (coupled-channel effects). Such
interplay is hard to simulate by the simple sum of two Breit-Wigner forms
since it does not take into account rescattering due to the coupled-channel
treatment.
The GRAAL collaboration finds no evidence for a narrow state around W=1.68
GeV. We also find no strong need for the narrow $P_{11}$ resonance
contribution to describe the asymmetry data. Taking contributions from the
established states into account our results are in close agreement with the
experimental data Bartalini:2007fg .
Figure 8: Calculated beam asymmetry. Experimental data are taken from
Bartalini:2007fg (GRAAL07).
### IV.4 $\gamma N\to\eta N$ above 1.89 GeV
Figure 9: (Color online) Differential $\eta p$ cross section as a function of
the scattering angle. The data are taken from CLAS 2009:Williams:2009yj and
CB-ELSA:Crede:2009zzb . Figure 10: (Color online) Differential $\eta p$ cross
section as a function of c.m. energy at fixed forward angles. Data are taken
from CLAS 2009:Williams:2009yj , CB-ELSA:Crede:2009zzb , and
MAMI2010:McNicoll:2010qk .
Since the MAMI measurements are available up to W=1.89 GeV the calculations in
the region W=1.89 …2.GeV are constrained by the combined data set constructed
out of the recent CLAS and CB-ELSA/TAPS Williams:2009yj ; Crede:2009zzb data.
Due to some inconsistencies between these two experiments Dey:2011rh ;
Sibirtsev:2010yj we did not try to fit the data above W=2.GeV but instead
extrapolate our calculation into the higher energies. In this region the
$t$-channel exchange starts to play a dominant role. One of the accepted
prescriptions is to use a Reggeized $t$-channel meson exchange as suggested in
Chiang:2002vq . We do not follow this procedure here but include all
$t$-channel exchanges into the interaction kernel. This allows for a
consistent unitary treatment of resonance and background contributions. The
calculated differential cross section is presented in Fig. (9) as a function
of the scattering angle. Except for the energy bin W = 2.097 GeV our results
are found to be in close agreement with the CLAS measurements. The major
contribution to the differential cross section at forward angles comes from
$\rho$\- and $\omega$-meson exchanges. The effect from the $\phi$-meson is
small due to the weakness of the $\phi NN$ coupling as dictated by the OZI
rule Okubo:1963fa ; Zweig:1964jf ; Iizuka:1966fk . We also checked for the
contributions from the Primakoff effect which is found to be negligible at
these energies.
It is interesting to compare our calculations with the data Williams:2009yj ;
Crede:2009zzb at forward angles plotted as a function of the c.m. energy, see
Fig. (10). The cusp due to the $\omega N$ production threshold is clearly seen
in our calculations around W=1.72 GeV. The quality of the data is still not
good enough to unambiguously resolve the cusp induced by the $\omega N$
threshold in the experimental data. Note, that the calculations are done
assuming a stable $\omega$-meson. Taking into account the final $\omega$-width
would smear out this effect. Since the $\omega N$ threshold lies 45 MeV above
the dip position ( W=1.68 GeV) we conclude that this effect cannot explain the
dip in the differential cross section. This conclusion is opposite to that
drawn in Anisovich:2011ka .
The discrepancy between the CLAS Williams:2009yj and CB-ELSA/TAPS data is
better seen at $\cos(\theta)=0.75$ whereas for $\cos(\theta)=0.85$ the
measurements are found to be in better agreement. One of the interesting
features observed in the recent CB-ELSA data is a sudden rise of the
differential cross section at W=1.92 GeV. The effect is more pronounced at
$\cos(\theta)=0.85...0.95$ and is absent at other scattering angles. This
phenomena might be attributed to sidefeeding from of one of the inelastic
channels (e.g. $\phi N$, $a_{0}(980)N$, $f_{0}(980)$, or $\eta^{\prime}N$).
However the problem with normalization inconsistencies between the CLAS and
CB-ELSA data should be solved first before any physical interpretation can be
given.
### IV.5 eta-photoproduction multipoles
The extracted $\gamma p\to\eta p$ multipoles are presented in Fig. 11. The
major contribution to the $E_{0+}$ multipole comes from the $S_{11}(1535)$
resonance. The second $S_{11}(1650)$ plays an important role in the region
W=1.6…1.7 GeV. We corroborate our previous results Shklyar:2006xw where only
a small effect from the spin-$\frac{5}{2}$ states has been found. A very small
signal from the $F_{15}(1680)$ resonance is seen in the $E_{3-}$ and $M_{3-}$
amplitudes at W=1.68 GeV.
It is interesting to note that the effect of $D_{13}(1520)$ is clearly seen in
the $E_{2-}$ and $M_{2-}$ though the overall contribution from this state
turns out to be small. The $M_{1-}$ multipole is affected by the Roper and
$P_{11}(1710)$ resonances leading to the rapid change in both real and
imaginary parts of the amplitude at W=1.7 GeV. In the region W=1.48…1.6 GeV
both the imaginary and the real parts of all multipoles with $l\neq 0$ are of
the order of magnitude smaller than $E_{0+}$ due to the strong dominant
contribution from $S_{11}(1535)$. However for higher energies the influence of
amplitudes with $l\neq 0$ becomes also important.
Figure 11: (Color online) $\gamma p\to\eta p$ multipoles extracted in the
present study.
## V Conclusion
We have performed a coupled-channel analysis of pion- and photon-induced
reactions including the recent eta-photoproduction data from the Crystal
Ball/MAMI collaboration. In the region W=1.89…2.0 GeV our solution is
constrained by the combined dataset built from the recent CLAS and CB-
ELSA/TAPS measurements. The dip in the differential cross-sections at W=1.68
GeV reported in McNicoll:2010qk is described in terms of an interference of
the $S_{11}(1535)$ and $S_{11}(1650)$ states. We stress that such an
interference also includes coupled-channel effects and rescattering which is
hard to simulate by a simple sum of two Breit-Wigner contributions. The
additional contribution at W=1.68 GeV comes from the $M_{1-}$ multipole where
the excitation of the $P_{11}(1710)$ leads to a rapid change of the real and
imaginary parts of the amplitude. We conclude that the cusp due to the $\omega
N$ threshold seen at 1.72 GeV is not important for the explanation of the dip
at W=1.68 GeV. However the quality of the data is still not sufficient to
resolve the threshold effect completely.
Above W=1.9 GeV the $t$-channel $\rho$\- and $\omega$-exchanges start to play
a dominant role in the calculations. The effect from the $\phi$-meson exchange
is less important because of the smallness of the $\phi NN$ coupling. We have
also checked for the contribution from the Primakoff-effect which is found to
be negligible. In the region W=1.9…2.2 GeV our calculations tend to be in
closer agreement with the CLAS data.
It is interesting to note that above W=1.92 GeV the cross sections of the CB-
ELSA/TAPS collaboration indicate a sudden rise from 0.2 $\mu$b up to 0.3
$\mu$b. The effect is observed only for scattering angles
$\cos(\theta)=0.85...0.95$. This phenomenon might be attributed to sidefeeding
from of one of the inelastic channels (e.g. $\phi N$, $a_{0}(980)N$,
$f_{0}(980)$, or $\eta^{\prime}N$). However the origin of the normalization
discrepancies between the CLAS and CB-ELSA/TAPS data should first be
understood before any physical interpretation can be given.
In the $\pi^{-}p\to\eta n$ reaction the main effect comes from three
resonances $S_{11}(1535)$, $S_{11}(1650)$, and $P_{11}(1710)$. Similar to eta-
photoproduction on the proton the overlap of the $S_{11}(1535)$ and
$S_{11}(1650)$ states produces a dip around W=1.68 GeV. For energies $W>1.68$
GeV the contribution from $P_{11}(1710)$ is found to be important. The above
reaction mechanism for the $(\gamma/\pi)N\to\eta N$ reaction is in line with
our early findings Shklyar:2006xw where the resonance like-structure in
$\eta$-photoproduction at W=1.68 GeV on the neutron was explained by the
excitations of the $S_{11}(1650)$, and $P_{11}(1710)$ resonances.
We conclude that further progress in understanding of $\eta$-meson production
would be hardly possible without new measurements of the $\pi N\to\eta N$
reaction. The experimental investigation of this reaction would help to
establish the resonance contributions to the $\eta$-photoproduction above
$W>1.6$ GeV. Finally, the study of the $\eta N$-channel with pion beams would
solve the question whether the observed phenomena in $\eta$ photoproduction
have their counterparts in $\pi N\to\eta N$ scattering.
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|
arxiv-papers
| 2012-06-23T17:03:16 |
2024-09-04T02:49:32.115093
|
{
"license": "Public Domain",
"authors": "V. Shklyar, H. Lenske and U. Mosel",
"submitter": "Vitaly Shklyar",
"url": "https://arxiv.org/abs/1206.5414"
}
|
1206.5418
|
# Superlattice formed by quantum-dot sheets: density of states and IR
absorption
F. T. Vasko fedirvas@buffalo.edu V. V. Mitin Department of Electrical
Engineering, University at Buffalo, Buffalo, NY 14260-1920, USA
###### Abstract
Low-energy continuous states of electron in heterosrtucture with periodically
placed quantum-dot sheets are studied theoretically. The Green’s function of
electron is governed by the Dyson equation with the self-energy function which
is determined the boundary conditions at quantum-dot sheets with weak damping
in low-energy region. The parameters of superlattice formed by quantum-dot
sheets are determined using of the short-range model of quantum dot. The
density of states and spectral dependencies of the anisotropic absorption
coefficient under mid-IR transitions from doped quantum dots into miniband
states of superlattice strongly depend on dot concentration and on period of
sheets. These dependencies can be used for characterization of the multi-layer
structure and they determine parameters of different optoelectronic devices
exploiting vertical transport of carriers through quantum-dot sheets.
###### pacs:
73.21.Cd, 73.21.La, 78.67.Pt
## I Introduction
Heterostructures formed by quantum dot (QD) sheets are widely investigated and
used in different devices, such as lasers, photodetectors, and solar cells,
see 1 ; 2 ; 3 for review. In such heterostructures, not only the additional
localized states of electrons captured into QDs should be taken into account
but also the continuous electronic states, which are subjected to reflections
on periodically placed QD sheets, should be modified significantly. Such a
periodical perturbation gives rise to a superlattice (SL) with energy spectrum
formed by gaps between allowed minibands. In contrast to the standard case, 3a
an additional damping of electronic states takes place due to scattering on
inhomogeneties of QD sheets stemming from a random in-plane distribution of
QDs. But such a damping appears to be weak for low-energy region. As a result,
SL effect should be essential near the edge of interband absorption in host
material, which is proportional to the density of states of SL, or under IR
transitions from doped QDs into miniband states. To the best of our knowledge,
these phenomena were not considered based on a simultaneous description of SL
minibands and damping effects in spite of the structures under consideration
are routinely used in different optoelectronic devices. At the same time the
opposite case of 3D ordering of the closely spaced QDs, when SL is formed as a
result of the tunneling mix between intra-QD states, was analyzed 4 and
demonstrated experimentally, see 5 ; 6 and references therein. Because of
this, it is important and timely to develop an adequate theory of low-energy
electrons interacting with the periodically-placed QD sheets and to study the
optical response of SL which can be used for characterization of structures
under consideration and for description of different optoelectronic devices.
In this paper we study low-energy electronic states, with energies in the
vicinity of the conduction band extremum, in heterosrtuctures formed by QD
sheets of period $l$ using the effective-mass equations for the Green’s
function averaged over randomly placed QDs in each sheet. In contrast to the
standard theoretical description based on the averaging over 3D or 2D space, 7
, here we perform the averaging over QD sheets with the identical statistical
characteristics. As a result, we obtain the inhomogeneous along SL axis Dyson
equation where the self-energy function can be replaced by the boundary
conditions at QD sheets. Since the damping of the low-energy states is weak,
one can consider SL which characteristics are determined by an effective
potential localized at the sheet positions, $z=nl$, $n=0,\pm 1,\ldots$. The
strength of this potential is determined by the concentration of QDs and shape
of QD potential. With respect to low-energy states, QD can be considered as a
short-range defect (which are widely investigated during past 50 years, see
Refs. 9) if the low-energy interval under consideration is smaller than the QD
binding energy.
The density of states in SL depends on the period $l$ and on the parameter
determined by a strength of QD’s potential described within the short-range
approximation. Spectral dependencies of interband absorption between the
heavy-hole and SL states are proportional to the density of states in
$c$-band. In addition, the anisotropic absorption coefficient, originated due
to mid-IR transitions from the doped QD ground state into the miniband states
of SL, is obtained through the QD concentration and the SL parameters. We
found that the efficiency of mid-IR photoexcitation is comparable to the
contribution of wetting layers formed under QD sheets 9 if doping levels are
the same. But the spectral dependencies are very different for these two
mechanisms. Thus, it is demonstrated that the results obtained can be used for
characterization of structure under consideration. It is more important that
the SL parameters determine a mechanism of vertical transport for underbarrier
electrons which is a key process in different optoelectronic devices
exploiting multi-QD sheets. Similar mechanism of transport through
underbarrier states of IR photodetectors formed by GaAs/AlGaAs-based SL was
considered in Ref. 11.
The paper is organized as follows. In Sect. II we describe the model of
periodical sheets formed by randomly placed QDs and evaluate the Green’s
function averaged over random positions of QDs. SL effects on the density of
states and on the process of anisotropic photoexcitation of QDs are considered
in Sect. III. List of assumptions used and concluding remarks are presented in
the last section. Appendix contains the justification of the effective SL
approach employed in the calculations performed.
## II Model
The electronic states near $c$-band extremum of heterostructure, which is
formed by QD sheets placed in host material, are described by the effective
mass Hamiltonian
$\hat{H}=\frac{\hat{p}^{2}}{2m}+\sum\limits_{rk}{u\left({{\bf r}-{\bf
R}_{rk}}\right)},$ (1)
where $\hat{\bf p}$ is the 3D momentum operator, $m$ is the effective mass,
and $u({\bf r}-{\bf R}_{rk})$ is the potential energy of QD placed at
coordinates ${\bf R}_{rk}=({\bf x}_{rk},rl)$. Here $r$ labels sheet ($r=0,\pm
1,\pm 2,\ldots$) placed with the period $l$ and $k$ stands for position of QD
over the $r$th sheet given by 2D random coordinate ${\bf x}_{rk}$
($k=1,2\ldots N$ where $N$ is number of QDs over each sheet with the
normalization area $L^{2}$). Electron of energy $E$ is described by the
Green’s function ${\cal G}_{E}\left({\bf r},{\bf r}^{\prime}\right)$ governed
by the equation
$\left({E+i\lambda-\hat{H}}\right){\cal G}_{E}\left({\bf r},{\bf
r}^{\prime}\right)=\delta\left({{\bf r}-{\bf r}^{\prime}}\right)$ (2)
with $\lambda\to+0$ and the 3D $\delta$-function $\delta(\Delta{\bf r})$.
Below we consider the averaged over all QD positions Green’s function
$G_{E}\left({\bf r},{\bf r}^{\prime}\right)=\left\langle{\cal G}_{E}\left({\bf
r},{\bf r}^{\prime}\right)\right\rangle$ where the averaging over $r$th sheet
is performed according to 7
$\left\langle\cdots\right\rangle_{r}=\frac{1}{L^{2N}}\int{d{\bf
x}_{r1}\cdots}\int{d{\bf x}_{rN}\cdots}$ (3)
and $\left\langle\ldots\right\rangle$ includes the averaging over all sheets.
Using the $({\bf p},z)$-representation ($\bf p$ is 2D momentum) one obtains
the Dyson equation governing the averaged Green’s function as follows
$\displaystyle
G_{Ep}\left(z,z^{\prime}\right)=g_{Ep}\left(z-z^{\prime}\right)~{}~{}~{}~{}~{}$
(4)
$\displaystyle+\int{dz_{1}}\int{dz_{2}g_{Ep}\left(z-z_{1}\right)}\Sigma_{Ep}\left({z_{1},z_{2}}\right)G_{Ep}\left(z_{2},z^{\prime}\right).$
Here $g_{Ep}\left(z-z^{\prime}\right)$ is the free Green’s function which is
governed by Eq. (2) with the Hamiltonian $\hat{p}^{2}/2m$, so that
$g_{Ep}(\Delta
z)=\frac{1}{\hbar}\sqrt{\frac{m}{{2(\varepsilon_{p}-E)}}}\exp\left(-\frac{\sqrt{2m(\varepsilon_{p}-E)}\Delta
z}{\hbar}\right),$ (5)
if $\varepsilon_{p}>E$ and the imaginary factor $i\sqrt{E-\varepsilon_{p}}$
should be used in (5) if $\varepsilon_{p}<E$. Within the self-consistent Born
approximation, the self-energy function
$\Sigma_{Ep}\left({z_{1},z_{2}}\right)$ in Eq.(4) is given by
$\displaystyle\Sigma_{Ep}\left({z_{1},z_{2}}\right)\simeq\frac{n_{QD}}{{L^{2}}}\sum\limits_{r{\bf
p}_{1}}u\left(\frac{{\bf p}-{\bf p}_{1}}{\hbar},z_{1}-rl\right)$ (6)
$\displaystyle\times G_{Ep_{1}}\left({z_{1},z_{2}}\right)u\left(\frac{{\bf
p}_{1}-{\bf p}}{\hbar},z_{2}-rl\right)+\ldots,$
where $u({\bf q},z)$ is the 2D Fourier transform of $u({\bf r})$ and $n_{QD}$
is the QD concentration over sheet which does not dependent on $r$, i.e. we
consider identical QD sheets.
Further, we restrict ourselves by the low-energy region where scattering on a
QD can be described by the short-range potential $u({\bf r})\approx
U\Delta({\bf r})$ with the form-factor $\Delta({\bf r})$ localized in volume
$\sim a^{3}$ ($a$ stands for the characteristic size of QD). We also neglect
high-order corrections to the self-energy function (6), see below the diagram
expansion of Fig. 4 and discussion in Appendix. Since the kernel (6) is
located near QD sheets with $z_{1,2}\sim rl$ and the Green’s functions vary
over scales $\hbar/\sqrt{2m|E-\varepsilon_{p}|}$, the integral equation (4) is
transformed into the finite-difference one:
$\displaystyle G_{Ep}(z,z^{\prime})=g_{Ep}(z-z^{\prime})$ (7)
$\displaystyle+\Lambda_{Ep}\sum\limits_{r}{g_{Ep}(z-rl)G_{Ep}(rl,z^{\prime})}.$
The self-energy function (6) is written here through the factor
$\Lambda_{Ep}=\frac{{n_{QD}}}{{L^{2}}}\sum\limits_{{\bf
p}_{1}}{G_{Ep_{1}}(rl,rl)}\left|{\int{d\Delta zu\left({\frac{{\bf p}-{\bf
p}_{1}}{\hbar},\Delta z}\right)}}\right|^{2},$ (8)
which is the same for any QD sheet [we moved $\Sigma_{r}\ldots$ from Eq. (6)
to Eq. (7)]. Instead of Eq. (7), one can determine $G_{Ep}(z,z^{\prime})$ from
Eq. (2) with the free Hamiltonian $\hat{p}^{2}/2m$ and describe the QD sheet
effect adding the boundary conditions
$\displaystyle\frac{{\hbar^{2}}}{2m}\left[\frac{d}{{dz}}G_{Ep}(z,z^{\prime})\right]_{z=rl-0}^{z=rl+0}=\Lambda_{Ep}G_{Ep}(rl,z^{\prime}),$
(9) $\displaystyle
G_{Ep}(z,z^{\prime})\left|{}_{z=rl-0}^{z=rl+0}\right.=0~{}~{}~{}~{}~{}~{}~{}~{}~{}$
at sheet positions $z=rl$. This result was evaluated after acting of the
operator $E+i\lambda-\hat{p}^{2}/2m$ on the integral Dyson equation (4) and
subsequent integration of the intergo-differential equation obtained over the
QD positions $(rl-0,rl+0)$.
Within the second-order Born approximation we use
$G_{Ep}\left(rl,rl\right)\simeq g_{Ep}(0)$ in the self-consistent equation
(8), see Ref. 8 for detais, and the momentum-independent factor $\Lambda_{E}$
in Eq. (9) takes the form
$\Lambda_{E}=\Lambda\left(1+i\sqrt{\frac{E}{\varepsilon_{a}}}\right),~{}~{}~{}~{}\Lambda\equiv\frac{{n_{QD}}}{2}U^{2}\overline{\rho}_{\varepsilon_{a}}.$
(10)
Here we estimate $\Lambda$ for the case of short-range defect within the
Koster-Slater approach 10 and $\overline{\rho}_{E}$ is the 3D density of
states which is taken at the cut-off energy
$\varepsilon_{a}\sim(\pi\hbar/a)^{2}/2m$. Since $E\ll\varepsilon_{a}$, damping
of low-energy states is weak and one can replace the complex boundary
condition (9) by the effective potential energy
$-\Lambda\sum_{r}\delta_{a}(z-rl)$ with $\delta_{a}(\Delta z)$ localized in
the interval $|\Delta z|<a$, so that in the framework of the effective SL
approach $G_{Ep}(z,z^{\prime})$ is governed by the one-dimensional equation:
$\displaystyle\left(E+i\lambda-\varepsilon_{p}-\hat{H}_{\bot}\right)G_{Ep}(z,z^{\prime})=\delta(z-z^{\prime}),$
(11)
$\displaystyle\hat{H}_{\bot}=\frac{\hat{p}_{z}^{2}}{2m}-\Lambda\sum\limits_{r}\delta_{a}\left(z-rl\right)$
with the electron effective mass in the GaAs matrix, $m$. Thus, the Green’s
function is expressed using the standard relation 7 between
$G_{Ep}(z,z^{\prime})$ and the solutions of the eigenstate problem for SL, 11
$\hat{H}_{\bot}\psi_{z}^{(np_{\bot})}=\varepsilon_{np_{\bot}}\psi_{z}^{(np_{\bot})}$.
The last equation determines the dispersion relations
$\varepsilon_{np_{\bot}}$ and the eigenfunctions $\psi_{z}^{(np_{\bot})}$.
Here $p_{\bot}$ is quasimomentum ($|p_{\bot}|<\pi\hbar/l$), $n$ labels
minibands, and the wavefunction takes form
$\psi_{z}^{(np_{\bot})}=\psi_{np_{\bot}}\left({e^{ik_{np_{\bot}}z}-R_{np_{\bot}}e^{-ik_{np_{\bot}}z}}\right),$
(12)
where the reflection coefficient and the normalization factor, $R_{np_{\bot}}$
and $\psi_{np_{\bot}}$, are expressed through $p_{\bot}$ and $k_{np_{\bot}}$.
12 The energy $\varepsilon_{np_{\bot}}=(\hbar k_{np_{\bot}})^{2}/2m$ is
founded from the dispersion equation
$\cos\frac{p_{\bot}l}{\hbar}=\cos k_{np_{\bot}}l-\frac{K}{k_{np_{\bot}}}\sin
k_{np_{\bot}}l,$ (13)
which is written through the characteristic wave vector, $K=\Lambda
m/\hbar^{2}\sim\pi^{3}n_{QD}a/2$, see Ref. 13 for details.
Figure 1: Miniband energy spectra, $E/\varepsilon_{l}$ versus
$p_{\bot}l/\hbar$, of the effective SL determined by Eq. (13) for $Kl=$1 (a),
2 (b), 4 (c), and 8 (d).
The dispersion relations for lower minibands determined by Eq. (13) are shown
in Fig. 1 for dimensionless parameter $Kl$ varied between 1 and 8 when the
transformation from the weakly-coupled SL (if $Kl\leq$2) to the tight-binding
regime of coupling (if $Kl>$4) takes place. The characteristic energy
$\varepsilon_{l}=(\pi\hbar/l)^{2}/2m$ is about 3.2 meV for SL of period $l=$40
nm. For SL formed by InAs QDs embedded by GaAs matrix $Kl\approx$3.1 if
$n_{QD}\simeq 5\times 10^{10}$ cm-2. As a result, minigaps exceed 5 meV for
the tight-binding regime, see Figs. 1c and 1d when dispersion laws are close
to cosine and sine dependencies, for odd and even $n$ respectively. For the
weakly-coupled SLs the dispersion laws are formed by parabolic curves modified
near $p_{\bot}l/\hbar=0,\pi$ with gaps $\sim$1 meV, see Figs. 1a and 1b. In
contrast to SL corresponding to under-barrier tunneling regime, 11 if
$Kl\leq$1.5 one obtains the lowest miniband at finite $p_{\bot}l/\hbar$ only,
as it is shown in Fig. 1a. This is because of absence of solution for Eq. (13)
at $p_{\bot}\to 0$ and $k_{np_{\bot}}l\ll 1$. Such a peculiarity change the
density of states and the edge of mid-IR absorption if $Kl\leq 1.5$, see Figs.
2a and 3a below.
## III Results
Using the model described above, we consider in this section the density of
states in SL formed by QD sheets, and calculate the absorption coefficient
under mid-IR photoexcitation from ground levels of doped QDs into miniband
states of SL.
### III.1 Density of states
The density of states is introduced through the averaged Green’s function by
the standard formula 7
$\displaystyle\rho_{E}=-\frac{2}{\pi L^{3}}{\rm Im}\int d{\bf
r}\left\langle{\cal G}_{E}\left({\bf r},{\bf r}\right)\right\rangle$ (14)
$\displaystyle\simeq\frac{2}{L^{3}}\sum\limits_{np_{\bot}{\bf
p}}\delta(E-\varepsilon_{p}-\varepsilon_{np_{\bot}}),$
where 2 is due to spin degeneracy and $L^{3}$ is the normalization volume. The
lower expression is obtained for the case of negligible damping in Eq. (10)
using of the effective SL approach determined by Eqs. (11)-(13), see the
energy spectra plotted in Fig. 1. The integration of $\delta$-function over
$\bf p$ gives the 2D density of states, $\rho_{2D}$, and after integration of
$\theta$-function over $p_{\bot}$ the density of states should be replaced by
constant if $E$ belongs to $\bar{n}$th gap. In $\bar{n}$th miniband (below
$\bar{n}$th gap), the integral over $p_{\bot}$ should be taken over the
interval $(0,p_{E})$ where $p_{E}$ is found as a root of the equation
$E=\varepsilon_{\bar{n}p_{E}}$. As a result, $\rho_{E}$ takes the form:
$\rho_{E}=\frac{\rho_{2D}}{l}\left\\{\begin{array}[]{*{20}c}\bar{n},&E\subset\bar{n}{\rm
th~{}gap}\\\ \bar{n}-1+p_{E}l/(\pi\hbar),&E\subset\bar{n}{\rm
th~{}band}\end{array}\right.$ (15)
and a shape of $\rho_{E}$ is determined by the gap-induced steps with
transitions between them determined by the miniband dispersion laws.
Figure 2: Normalized density of states $\rho_{E}l/\rho_{2D}$ versus
$E/\varepsilon_{l}$ given by Eq. (15) for the parameters used in panels (a-d)
of Fig. 1. Dotted curve in upper panel corresponds to the 3D density of states
$\propto\sqrt{E}$ if SL effect is negligible, $\Lambda\to 0$.
In Fig. 2 we plot the dimensionless density of states, in units $\rho_{2D}/l$,
for the same parameters as in Fig. 1. For the weak coupling regime, the jump
of $\rho_{E}$ at $E\to 0$ appears due to the cut-off of the lowest miniband at
finite $p_{\bot}l/\hbar$, c. f. Figs. 1a and 2a at $E/\varepsilon_{l}\leq 2$.
With increasing of $Kl$ under transition to the tight-binding regime, the
energy-independent gap contributions to the density of states increase and
$\rho_{E}$ between these steps is transformed from $\propto\sqrt{E}$
dependency shown by dotted curve in Fig. 2a to the arccosine dependencies. In
addition, the bottom of lowest subband is shifted to energies
$\sim\varepsilon_{l}$. Since $\rho_{E}$ is connected directly to the shape of
interband optical spectra, see Ref. 13, the step-like dependencies permit one
to extract $Kl$ value which determine the bandstructure of SL according to Eq.
(13).
Let us compare the energy scale of SL effect, determined by $\varepsilon_{l}$,
and the SL effect due to the wetting layer contribution analyzed in Refs. 10.
For the parameters given at the end of Sect. II, one obtains that the
contribution of QD sheet with $n_{QD}=5\times 10^{10}$ cm-2 is reduced $\sim$2
times in comparison with the wetting layer effect if levels of electron doping
are the same. Thus, an interplay of both mechanisms should take place for
$n_{QD}\geq 10^{11}$ cm-2. For such a case, the interband optical spectra
should be dependent on both the QDs contributions and the wetting layer
contributions.
### III.2 Photoionization
The anisotropic absorption coefficients $\alpha^{||}_{\omega}$ and
$\alpha^{\bot}_{\omega}$ are determined from the general Kubo formula as
follows:
$\displaystyle\alpha^{||,\bot}_{\omega}=\frac{8(\pi
e)^{2}}{\sqrt{\epsilon}c\omega
L^{3}}\sum\limits_{\delta\delta^{\prime}}\left[f(\varepsilon_{\delta})-f(\varepsilon_{\delta}+\hbar\omega)\right]$
$\displaystyle\times\left|(\delta|{\bf e}_{||,\bot}\cdot\hat{\bf
v}|\delta^{\prime})\right|^{2}\delta\left(\varepsilon_{\delta}-\varepsilon_{\delta^{\prime}}+\hbar\omega\right),$
(16)
where $\epsilon$ is the dielectric permittivity of the host semiconductor and
the matrix element $\left|(\delta|{\bf e}_{||,\bot}\cdot\hat{\bf
v}|\delta^{\prime})\right|^{2}$ corresponds to transitions between $\delta$\-
and $\delta^{\prime}$-states of energies $\varepsilon_{\delta}$ and
$\varepsilon_{\delta^{\prime}}$ under radiation with polarization orts ${\bf
e}_{||,\bot}$. We use the equilibrium distributions
$f(\varepsilon_{\delta})\to 1$ and $f(\varepsilon_{\delta}+\hbar\omega)\to 0$
because the only localized states are populated at temperatures lower the
binding energy $|E_{0}|$. Due to the in-plane isotropy of the problem, we
separate the cases of $s$\- and $p$-polarized radiation corresponding to the
polarization orts ${\bf e}_{\|}$ and ${\bf e}_{z}$. Neglecting the overlap
between QD states and taking the ground state wave functions $\Psi_{P}$ in the
momentum representation ($\bf P$ is 3D momentum) we transform Eq. (16) into:
$\displaystyle\left|{\begin{array}[]{*{20}c}\alpha_{\omega}^{\|}\\\
\alpha_{\omega}^{\bot}\\\ \end{array}}\right|=-\frac{4\pi
e^{2}}{\sqrt{\epsilon}c\omega m^{2}L^{9}}\sum\limits_{{\bf
PP}^{\prime}}\Psi_{P}\Psi_{P^{\prime}}^{*}\int d{\bf r}\int d{\bf r}^{\prime}$
(19) $\displaystyle\times e^{i({\bf Pr}-{\bf P}^{\prime}{\bf
r}^{\prime})/\hbar}\left|\begin{array}[]{*{20}c}({\bf e}_{\|}{\bf P})({\bf
e}_{\|}{\bf P}^{\prime})\\\ {\hat{p}}_{z}{\hat{p}}_{z^{\prime}}^{+}\\\
\end{array}\right|K_{\Delta{\bf p},E_{0}+\hbar\omega}({\bf r}^{\prime},{\bf
r}).~{}~{}~{}$ (22)
The contribution of miniband states are described here through the average of
the exact Green’s function ${\cal G}_{E}({\bf r}^{\prime},{\bf r})$ with the
exponential factor corresponding to random QD positions (here $\Delta{\bf
p}\equiv{\bf P}-{\bf P^{\prime}}$):
$K_{\Delta{\bf p},E}({\bf r}^{\prime},{\bf
r})=\left\langle{\sum\limits_{rk}{e^{i\Delta{\bf pR}_{rk}/\hbar}{\rm Im}{\cal
G}_{E}({\bf r}^{\prime},{\bf r})}}\right\rangle$ (23)
which is analyzed in the Appendix. Within the low-order approach, the
correlation function (18) takes the form:
$K_{\Delta{\bf p},E}({\bf r}^{\prime},{\bf r})\approx
N_{QD}\frac{L}{l}\delta_{\Delta{\bf p},0}{\rm Im}G_{E}({\bf r}^{\prime},{\bf
r}),$ (24)
where $N_{QD}L/l$ is the total number of QDs in the normalization volume
$L^{3}$ and the averaged Green’s function $G_{E}({\bf r}^{\prime},{\bf r})$
was considered in Sect. II.
Figure 3: (Color online) Spectral dependencies of dimensionless absorption
coefficients determined by Eqs. (20) and (21) for the same conditions as in
Figs. 1 and 2. Solid and dashed curves correspond to the perpendicular and
parallel polarizations, respectively. Dotted curves correspond to the case
$\Lambda\to 0$, when SL effect is negligible.
Using the ground state wave function $\Psi_{P}$ written in the Koster-Slater
approach 10 and neglecting the damping correction in Eq. (10) we transform
Eq. (17) as follows
$\displaystyle\left|{\begin{array}[]{*{20}c}{\alpha_{\omega}^{||}}\\\
{\alpha_{\omega}^{\bot}}\\\ \end{array}}\right|=\frac{{(2\pi
e)^{2}n_{QD}}}{{\sqrt{\epsilon}c\omega m^{2}lL^{3}}}\sum\limits_{{\bf
p}p_{\bot}}{\left|\Psi_{P}\right|}^{2}\left|{\begin{array}[]{*{20}c}p^{2}/2\\\
{p_{\bot}^{2}}\end{array}}\right|~{}~{}~{}$ (29)
$\displaystyle\times\sum_{n\bar{p}_{\bot}}\left|{\frac{2}{l}\int\limits_{-l}^{l}{dze^{-ip_{\bot}z/\hbar}\psi_{z}^{(n\bar{p}_{\bot})}}}\right|^{2}\delta\left({\hbar\Delta\omega-\varepsilon_{n\bar{p}_{\bot}}-\varepsilon_{p}}\right).$
Here ${\bf P}\equiv({\bf p},p_{\bot})$ and we have replaced
$G_{Ep}(z^{\prime},z)$ from Eq. (19) using the wave function (12). In the
expressions for $\alpha_{\omega}^{||,\bot}$ integrals over $\bf p$-plane and
over $z$ are taken analytically and the spectral dependencies of IR absorption
are obtained after the double numerical integrations over the transverse
momenta $p_{\bot}$ and $\bar{p}_{\bot}$. The dimensionless spectral
dependencies are plotted in Fig. 3 for the same conditions as in Figs. 1 and
2. The characteristic absorption $\alpha_{0}$ is given by
$\alpha_{0}=\frac{(4e)^{2}n_{QD}}{c\sqrt{\epsilon
m|E_{0}|/2}}\left(\frac{\varepsilon_{l}}{E_{0}}\right)^{2}$ (30)
and $\alpha_{0}\sim$3 cm-1 for the above listed parameters. Thus, for the
maximal absorption, when $\hbar\Delta\omega/\varepsilon_{l}\sim$20 - 30 or
$\hbar\Delta\omega\sim|E_{0}|$, one obtains $\alpha_{max}\sim$45 cm-1. Since
$\alpha_{\omega}^{||,\bot}\propto n_{QD}/l^{4}$, the maximal absorption
increases up to $\alpha_{max}\geq 10^{3}$ cm-1 if $n_{QD}>10^{11}$ cm-2 and
$l\simeq$20 nm; an approximation of low QD concentration remains valid for
such a set of parameters. Further increase of $\alpha_{max}$ is possible in
the case of heavily doped SL, with a few electrons captured in QD.
Anisotropy of absorption is about 20% without any strong dependency on
effective potential, c. f. Figs. 3a - 3d where parameter $Kl$ varies from 1 to
8. Peculiarities of miniband spectra are visible clearly in
$\alpha_{\omega}^{\bot}$ starting from $Kl\geq$2 while $\alpha_{\omega}^{\|}$
does not show any peculiarities at the edges of minibands. This is due to
different selection rules for transverse and longitudinally polarized
excitations: in the last case, transitions are forbidden at edges of minibands
and the spectral dependencies rmain smooth. In addition, Fig. 1a shows a jump
of $\alpha_{\omega}^{\bot}$ at $\hbar\Delta\omega=0$ which is similar to the
jump of the density of states in Fig. 2a (we do not consider IR transitions
into shallow underbarrier states at $\hbar\Delta\omega<0$). In Figs. 3b-d,
shifts of absorption edges to finite $\hbar\Delta\omega>0$ take place due to
lower miniband shifts, see Figs. 1b-d and 2b-d.
## IV Conclusions
In summary, we have developed the theory of the superlattice formed by
periodically placed quantum dot sheets. It was found that the damping due to
random in-plane positions of dots is weak and effect of the sheets on
electronic states can be described using of the effective boundary conditions.
Within this approach we have demonstrated that the miniband density of states,
which describes the interband absorption, and spectra of mid-IR
photoexcitation of doped quantum dots into minibands strongly depend on
parameters of quantum dot sheets. Visible anisotropy of the absorption
coefficient is also found, with transverse absorption which is strongly
modulated by the miniband spectrum of SL.
Now we discuss the main assumptions in the calculations performed. We
restricted ourselves by the vicinity of $c$-band using the effective-mass
approach in Eq. (1) and in further consideration of the photoionization
process. In order to describe the energy intervals comparable to the gap, one
needs to use the multi-band $\bf kp$-Hamiltonian for more detailed description
of QD states. 13 We consider the case of low QD concentration ($n_{QD}/l\sim
10^{15}$ cm-3 in our numerical estimates) and the electron-electron
interaction effect on the energy spectrum; thus, the IR-absorption should be
weak. Numerical estimates for the SL parameters were performed here based on
simplified description of QD as an isotropic short-range defect with the
binding energy corresponding to typical QD. This approach gives approximate SL
parameters only and a more precise description should be based on a numerical
solution of the self-consistent Dyson equation taking into account a real
potential of QD. 1 ; 13 Because parameters of QD sheet (materials,
concentration, and shape of QD) can be very different, such a consideration
should be performed for different specific cases (e.g. for Ge/Si-based or
AIIBVI-based QD sheets, for review see Ref. 16).
To conclude, we believe that the results obtained will stimulate an
investigation of underbarrier vertical transport of carriers in order to
verify SL effect on electronic properties of structures formed by QD sheets.
The spectral and polarization dependencies of the mid-IR photoexcitation are
convenient for direct measurements because the valence band states are not
essential. These results should be important for description of different
devices utilizing periodical QD sheet structures.
## ACKNOWLEDGMENT
This work was supported by the AFOSR.
Figure 4: Self-consistent Dyson equation for averaged Green’s function
$G_{Ep}(z,z^{\prime})$ and the self-energy function
$\Sigma_{Ep}\left(z_{1},z_{2}\right)$ shown in upper and lower lines,
respectively.
*
## Appendix A
In order to estimate the corrections beyond the effective potential approach
used in Eqs. (9)-(11) we consider here the method of calculations in more
details. Using the ${\bf p}z$-representation, one obtains the self-consistent
Dyson equation (4) for the averaged Green’s function $G_{Ep}(z,z^{\prime})$
shown by a bold line as it is plotted in Fig. 4. Within the second-order Born
approximation, we use the free Green’s function in the self-energy function
(6) given by the first diagram of the set for $\Sigma_{Ep}$ shown in the lower
line of Fig. 4. The next corrections in this set can be neglected under the
standard condition 7
$E\gg|\Sigma_{Ep}|\simeq\Lambda$ (31)
and we arrive to Eq. (7) using the free Green’s function in $\Sigma_{Ep}$
determined by Eq. (6).
Figure 5: Diagram expansion for correlation function ${\cal K}_{\Delta{\bf
p}E}({\bf r},{\bf r^{\prime}})$ written through the diagram set for vertex
part shown in lower line.
More complicate consideration is necessary for the correlation function
$K_{\Delta{\bf p}E}({\bf r},{\bf r^{\prime}})$ appearing in Eq. (17) because
of the random factor $\exp(i\Delta{\bf pR}_{rk}/\hbar)$ describing positions
of QDs. Instead Eq. (18) it is convenient to consider the generalized
expression
${\cal K}_{\Delta{\bf p},E}({\bf r}^{\prime},{\bf
r})=\left\langle{\sum\limits_{rk}{e^{i\Delta{\bf pR}_{rk}/\hbar}{\cal
G}_{E}({\bf r}^{\prime},{\bf r})}}\right\rangle$ (32)
which is shown in Fig. 5. Here a dotted curve corresponds to the averaged
factor
$\left\langle{\sum\limits_{r_{1}k_{1}r_{2}k_{2}}{\exp\left({-\frac{i}{\hbar}\Delta{\bf
p}\cdot{\bf R}_{r_{1}k_{1}}}\right)}{\rm}u\left({{\bf r}-{\bf
R}_{r_{2}k_{2}}}\right)}\right\rangle,$ (33)
while dashed curves in Figs. 4 and 5 stand for the paired QD potentials. After
summation over all reducible diagrams, $K_{\Delta{\bf p}E}({\bf r},{\bf
r^{\prime}})$ is written through the averaged Green’s function and the vertex
part which is given by the set shown in the lower line of Fig. 5 with the
initial vortex determined from (A.3) as follows
$\displaystyle\gamma_{\Delta{\bf p}}({\bf r}_{1},{\bf
r}_{2})=n_{QD}\sum_{r}u\left(-\frac{\Delta{\bf p}}{\hbar},z_{1}-rl\right)$
(34) $\displaystyle\times e^{-\frac{i}{\hbar}(\Delta{\bf
px}_{1}+rp_{\bot}l)}\delta({\bf r}_{1}-{\bf r}_{2}).$
The first correction to Eq. (19) appears, if we use (A.4), as the vertex part
in the diagram expansion for correlation function shown in Fig. 5. Performing
straightforward calculations under the condition (A.1), one obtains that this
correction and next contributions are negligible in comparison with Eq. (19).
## References
* (1) D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures (J. Wiley and Sons, New York, 1999); A.A. Lagatsky, C.G. Leburn, C.T.A. Brown, W. Sibbett, S.A. Zolotovskaya, and E.U. Rafailov, Progr. in Quantum Electronics, 34, 1 (2010).
* (2) A.V. Barve, S.J. Lee, S.K. Noh, and S. Krishna, Laser and Photonics Reviews, 4, 738 (2010); A. Rogalski, J. Antoszewski, and L. Faraone, J. of Appl. Phys. 105, 091101 (2009); S. D. Gunapala, S. V. Bandara, S. B. Rafol, and D. Z. Ting, Semiconductors and Semimetals 84, 59 (2011).
* (3) A. J. Nozik, M. C. Beard, J. M. Luther, M. Law, R. J. Ellingson, and J. C. Johnson, Chem. Rev. 110, 6873 (2010); E. U. Rafailov, M. A. Cataluna, and W. Sibbett, Nature Photonics 1, 395 (2007).
* (4) M. Steslicka, R. Kucharczyk, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, and S.G. Davison, Surf. Sci. Rep. 47, 93 (2002).
* (5) D. L. Nika, E. P. Pokatilov, Q. Shao, and A. A. Balandin, Phys. Rev. B 76, 125417 (2007); O. L. Lazarenkova and A. A. Balandin, Phys. Rev. B 66, 245319 (2002).
* (6) M. Buljan, U. V. Desnica, M. Ivanda, N. Radic, P. Dubcek, G. Drazic, K. Salamon, S. Bernstorff, and V. Holy, Phys. Rev. B 79, 035310 (2009); D. Grutzmacher, T. Fromherz, C. Dais, J. Stangl, E. Muller, Y. Ekinci, H. H. Solak, H. Sigg, R. T. Lechner, E. Wintersberger, S. Birner, V. Holy, and G. Bauer, Nano Letters 7, 3150 (2007); M. V. Artemyev, A. I. Bibik, L. I. Gurinovich, S. V. Gaponenko, and U. Woggon, Phys. Rev. B 60, 1504 (1999).
* (7) S. Kiravittaya, A. Rastelli, and O. G. Schmidt, Rep. Prog. Phys. 72, 046502 (2009).
* (8) G. D. Mahan, _Many-Particle Physics_ (Plenum Press, N.Y., 1990); F. T. Vasko and O. E. Raichev, _Quantum Kinetic Theory and Applications_ (Springer, New York, 2005).
* (9) G. Lukovsky, Solid State Commun. 3, 299 (1965); A. M. Stoneham, Theory of Defects in Solids, (Oxford Univ. Press, Oxford, 2001).
* (10) R. Pickenhain, H. Schmidt, and V. Gottschalch, J. Appl. Phys. 88, 948 (2000); F. T. Vasko and V. V. Mitin, arXiv:1110.0744.
* (11) S. D. Gunapala, B. F. Levine, and N. Chand, J. of Appl. Phys. 70, 305 (1991); M. Helm, Semicond. Sci. Technol. 10, 557 (1995).
* (12) Using the Koster-Slater approach 8 one can find the binding energy of the short-range defect, $E_{0}$, from the equation $1+U\overline{\rho}_{\varepsilon_{a}}=\pi U\overline{\rho}_{E_{0}}/2$ and the normailzed wave function of the ground state $\Psi_{P}$ used in Eqs. (17) and (20) is given by $\Psi_{P}=\frac{2\sqrt{|E_{0}|/2\rho_{|E_{0}|}}}{|E_{0}|+\varepsilon_{P}}.$
* (13) M. Herman, Semiconductor Superlattices, (Academie-Verlag, Berlin, 1986); F. T. Vasko and A. Kuznetsov, Electronic States and Optical Transitions in Semiconductor Heterostructures, (Springer, New York, 1998).
* (14) According to Ref. 13, the explicite expressions for the reflection coefficient, $R_{np_{\bot}}$, and the normalization factor, $\left|\psi_{np_{\bot}}\right|^{2}$, are written as $R_{np_{\bot}}=\frac{\exp(ip_{\bot}l/\hbar)-\exp(ik_{np_{\bot}}l)}{\exp(ip_{\bot}l/\hbar)-\exp(-ik_{np_{\bot}}l)}$ and $\displaystyle 2l\left|\psi_{np_{\bot}}\right|^{2}=[1-\cos(p_{\bot}l/\hbar+k_{np_{\bot}}l)]~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle\times\left\\{1-\frac{\sin(2k_{np_{\bot}}l)}{2k_{np_{\bot}}l}-\cos\frac{p_{\bot}l}{\hbar}\left[\cos(k_{np_{\bot}}l)-\frac{\sin(k_{np_{\bot}}l)}{k_{np_{\bot}}l}\right]\right\\}^{-1}.$
* (15) O. Stier, M. Grundmann, and D. Bimberg, Phys. Rev. B 59, 5688 (1999); F. Boxberg and J. Tulkki, Rep. Prog. Phys. 70, 1425 (2007).
* (16) A.D. Yoffe, Adv. in Physics, 50, 1 (2001).
|
arxiv-papers
| 2012-06-23T17:45:31 |
2024-09-04T02:49:32.125896
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. T. Vasko and V. V. Mitin",
"submitter": "Fedir Vasko T",
"url": "https://arxiv.org/abs/1206.5418"
}
|
1206.5433
|
# On the families of $q$-Euler numbers and polynomials and their applications
Serkan Aracı University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com.tr ,
Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr and
Hassan Jolany School of Mathematics, Statistics and Computer Science,
University of Tehran, Iran hassan.jolany@khayam.ut.ac.ir
###### Abstract.
In the present paper, we investigate special generalized $q$-Euler numbers and
polynomials. Some earlier results of T. Kim in terms of $q$-Euler polynomials
with weight $\alpha$ can be deduced. For presentation of our formulas we apply
the method of generating function and $p$-adic $q$-integral representation on
$\mathbb{Z}_{p}$. We summarize our results as follows. In section 2, by using
combinatorial techniques we present two formulas for $q$-Euler numbers with
weight $\alpha$. In section 3, we derive distribution formula (Multiplication
Theorem) for Dirichlet type of $q$-Euler numbers and polynomials with weight
$\alpha$. Moreover we define partial Dirichlet type zeta function and
Dirichlet $q$-$L$-function, and obtain some interesting combinatorial
identities for interpolating our new definitions. In addition, we derive
behavior of the Dirichlet type of $q$-Euler $L$-function with weight $\alpha$,
$\mathcal{L}_{q}^{\chi}\left(s,x\mid\alpha\right)$ at $s=0$. Furthermore by
using second kind stirling numbers, we obtain an explicit formula for
Dirichlet type $q$-Euler numbers with weight $\alpha$, and $\beta$. Moreover a
novel formula for $q$-Euler-Zeta function with weight $\alpha$ in terms of
nested series of $\widetilde{\zeta}_{E,q}\left(n\mid\alpha\right)$ is derived
. In section 4, by introducing $p$-adic Dirichlet type of $q$-Euler measure
with weight $\alpha$, and $\beta$, we obtain some combinatorial relations,
which interpolate our previous results. In section 5, which is the main
section of our paper. As an application, we introduce a novel concept of
dynamics of the zeros of analytically continued $q$-Euler polynomials with
weight $\alpha$.
###### Key words and phrases:
Euler numbers and polynomials, $q$-Euler numbers and polynomials, weighted$\
q$-Euler numbers and polynomials, weighted $q$-Euler-Zeta function, $p$-adic
$q$-integral on $\mathbb{Z}_{p}$.
###### 2000 Mathematics Subject Classification:
Primary 05A10, 11B65; Secondary 11B68, 11B73.
## 1\. Introduction
In this paper, we use notations like $\mathbb{N}$, $\mathbb{R}$ and
$\mathbb{C}$, where $\mathbb{N}$ denotes the set of natural numbers,
$\mathbb{R}$ denotes the field of real numbers and $\mathbb{C}$ also denotes
the set of complex numbers. When one talks of $q$-extension, $q$ is variously
considered as an indeterminate, a complex number or a $p$-adic number.
Throughout this paper, we will assume that $q\in\mathbb{C}$ with
$\left|q\right|<1$. The $q$-symbol $\left[x:q\right]$ denotes as
$\left[x:q\right]=\frac{q^{x}-1}{q-1}\text{.}$
Originally, $q$-Euler numbers and polynomials were introduced by L. Carlitz in
1948 and gave properties of this polynomials (see [20], [21]). Recently,
Taekyun Kim, by using $p$-adic $q$-integral in the $p$-adic integers ring, has
added a weight to $q$-Bernoulli numbers and polynomials and gave surprising
and fascinating identities of them (for details, see [8]). The $q$-Bernoulli
numbers and polynomials with weight $\alpha$ are related to weighted
$q$-Bernstein polynomials which is shown by Kim (for details, see [7]). These
polynomials have surprising properties in Analytic Numbers Theory and in
$p$-adic analysis, especially, in Mathematical physics. Ryoo also constructed
$q$-Euler numbers and polynomials with weight $\alpha$ and introduced some
properties of $q$-Euler numbers and polynomials with weight $\alpha$ in ”A
note on the weighted $q$-Euler numbers and polynomials with weight $\alpha$,
Advanced Studies Contemporary Mathematics 21 (2011), No. 1, 47-54.”
Analytic continuation of $q$-Euler numbers and polynomials was investigated by
Kim in [1]. In previous paper, Araci $et$ $al$. also considered analytic
continuation of weighted $q$-Genocchi numbers and polynomials and introduced
some interesting ideas (for detail, see [26]). In this article, we also
specify analytic continuation of weighted $q$-Euler numbers and polynomials.
Also, we give some interesting identities by using generating function of
Ryoo’s weighted $q$-Euler polynomials.
Because in the literature of our present paper we use of $p$-adic Arithmetic
and $p$-adic numbers, so we need to give short review on $p$-adic numbers.
Historically the $p$-adic numbers were introduced by K. Hensel in 1908 in his
book Theorie der algebraíschen Zahlen, Leipzig, 1908 (for more informations on
this subject, see [19]).
Let $p$ be a prime number, fixed once and for all. If $x$ is any rational
number other than 0, we can write $x$ in the form $x=p^{n}\frac{a}{b}$ , where
$a,b\in\mathbb{Z}$ are relatively prime to $p$ and $n\in\mathbb{Z}$. We now
define
$|x|_{p}=p^{-n}\text{ and }|0|_{p}=0,\text{and }ord_{p}(x)=n\text{ and
}ord_{p}(0)=+\infty\text{.}$
They satisfy the following properties,
$\displaystyle|x|_{p}\geq 0,\text{ }|x|_{p}=0\text{ if and only if }x=0$
$\displaystyle|x+y|_{p}\leq\max\\{|x|_{p},|y|_{p}\\}\text{ }(\text{the strong
triangle inequality})$ with
$\displaystyle|x+y|_{p}=\max\\{|x|_{p},|y|_{p}\\}\text{ if
}|x|_{p}\neq|y|_{p}(\text{the isosceles triangle principle})$
$\displaystyle|x.y|_{p}=|x|_{p}.|y|_{p}$
$|x|_{p}$ is called the $p$-adic valuation. Ostrowski proved that each
nontrivial valuation on the field of rational numbers is equivalent either to
the absolute value function or to some $p$-adic valuation. The completion of
the field $\mathbb{Q}$ of rational numbers with respect to the p-adic
valuation $|.|_{p}$ is called the field of $p$-adic numbers and will be
denoted $\mathbb{Q}_{p}$. The set
$\mathbb{Z}_{p}=\\{x\in\mathbb{Q}_{p}\mid|x|_{p}\leq 1\\}$
is the ring of $p$-adic integers. It can be easily proved that each $p$-adic
number $x$ can be written in the form
$x=\sum_{n=-f}^{\infty}a_{n}p^{n}$
where each $a_{n}$ is one of the elements $0,1,\cdot\cdot\cdot,p-1$, and
$f\in\mathbb{Z}$. This is called the Hensel representation of $p$-adic
numbers. With this representation, one obtain for $x\in\mathbb{Q}_{p}$,
$ord_{p}(x)=+\infty$ if $a_{i}=0$ for all $i$ and
$ord_{p}(x)=min\\{s|a_{s}\neq 0\\}$, otherwise.
Moreover we can write
$|x|_{p}=p^{-ord_{p}(x)}\text{.}$
## 2\. Properties of the $q$-Euler Numbers and polynomials with weight
$\alpha$
For $\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, the
weighted $q$-Euler polynomials are given as:
For $x\in\mathbb{C}$,
(2.1)
$\sum_{n=0}^{\infty}\widetilde{E}_{n,q}\left(x\mid\alpha\right)\frac{t^{n}}{n!}=\left[2:q\right]\sum_{n=0}^{\infty}\left(-1\right)^{n}q^{n}e^{t\left[n+x:q^{\alpha}\right]}\text{.}$
As a special case, substituting $x=0$ into (2.1),
$\widetilde{E}_{n,q}\left(0\mid\alpha\right):=\widetilde{E}_{n,q}\left(\alpha\right)$
are called weighted $q$-Euler numbers. By (2.1), we readily derive the
following
(2.2)
$\widetilde{E}_{n,q}\left(x\mid\alpha\right)=\frac{\left[2:q\right]}{\left[\alpha:q\right]^{n}\left(1-q\right)^{n}}\sum_{l=0}^{n}\binom{n}{l}\left(-1\right)^{l}\frac{q^{\alpha
lx}}{1+q^{\alpha l+1}}\text{,}$
where $\binom{n}{l}$ is the binomial coefficient. By expression (2.1), we see
that
(2.3) $\widetilde{E}_{n,q}\left(x\mid\alpha\right)=q^{-\alpha
x}\left(q^{\alpha
x}\widetilde{E}_{q}\left(\alpha\right)+\left[x:q^{\alpha}\right]\right)^{n}\text{,}$
with the usual convention of replacing
$\left(\widetilde{E}_{q}\left(\alpha\right)\right)^{n}$ by
$\widetilde{E}_{n,q}\left(\alpha\right)$ (for details, see [14]).
Let $\widetilde{H}_{q}^{\left(\alpha\right)}\left(x,t\right)$ be the
generating function of weighted $q$-Euler polynomials as follows:
(2.4)
$\widetilde{H}_{q}^{\left(\alpha\right)}\left(x,t\right)=\sum_{n=0}^{\infty}\widetilde{E}_{n,q}\left(x\mid\alpha\right)\frac{t^{n}}{n!}\text{.}$
Then, we easily notice that
(2.5)
$\widetilde{H}_{q}^{\left(\alpha\right)}\left(x,t\right)=\left[2:q\right]\sum_{n=0}^{\infty}\left(-1\right)^{n}q^{n}e^{t\left[n+x:q^{\alpha}\right]}\text{.}$
From expressions (2.4) and (2.5), we procure the followings:
For $k$ (=even) and
$n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, we have
(2.6)
$\widetilde{E}_{n,q}\left(\alpha\right)-q^{k}\widetilde{E}_{n,q}\left(k\mid\alpha\right)=\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{l}\left[l:q^{\alpha}\right]^{n}\text{.}$
For $k$ (=odd) and
$n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, we have
(2.7)
$q^{k}\widetilde{E}_{n,q}\left(k\mid\alpha\right)+\widetilde{E}_{n,q}\left(\alpha\right)=\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{l}\left[l:q^{\alpha}\right]^{n}\text{.}$
Via Eq. (2.5), we easily obtain the following:
(2.8) $\widetilde{E}_{n,q}\left(x\mid\alpha\right)=q^{-\alpha
x}\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\widetilde{E}_{k,q}\left(\alpha\right)\left[x:q^{\alpha}\right]^{n-k}\text{.}$
From (2.6)-(2.8), we get the following theorem.
###### Theorem 1.
Let $k$ be even positive integer. Then we have
(2.9)
$\displaystyle\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{l}\left[l:q^{\alpha}\right]^{n}$
$\displaystyle=\left(1-q^{k\left(1-\alpha+\alpha
n\right)}\right)\widetilde{E}_{n,q}\left(\alpha\right)-q^{k\left(1-\alpha\right)}\sum_{l=0}^{n-1}\binom{n}{l}q^{\alpha
lk}\widetilde{E}_{l,q}\left(\alpha\right)\left[k:q^{\alpha}\right]^{n-l}\text{.}$
###### Theorem 2.
Let $k$ be an odd positive integer. Then, we procure the following
(2.10)
$\displaystyle\left[2:q\right]\sum_{l=0}^{k-1}\left(-1\right)^{l}q^{l}\left[l:q^{\alpha}\right]^{n}$
$\displaystyle=\left(q^{k\left(1-\alpha+\alpha
n\right)}+1\right)\widetilde{E}_{n,q}\left(\alpha\right)+q^{k\left(1-\alpha\right)}\sum_{l=0}^{n-1}\binom{n}{l}q^{\alpha
lk}\widetilde{E}_{l,q}\left(\alpha\right)\left[k:q^{\alpha}\right]^{n-l}\text{.}$
## 3\. $q$-Euler-Zeta function with weight $\alpha$
The familiar Euler polynomials are defined by
(3.1)
$\frac{2}{e^{t}+1}e^{xt}=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!},\text{
}\left|t\right|<\pi\text{ cf. \cite[cite]{[\@@bibref{}{kim 4}{}{}]}.}$
For $s\in\mathbb{C}$, $x\in\mathbb{R}$ with $0\leq x<1$, Euler-Zeta function
is given by
(3.2)
$\zeta_{E}\left(s,x\right)=2\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}}{\left(m+x\right)^{s}}\text{,
}$
and
(3.3)
$\zeta_{E}\left(s\right)=\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}}{m^{s}}\text{.}$
By expressions (3.1), (3.2) and (3.3), Euler-Zeta functions are related to the
Euler numbers as follows:
$\zeta_{E}\left(-n\right)=E_{n}\text{.}$
Moreover, it is simple to see
$\zeta_{E}\left(-n,x\right)=E_{n}\left(x\right)\text{.}$
The weighted $q$-Euler Hurwitz-Zeta type function is defined by
$\widetilde{\zeta}_{E,q}\left(s,x\mid\alpha\right)=\left[2:q\right]\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m+x:q^{\alpha}\right]^{s}}\text{
.}$
Similarly, weighted $q$-Euler-Zeta function is given by
$\widetilde{\zeta}_{E,q}\left(s\mid\alpha\right)=\left[2:q\right]\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m:q^{\alpha}\right]^{s}}\text{.}$
For $n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$, we
have
$\widetilde{\zeta}_{E,q}\left(-n\mid\alpha\right)=\widetilde{E}_{n,q}\left(\alpha\right)\text{
(see \cite[cite]{[\@@bibref{}{Ryoo}{}{}]}).}$
We now consider the function $\widetilde{E}_{q}\left(n:\alpha\right)$ as the
analytic continuation of weighted $q$-Euler numbers. All the weighted
$q$-Euler numbers agree with $\widetilde{E}_{q}\left(n:\alpha\right)$, the
analytic continuation of weighted $q$-Euler numbers evaluated at $n$. For
$n\geq 0$,
$\widetilde{E}_{q}\left(n:\alpha\right)=\widetilde{E}_{n,q}\left(\alpha\right)$.
We can now state $\widetilde{E}{\acute{}}_{q}\left(s:\alpha\right)$ in terms
of $\widetilde{\zeta}{\acute{}}_{E,q}\left(s\mid\alpha\right)$, the derivative
of $\widetilde{\zeta}_{E,q}\left(s:\alpha\right)$
$\widetilde{E}_{q}\left(s:\alpha\right)=\widetilde{\zeta}_{E,q}\left(-s\mid\alpha\right)\text{,
}\widetilde{E}{\acute{}}_{q}\left(s:\alpha\right)=\widetilde{\zeta}{\acute{}}_{E,q}\left(-s\mid\alpha\right)\text{.}$
For $n,\alpha\in\mathbb{N}\mathop{\textstyle\bigcup}\left\\{0\right\\}$
$\text{
}\widetilde{E}{\acute{}}_{q}\left(2n:\alpha\right)=\widetilde{\zeta}{\acute{}}_{E,q}\left(-2n\mid\alpha\right)\text{.}$
This is suitable for the differential of the functional equation and so
supports the coherence of $\widetilde{E}_{q}\left(s:\alpha\right)$ and
$\widetilde{E}{\acute{}}_{q}\left(s:\alpha\right)$ with
$\widetilde{E}_{n,q}\left(\alpha\right)$ and
$\widetilde{\zeta}_{E,q}\left(s\mid\alpha\right)$. From the analytic
continuation of weighted $q$-Euler numbers, we derive as follows:
$\widetilde{E}_{q}\left(s:\alpha\right)=\widetilde{\zeta}_{E,q}\left(-s\mid\alpha\right)\text{
and
}\widetilde{E}_{q}\left(-s:\alpha\right)=\widetilde{\zeta}_{E,q}\left(s\mid\alpha\right)\text{.}$
Moreover, we derive the following:
For $n\in\mathbb{N}$
$\widetilde{E}_{-n,q}\left(\alpha\right)=\widetilde{E}_{q}\left(-n:\alpha\right)=\widetilde{\zeta}_{E,q}\left(n\mid\alpha\right)\text{.}$
The curve $\widetilde{E}_{q}\left(s:a\right)$ runs through the points
$\widetilde{E}_{-s,q}\left(\alpha\right)$ and grows $\sim n$ asymptotically
$\left(-n\right)\rightarrow-\infty$. The curve
$\widetilde{E}_{q}\left(s:a\right)$ runs through the point
$\widetilde{E}_{q}\left(-n:a\right)$. Then, we procure the following:
$\displaystyle\lim_{n\rightarrow\infty}\widetilde{E}_{q}\left(-n:\alpha\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\widetilde{\zeta}_{E,q}\left(n\mid\alpha\right)=\lim_{n\rightarrow\infty}\left(\left[2:q\right]\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m:q^{\alpha}\right]^{n}}\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\left(-q\left[2:q\right]+\left[2:q\right]\sum_{m=2}^{\infty}\frac{\left(-1\right)^{m}q^{m}}{\left[m:q^{\alpha}\right]^{n}}\right)=-q^{2}\left[2:q^{-1}\right]\text{.}$
From this, we note that
$\widetilde{E}_{q}\left(-n:\alpha\right)=\widetilde{\zeta}_{E,q}\left(n\mid\alpha\right)\mapsto\widetilde{E}_{q}\left(-s:\alpha\right)=\widetilde{\zeta}_{E,q}\left(s\mid\alpha\right)\text{.}$
Notations: Assume that $p$ be a fixed odd prime number. Throughout this paper
we use the following notations. By $\mathbb{Z}_{p}$ we denote the ring of
$p$-adic rational integers, $\mathbb{Q}$ denotes the field of rational
numbers, $\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$. The $p$-adic absolute
value is defined by $\left|p\right|_{p}=\frac{1}{p}$. In this paper we assume
$\left|q-1\right|_{p}<1$ as an indeterminate. Let
$UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly differentiable
functions on $\mathbb{Z}_{p}$. For a positive integer $d$ with
$\left(d,p\right)=1$, set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{d}=\lim_{\overleftarrow{n}}\mathbb{Z}/dp^{n}\mathbb{Z}\text{,}$
$\displaystyle X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$
and
$a+dp^{n}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{n}\right)\right\\}\text{,}$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{n}$.
Firstly, for introducing fermionic $p$-adic $q$-integral, we need some basic
information which we state here. A measure on $\mathbb{Z}_{p}$ with values in
a $p$-adic Banach space $B$ is a continuous linear map
$f\mapsto\int f(x)\mu=\int_{\mathbb{Z}_{p}}f(x)\mu(x)$
from $C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$, (continuous function on
$\mathbb{Z}_{p}$ ) to $B$. We know that the set of locally constant functions
from $\mathbb{Z}_{p}$ to $\mathbb{Q}_{p}$ is dense in
$C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$ so.
Explicitly, for all $f\in C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$, the locally
constant functions
$f_{n}=\sum_{i=0}^{p^{n}-1}f(i)1_{i+p^{n}\mathbb{Z}_{p}}\rightarrow f\text{ in
}C^{0}$
Now, set
$\mu(i+p^{n}\mathbb{Z}_{p})=\int_{\mathbb{Z}_{p}}1_{i+p^{n}\mathbb{Z}_{p}}\mu$.
Then $\int_{\mathbb{Z}_{p}}f\mu$, is given by the following Riemannian sum
$\int_{\mathbb{Z}_{p}}f\mu=\lim_{n\rightarrow\infty}\sum_{i=0}^{p^{n}-1}f(i)\mu{(i+p^{n}\mathbb{Z}_{p})}$
T. Kim introduced $\mu$ as follows:
$\mu_{-q}(a+p^{n}\mathbb{Z}_{p})=\frac{(-q)^{a}}{[p^{n}]_{-q}}$
So, for $f\in UD\left(\mathbb{Z}_{p}\right)$, the fermionic $p$-adic
$q$-integral on $\mathbb{Z}_{p}$ is defined by Kim as follows:
$\displaystyle I_{-q}\left(f\right)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}f\left(\eta\right)d\mu_{-q}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}:-q\right]}\sum_{\eta=0}^{p^{n}-1}q^{\eta}f\left(\eta\right)\left(-1\right)^{\eta}\text{.}$
Let $\chi$ be the Dirichlet’s character with conductor $d$ (=
odd)$\in\mathbb{N}$ and let us take
$f\left(\eta\right)=\chi\left(\eta\right)\left[x+\eta:q^{\alpha}\right]^{n}$,
then we define Dirichlet’s type of $q$-Euler numbers and polynomials with
weight $\alpha$ as follows:
(3.5)
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha\right)=\int_{\mathbb{Z}_{p}}\chi\left(\eta\right)\left[x+\eta:q^{\alpha}\right]^{n}d\mu_{-q}\left(\eta\right)\text{.}$
From (3), we have the following well-known equality.
(3.6)
$q^{d}\int_{\mathbb{Z}_{p}}f\left(\eta+d\right)d\mu_{-q}\left(\eta\right)+\left(-1\right)^{d-1}\int_{\mathbb{Z}_{p}}f\left(\eta\right)d\mu_{-q}\left(\eta\right)=\left[2:q\right]\sum_{l=0}^{d-1}q^{l}\left(-1\right)^{d-1-l}f\left(l\right)\text{.}$
By expressions of (3.5) and (3.6), for $d$ $($=odd$)$ positive integer, we
have the following
(3.7)
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha\right)=\frac{\left[2:q\right]}{\left[\alpha:q\right]^{n}\left(1-q\right)^{n}}\sum_{j=0}^{n}\binom{n}{j}\left(-1\right)^{j}q^{\alpha
jx}\sum_{l=0}^{d-1}\chi\left(l\right)q^{l}\left(-1\right)^{l}\frac{q^{\alpha
jl}}{q^{\left(\alpha j+1\right)d}+1}\text{.}$
Substituting $x=0$ in (3.7),
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(0\mid\alpha\right):=$
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(\alpha\right)$ are called Dirichlet
type of $q$-Euler numbers with weight $\alpha$. That is, we easily derive the
following
(3.8)
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(\alpha\right)=\frac{\left[2:q\right]}{\left(1-q^{\alpha}\right)^{n}}\sum_{j=0}^{n}\binom{n}{j}\left(-1\right)^{j}\sum_{l=0}^{d-1}\chi\left(l\right)\left(-1\right)^{l}\frac{q^{\left(\alpha
j+1\right)l}}{q^{\left(\alpha j+1\right)d}+1}\text{.}$
###### Theorem 3.
Let $\chi$ be Dirichlet’s character and for any $n\in\mathbb{N}^{\ast}$. Then
we have
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha\right)=\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha\right)\left[x:q^{\alpha}\right]^{n-k}\text{.}$
###### Proof.
By using (3.5) and (3.8), becomes
$\displaystyle\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(\eta\right)\left[x+\eta:q^{\alpha}\right]^{n}d\mu_{-q}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(\eta\right)\left(\left[x:q^{\alpha}\right]+q^{\alpha
x}\left[\eta:q^{\alpha}\right]\right)^{n}d\mu_{-q}\left(\eta\right)\text{.}$
From this, by using binomial theorem, we can write the following
$\displaystyle\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\left[x:q^{\alpha}\right]^{n-k}\int_{\mathbb{Z}_{p}}\chi\left(\eta\right)\left[\eta:q^{\alpha}\right]^{k}d\mu_{-q}\left(\eta\right)$
$\displaystyle=$ $\displaystyle\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\left[x:q^{\alpha}\right]^{n-k}\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha\right)\text{.}$
Thus, we complete the proof of the theorem.
###### Theorem 4.
The following identity
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(dx\mid\alpha\right)=\frac{\left[d:q^{\alpha}\right]}{\left[d:-q\right]}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{a}\widetilde{E}_{n,q^{d}}\left(x+\frac{a}{d}\mid\alpha\right)$
holds true.
###### Proof.
To prove this, we compute as follows:
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[dp^{n}:-q\right]}\sum_{y=0}^{dp^{n}-1}\left(-q\right)^{y}\chi\left(y\right)\left[x+y:q^{\alpha}\right]^{n}$
$\displaystyle=$
$\displaystyle\frac{1}{\left[d:-q\right]}\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}:-q^{d}\right]}\sum_{y=0}^{p^{n}-1}\sum_{a=0}^{d-1}\left(-q\right)^{a+dy}\chi\left(a+dy\right)\left[x+a+dy:q^{\alpha}\right]^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[d:q^{\alpha}\right]}{\left[d:-q\right]}\sum_{a=0}^{d-1}\left(-q\right)^{a}\chi\left(a\right)\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{-q^{d}}}\sum_{y=0}^{p^{n}-1}\left(-q\right)^{dy}\left[\frac{x+a}{d}+y:q^{d\alpha}\right]^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[d:q^{\alpha}\right]}{\left[d:-q\right]}\sum_{a=0}^{d-1}\left(-q\right)^{a}\chi\left(a\right)\widetilde{E}_{n,q^{d}}\left(\frac{x+a}{d}\mid\alpha\right)\text{.}$
So, we get the desired result and proof is complete.
By (3.7), we procure the following:
(3.9)
$\sum_{n=0}^{\infty}\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha\right)\frac{t^{n}}{n!}=\left[2:q\right]\sum_{m=0}^{\infty}q^{m}\chi\left(m\right)\left(-1\right)^{m}e^{t\left[x+m:q^{\alpha}\right]}\text{.}$
By applying derivative operator of order $k$ as
$\frac{d^{k}}{dt^{k}}\mid_{t=0}$, we have the following
$\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(x\mid\alpha\right)=\left[2:q\right]\sum_{m=0}^{\infty}q^{m}\chi\left(m\right)\left(-1\right)^{m}\left[x+m:q^{\alpha}\right]^{k}\text{.}$
That is, we can define Dirichlet $q$-$L$-function as follows:
(3.10)
$\mathcal{L}_{q}^{\chi}\left(s,x\mid\alpha\right)=\left[2:q\right]\sum_{m=0}^{\infty}\frac{q^{m}\chi\left(m\right)\left(-1\right)^{m}}{\left[x+m:q^{\alpha}\right]^{s}}.$
###### Lemma 1.
The following equality holds true:
$\mathcal{L}_{q}^{\chi}\left(-k,x\mid\alpha\right)=\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(x\mid\alpha\right)\text{.}$
###### Proof.
Substituting $s=-k$ into (3.10), we arrive at the desired result.
Now also, we define partial Dirichlet type zeta function as follows:
(3.11)
$\mathcal{H}_{q}^{\chi}\left(s:x:a:F\mid\alpha\right)=\left[2:q\right]\sum_{m\equiv
a\left(\mathop{\mathrm{m}od}F\right)}^{\infty}\frac{q^{m}\chi\left(m\right)\left(-1\right)^{m}}{\left[x+m:q^{\alpha}\right]^{s}}\text{.}$
Now, for interpolating partial Dirichlet type zeta function, we rewrite it in
terms of weighted $q$-Euler Hurwitz-Zeta function as follows.
###### Theorem 5.
For $F\equiv 1(\mathop{\mathrm{m}od}2)$, then the following equality holds
true:
(3.12)
$\mathcal{H}_{q}^{\chi}\left(s:x:a:F\mid\alpha\right)=\frac{\left[2:q\right]q^{a}\left(-1\right)^{a}\chi\left(a\right)}{\left[F:q^{\alpha}\right]^{s}}\widetilde{\zeta}_{E,q^{F}}\left(s,\frac{x+a}{F}\mid\alpha\right)\text{.}$
###### Proof.
By expression of (3.11), we compute as follows:
$\displaystyle\mathcal{H}_{q}^{\chi}\left(s:x:a:F\mid\alpha\right)$
$\displaystyle=$ $\displaystyle\left[2:q\right]\sum_{m\equiv
a\left(\mathop{\mathrm{m}od}F\right)}^{\infty}\frac{q^{m}\chi\left(m\right)\left(-1\right)^{m}}{\left[x+m:q^{\alpha}\right]^{s}}$
$\displaystyle=$
$\displaystyle\left[2:q\right]\sum_{m=0}^{\infty}\frac{q^{mF+a}\chi\left(mF+a\right)\left(-1\right)^{mF+a}}{\left[x+mF+a:q^{\alpha}\right]^{s}}$
$\displaystyle=$
$\displaystyle\frac{\left[2:q\right]q^{a}\left(-1\right)^{a}\chi\left(a\right)}{\left[F:q^{\alpha}\right]^{s}}\sum_{m=0}^{\infty}\frac{\left(q^{F}\right)^{m}\left(-1\right)^{m}}{\left[\frac{x+a}{F}+m:q^{F\alpha}\right]^{s}}\text{.}$
Thus, we arrive at the desired result.
If we put $s=-n$ into (3.12), then, we can write partial Dirichlet type Zeta
function in terms of weighted $q$-Euler numbers
(3.13)
$\mathcal{H}_{q}^{\chi}\left(-n:x:a:F\mid\alpha\right)=\left[2:q\right]q^{a}\left(-1\right)^{a}\chi\left(a\right)\left[F:q^{\alpha}\right]^{n}\widetilde{E}_{n,q^{F}}\left(\frac{x+a}{F}\mid\alpha\right)\text{.}$
###### Theorem 6.
The following identity
(3.14)
$\mathcal{H}_{q}^{\chi}\left(s:x:a:F\mid\alpha\right)=\frac{\left[2:q\right]q^{a}\left(-1\right)^{a}\chi\left(a\right)}{\left[x+a:q^{\alpha}\right]^{s}}\sum_{k=0}^{\infty}q^{\alpha
k\left(x+a\right)}\binom{-s}{k}\left(\frac{\left[F:q^{\alpha}\right]}{\left[x+a:q^{\alpha}\right]}\right)^{k}\widetilde{E}_{k,q^{F}}$
holds true.
###### Proof.
Taking $n=-s$ into (3.13) and some manipulation by using combinatorial
techniques, we can reach to the proof of the theorem.
If we substitute $s=-n$ into (3.14), then, (3.14) reduces to (3.13). Now also,
we give the following theorem.
###### Theorem 7.
Let $d\equiv 1(\mathop{\mathrm{m}od}2)$, then, we have
$\mathcal{L}_{q}^{\chi}\left(s,x\mid\alpha\right)=\frac{\left[2:q\right]}{\left[2:q^{d}\right]\left[d:q^{\alpha}\right]^{s}}\sum_{l=0}^{d-1}\left(-1\right)^{l}\chi\left(l\right)q^{l}\widetilde{\zeta}_{E,q^{d}}\left(s,\frac{x+l}{d}\mid\alpha\right)\text{.}$
###### Proof.
By using (3.10), we compute as follows:
$\displaystyle\mathcal{L}_{q}^{\chi}\left(s,x\mid\alpha\right)$
$\displaystyle=$
$\displaystyle\left[2:q\right]\sum_{m=0}^{\infty}\sum_{l=0}^{d-1}\frac{q^{l+md}\chi\left(l+md\right)\left(-1\right)^{l+md}}{\left[x+l+md:q^{\alpha}\right]^{s}}$
$\displaystyle=$
$\displaystyle\frac{\left[2:q\right]}{\left[2:q^{d}\right]\left[d:q^{\alpha}\right]^{s}}\sum_{l=0}^{d-1}\left(-1\right)^{l}\chi\left(l\right)q^{l}\widetilde{\zeta}_{E,q^{d}}\left(s,\frac{x+l}{d}\mid\alpha\right)\text{.}$
Thus, we prove the above theorem.
By means of the above theorem and (3.12), we have the following corollary.
###### Corollary 1.
The following equality
(3.15)
$\mathcal{L}_{q}^{\chi}\left(s,x\mid\alpha\right)=\frac{1}{\left[2:q^{d}\right]}\sum_{l=0}^{d-1}\mathcal{H}_{q}^{\chi}\left(s:x:l:d\mid\alpha\right)$
holds true.
By (3.14) and (3.15), we have the following corollary.
###### Corollary 2.
The following nice identity
$\mathcal{L}_{q}^{\chi}\left(s,x\mid\alpha\right)=\frac{\left[2:q\right]}{\left[2:q^{d}\right]}\sum_{l=0}^{d-1}\frac{\left(-1\right)^{l}\chi\left(l\right)}{\left[x+l:q^{\alpha}\right]^{s}}\sum_{k=0}^{\infty}q^{\alpha
k\left(x+l\right)+l}\binom{-s}{k}\widetilde{E}_{k,q^{F}}\left(\frac{\left[F:q^{\alpha}\right]}{\left[x+l:q^{\alpha}\right]}\right)^{k}$
holds true.
By using (3.13) and (3.15), we derive behavior of the Dirichlet type of
$q$-Euler $L$-function with weight $\alpha$ at $s=0$ as follows:
###### Theorem 8.
The following identity holds true:
$\mathcal{L}_{q}^{\chi}\left(0,x\mid\alpha\right)=\frac{\left[2:q\right]}{\left[2:q^{d}\right]}\sum_{l=0}^{d-1}\left(-1\right)^{l}\chi\left(l\right)q^{l}\text{.}$
Now also, we define Dirichlet type $q$-Euler polynomials with weight $\alpha$
and $\beta$ with the following expression
(3.16)
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha:\beta\right)=\int_{\mathbb{Z}_{p}}\left[x+\eta:q^{\alpha}\right]^{n}\chi\left(\eta\right)d\mu_{-q^{\beta}}\left(\eta\right)\text{.}$
Taking $x=0$ into (3.16), we have
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(0\mid\alpha:\beta\right):=\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(\alpha:\beta\right)$
which is called Dirichlet type $q$-Euler numbers. Then, by (3.16), we easily
derive the following
(3.17)
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha:\beta\right)=\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
lx}\widetilde{\mathcal{E}}_{l,q}^{\chi}\left(\alpha:\beta\right)\left[x:q^{\alpha}\right]^{n-l}\text{.}$
###### Theorem 9.
The following equality holds true:
$\displaystyle\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha:\beta\right)$
$\displaystyle=$
$\displaystyle\sum_{l=0}^{n}\binom{n}{l}\widetilde{\mathcal{E}}_{l,q}^{\chi}\left(\alpha:\beta\right)\sum_{j=0}^{l}\binom{l}{j}\left(q^{\alpha}-1\right)^{j}\left(n-l+j\right)!\left(-1\right)^{n-l+j}$
$\displaystyle\times\sum_{m,n=0}^{\infty}\binom{n-l+j+m-1}{m}\alpha^{n}q^{\alpha
m}\left(\log
q\right)^{n}\mathcal{S}\left(n,n-l+j\right)\frac{x^{n}}{n!}\text{.}$
###### Proof.
To prove this, by applying (3.17), we easily discover the following assertion
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha:\beta\right)=\sum_{l=0}^{n}\binom{n}{l}\widetilde{\mathcal{E}}_{l,q}^{\chi}\left(\alpha:\beta\right)\sum_{j=0}^{l}\binom{l}{j}\left(q^{\alpha}-1\right)^{j}\left[x:q^{\alpha}\right]^{n-l+j}\text{.}$
The Second kind Stirling numbers are defined by means of the following
generating function.
(3.18)
$\sum_{n=0}^{\infty}\mathcal{S}\left(n,k\right)\frac{t^{n}}{n!}=\frac{\left(e^{t}-1\right)^{k}}{k!}$
(for details on this subject, see [12]). $t$ replace by $\alpha x\log q$ in
(3.18), then, we easily derive the following
(3.19)
$\left[x:q^{\alpha}\right]^{k}=k!\left(-1\right)^{k}\sum_{m,n=0}^{\infty}\binom{k+m-1}{m}\alpha^{n}q^{\alpha
m}\left(\log q\right)^{n}\mathcal{S}\left(n,k\right)\frac{x^{n}}{n!}\text{,}$
where $\sum_{m,n=0}^{\infty}=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}$. Thus, by
(3.18) and (3.19), we get the desired result and proof is complete.
###### Theorem 10.
The following equality
$\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha:\beta\right)=\frac{\left[d:q^{\alpha}\right]^{n}}{\left[d:-q^{\beta}\right]}\sum_{a=0}^{d-1}\left(-q\right)^{a}\chi\left(a\right)\widetilde{E}_{n,q^{d}}\left(\frac{x+a}{d}\mid\alpha:\beta\right)$
holds true.
###### Proof.
By applying the $p$-adic integral representation on the Dirichlet type of
$q$-Euler polynomials with weight $\alpha$ and $\beta$, we compute as follows:
$\displaystyle\widetilde{\mathcal{E}}_{n,q}^{\chi}\left(x\mid\alpha:\beta\right)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(\eta\right)\left[x+\eta:q^{\alpha}\right]^{n}d\mu_{-q^{\beta}}\left(\eta\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{1}{\left[dp^{n}:-q^{\beta}\right]}\sum_{y=0}^{dp^{n}-1}\left(-q\right)^{y}\chi\left(y\right)\left[x+y:q^{\alpha}\right]$
$\displaystyle=$
$\displaystyle\frac{1}{\left[d:-q^{\beta}\right]}\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}:-q^{d\beta}\right]}\sum_{y=0}^{p^{n}-1}\sum_{a=0}^{d-1}\left(-q\right)^{a+dy}\chi\left(a+dy\right)\left[x+a+dy:q^{\alpha}\right]^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[d:q^{\alpha}\right]^{n}}{\left[d:-q^{\beta}\right]}\sum_{a=0}^{d-1}\left(-q\right)^{a}\chi\left(a\right)\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}:-q^{d}\right]}\sum_{y=0}^{p^{n}-1}\left(-q^{d\beta}\right)^{y}\left[\frac{x+a}{d}+y:q^{d\alpha}\right]^{n}$
$\displaystyle=$
$\displaystyle\frac{\left[d:q^{\alpha}\right]^{n}}{\left[d:-q^{\beta}\right]}\sum_{a=0}^{d-1}\left(-q\right)^{a}\chi\left(a\right)\widetilde{E}_{n,q^{d}}\left(\frac{x+a}{d}\mid\alpha:\beta\right)\text{.}$
Here, $\widetilde{E}_{n,q^{d}}\left(\frac{x+a}{d}\mid\alpha:\beta\right)$ is
defined by Ryoo in [16], which is called $q$-Euler polynomials with weight
$\left(\alpha,\beta\right)$. As a result, we have the proof of the theorem.
## 4\. On $p$-adic Dirichlet type of $q$-Euler measure with weight $\alpha$
and $\beta$
Now, we introduce a map
$\mu_{k,q}^{\left(\alpha,\beta\right)}\left(a+p^{n}\mathbb{Z}_{p}\right)$ on
the balls in $\mathbb{Z}_{p}$ as follows:
(4.1)
$\mathcal{\mu}_{k,q}^{\left(\alpha,\beta\right)}\left(a+p^{n}\mathbb{Z}_{p}\mid\chi\right)=\frac{\left[p^{n}:q^{\alpha}\right]^{k}}{\left[p^{n}:-q^{\beta}\right]}\chi\left(a\right)\left(-1\right)^{a}q^{a}f_{k,p^{n}}\left(\frac{\left\\{a\right\\}_{n}}{p^{n}}\mid\alpha:\beta\right)$
where $\left\\{a\right\\}_{n}\equiv a\left(\mathop{\mathrm{m}od}p^{n}\right)$.
###### Theorem 11.
Let $\alpha$, $k\in\mathbb{N}$. Then we specify that
$\mathcal{\mu}_{k,q}^{\left(\alpha,\beta\right)}$ is $p$-adic measure on
$\mathbb{Z}_{p}$ if and only if
$f_{k,q^{p^{n}}}\left(\frac{a}{p^{n}}\mid\alpha:\beta\right)=\frac{\left[p^{n}:q^{p\alpha}\right]^{k}}{\left[p^{n}:-q^{p\beta}\right]}\sum_{b=0}^{p-1}\left(-1\right)^{b}q^{bp^{n}}f_{k,\left(q^{p^{n}}\right)^{p}}\left(\frac{\frac{a}{p^{n}}+b}{p}\mid\alpha:\beta\right)\text{.}$
###### Proof.
By similar method in [23], we can state the proof of this theorem. Therefore,
we omit it.
We now set as follows:
(4.2)
$f_{k,q^{p^{n}}}\left(\frac{a}{p^{n}}\mid\alpha:\beta\right)=\widetilde{E}_{n,q^{p^{n}}}\left(\frac{a}{p^{n}}\mid\alpha:\beta\right)\text{.}$
From (4.1) and (4.2), we easily see
(4.3)
$\mathcal{\mu}_{k,q}^{\left(\alpha,\beta\right)}\left(a+p^{n}\mathbb{Z}_{p}\mid\chi\right)=\frac{\left[p^{n}:q^{\alpha}\right]^{k}}{\left[p^{n}:-q^{\beta}\right]}\chi\left(a\right)\left(-1\right)^{a}q^{a}\widetilde{E}_{k,q^{p^{n}}}\left(\frac{a}{p^{n}}\mid\alpha:\beta\right)\text{.}$
By (2.1) and (4.3), then, we have the following theorem.
###### Theorem 12.
For $\alpha$,$k\in\mathbb{N}$, we have
$\int_{X}d\mu_{k,q}^{\left(\alpha,\beta\right)}\left(x\mid\chi\right)=\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha:\beta\right)\text{.}$
###### Proof.
By using combinatorial techniques, we compute as follows:
$\displaystyle\int_{X}d\mu_{k,q}^{\left(\alpha,\beta\right)}\left(x\mid\chi\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{\left[dp^{n}:q^{\alpha}\right]^{k}}{\left[dp^{n}:-q^{\beta}\right]}\sum_{x=0}^{dp^{n}-1}\chi\left(x\right)\left(-1\right)^{x}q^{x}\widetilde{E}_{k,q^{dp^{n}}}\left(\frac{x}{dp^{n}}\mid\alpha:\beta\right)$
$\displaystyle=$
$\displaystyle\frac{\left[d:q^{\alpha}\right]^{k}}{\left[d:-q^{\beta}\right]}\sum_{a=0}^{d-1}\left(-1\right)^{a}q^{a}\chi\left(a\right)\lim_{n\rightarrow\infty}\frac{\left[p^{n}:q^{d\alpha}\right]^{k}}{\left[p^{n}:-q^{d\beta}\right]}\sum_{x=0}^{p^{n}-1}\left(-1\right)^{x}q^{dx}\widetilde{E}_{k,\left(q^{d}\right)^{p^{n}}}\left(\frac{\frac{a}{d}+x}{p^{n}}\mid\alpha:\beta\right)$
$\displaystyle=$
$\displaystyle\frac{\left[d:q^{\alpha}\right]^{k}}{\left[d:-q^{\beta}\right]}\sum_{a=0}^{d-1}\left(-1\right)^{a}q^{a}\chi\left(a\right)\widetilde{E}_{n,q^{d}}\left(\frac{a}{d}\mid\alpha:\beta\right)$
so,we obtain the desired result.
###### Theorem 13.
For any $k\in\mathbb{N}$, we get
$\int_{pX}d\mu_{k,q}^{\left(\alpha,\beta\right)}\left(x\mid\chi\right)=\chi\left(p\right)\frac{\left[p:q^{\alpha}\right]}{\left[p:-q^{\beta}\right]}\widetilde{\mathcal{E}}_{k,q^{p}}^{\chi}\left(\alpha:\beta\right)\text{.}$
###### Proof.
From (2.1) and (4.3), we derive the followings assertions
$\displaystyle\int_{pX}d\mu_{k,q}^{\left(\alpha,\beta\right)}\left(x\mid\chi\right)$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\frac{\left[dp^{n+1}:q^{\alpha}\right]^{k}}{\left[dp^{n+1}:-q^{\beta}\right]}\sum_{x=0}^{dp^{n}-1}\chi\left(px\right)\left(-1\right)^{px}q^{px}\widetilde{E}_{k,q^{dp^{n}}}\left(\frac{px}{dp^{n+1}}\mid\alpha:\beta\right)$
$\displaystyle=$
$\displaystyle\chi\left(p\right)\frac{\left[p:q^{\alpha}\right]}{\left[p:-q^{\beta}\right]}\frac{\left[d:q^{p\alpha}\right]^{k}}{\left[d:-q^{p\beta}\right]}\sum_{a=0}^{d-1}\left\\{\begin{array}[]{c}\left(-1\right)^{a}q^{pa}\chi\left(a\right)\lim_{n\rightarrow\infty}\frac{\left[p^{n}:q^{dp\alpha}\right]^{k}}{\left[p^{n}:-q^{pd\beta}\right]}\\\
\times\sum_{x=0}^{p^{n}-1}\left(-1\right)^{x}q^{pdx}\widetilde{E}_{k,\left(q^{d}\right)^{p^{n}}}\left(\frac{dp\left(\frac{a}{d}+x\right)}{pdp^{n}}\mid\alpha:\beta\right)\end{array}\right\\}$
$\displaystyle=$
$\displaystyle\chi\left(p\right)\frac{\left[p:q^{\alpha}\right]}{\left[p:-q^{\beta}\right]}\frac{\left[d:q^{p\alpha}\right]^{k}}{\left[d:-q^{p\beta}\right]}\sum_{a=0}^{d-1}\left(-1\right)^{a}q^{pa}\chi\left(a\right)\widetilde{E}_{k,q^{pd}}\left(\frac{a}{d}\mid\alpha:\beta\right)$
$\displaystyle=$
$\displaystyle\chi\left(p\right)\frac{\left[p:q^{\alpha}\right]}{\left[p:-q^{\beta}\right]}\widetilde{\mathcal{E}}_{k,q^{p}}^{\chi}\left(\alpha:\beta\right)\text{.}$
Thus, we get the desired result and proof is complete.
By the same method which we used in above theorem, by a little bit
manipulations we can state the following theorem.
###### Theorem 14.
For $c\left(\neq 1\right)\in X^{\ast}$
, we have
$\int_{pX}d\mu_{k,q^{\frac{1}{c}}}^{\left(\alpha,\beta\right)}\left(cx\mid\chi\right)=\chi\left(\frac{p}{c}\right)\frac{\left[p:q^{\frac{\alpha}{c}}\right]}{\left[p:\left(-q^{\beta}\right)^{\frac{1}{c}}\right]}\widetilde{\mathcal{E}}_{k,q^{\frac{p}{c}}}^{\chi}\left(\alpha:\beta\right)\text{.}$
###### Theorem 15.
For $c\left(\neq 1\right)\in X^{\ast}$
, we have
$\int_{X}d\mu_{k,q^{\frac{1}{c}}}^{\left(\alpha,\beta\right)}\left(cx\mid\chi\right)=\chi\left(\frac{1}{c}\right)\widetilde{\mathcal{E}}_{k,q^{\frac{1}{c}}}^{\chi}\left(\alpha:\beta\right)\text{.}$
We can define the following identity:
$\mathcal{\mu}_{k,c,q}^{\left(\alpha,\beta\right)}\left(U\mid\chi\right)=\mathcal{\mu}_{k,q}^{\left(\alpha,\beta\right)}\left(U\mid\chi\right)-c^{-1}\frac{\left[c^{-1}:q^{\alpha}\right]^{k}}{\left[c^{-1}:-q^{\beta}\right]}\mathcal{\mu}_{k,q^{\frac{1}{c}}}^{\left(\alpha,\beta\right)}\left(cU\mid\chi\right)$
here $U$ is any compact open subset of $\mathbb{Z}_{p}$, it can be written as
a finite disjoint union of sets
$U=\overset{k}{\underset{j=1}{\cup}}\left(a_{j}+p^{n}\mathbb{Z}_{p}\right),$
where $n\in\mathbb{N}$ and $a_{1},a_{2},...,a_{k}\in\mathbb{Z}$ with $0\leq
a_{i}<p^{n}$ for $i=1,2,...,k$.
###### Theorem 16.
For $c\left(\neq 1\right)\in X^{\ast}$
, we procure the following
$\int_{X^{\ast}}d\mu_{k,c,q}^{\left(\alpha,\beta\right)}\left(cx\mid\chi\right)=(1-\chi^{p})\left(1-c^{-1}\chi^{c^{-1}}\right)\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha:\beta\right)$
where the operator $\chi^{y}:=\chi^{y,k,\alpha;q}$ on $f\left(q\right)$ is
defined by
$\chi^{y}f\left(q\right)=\chi^{y,k,\alpha;q}f\left(q\right)=\frac{\left[y:q^{\alpha}\right]}{\left[y:-q^{\beta}\right]}\chi\left(y\right)f\left(q^{y}\right)$
That is, we can write
$\chi^{x,k,\alpha;q}\circ\chi^{y,k,\alpha;q}f\left(q\right)=\chi^{xy,k,\alpha;q}f\left(q\right)=\chi^{xy}f\left(q\right)\text{.}$
###### Proof.
To prove this, we assume that
$f\left(q\right)=\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha:\beta\right)$.
Then, we get
$\left\\{\begin{array}[]{c}\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha:\beta\right)-\chi\left(p\right)\frac{\left[p:q^{\alpha}\right]}{\left[p:-q^{\beta}\right]}\widetilde{\mathcal{E}}_{k,q^{p}}^{\chi}\left(\alpha:\beta\right)-c^{-1}\frac{\left[c^{-1}:q^{\alpha}\right]^{k}}{\left[c^{-1}:-q^{\beta}\right]}\\\
\times\chi\left(\frac{1}{c}\right)\widetilde{\mathcal{E}}_{k,q^{\frac{1}{c}}}^{\chi}\left(\alpha:\beta\right)+\chi\left(\frac{p}{c}\right)\frac{\left[p:q^{\frac{\alpha}{c}}\right]}{\left[p:\left(-q^{\beta}\right)^{\frac{1}{c}}\right]}\widetilde{\mathcal{E}}_{k,q^{\frac{p}{c}}}^{\chi}\left(\alpha:\beta\right)\end{array}\right\\}$
From this, we derive the following
$(1-\chi^{p})\left(1-c^{-1}\chi^{c^{-1}}\right)\widetilde{\mathcal{E}}_{k,q}^{\chi}\left(\alpha:\beta\right)$
Then, we complete the proof of theorem.
## 5\.
Analytic continuation of $q$-Euler Polynomials with weight $\alpha$
The concept of analytic continuation just means enlarging the domain without
giving up the property of being differentiable, i.e. holomorphic or
meromorphic. More precisely Let $f_{1}$ and $f_{2}$ be analytic functions on
domains $\Omega_{1}$ and $\Omega_{2},$ respectively, and suppose that the
intersection $\Omega_{1}\cap\Omega_{2}$ is not empty and that$f_{1}=f_{2}$ on
$\Omega_{1}\cap\Omega_{2}$. Then $f_{2}$ is called an analytic continuation of
$f_{1}$ to $\Omega_{2}$, and vice versa . Moreover, if it exists, the analytic
continuation of $f_{1}$ to $\Omega_{2}$ is unique. By means of analytic
continuation, starting from a representation of a function by any one power
series, any number of other power series can be found which together define
the value of the function at all points of the domain. Furthermore, any point
can be reached from a point without passing through a singularity of the
function, and the aggregate of all the power series thus obtained constitutes
the analytic expression of the function. So we are ready to state analytic
continuation of $q$-Euler polynomials with weight $\alpha$ as follows.
For coherence with the redefinition of
$\widetilde{E}_{n,q}\left(\alpha\right)=\widetilde{E}_{q}\left(n:\alpha\right)$,
we have
$\widetilde{E}_{n,q}\left(x\mid\alpha\right)=q^{-\alpha
x}\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\widetilde{E}_{k,q}\left(\alpha\right)\left[x:q^{\alpha}\right]^{n-k}\text{.}$
Let $\Gamma\left(s\right)$ be Euler-gamma function. Then the analytic
continuation can be get as
$\displaystyle n$ $\displaystyle\mapsto$ $\displaystyle s\in\mathbb{R}\text{,
}x\mapsto w\in\mathbb{C}\text{,}$
$\displaystyle\widetilde{E}_{n,q}\left(\alpha\right)$ $\displaystyle\mapsto$
$\displaystyle\widetilde{E}_{q}\left(k+s-\left[s\right]:\alpha\right)=\widetilde{\zeta}_{E,q}\left(-\left(k+s-\left[s\right]\right)\mid\alpha\right)\text{,}$
$\displaystyle\binom{n}{k}$ $\displaystyle=$
$\displaystyle\frac{\Gamma\left(n+1\right)}{\Gamma\left(n-k+1\right)\Gamma\left(k+1\right)}\mapsto\frac{\Gamma\left(s+1\right)}{\Gamma\left(1+k+\left(s-\left[s\right]\right)\right)\Gamma\left(1+\left[s\right]-k\right)}$
$\displaystyle\widetilde{E}_{s,q}\left(w\mid\alpha\right)$
$\displaystyle\mapsto$
$\displaystyle\widetilde{E}_{q}\left(s,w:\alpha\right)=q^{-\alpha
w}\sum_{k=-1}^{\left[s\right]}\frac{\Gamma\left(s+1\right)\widetilde{E}_{q}\left(k+\left(s-\left[s\right]\right):\alpha\right)q^{\alpha
w\left(k+\left(s-\left[s\right]\right)\right)}}{\Gamma\left(1+k+\left(s-\left[s\right]\right)\right)\Gamma\left(1+\left[s\right]-k\right)}\left[w:q^{\alpha}\right]^{\left[s\right]-k}$
$\displaystyle=$ $\displaystyle q^{-\alpha
w}\sum_{k=0}^{\left[s\right]+1}\frac{\Gamma\left(s+1\right)\widetilde{E}_{q}\left(-1+k+\left(s-\left[s\right]\right):\alpha\right)q^{\alpha
w\left(k-1+\left(s-\left[s\right]\right)\right)}}{\Gamma\left(k+\left(s-\left[s\right]\right)\right)\Gamma\left(2+\left[s\right]-k\right)}\left[w:q^{\alpha}\right]^{\left[s\right]+1-k}\text{.}$
Here $\left[s\right]$ gives the integer part of s, and so $s-\left[s\right]$
gives the fractional part.
Deformation of the curve $\widetilde{E}_{q}\left(1,w:\alpha\right)$ into the
curve of $\widetilde{E}_{q}\left(2,w:\alpha\right)$ is by means of the real
analytic cotinuation $\widetilde{E}_{q}\left(s,w:\alpha\right)$, $1\leq s\leq
2$, $-0.5\leq w\leq 0.5$.
###### Acknowledgement 1.
The third author would like to thank the Association SARA-GHU à Marseille for
their hospitality during his stay there, when the work for this paper was done
and dedicated this paper to Neda Agha-Soltan.
## References
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* [2] T. Kim, On explicit formulas of $p$-adic $q$-$L$-functions, Kyushu J. Math. 43 (1994) 73–86.
* [3] T. Kim, On $p$-adic interpolating function for $q$-Euler numbers and its derivatives, J. Math. Anal. Appl. 339 (2008) 598–608.
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* [5] T. Kim, On a $q$-analogue of the $p$-adic $\log$ gamma functions and related integrals, Journal of Number Theory 76 (1999) 320-329.
* [6] T. Kim, On the analogs of Euler numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, J. Math. Anal. Appl. 331 (2007) 779–792.
* [7] T. Kim, A. Bayad, Y-H. Kim, A study on the $p$-adic $q$-integrals representation on $\mathbb{Z}_{p}$ associated with the weighted $q$-Bernstein and $q$-Bernoulli polynomials, Journal of Inequalities and Applications, Article ID 513821, 8 pages, doi:10.1155/2011/513821.
* [8] T. Kim, On the weighted $q$-Bernoulli numbers and polynomials, Advanced Studies in Contemporary Mathematics 21 (2011), no. 2, p. 207-215.
* [9] T. Kim, $q$-Volkenborn integration, Russ. J. Math. phys. 9 (2002), page 288-299.
* [10] T. Kim, An invariant $p$-adic $q$-integrals on $\mathbb{Z}_{p}$, Applied Mathematics Letters, vol. 21, pp. 105-108, 2008.
* [11] T. Kim, $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals, J. Nonlinear Math. Phys., 14 (2007), No. 1, 15–27.
* [12] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-Stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russ. J. Math. Phys., 16 (2009), No.4, 484–491.
* [13] T. Kim, Analytic continuation of multiple $q$-zeta functions and their values at negative integers, Russ. J. Math. Phys. 11 (1) (2004) 71-76.
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* [22] J. J. Seo and S. Araci, Some Properties involving the higher order $q$-Genocchi numbers and polynomials with weight ($\alpha,\beta$) via the $p$-adic $q$-integral on $\mathbb{Z}_{p}$, Journal of Chungcheon Mathematical Society, Vol 24, No. 4, December 2011\.
* [23] H. Jolany, S. Araci, M. Acikgoz and J. J. Seo, A note on the generalized $q$-Genocchi measure with weight $\alpha$, Bol. Soc. Paran. Math. v. 31 1 (2013): pp. 17-27 (in press).
* [24] S. Araci, D. Erdal, J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [25] S. Araci, M. Acikgoz and J. J. Seo, A study on the weighted $q$-Genocchi numbers and polynomials with their interpolation function, Honam Mathematical J. 34 (2012), No. 1, pp. 11-18.
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* [27] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, accepted in Bulletin of the Korean Mathematical Society.
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|
arxiv-papers
| 2012-06-23T19:24:41 |
2024-09-04T02:49:32.133472
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Hassan Jolany",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1206.5433"
}
|
1206.5461
|
# Some new identities on the ($h,q$)-Genocchi numbers and polynomials with
weight $\alpha$
S. Araci University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , E. Cetin Uludag
University, Faculty of Arts and Science, Department of Mathematics, Bursa,
Turkey elifc2@hotmail.com , M. Acikgoz University of Gaziantep, Faculty of
Science and Arts, Department of Mathematics, 27310 Gaziantep, TURKEY
acikgoz@gantep.edu.tr and I. N. Cangul Uludag University, Faculty of Arts and
Science, Department of Mathematics, Bursa, Turkey ncangul@gmail.com
###### Abstract.
We give some new identities for ($h,q$)-Genocchi numbers and polynomials by
means of the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and the
weighted $q$-Bernstein polynomials.
2000 Mathematics Subject Classification. 05A10, 11B65, 28B99, 11B68, 11B73.
Keywords and phrases. ($h$,$q$)-Genochhi numbers and polynomials with weight
$\alpha$, weighted Bernstein polynomials, fermionic $p$-adic $q$-integral on
$\mathbb{Z}_{p}$.
## 1\. Introduction and Notations
Let $p$ be a fixed odd prime number. Throughout this paper we use the
following notations. By $\mathbb{Z}_{p}$ we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$. The $p$-adic absolute
value is defined by
$\left|p\right|_{p}=\frac{1}{p}\text{.}$
In this paper, we assume $\left|q-1\right|_{p}<1$ as an indeterminate. Let
$UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly differentiable
functions on $\mathbb{Z}_{p}$. For $f\in UD\left(\mathbb{Z}_{p}\right)$, the
fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ is defined by T. Kim:
(1.1)
$I_{-q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\xi\right)d\mu_{-q}\left(\xi\right)=\lim_{N\rightarrow\infty}\sum_{\xi=0}^{p^{N}-1}q^{\xi}f\left(\xi\right)\left(-1\right)^{\xi}$
(for more informations on this subject, see [29], [30] and [31]).
From (1.1), we have well known the following equality:
(1.2)
$qI_{-q}\left(f_{1}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}f\left(0\right)$
here $f_{1}\left(x\right):=f\left(x+1\right)$ (for details, see[2-40]).
Let $C\left(\left[0,1\right]\right)$ be the space of continuous functions on
$\left[0,1\right]$. For $C\left(\left[0,1\right]\right)$, the weighted
$q$-Bernstein operator for $f$ is defined by
$\mathcal{B}_{n,q}^{\left(\alpha\right)}\left(f,x\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)B_{k,n}^{\left(\alpha\right)}\left(x\mid
q\right)=\sum_{k=0}^{n}f\left(\frac{k}{n}\right)\binom{n}{k}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k}$
where $n,$ $k\in\mathbb{N}^{\ast}$. Here
$B_{k,n}^{\left(\alpha\right)}\left(x\mid q\right)$ is called weighted
$q$-Bernstein polynomials, which are defined by
(1.3) $B_{k,n}^{\left(\alpha\right)}\left(x\mid
q\right)=\binom{n}{k}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k}\text{,
}x\in\left[0,1\right]$
(for more informations on this subject, see [3], [33], [39] and [40]).
As is well known, the ordinary Genocchi polynomials are defined by menas of
the following generating function:
(1.4)
$\sum_{n=0}^{\infty}G_{n}\left(x\right)\frac{t^{n}}{n!}=e^{G\left(x\right)t}=\frac{2t}{e^{t}+1}e^{xt}\text{.}$
where the usual convention about replacing $G^{n}\left(x\right)$ by
$G_{n}\left(x\right)$. For $x=0$ in (1.4), we have to
$G_{n}\left(0\right):=G_{n}$, which is called Genocchi numbers. Then, we can
write the following
(1.5)
$e^{Gt}=\sum_{n=0}^{\infty}G_{n}\frac{t^{n}}{n!}=\frac{2t}{e^{t}+1}\text{.}$
In [4], the $q$-Genocchi numbers are given as
$G_{0,q}=0\text{ and
}q\left(qG_{q}+1\right)^{n}+G_{n,q}=\left\\{\begin{array}[]{cc}\left[2\right]_{q}&\text{if
}n=1\\\ 0&\text{if }n\neq 1\end{array}\right.$
where the usual convention about replacing $\left(G_{q}\right)^{n}$ by
$G_{n,q}$.
For any $n\in\mathbb{N}^{\ast}$, the ($h,q$)-Genocchi numbers are defined by
$G_{0,q}^{\left(h\right)}=0\text{ and
}q^{h-1}\left(qG_{q}^{\left(h\right)}+1\right)^{n}+G_{n,q}^{\left(h\right)}=\left\\{\begin{array}[]{cc}\left[2\right]_{q}&\text{if
}n=1\\\ 0&\text{if }n\neq 1\end{array}\right.$
where the usual convention about replacing
$\left(G_{q}^{\left(h\right)}\right)^{n}$ by $G_{n,q}^{\left(h\right)}$ (for
details, see [11]).
Recently, Araci $et$ $al$. are defined the $(h,q)$-Genocchi numbers with
weight $\alpha$ by
(1.6)
$\frac{\widetilde{G}_{n+1,q}^{\left(\alpha,h\right)}\left(x\right)}{n+1}=\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)\text{.}$
By (1.6), we have the following identity
(1.7)
$\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(x\right)=\sum_{k=0}^{n}\binom{n}{k}q^{\alpha
kx}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left[x\right]_{q^{\alpha}}^{n-k}=q^{-\alpha
x}\left(q^{\alpha
x}\widetilde{G}_{q}^{\left(\alpha,h\right)}+\left[x\right]_{q^{\alpha}}\right)^{n}$
where the usual convention about replacing
$\left(\widetilde{G}_{q}^{\left(\alpha,h\right)}\right)^{n}$ by
$\widetilde{G}_{n,q}^{\left(\alpha,h\right)}$ is used (for details, [5]).
In this paper, we derive some new properties ($h,q$)-Genocchi numbers and
polynomials from the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$.
Also, we show that these type polynomials are related to ($h,q$)-Genocchi
numbers and polynomials.
## 2\. On the $\left(h,q\right)$-Genocchi numbers and polynomials
In this section, we consider the ($h,q$)-Genocchi numbers and polynomials by
using fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ and the weighted
$q$-Bernstein polynomials. We can now start the following expression.
In [5], we have the ($h,q$)-Genocchi numbers as follows: For
$\alpha\in\mathbb{N}^{\ast}$ and $n,h\in\mathbb{N}$,
(2.1) $\widetilde{G}_{0,q}^{\left(\alpha,h\right)}=0\text{ and
}q^{h}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(1\right)+\widetilde{G}_{n,q}^{\left(\alpha,h\right)}=\left\\{\begin{array}[]{cc}\left[2\right]_{q}&\text{if
}n=1,\\\ 0&\text{if }n\neq 1.\end{array}\right.$
By (1.7) and (2.1), we have the following corollary.
###### Corollary 1.
For $\alpha\in\mathbb{N}^{\ast}$ and $n,h\in\mathbb{N}$, then we have
(2.2) $\widetilde{G}_{0,q}^{\left(\alpha,h\right)}=0\text{ and
}q^{h-\alpha}\left(q^{\alpha}\widetilde{G}_{q}^{\left(\alpha,h\right)}+1\right)^{n}+\widetilde{G}_{n,q}^{\left(\alpha,h\right)}=\left\\{\begin{array}[]{cc}\left[2\right]_{q}&\text{if
}n=1,\\\ 0&\text{if }n\neq 1.\end{array}\right.$
By (1.6), we get symmetric property that
$\displaystyle\frac{\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}\left(1-x\right)}{n+1}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(1-h\right)\xi}\left[1-x+\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q^{-1}}\left(\xi\right)$
$\displaystyle=$ $\displaystyle\left(-1\right)^{n}q^{h+\alpha
n-1}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[x+\xi\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
From this, we state the following theorem.
###### Theorem 1.
The following identity
(2.3)
$\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}\left(1-x\right)=\left(-1\right)^{n}q^{h+\alpha
n-1}\widetilde{G}_{n+1,q}^{\left(\alpha,h\right)}\left(x\right)$
is true.
By using (1.7), (2.1) and (2.2), we compute as follows:
$\displaystyle
q^{2\alpha}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(2\right)$
$\displaystyle=$
$\displaystyle\left(q^{2\alpha}\widetilde{G}_{q}^{\left(\alpha,h\right)}+\left[2\right]_{q^{\alpha}}\right)^{n}$
$\displaystyle=$ $\displaystyle\sum_{l=0}^{n}\binom{n}{l}q^{\alpha
l}\left(q^{\alpha}\widetilde{G}_{q}^{\left(\alpha,h\right)}+1\right)^{l}$
$\displaystyle=$ $\displaystyle
nq^{2\alpha-h}\left(\left[2\right]_{q}-\widetilde{G}_{1,q}^{\left(\alpha,h\right)}\right)-q^{\alpha-h}\sum_{l=2}^{n}\binom{n}{l}q^{\alpha
l}\widetilde{G}_{l,q}^{\left(\alpha,h\right)}$ $\displaystyle=$ $\displaystyle
nq^{2\alpha-h}\left[2\right]_{q}+q^{2\alpha-2h}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\text{
if }n>1.$
After the above applications, we procure the following theorem.
###### Theorem 2.
For $n>1$, then we have
$\widetilde{G}_{n,q}^{\left(\alpha,h\right)}\left(2\right)=nq^{-h}\left[2\right]_{q}+q^{-2h}\widetilde{G}_{n,q}^{\left(\alpha,h\right)}.$
We need the following equality for sequel of this paper:
(2.5)
$\left[1-x\right]_{q^{-\alpha}}^{n}=\left(\frac{1-q^{-\alpha\left(1-x\right)}}{1-q^{-\alpha}}\right)^{n}=\left(-1\right)^{n}q^{n\alpha}\left[x-1\right]_{q^{\alpha}}^{n}\text{.}$
Now also, by using (2.5), we consider the following
$\displaystyle
q^{h-1}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}q^{h+n\alpha-1}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[\xi-1\right]_{q^{\alpha}}^{n}d\mu_{-q}\left(\xi\right)$
$\displaystyle=$
$\displaystyle\left(-1\right)^{n}q^{h+n\alpha-1}\frac{\widetilde{G}_{n+1,q}^{\left(\alpha,h\right)}\left(-1\right)}{n+1}\text{.}$
By considering last identity and (2.3), we get the following theorem.
###### Theorem 3.
The following identity holds true:
(2.6)
$\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\left(\xi+1\right)}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q}\left(\xi\right)=\frac{\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}\left(2\right)}{n+1}\text{.}$
From (2.6), we have the following
$\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q}\left(\xi\right)=\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}}{n+1}\text{.}$
Thus, we obtain the following theorem.
###### Theorem 4.
The following identity
(2.7)
$\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)\xi}\left[1-\xi\right]_{q^{-\alpha}}^{n}d\mu_{-q}\left(\xi\right)=\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}}{n+1}$
is true.
## 3\. Some new identities on the $\left(h,q\right)$-Genocchi numbers
In this section, we introduce new identities of the ($h,q$)-Genocchi numbers,
that is, we derive some interesting and worthwhile relations for studying in
Theory of Analytic Numbers.
For $x\in\left[0,1\right]$, we give definition of weighted $q$-Bernstein
polynomials as follows:
(3.1) $B_{k,n}^{\left(\alpha\right)}\left(x\mid
q\right)=\binom{n}{k}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k}\text{,
where }n,k\in\mathbb{Z}_{+}\text{.}$
By expression of (3.1), we have the properties of symmetry of weighted
$q$-Bernstein polynomials as follows:
(3.2) $B_{k,n}^{\left(\alpha\right)}\left(x\mid
q\right)=B_{n-k,n}^{\left(\alpha\right)}\left(1-x\mid\frac{1}{q}\right)\text{,
(for details, see \cite[cite]{[\@@bibref{}{kim 19}{}{}]}).}$
Thus, (2.7), (3.1) and (3.2), we see that
$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}B_{k,n}^{\left(\alpha\right)}\left(x\mid
q\right)d\mu_{-q}\left(x\right)=\binom{n}{k}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[1-x\right]_{q^{-\alpha}}^{n-l}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n-l+1}\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}}{n+1}&\text{if
}k=0,\\\
\binom{n}{k}\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n-l+1}\right\\}&\text{if
}k\neq 0.\end{array}\right.$
On the other hand, for $n$, $k\in\mathbb{Z}_{+}$ with $n>k$, we compute
$\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}B_{k,n}^{\left(\alpha\right)}\left(x\mid
q\right)d\mu_{-q}\left(x\right)$ $\displaystyle=$
$\displaystyle\binom{n}{k}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[x\right]_{q^{\alpha}}^{k}\left[1-x\right]_{q^{-\alpha}}^{n-k}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[x\right]_{q^{\alpha}}^{l+k}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\binom{n}{k}\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}\frac{\widetilde{G}_{l+k+1,q}^{\left(\alpha,h\right)}}{l+k+1}\text{.}$
Equating $I_{1}$ and $I_{2}$, then we have the following theorem.
###### Theorem 5.
The following identity holds true:
$\sum_{l=0}^{n-k}\binom{n-k}{l}\left(-1\right)^{l}\frac{\widetilde{G}_{l+k+1,q}^{\left(\alpha,h\right)}}{l+k+1}=\left\\{\begin{array}[]{cc}\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n+1,q^{-1}}^{\left(\alpha,h\right)}}{n+1}&\text{if
}k=0,\\\
\sum_{l=0}^{k}\binom{k}{l}\left(-1\right)^{k+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n-l+1}\right\\}&\text{if
}k\neq 0.\end{array}\right.$
Let $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k$. Then, we derive
the followings
(3.6) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}B_{k,n_{1}}^{\left(\alpha\right)}\left(x\mid
q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(x\mid
q\right)d\mu_{-q}\left(x\right)$ $\displaystyle=$
$\displaystyle\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[1-x\right]_{q^{-\alpha}}^{n_{1}+n_{2}-l}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left(\binom{n_{1}}{k}\binom{n_{2}}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left(\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}-l+1}\right)\right)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}+1,q^{-1}}^{\left(\alpha,h\right)}}{n+1}&\text{if
}k=0,\\\
\binom{n}{k}\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}-l+1}\right\\}&\text{if
}k\neq 0.\end{array}\right.$
In other words, by using the binomial theorem, we can derive the following
equation.
$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}B_{k,n_{1}}^{\left(\alpha\right)}\left(x\mid
q\right)B_{k,n_{2}}^{\left(\alpha\right)}\left(x\mid
q\right)d\mu_{-q}\left(x\right)$ $\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left[x\right]_{q^{\alpha}}^{2k+l}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{2}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\frac{\widetilde{G}_{l+2k+1,q}^{\left(\alpha,h\right)}}{l+2k+1}.$
Combining $I_{3}$ and $I_{4}$, we state the following theorem.
###### Theorem 6.
For $n_{1},n_{2},k\in\mathbb{Z}_{+}$ with $n_{1}+n_{2}>2k,$ we have
$\displaystyle\sum_{l=0}^{n_{1}+n_{2}-2k}\binom{n_{1}+n_{2}-2k}{l}\left(-1\right)^{l}\frac{\widetilde{G}_{l+2k+1,q}^{\left(\alpha,h\right)}}{l+2k+1}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}+1}&\text{if
}k=0,\\\
\sum_{l=0}^{2k}\binom{2k}{l}\left(-1\right)^{2k+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}-l+1}\right\\}&\text{if
}k\neq 0.\end{array}\right.$
For $x\in\mathbb{Z}_{p}$ and $s\in\mathbb{N}$ with $s\geq 2,$ let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk$. Then
we take the fermionic $p$-adic $q$-integral on $\mathbb{Z}_{p}$ for the
weighted $q$-Bernstein polynomials of degree $n$ as follows:
$\displaystyle I_{5}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left\\{\mathop{\textstyle\prod}\limits_{i=1}^{s}B_{k,n_{i}}^{\left(\alpha\right)}\left(x\mid
q\right)\right\\}d\mu_{-q}\left(x\right)$ $\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\int_{\mathbb{Z}_{p}}\left[x\right]_{q^{\alpha}}^{sk}\left[1-x\right]_{q^{-\alpha}}^{n_{1}+n_{2}+...+n_{s}-sk}q^{\left(h-1\right)x}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{l+sk}\int_{\mathbb{Z}_{p}}\left[1-x\right]_{q^{-\alpha}}^{n_{1}+n_{2}+...+n_{s}-l}q^{\left(h-1\right)x}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}+...+n_{s}+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}+...+n_{s}+1}&\text{if
}k=0,\\\
\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}+...+n_{s}-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}+...+n_{s}-l+1}\right\\}&\text{if
}k\neq 0.\end{array}\right.$
On the other hand, from the definition of weighted $q$-Bernstein polynomials
and the binomial theorem, we easily get
$\displaystyle I_{6}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{Z}_{p}}q^{\left(h-1\right)x}\left\\{\mathop{\textstyle\prod}\limits_{i=1}^{s}B_{k,n_{i}}^{\left(\alpha\right)}\left(x\mid
q\right)\right\\}d\mu_{-q}\left(x\right)$ $\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\int_{\mathbb{Z}_{p}}\left[x\right]_{q^{\alpha}}^{sk+l}q^{\left(h-1\right)x}d\mu_{-q}\left(x\right)$
$\displaystyle=$
$\displaystyle\mathop{\displaystyle\prod}\limits_{i=1}^{s}\binom{n_{i}}{k}\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\frac{\widetilde{G}_{l+sk+1,q}^{\left(\alpha,h\right)}}{l+sk+1}\text{.}$
Equating $I_{5}$ and $I_{6}$, we discover the following theorem.
###### Theorem 7.
For $s\in\mathbb{N}$ with $s\geq 2$, let
$n_{1},n_{2},...,n_{s},k\in\mathbb{Z}_{+}$ with $\sum_{l=1}^{s}n_{l}>sk.$
Then, we have
$\displaystyle\sum_{l=0}^{n_{1}+...+n_{s}-sk}\binom{\sum_{d=1}^{s}\left(n_{d}-k\right)}{l}\left(-1\right)^{l}\frac{\widetilde{G}_{l+sk+1,q}^{\left(\alpha,h\right)}}{l+sk+1}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}+...+n_{s}+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}+...+n_{s}+1}&\text{if
}k=0,\\\
\sum_{l=0}^{sk}\binom{sk}{l}\left(-1\right)^{sk+l}\left\\{\left[2\right]_{q}+q^{h+1}\frac{\widetilde{G}_{n_{1}+n_{2}+...+n_{s}-l+1,q^{-1}}^{\left(\alpha,h\right)}}{n_{1}+n_{2}+...+n_{s}-l+1}\right\\}&\text{if
}k\neq 0.\end{array}\right.$
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|
arxiv-papers
| 2012-06-24T03:37:27 |
2024-09-04T02:49:32.145036
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Elif Cetin, Mehmet Acikgoz and Ismail Naci Cangul",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1206.5461"
}
|
1206.5504
|
# Singular quadratic Lie superalgebras
Minh Thanh Duong, Rosane Ushirobira Minh Thanh Duong, Department of Physics,
University of Education of Ho Chi Minh city, 280 An Duong Vuong, Ho Chi Minh
city, Vietnam. Rosane Ushirobira, Institut de Mathématiques de Bourgogne,
Université de Bourgogne, B.P. 47870, F-21078 Dijon Cedex & Non-A Inria Lille -
Nord Europe, France thanhdmi@hcmup.edu.vn Rosane.Ushirobira@u-bourgogne.fr
###### Abstract.
In this paper, we give a generalization of results in [PU07] and [DPU10] by
applying the tools of graded Lie algebras to quadratic Lie superalgebras. In
this way, we obtain a numerical invariant of quadratic Lie superalgebras and a
classification of singular quadratic Lie superalgebras, i.e. those with a
nonzero invariant. Finally, we study a class of quadratic Lie superalgebras
obtained by the method of generalized double extensions.
###### Key words and phrases:
Quadratic Lie superalgebras. Super Poisson bracket. Invariant. Double
extensions. Generalized double extensions. Adjoint orbits.
###### 2000 Mathematics Subject Classification:
15A63, 17B05, 17B30, 17B70
## 0\. Introduction
Throughout the paper, the base field is $\mathbb{C}$ and all vector spaces are
complex and finite-dimensional. We denote the ring $\mathbb{Z}/2\mathbb{Z}$ by
$\mathbb{Z}_{2}$ as in superalgebra theory.
Let us begin with a $\mathbb{Z}_{2}$-graded vector space
$\mathfrak{g}=\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
We denote by
$\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$ the
Grassmann algebra of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$, that
is, the algebra of alternating multilinear forms on
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ equipped with the wedge
product and by
$\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$ the
algebra of symmetric multilinear forms on
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
We say that $\mathfrak{g}$ is a quadratic $\mathbb{Z}_{2}$-graded vector space
if it is endowed with a non-degenerate even supersymmetric bilinear form $B$
(that is, $B$ is symmetric on
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$, skew-symmetric on
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ and
$B(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$).
In addition, if there is a Lie superalgebra structure $[\cdot,\cdot]$ on
$\mathfrak{g}$ such that $B$ is invariant, i.e. $B([X,Y],Z)=B(X,[Y,Z])$ for
all $X,Y,Z\in\mathfrak{g}$, then $\mathfrak{g}$ is called a quadratic (or
orthogonal or metrised) Lie superalgebra.
Algebras endowed with an invariant bilinear form appear in many areas of
Mathematics and Physics and they are a remarkable algebraic object. A
structural theory of quadratic Lie algebras, based on the notion of a double
extension (a combination of a central extension and a semi-direct product),
was introduced by V. Kac [Kac85] in the solvable case and by A. Medina and P.
Revoy [MR85] in the general case. Another interesting construction, the
$T^{*}$-extension, based on the notion of a generalized semi-direct product of
a Lie algebra and its dual space was given by M. Bordemann [Bor97] for
solvable quadratic Lie algebras. Both notions have been generalized for
quadratic Lie superalgebras in papers by H. Benamor and S. Benayadi [BB99] and
by I. Bajo, S. Benayadi and M. Bordemann [BBB].
A third approach, based on the concept of super Poisson bracket, was
introduced in [PU07], providing several interesting properties of quadratic
Lie algebras: the authors consider $(\mathfrak{g},[\cdot,\cdot],B)$ a non-
Abelian quadratic Lie algebra and define a 3-form $I$ on $\mathfrak{g}$ by
$I(X,Y,Z)=B([X,Y],Z),\ \forall\ X,Y,Z\in\mathfrak{g}.$
Then $I$ is nonzero and $\\{I,I\\}=0$, where $\\{\cdot,\cdot\\}$ is the super
Poisson bracket defined on the ($\mathbb{Z}$-graded) Grassmann algebra
$\operatorname{Alt}(\mathfrak{g})$ of $\mathfrak{g}$, by
$\\{\Omega,\Omega^{\prime}\\}=(-1)^{\operatorname{deg}_{\mathbb{Z}}(\Omega)+1}\sum_{j=1}^{n}\operatorname{\iota}_{X_{j}}(\Omega)\wedge\operatorname{\iota}_{X_{j}}(\Omega^{\prime}),\
\forall\ \Omega,\ \Omega^{\prime}\in\operatorname{Alt}(\mathfrak{g})$
with $\\{X_{1},\dots,X_{n}\\}$ a fixed orthonormal basis of $\mathfrak{g}$.
Conversely, given a quadratic vector space $(\mathfrak{g},B)$ and a nonzero
3-form $I\in\operatorname{Alt}^{3}(\mathfrak{g})$ satisfying $\\{I,I\\}=0$,
then there is a non-Abelian Lie algebra structure on $\mathfrak{g}$ such that
$B$ is invariant.
The element $I$ carries some useful information about corresponding quadratic
Lie algebras. For instance, when $I$ is decomposable and nonzero,
corresponding quadratic Lie algebras are called elementary quadratic Lie
algebras and they are exhaustively classified in [PU07]. This classification
is based on basic properties of quadratic forms and the super Poisson bracket.
In this case, $\dim([\mathfrak{g},\mathfrak{g}])=3$ and coadjoint orbits have
dimension at most 2\. In [DPU10], the authors consider further a notion that
is called the $\operatorname{dup}$-number of a non-Abelian quadratic Lie
algebra $\mathfrak{g}$. It is defined by
$\operatorname{dup}(\mathfrak{g})=\dim\left(\\{\alpha\in\mathfrak{g}^{*}\mid\
\alpha\wedge I=0\\}\right)$ where $\mathfrak{g}^{*}$ is the dual space of
$\mathfrak{g}$. The dup-number receives values 0, 1 or 3 and it measures the
decomposability of $I$. For instance, $I$ is decomposable if and only if
$\operatorname{dup}(\mathfrak{g})=3$. Moreover, it is also a numerical
invariant of quadratic Lie algebras under Lie algebra isomorphisms, meaning
that if $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are isomorphic quadratic
Lie algebras then
$\operatorname{dup}(\mathfrak{g})=\operatorname{dup}(\mathfrak{g}^{\prime})$.
Its proof is rather non-trivial. It is obtained through a description of the
space generated by invariant symmetric bilinear forms on a quadratic Lie
algebra with nonzero dup-number. Such quadratic Lie algebra is called a
singular. An unexpected property is that there are many non-degenerate
invariant symmetric bilinear forms on a singular quadratic Lie algebra. Though
they can be linearly independent all of them are equivalent in the solvable
case, i.e. two solvable singular quadratic Lie algebras with same Lie algebra
structure are isometrically isomorphic (or i-isomorphic, for short). In other
words, isomorphic and i-isomorphic notions are equivalent on solvable singular
quadratic Lie algebras. Another remarkable result is that all singular
quadratic Lie algebras are classified up to isomorphism by $O(n)$-adjoint
orbits of the Lie algebra $\mathfrak{o}(n)$.
The purpose of this paper is to give a interpretation of this last approach
for quadratic Lie superalgebras. We combine it with the notion of double
extension as it was done for quadratic Lie algebras [DPU10]. Further, we use
the notion of generalized double extension. In result, we obtain a rather
colorful picture of quadratic Lie superalgebras.
Let us give some details of our main results. First, let $\mathfrak{g}$ be a
quadratic $\mathbb{Z}_{2}$-graded vector space. Recall that
$\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$ and
$\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$ are
$\mathbb{Z}$-graded algebras and this gradation can be used to define a
$\mathbb{Z}\times\mathbb{Z}_{2}$-gradation on each algebra. Consider then the
$\mathbb{Z}\times\mathbb{Z}_{2}$-graded super-exterior algebra of
$\mathfrak{g}^{*}$ defined by
$\mathscr{E}(\mathfrak{g})=\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\
\underset{\mathbb{Z}\times\mathbb{Z}_{2}}{\otimes}\
\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$
with the natural super-exterior product
$(\Omega\otimes F)\wedge(\Omega^{\prime}\otimes
F^{\prime})=(-1)^{\operatorname{deg}_{\mathbb{Z}}(F)\operatorname{deg}_{\mathbb{Z}}(\Omega^{\prime})}(\Omega\wedge\Omega^{\prime})\otimes
FF^{\prime},$
for all $\Omega,\Omega^{\prime}$ in
$\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$ and
$F,F^{\prime}$ in
$\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$. It is
clear that $\mathscr{E}$ is commutative and associative. For more details of
the algebra $\mathscr{E}(\mathfrak{g})$ the reader should refer to [Sch79],
[BP89] or [MPU09].
In [MPU09], the authors use the super $\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson
bracket $\\{\cdot,\cdot\\}$ on the super-exterior algebra
$\mathscr{E}(\mathfrak{g})$ defined as follows:
$\\{\Omega\otimes F,\Omega^{\prime}\otimes
F^{\prime}\\}=(-1)^{\operatorname{deg}_{\mathbb{Z}}(F)\operatorname{deg}_{\mathbb{Z}}(\Omega^{\prime})}\left(\\{\Omega,\Omega^{\prime}\\}\otimes
FF^{\prime}+(\Omega\wedge\Omega^{\prime})\otimes\\{F,F^{\prime}\\}\right),$
for all $\Omega$,
$\Omega^{\prime}\in\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$,
$F$,
$F^{\prime}\in\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$.
In Section 1, we will recall some simple properties of the super
$\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket that are necessary for our
purpose.
It is easy to check that for a quadratic Lie superalgebra $(\mathfrak{g},B)$,
if we define a trilinear form $I$ on $\mathfrak{g}$ by
$I(X,Y,Z)=B([X,Y],Z),\ \forall\ X,Y,Z\in\mathfrak{g}$
then $I\in\mathscr{E}^{(3,\overline{0})}(\mathfrak{g})$. Therefore, it seems
to be natural to ask: when does $\\{I,I\\}=0$? We shall give an affirmative
answer to this in Proposition 1.11. Moreover, similarly to the Lie algebra
case, we show that non-Abelian quadratic Lie superalgebra structures on a
quadratic $\mathbb{Z}_{2}$-graded vector space $(\mathfrak{g},B)$ are in one-
to-one correspondence with nonzero elements $I$ in
$\mathscr{E}^{(3,\overline{0})}(\mathfrak{g})$ satisfying $\\{I,I\\}=0$.
In Section 2, we introduce the notion of $\operatorname{dup}$-number for a
non-Abelian quadratic Lie superalgebra $\mathfrak{g}$:
$\operatorname{dup}(\mathfrak{g})=\dim\left(\\{\alpha\in\mathfrak{g}^{*}\mid\
\alpha\wedge I=0\\}\right)$
and consider the set of quadratic Lie superalgebras with nonzero
$\operatorname{dup}$-number: the set of singular quadratic Lie superalgebras.
Similarly to quadratic Lie algebras, if $\mathfrak{g}$ is non-Abelian then
$\operatorname{dup}(\mathfrak{g})\in\\{0,1,3\\}$. Thanks to Lemma 2.1, if
$\operatorname{dup}(\mathfrak{g})=3$ then
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ is a central ideal of
$\mathfrak{g}$ and $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is an
elementary quadratic Lie algebra. Therefore, we focus on singular quadratic
Lie superalgebras $\mathfrak{g}$ with $\operatorname{dup}(\mathfrak{g})=1$. We
call them singular quadratic Lie superalgebras of type $\mathsf{S}_{1}$.
However, differently than the Lie algebra case, the element $I$ may be
decomposable.
We list in Section 3 all non-Abelian reduced quadratic Lie superalgebras with
$I$ decomposable (see Definition 2.5 for the definition of a reduced quadratic
Lie superalgebra). We call them elementary quadratic Lie superalgebras. In
this case, $\operatorname{dup}(\mathfrak{g})$ is nonzero. In particular, if
$\operatorname{dup}(\mathfrak{g})=3$ then $\mathfrak{g}$ is an elementary
quadratic Lie algebra. If $\operatorname{dup}(\mathfrak{g})=1$ then we obtain
three quadratic Lie superalgebras with 2-dimensional even part. Actually, we
prove in Proposition 4.1 that if $\mathfrak{g}$ is a non-Abelian quadratic Lie
superalgebra with 2-dimensional even part then
$\operatorname{dup}(\mathfrak{g})=1$.
Section 4 details a study of quadratic Lie superalgebras with 2-dimensional
even part. We apply the concept of double extension as in [DPU10] with a
little change by replacing a quadratic vector space by a symplectic vector
space and keeping the other conditions (see Definition 4.6). Then we obtain a
structure that we still call a double extension and one has (Proposition 4.8):
Theorem 1:
A quadratic Lie superalgebra has a 2-dimensional even part if and only if it
is a double extension.
By a very similar process as in [DPU10] for solvable singular quadratic Lie
algebras, a classification of quadratic Lie superalgebras with 2-dimensional
even part up to isomorphism is given as follows. Let $\mathscr{S}(2+2n)$ be
the set of such structures on $\mathbb{C}^{2+2n}$. We call an algebra
$\mathfrak{g}\in\mathscr{S}(2+2n)$ diagonalizable (resp. invertible) if it is
the double extension by a diagonalizable (resp. invertible) map. Denote the
subsets of nilpotent elements, diagonalizable elements and invertible elements
in $\mathscr{S}(2+2n)$, respectively by $\mathscr{N}(2+2n)$,
$\mathscr{D}(2+2n)$ and by $\mathscr{S}_{\mathrm{inv}}(2+2n)$. Denote by
$\widehat{{\mathscr{N}}}(2+2n)$, $\widehat{{\mathscr{D}}}(2+2n)$,
$\widehat{\mathscr{S}_{\mathrm{inv}}}(2+2n)$ the sets of isomorphic classes in
$\mathscr{N}(2+2n)$, $\mathscr{D}(2+2n)$, $\mathscr{S}_{\mathrm{inv}}(2+2n)$
respectively and by $\widehat{{\mathscr{D}}}_{\mathrm{red}}(2+2n)$ the subset
of $\widehat{{\mathscr{D}}}(2+2n)$ including reduced ones. Also, we denote by
$\mathbb{P}^{1}(\mathfrak{sp}(2n))$ the projective space of
$\mathfrak{sp}(2n)$ with the action induced by the
$\operatorname{Sp}(2n)$-adjoint action on $\mathfrak{sp}(2n)$. Then we have
the classification result (Propositions 4.13 and 4.14 and Appendix):
Theorem 2:
* (i)
Let $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ be elements in
$\mathscr{S}(2+2n)$. Then $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are
isometrically isomorphic and if and only if they are isomorphic.
* (ii)
There is a bijection between $\widehat{{\mathscr{N}}}(2+2n)$ and the set of
nilpotent $\operatorname{Sp}(2n)$-adjoint orbits of $\mathfrak{sp}(2n)$. It
induces a bijection between $\widehat{{\mathscr{N}}}(2+2n)$ and the set of
partitions $\mathcal{P}_{-1}(2n)$ of $2n$ in which odd parts occur with even
multiplicity.
* (iii)
There is a bijection between $\widehat{{\mathscr{D}}}(2+2n)$ and the set of
semisimple $\operatorname{Sp}(2n)$-orbits of $\
\mathbb{P}^{1}(\mathfrak{sp}(2n))$.
* (iv)
There is a bijection between $\widehat{\mathscr{S}_{\mathrm{inv}}}(2+2n)$ and
the set of invertible $\operatorname{Sp}(2n)$-orbits of $\
\mathbb{P}^{1}(\mathfrak{sp}(2n))$.
* (v)
There is a bijection between $\widehat{\mathscr{S}}(2+2n)$ and the set of
$\operatorname{Sp}(2n)$-orbits of $\mathbb{P}^{1}(\mathfrak{sp}(2n))$.
As for quadratic Lie algebras, we have the notion of quadratic dimension of a
quadratic Lie superalgebra. In the case $\mathfrak{g}$ is a quadratic Lie
superalgebra having a 2-dimensional even part, we can compute its quadratic
dimension as follows:
$d_{q}(\mathfrak{g})=2+\displaystyle\frac{(\dim(\mathscr{Z}(\mathfrak{g})-1))(\dim(\mathscr{Z}(\mathfrak{g})-2)}{2}.$
where $\mathscr{Z}(\mathfrak{g})$ is the center of $\mathfrak{g}$. It
indicates that there are many non-degenerate invariant even supersymmetric
bilinear forms on a quadratic Lie superalgebra with 2-dimensional even part
but by Theorem 2 (i), all of them are equivalent.
Section 5 contains more results on a singular quadratic Lie superalgebra
$(\mathfrak{g},B)$ of type $\mathsf{S}_{1}$, that is, those with 1-valued
$\operatorname{dup}$-number. The first result is that
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is solvable and so
$\mathfrak{g}$ is solvable. Moreover, by Definition 5.3 and Lemma 5.5, the Lie
superalgebra $\mathfrak{g}$ can be realized as the double extension of a
quadratic $\mathbb{Z}_{2}$-graded vector space
$\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
by a map
${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}\in\mathfrak{o}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})\oplus\mathfrak{sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$.
Denote by $\mathcal{L}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$
(resp. $\mathcal{L}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$) the set
of endomorphisms of $\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$ (resp.
$\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$). We give isomorphic and
i-isomorphic characterizations of two singular quadratic Lie superalgebras of
type $\mathsf{S}_{1}$ as follows (Proposition 5.7).
Theorem 3:
Let $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ be two double extensions of
$\mathfrak{q}$ by ${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}$ and
$\overline{C^{\prime}}=\overline{C_{0}^{\prime}}+\overline{C_{1}^{\prime}}$,
respectively. Assume that ${\overline{C}}_{1}$ is nonzero. Then
1. (1)
there exists a Lie superalgebra isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exist invertible maps
$P\in\mathcal{L}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$,
$Q\in\mathcal{L}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$ and a
nonzero $\lambda\in\mathbb{C}$ such that
* (i)
$\overline{C_{0}^{\prime}}=\lambda P{\overline{C}}_{0}P^{-1}$ and
$P^{*}P{\overline{C}}_{0}={\overline{C}}_{0}$.
* (ii)
$\overline{C_{1}^{\prime}}=\lambda Q{\overline{C}}_{1}Q^{-1}$ and
$Q^{*}Q{\overline{C}}_{1}={\overline{C}}_{1}$.
where $P^{*}$ and $Q^{*}$ are the adjoint maps of $P$ and $Q$ with respect to
$B|_{\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\times\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}}$
and
$B|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}\times\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$.
2. (2)
there exists an i-isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there is a nonzero
$\lambda\in\mathbb{C}$ such that $\overline{C_{0}^{\prime}}$ is in the
$\operatorname{O}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$-adjoint
orbit through $\lambda{\overline{C}}_{0}$ and $\overline{C_{1}^{\prime}}$ is
in the
$\operatorname{Sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$-adjoint
orbit through $\lambda{\overline{C}}_{1}$.
We recall a remarkable result in [DPU10] that two solvable singular quadratic
Lie algebras are i-isomorphic if and only if they are isomorphic. A similar
situation occurs for two quadratic Lie superalgebras with 2-dimensional even
part as in Theorem 2. Therefore, there is a very natural question: is this
result also true for two singular quadratic Lie superalgebras? We have an
affirmative answer as follows (Proposition 5.15 for
$\operatorname{dup}(\mathfrak{g})=1$ and [DPU10] for
$\operatorname{dup}(\mathfrak{g})=3$):
Theorem 4:
Let $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ be two solvable singular
quadratic Lie superalgebras. Then $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$
are i-isomorphic if and only if they are isomorphic.
We close the problem on singular quadratic Lie superalgebras with an assertion
that (Proposition 5.16):
Theorem 5:
The $\operatorname{dup}$-number is invariant under Lie superalgebra
isomorphism.
As a consequence of its proof, we obtain a formula for the quadratic dimension
of reduced singular quadratic Lie superalgebras of type $\mathsf{S}_{1}$
having
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq\\{0\\}$.
In the last Section, we study the structure of a quadratic Lie superalgebra
$\mathfrak{g}$ such that its element $I$ has the form:
$I=J\wedge p$
where $p\in\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}^{*}$ is nonzero
and
$J\in\operatorname{Alt}{}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}{}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
is indecomposable. We call $\mathfrak{g}$ a quasi-singular quadratic Lie
superalgebra. With the notion of generalized double extension given by I.
Bajo, S. Benayadi and M. Bordemann in [BBB], we prove that (Corollary 6.5 and
Proposition 6.9):
Theorem 6:
A quasi-singular quadratic Lie superalgebra is a generalized double extension
of a quadratic $\mathbb{Z}_{2}$-graded vector space. This superalgebra is
2-nilpotent.
In the Appendix, we recall fundamental results in the classification of
$\operatorname{O}(m)$-adjoint orbits of $\mathfrak{o}(m)$ and
$\operatorname{Sp}(2n)$-adjoint orbits of $\mathfrak{sp}(2n)$. The
classification of nilpotent and semisimple orbits is well-known. We further
give here the classification of invertible orbits, i.e. orbits of isomorphisms
in $\mathfrak{o}(m)$ and $\mathfrak{sp}(2n)$. By the Fitting decomposition, we
obtain a complete classification in the general case.
Many concepts used in this paper are generalizations of the quadratic Lie
algebra case. We do not recall their original definitions here. For more
details the reader can refer to [PU07] and [DPU10].
###### Acknowledgments.
We would like to thank D. Arnal and R. Yu for many valuable remarks and
stimulating discussions concerning the first version of this article.
Moreover, we would like to thank S. Benayadi for very interesting suggestions
for the improvement of Section 5.
This article is dedicated to our admirable mentor Georges Pinczon (1948 –
2010). He suggested the main idea in Section 1 and discussed results in
Sections 2 and 3.
## 1\. Applications of graded Lie algebras to quadratic Lie superalgebras
Let
$\mathfrak{g}=\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
be a $\mathbb{Z}_{2}$-graded vector space. We call
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ respectively the even and
the odd part of $\mathfrak{g}$. We begin by reviewing the construction of the
super-exterior algebra of the dual space $\mathfrak{g}^{*}$ of $\mathfrak{g}$.
Then we define the super $\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket on
$\mathfrak{g}^{*}$ (for more details, see [MPU09] and [Sch79]).
### 1.1. The super-exterior algebra of $\mathfrak{g}^{*}$
Denote by
$\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$ the
algebra of alternating multilinear forms on
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and by
$\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$ the
algebra of symmetric multilinear forms on
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$. Recall that
$\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$ is the
exterior algebra of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$ and
$\operatorname{Sym}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$ is the
symmetric algebra of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}^{*}$.
These algebras are $\mathbb{Z}$-graded algebras. We define a
$\mathbb{Z}\times\mathbb{Z}_{2}$-gradation on
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$ and on
$\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$ by
$\operatorname{Alt}^{(i,\overline{0})}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})=\operatorname{Alt}^{i}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}),\quad\operatorname{Alt}^{(i,\overline{1})}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})=\\{0\\}$
$\text{and}\quad\operatorname{Sym}^{(i,\overline{i})}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})=\operatorname{Sym}^{i}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}),\quad\operatorname{Sym}^{(i,\overline{j})}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})=\\{0\\}\quad\text{if}\quad\overline{i}\neq\overline{j},$
where $i,j\in\mathbb{Z}$ and $\overline{i},\overline{j}$ are respectively the
residue classes modulo 2 of $i$ and $j$.
The super-exterior algebra of $\mathfrak{g}^{*}$ is the
$\mathbb{Z}\times\mathbb{Z}_{2}$-graded algebra defined by:
$\mathscr{E}(\mathfrak{g})=\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\underset{\mathbb{Z}\times\mathbb{Z}_{2}}{\otimes}\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
endowed with the super-exterior product on $\mathscr{E}(\mathfrak{g})$:
$(\Omega\otimes F)\wedge(\Omega^{\prime}\otimes
F^{\prime})=(-1)^{f\omega^{\prime}}(\Omega\wedge\Omega^{\prime})\otimes
FF^{\prime},$
for all
$\Omega\in\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$\Omega^{\prime}\in\operatorname{Alt}^{\omega^{\prime}}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$F\in\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$,
$F^{\prime}\in\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
Remark that the $\mathbb{Z}\times\mathbb{Z}_{2}$-gradation on
$\mathscr{E}(\mathfrak{g})$ is given by:
$\text{if }A=\Omega\otimes
F\in\operatorname{Alt}^{\omega}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})\
\text{ with }\omega,\ f\in\mathbb{Z},\ \text{ then
}A\in\mathscr{E}^{(\omega+f,\overline{f})}(\mathfrak{g}).$
So, in terms of the $\mathbb{Z}$-gradations of
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$ and
$\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$, we
have:
$\mathscr{E}^{n}(\mathfrak{g})=\bigoplus^{n}_{m=0}\left(\operatorname{Alt}^{m}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{n-m}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})\right)$
and in terms of the $\mathbb{Z}_{2}$-gradations,
$\mathscr{E}_{{\scriptscriptstyle{\overline{0}}}}(\mathfrak{g})=\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\otimes\left(\underset{j\geq
0}{\oplus}\operatorname{Sym}^{2j}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})\right)\
\text{ and }\
\mathscr{E}_{{\scriptscriptstyle{\overline{1}}}}(\mathfrak{g})=\operatorname{Alt}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\otimes\left(\underset{j\geq
0}{\oplus}\operatorname{Sym}^{2j+1}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})\right).$
Notice that the graded vector space $\mathscr{E}(\mathfrak{g})$ endowed with
this product is a commutative and associative graded algebra.
Another equivalent construction is given in [BP89]:
$\mathscr{E}(\mathfrak{g})$ is the graded algebra of super-antisymmetric
multilinear forms on $\mathfrak{g}$. The algebras
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$ and
$\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$ are
regarded as subalgebras of $\mathscr{E}(\mathfrak{g})$ by identifying
$\Omega:=\Omega\otimes 1$, $F:=1\otimes F$, and the tensor product
$\Omega\otimes F=(\Omega\otimes 1)\wedge(1\otimes F)$ for all
$\Omega\in\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$F\in\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
### 1.2. The super $\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket on
$\mathscr{E}(\mathfrak{g}^{*})$
Let us assume that the vector space $\mathfrak{g}$ is equipped with a non-
degenerate even supersymmetric bilinear form $B$. That means
$B(X,Y)=(-1)^{xy}B(Y,X)$
for all homogeneous $X\in\mathfrak{g}_{x}$, $Y\in\mathfrak{g}_{y}$ and
$B(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$.
In this case, $\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$ must be
even and $\mathfrak{g}$ is also called a quadratic $\mathbb{Z}_{2}$-graded
vector space.
The Poisson bracket on
$\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$ and
the super Poisson bracket on
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$ are
defined as follows. Let $\mathcal{B}=\\{X_{1},...,X_{n},Y_{1},...,Y_{n}\\}$ be
a Darboux basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$, meaning
that $B(X_{i},X_{j})=B(Y_{i},Y_{j})=0$ and $B(X_{i},Y_{j})=\delta_{ij}$, for
all $1\leq i,j\leq n$. Let $\\{p_{1},...,p_{n},q_{1},...,q_{n}\\}$ be its dual
basis. Then the algebra
$\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
regarded as the polynomial algebra
$\mathbb{C}[p_{1},...,p_{n},q_{1},...,q_{n}]$ is equipped with the Poisson
bracket:
$\\{F,G\\}=\sum_{i=1}^{n}\left(\displaystyle\frac{\partial F}{\partial
p_{i}}\displaystyle\frac{\partial G}{\partial
q_{i}}-\displaystyle\frac{\partial F}{\partial
q_{i}}\displaystyle\frac{\partial G}{\partial p_{i}}\right),\ \text{for all}\
F,G\in\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}).$
It is well-known that the algebra
$\left(\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}),\\{\cdot,\cdot\\}\right)$
is a Lie algebra. Now, let
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and denote by
$\iota_{X}$ the derivation of
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$
defined by:
$\iota_{X}(\Omega)(Z_{1},\dots,Z_{k})=\Omega(X,Z_{1},\dots,Z_{k}),\ \forall\
\Omega\in\operatorname{Alt}^{k+1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}),\
X,Z_{1},\dots,Z_{k}\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\ (k\geq
0),$
and $\iota_{X}(1)=0$. Let $\\{Z_{1},\dots,Z_{m}\\}$ be a fixed orthonormal
basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$. The super Poisson
bracket on the algebra
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$ is
defined by (see [PU07] for details):
$\\{\Omega,\Omega^{\prime}\\}=(-1)^{k+1}\sum_{j=1}^{m}\operatorname{\iota}_{Z_{j}}(\Omega)\wedge\operatorname{\iota}_{Z_{j}}(\Omega^{\prime}),\
\forall\
\Omega\in\operatorname{Alt}^{k}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}),\
\Omega^{\prime}\in\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}).$
Remark that the definitions above do not depend on the choice of the basis.
Next, for any
$\Omega\in\operatorname{Alt}^{k}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
we define the map $\operatorname{ad}_{\tt P}(\Omega)$ by
$\operatorname{ad}_{\tt
P}(\Omega)\left(\Omega^{\prime}\right)=\\{\Omega,\Omega^{\prime}\\},\ \forall\
\Omega^{\prime}\in\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}).$
It is easy to check that $\operatorname{ad}_{\tt P}(\Omega)$ is a super-
derivation of degree $k-2$ of the algebra
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$. One
has:
$\operatorname{ad}_{\tt
P}(\Omega)\left(\\{\Omega^{\prime},\Omega^{\prime\prime}\\}\right)=\\{\operatorname{ad}_{\tt
P}(\Omega)(\Omega^{\prime}),\Omega^{\prime\prime}\\}+(-1)^{kk^{\prime}}\\{\Omega^{\prime},\operatorname{ad}_{\tt
P}(\Omega)(\Omega^{\prime\prime})\\},$
for all
$\Omega^{\prime}\in\operatorname{Alt}^{k^{\prime}}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$\Omega^{\prime\prime}\in\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$.
Therefore
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$ is a
graded Lie algebra for the super-Poisson bracket.
###### Definition 1.1.
[MPU09]
The super $\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket on
$\mathscr{E}(\mathfrak{g})$ is given by:
$\\{\Omega\otimes F,\Omega^{\prime}\otimes
F^{\prime}\\}=(-1)^{f\omega^{\prime}}\left(\\{\Omega,\Omega^{\prime}\\}\otimes
FF^{\prime}+(\Omega\wedge\Omega^{\prime})\otimes\\{F,F^{\prime}\\}\right),$
for all
$\Omega\in\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$\Omega^{\prime}\in\operatorname{Alt}^{\omega^{\prime}}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$F\in\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}){}^{f}$,
$F^{\prime}\in\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
By a straightforward computation, it is easy to obtain the following result:
###### Proposition 1.2.
The algebra $\mathscr{E}(\mathfrak{g})$ is a graded Lie algebra with the super
$\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket. More precisely, for all
$A\in\mathscr{E}^{(a,b)}(\mathfrak{g})$,
$A^{\prime}\in\mathscr{E}^{(a^{\prime},b^{\prime})}(\mathfrak{g})$ and
$A^{\prime\prime}\in\mathscr{E}^{(a^{\prime\prime},b^{\prime\prime})}(\mathfrak{g})$:
1. (1)
$\\{A^{\prime},A\\}=-(-1)^{aa^{\prime}+bb^{\prime}}\\{A,A^{\prime}\\}.$
2. (2)
$\\{\\{A,A^{\prime}\\},A^{\prime\prime}\\}=\\{A,\\{A^{\prime},A^{\prime\prime}\\}\\}-(-1)^{aa^{\prime}+bb^{\prime}}\\{A^{\prime},\\{A,A^{\prime\prime}\\}\\}.$
Moreover, one has $\\{A,A^{\prime}\wedge
A^{\prime\prime}\\}=\\{A,A^{\prime}\\}\wedge
A^{\prime\prime}+(-1)^{aa^{\prime}+bb^{\prime}}A^{\prime}\wedge\\{A,A^{\prime\prime}\\}.$
### 1.3. Super-derivations
Denote by $\mathcal{L}(\mathscr{E}(\mathfrak{g}))$ the vector space of
endomorphisms of $\mathscr{E}(\mathfrak{g})$. Let $\operatorname{ad}_{\tt
P}(A):=\\{A,.\\}$, for all $A\in\mathscr{E}(\mathfrak{g})$. Then
$\operatorname{ad}_{\tt P}(A)\in\mathcal{L}(\mathscr{E}(\mathfrak{g}))$ and:
$\operatorname{ad}_{\tt P}(\\{A,A^{\prime}\\})=\operatorname{ad}_{\tt
P}(A)\circ\operatorname{ad}_{\tt
P}(A^{\prime})-(-1)^{aa^{\prime}+bb^{\prime}}\operatorname{ad}_{\tt
P}(A^{\prime})\circ\operatorname{ad}_{\tt P}(A)$
for all $A$,
$A^{\prime}\in\mathscr{E}^{(a^{\prime},b^{\prime})}(\mathfrak{g})$. The space
$\mathcal{L}(\mathscr{E}(\mathfrak{g}))$ is naturally
$\mathbb{Z}\times\mathbb{Z}_{2}$-graded as follows:
$\deg(F)=(n,d),\ n\in\mathbb{Z},\ d\in\mathbb{Z}_{2}\text{ if
}\deg(F(A))=(n+a,d+b),\ \text{where}\ A\in\mathscr{E}^{(a,b)}(\mathfrak{g}).$
We denote by $\operatorname{End}^{n}_{f}(\mathscr{E}(\mathfrak{g}))$ the
subspace of endomorphisms of degree $(n,f)$ of
$\mathcal{L}(\mathscr{E}(\mathfrak{g}))$. It is clear that if
$A\in\mathscr{E}^{(a,b)}(\mathfrak{g})$ then $\operatorname{ad}_{\tt P}(A)$
has degree $(a-2,b)$. Moreover, it is known that
$\mathcal{L}(\mathscr{E}(\mathfrak{g}))$ is also a graded Lie algebra,
frequently denoted by $\mathfrak{gl}(\mathscr{E}(\mathfrak{g}))$ and equipped
with the Lie super-bracket:
$[F,G]=F\circ G-(-1)^{np+fg}G\circ F,\ \ \forall\
F\in\operatorname{End}^{n}_{f}(\mathscr{E}(\mathfrak{g})),\
G\in\operatorname{End}^{p}_{g}(\mathscr{E}(\mathfrak{g})).$
Therefore, by Proposition 1.2, we obtain that $\operatorname{ad}_{\tt P}$ is a
graded Lie algebra homomorphism from $\mathscr{E}(\mathfrak{g})$ onto
$\mathfrak{gl}(\mathscr{E}(\mathfrak{g}))$. In other words, one has:
$\operatorname{ad}_{\tt P}(\\{A,A^{\prime}\\})=[\operatorname{ad}_{\tt
P}(A),\operatorname{ad}_{\tt P}(A^{\prime})],\ \ \forall\
A,A^{\prime}\in\mathscr{E}(\mathfrak{g}).$
###### Definition 1.3.
A homogeneous endomorphism $D\in\mathfrak{gl}(\mathscr{E}(\mathfrak{g}))$ of
degree $(n,d)$ is called a super-derivation of degree $(n,d)$ of
$\mathscr{E}(\mathfrak{g})$ (for the super-exterior product) if it satisfies
the following condition:
$D(A\wedge A^{\prime})=D(A)\wedge A^{\prime}+(-1)^{na+db}A\wedge
D(A^{\prime}),\ \forall\ A\in\mathscr{E}^{(a,b)}(\mathfrak{g}),\
A^{\prime}\in\mathscr{E}(\mathfrak{g}).$
Denote by $\mathscr{D}^{n}_{d}(\mathscr{E}(\mathfrak{g}))$ the space of super-
derivations of degree $(n,d)$ of $\mathscr{E}(\mathfrak{g})$ then we obtain a
$\mathbb{Z}\times\mathbb{Z}_{2}$-gradation of the space of super-derivations
$\mathscr{D}(\mathscr{E}(\mathfrak{g}))$ of $\mathscr{E}(\mathfrak{g})$:
$\mathscr{D}(\mathscr{E}(\mathfrak{g}))=\bigoplus_{(n,d)\in\mathbb{Z}\times\mathbb{Z}_{2}}\mathscr{D}^{n}_{d}(\mathscr{E}(\mathfrak{g}))$
and $\mathscr{D}(\mathscr{E}(\mathfrak{g}))$ becomes a graded subalgebra of
$\mathfrak{gl}(\mathscr{E}(\mathfrak{g}))$ [NR66]. Moreover, the last formula
in Proposition 1.2 affirms that $\operatorname{ad}_{\tt
P}(A)\in\mathscr{D}(\mathscr{E}(\mathfrak{g}))$, for all
$A\in\mathscr{E}(\mathfrak{g})$.
Another example of a super-derivation in
$\mathscr{D}(\mathscr{E}(\mathfrak{g}))$ is given in [BP89] as follows. Let
$X\in\mathfrak{g}_{x}$ be a homogeneous element in $\mathfrak{g}$ of degree
$x$ and define the endomorphism $\operatorname{\iota}_{X}$ of
$\mathscr{E}(\mathfrak{g})$ by
$\operatorname{\iota}_{X}(A)(X_{1},...,X_{a-1})=(-1)^{xb}A(X,X_{1},...,X_{a-1})$
for all $A\in\mathscr{E}^{(a,b)}(\mathfrak{g}),\ X_{1},\dots,X_{a-1}\in V$.
Then one has
$\operatorname{\iota}_{X}(A\wedge
A^{\prime})=\operatorname{\iota}_{X}(A)\wedge
A^{\prime}+(-1)^{-a+xb}A\wedge\operatorname{\iota}_{X}(A^{\prime})$
holds for all $A\in\mathscr{E}^{(a,b)}(\mathfrak{g}),\
A^{\prime}\in\mathscr{E}(\mathfrak{g})$. It means that
$\operatorname{\iota}_{X}$ is a super- derivation of
$\mathscr{E}(\mathfrak{g})$ of degree $(-1,x)$. The proof of the following
Lemma is straightforward:
###### Lemma 1.4.
Let
$X_{\scriptscriptstyle{\overline{0}}}\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$
and
$X_{\scriptscriptstyle{\overline{1}}}\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
Then, for all $\Omega\otimes
F\in\operatorname{Alt}^{\omega}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$:
1. (1)
$\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}}(\Omega\otimes
F)=\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}}(\Omega)\otimes
F$,
2. (2)
$\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}}(\Omega\otimes
F)=(-1)^{\omega}\Omega\otimes\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}}(F)$.
###### Remark 1.5.
1. (1)
If
$\Omega\in\operatorname{Alt}^{\omega}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$
then
$\operatorname{\iota}_{X}(\Omega)(X_{1},\dots,X_{\omega-1})=\Omega(X,X_{1},\dots,X_{\omega-1})$
for all
$X,X_{1},\dots,X_{\omega-1}\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$.
That coincides with the previous definition of $\operatorname{\iota}_{X}$ on
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$.
2. (2)
Let $X$ be an element in a fixed Darboux basis of
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ and
$p\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}^{*}$ be its dual form. By
the Corollary II.1.52 in [Gie04] one has:
$\operatorname{\iota}_{X}(p^{n})(X^{n-1})=(-1)^{n}p^{n}(X^{n})=(-1)^{n}(-1)^{n(n-1)/2}n!.$
Moreover, $\displaystyle\frac{\partial p^{n}}{\partial
p}(X^{n-1})=n(p^{n-1})(X^{n-1})=(-1)^{(n-1)(n-2)/2}n!$. It implies that
$\operatorname{\iota}_{X}(p^{n})(X^{n-1})=-\displaystyle\frac{\partial
p^{n}}{\partial p}(X^{n-1}).$
Since each
$F\in\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
can be regarded as a polynomial in the variable $p$, one has the following
property:
$\operatorname{\iota}_{X}(F)=-\displaystyle\frac{\partial F}{\partial p},\
\forall\
F\in\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}).$
The super-derivations $\operatorname{\iota}_{X}$ play an important role in the
description of the space $\mathscr{D}(\mathscr{E}(\mathfrak{g}))$ (for
details, see [Gie04]). For instance, they can used to express the super-
derivation $\operatorname{ad}_{\tt P}(A)$ defined above:
###### Proposition 1.6.
Fix an orthonormal basis
$\\{X_{\scriptscriptstyle{\overline{0}}}^{1},\dots,X_{\scriptscriptstyle{\overline{0}}}^{m}\\}$
of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and a Darboux basis
$\mathcal{B}=\\{X_{\scriptscriptstyle{\overline{1}}}^{1},\dots,X_{\scriptscriptstyle{\overline{1}}}^{n},Y_{\scriptscriptstyle{\overline{1}}}^{1},\dots,Y_{\scriptscriptstyle{\overline{1}}}^{n}\\}$
of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$. Then the super
$\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket on
$\mathscr{E}(\mathfrak{g})$ is given by:
$\displaystyle\\{A,A^{\prime}\\}$ $\displaystyle=$
$\displaystyle(-1)^{\omega+f+1}\sum^{m}_{j=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(A)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(A^{\prime})$
$\displaystyle+(-1)^{\omega}\sum^{n}_{k=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(A)\wedge\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(A^{\prime})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(A)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(A^{\prime})\right)$
for all
$A\in\operatorname{Alt}^{\omega}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
and $A^{\prime}\in\mathscr{E}(\mathfrak{g})$.
###### Proof.
Let $A=\Omega\otimes
F\in\operatorname{Alt}^{\omega}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
and $A^{\prime}=\Omega^{\prime}\otimes
F^{\prime}\in\operatorname{Alt}^{\omega^{\prime}}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{f^{\prime}}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
The super $\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket of $A$ and
$A^{\prime}$ is defined by:
$\\{A,A^{\prime}\\}=(-1)^{f\omega^{\prime}}\left(\\{\Omega,\Omega^{\prime}\\}\otimes
FF^{\prime}+(\Omega\wedge\Omega^{\prime})\otimes\\{F,F^{\prime}\\}\right).$
By the definition of the super Poisson bracket on
$\operatorname{Alt}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$
combined with Lemma 1.4 (1), one has
$\displaystyle\\{\Omega,\Omega^{\prime}\\}\otimes FF^{\prime}$
$\displaystyle=(-1)^{\omega+1}\sum^{m}_{j=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\Omega)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\Omega^{\prime})\right)\otimes
FF^{\prime}$
$\displaystyle=(-1)^{f(\omega^{\prime}-1)+\omega+1}\sum^{m}_{j=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\Omega)\otimes
F\right)\wedge\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\Omega^{\prime})\otimes
F^{\prime}\right)$
$\displaystyle=(-1)^{f\omega^{\prime}+\omega+f+1}\sum^{m}_{j=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(A)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(A^{\prime}).$
Let $\\{p_{1},\dots,p_{n},q_{1},\dots,q_{n}\\}$ be the dual basis of
$\mathcal{B}$. By Remark 1.5 (2), the Poisson bracket on
$\operatorname{Sym}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$ can be
expressed by:
$\\{F,F^{\prime}\\}=\sum_{k=1}^{n}\left(\displaystyle\frac{\partial
F}{\partial p_{k}}\displaystyle\frac{\partial F^{\prime}}{\partial
q_{k}}-\displaystyle\frac{\partial F}{\partial
q_{k}}\displaystyle\frac{\partial F^{\prime}}{\partial
p_{k}}\right)=\sum^{n}_{k=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(F)\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(F^{\prime})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(F)\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(F^{\prime})\right).$
Combined with Lemma 1.4 (2), we obtain
$(\Omega\wedge\Omega^{\prime})\otimes\\{F,F^{\prime}\\}=(\Omega\wedge\Omega^{\prime})\otimes\sum^{n}_{k=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(F)\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(F^{\prime})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(F)\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(F^{\prime})\right)$
$=(-1)^{(f-1)\omega^{\prime}}\sum^{n}_{k=1}\left(\left(\Omega\otimes\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(F)\right)\wedge\left(\Omega^{\prime}\otimes\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(F^{\prime})\right)-\left(\Omega\otimes\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(F)\right)\wedge\left(\Omega^{\prime}\otimes\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(F^{\prime})\right)\right)$
$=(-1)^{f\omega^{\prime}+\omega}\sum^{n}_{k=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(A)\wedge\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(A^{\prime})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(A)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(A^{\prime})\right).$
The result follows.
∎
Since the bilinear form $B$ is non- degenerate and even, then there is an
(even) isomorphism $\phi$ from $\mathfrak{g}$ onto $\mathfrak{g}^{*}$ defined
by $\phi(X)(Y)=B(X,Y)$, for all $X$, $Y\in\mathfrak{g}$.
###### Corollary 1.7.
The following expressions:
1. (1)
$\\{\alpha,A\\}=\operatorname{\iota}_{\phi^{-1}(\alpha)}(A)$,
2. (2)
$\\{\alpha,\alpha^{\prime}\\}=B(\phi^{-1}(\alpha),\phi^{-1}(\alpha^{\prime}))$,
hold for all $\alpha,\alpha^{\prime}\in\mathfrak{g}^{*},\
A\in\mathscr{E}(\mathfrak{g})$.
###### Proof.
1. (1)
We apply Proposition 1.6, respectively for
$\alpha=(X_{\scriptscriptstyle{\overline{0}}}^{i})^{*}=\phi(X_{\scriptscriptstyle{\overline{0}}}^{i})$,
$i=1,\dots,m$,
$\alpha=(Y_{\scriptscriptstyle{\overline{1}}}^{l})^{*}=\phi(X_{\scriptscriptstyle{\overline{1}}}^{l})$
and
$\alpha=(-X_{\scriptscriptstyle{\overline{1}}}^{l})^{*}=\phi(Y_{\scriptscriptstyle{\overline{1}}}^{l})$,
$l=1,\dots,n$ to obtain the result.
2. (2)
Let $\alpha\in\mathfrak{g}_{x}^{*},\
\alpha^{\prime}\in\mathfrak{g}_{x^{\prime}}^{*}$ be homogeneous forms in
$\mathfrak{g}^{*}$, one has
$\displaystyle\hskip
28.45274pt\\{\alpha,\alpha^{\prime}\\}=\operatorname{\iota}_{\phi^{-1}(\alpha)}(\alpha^{\prime})=(-1)^{xx^{\prime}}\alpha^{\prime}(\phi^{-1}(\alpha))=(-1)^{xx^{\prime}}B(\phi^{-1}(\alpha^{\prime}),\phi^{-1}(\alpha))$
$\displaystyle=B(\phi^{-1}(\alpha),\phi^{-1}(\alpha^{\prime})).$
∎
In this section, Proposition 1.6 and Corollary 1.7 are enough for our purpose.
But as a consequence of Lemma 6.9 in [PU07], one has a more general result of
Proposition 1.6 as follows:
###### Proposition 1.8.
Let
$\\{X_{\scriptscriptstyle{\overline{0}}}^{1},\dots,X_{\scriptscriptstyle{\overline{0}}}^{m}\\}$
be a basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$\\{\alpha_{1},\dots,\alpha_{m}\\}$ its dual basis. Let
$\\{Y_{\scriptscriptstyle{\overline{0}}}^{1},\dots,Y_{\scriptscriptstyle{\overline{0}}}^{m}\\}$
be the basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ defined by
$Y_{\scriptscriptstyle{\overline{0}}}^{i}=\phi^{-1}(\alpha_{i})$. Set
$\mathcal{B}=\\{X_{\scriptscriptstyle{\overline{1}}}^{1},\dots,X_{\scriptscriptstyle{\overline{1}}}^{n},Y_{\scriptscriptstyle{\overline{1}}}^{1},\dots,Y_{\scriptscriptstyle{\overline{1}}}^{n}\\}$
be a Darboux basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$. Then
the super $\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket on
$\mathscr{E}(\mathfrak{g})$ is given by
$\displaystyle\\{A,A^{\prime}\\}$ $\displaystyle=$
$\displaystyle(-1)^{\omega+f+1}\sum^{m}_{i,j=1}B(Y_{\scriptscriptstyle{\overline{0}}}^{i},Y_{\scriptscriptstyle{\overline{0}}}^{j})\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{i}}(A)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(A^{\prime})$
$\displaystyle+(-1)^{\omega}\sum^{n}_{k=1}\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(A)\wedge\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(A^{\prime})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(A)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(A^{\prime})\right)$
for all
$A\in\operatorname{Alt}^{\omega}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{f}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
and $A^{\prime}\in\mathscr{E}(\mathfrak{g})$.
### 1.4. Super-antisymmetric linear maps
Consider the vector space
$\mathcal{E}=\bigoplus_{n\in\mathbb{Z}}\limits\mathcal{E}^{n},$
where $\mathcal{E}^{n}=\\{0\\}$ if $n\leq-2$, $\mathcal{E}^{-1}=\mathfrak{g}$
and $\mathcal{E}^{n}$ is the space of super-antisymmetric $n+1$-linear
mappings from $\mathfrak{g}^{n+1}$ onto $\mathfrak{g}$. Each of the subspaces
$\mathcal{E}^{n}$ is $\mathbb{Z}_{2}$-graded then the space $\mathcal{E}$ is
$\mathbb{Z}\times\mathbb{Z}_{2}$-graded by
$\mathcal{E}=\bigoplus^{n\in\mathbb{Z}}_{f\in\mathbb{Z}_{2}}\limits\mathcal{E}^{n}_{f}.$
There is a natural isomorphism between the spaces $\mathcal{E}$ and
$\mathscr{E}(\mathfrak{g})\otimes\mathfrak{g}$. Moreover, $\mathcal{E}$ is a
graded Lie algebra, called the graded Lie algebra of $\mathfrak{g}$. It is
isomorphic to $\mathscr{D}(\mathscr{E}(\mathfrak{g}))$ by the graded Lie
algebra isomorphism $D$ such that if $F=\Omega\otimes
X\in\mathcal{E}^{n}_{\omega+x}$ then
$D_{F}=-(-1)^{x\omega}\Omega\wedge\iota_{X}\in\mathscr{D}^{n}_{\omega+x}(\mathscr{E}(\mathfrak{g}))$.
For more details on the Lemma below, see for instance, [BP89] and [Gie04].
###### Lemma 1.9.
Fix $F\in\mathcal{E}^{1}_{\scriptscriptstyle{\overline{0}}}$, denote by
$d=D_{F}$ and define the product $[X,Y]=F(X,Y)$, for all $X,Y\in\mathfrak{g}$.
Then one has
1. (1)
$d(\phi)(X,Y)=-\phi([X,Y])$, for all $X,Y\in\mathfrak{g},\
\phi\in\mathfrak{g}^{*}.$
2. (2)
The product $[~{},~{}]$ becomes a Lie super-bracket if and only if $d^{2}=0$.
In this case, $d$ is called a super-exterior differential of
$\mathscr{E}(\mathfrak{g})$.
### 1.5. Quadratic Lie superalgebras
The construction of graded Lie algebras and the super
$\mathbb{Z}\times\mathbb{Z}_{2}$-Poisson bracket above can be applied to the
theory of quadratic Lie superalgebras. This later is regarded as a graded
version of the quadratic Lie algebra case and we obtain then similar results.
###### Definition 1.10.
A quadratic Lie superalgebra $(\mathfrak{g},B)$ is a $\mathbb{Z}_{2}$-graded
vector space $\mathfrak{g}$ equipped with a non-degenerate even supersymmetric
bilinear form $B$ and a Lie superalgebra structure such that $B$ is invariant,
i.e. $B([X,Y],Z)=B(X,[Y,Z])$, for all $X$, $Y$, $Z\in\mathfrak{g}$.
###### Proposition 1.11.
Let $(\mathfrak{g},B)$ be a quadratic Lie superalgebra and define a trilinear
form $I$ on $\mathfrak{g}$ by
$I(X,Y,Z)=B([X,Y],Z),\ \forall\ X,Y,Z\in\mathfrak{g}.$
Then one has
1. (1)
$I\in\mathscr{E}^{(3,\overline{0})}(\mathfrak{g})=\operatorname{Alt}^{3}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\oplus\left(\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})\right).$
2. (2)
$d=-\operatorname{ad}_{\tt P}(I).$
3. (3)
$\\{I,I\\}=0.$
###### Proof.
The assertion (1) follows clearly from the properties of $B$. Note that
$B([\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}],\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=B([\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}],\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$.
For (2), fix an orthonormal basis
$\\{X_{\scriptscriptstyle{\overline{0}}}^{1},\dots,X_{\scriptscriptstyle{\overline{0}}}^{m}\\}$
of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and a Darboux basis
$\\{X_{\scriptscriptstyle{\overline{1}}}^{1},\dots,X_{\scriptscriptstyle{\overline{1}}}^{n},\linebreak
Y_{\scriptscriptstyle{\overline{1}}}^{1},\dots,Y_{\scriptscriptstyle{\overline{1}}}^{n}\\}$
of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$. Let
$\\{\alpha_{1},\dots,\alpha_{m}\\}$ and
$\\{\beta_{1},\dots,\beta_{n},\gamma_{1},\dots,\gamma_{n}\\}$ be their dual
basis, respectively. Then for all $X,Y\in\mathfrak{g},\ i=1,...,m,\ l=1,...,n$
we have:
$\displaystyle\operatorname{ad}_{\tt P}(I)(\alpha_{i})(X,Y)$
$\displaystyle=\left(\sum^{m}_{j=1}\limits\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\alpha_{i})-\sum^{n}_{k=1}\limits\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(\alpha_{i})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(\alpha_{i})\right)\right)(X,Y)$
$\displaystyle=\left(\sum^{m}_{j=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\alpha_{i})\right)(X,Y)=\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{i}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{i}}(\alpha_{i})\right)(X,Y)$
$\displaystyle=B(X_{\scriptscriptstyle{\overline{0}}}^{i},[X,Y])=\alpha_{i}([X,Y])=-d(\alpha_{i})(X,Y),$
$\displaystyle\operatorname{ad}_{\tt P}(I)(\beta_{l})(X,Y)$
$\displaystyle=\left(\sum^{m}_{j=1}\limits\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(\beta_{l})-\sum^{n}_{k=1}\limits\left(\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(\beta_{l})-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(\beta_{l})\right)\right)(X,Y)$
$\displaystyle=\left(\sum^{n}_{k=1}\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(\beta_{l})\right)(X,Y)=\left(\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{l}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{l}}(\beta_{l})\right)(X,Y)$
$\displaystyle=-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{l}}(I)(X,Y)=-B(Y_{\scriptscriptstyle{\overline{1}}}^{l},[X,Y])=\beta_{l}([X,Y])=-d(\beta_{l})(X,Y).$
Similarly, $\operatorname{ad}_{\tt P}(I)(\gamma_{l})=-d(\gamma_{l})$ for
$1\leq l\leq n$. Therefore, $d=-\operatorname{ad}_{\tt P}(I)$.
Moreover, $\operatorname{ad}_{\tt P}(\\{I,I\\})=[\operatorname{ad}_{\tt
P}(I),\operatorname{ad}_{\tt P}(I)]=[d,d]=2d^{2}=0$. Therefore, for all $1\leq
i\leq m$, $1\leq j,k\leq n$ one has $\\{\alpha_{i},\\{I,I\\}\\}$ =
$\\{\beta_{j},\\{I,I\\}\\}$ = $\\{\gamma_{k},\\{I,I\\}\\}=0$. Those imply
$\iota_{X}\left(\\{I,I\\}\right)=0$ for all $X\in\mathfrak{g}$ and hence, we
obtain $\\{I,I\\}=0$.
∎
Conversely, let $\mathfrak{g}$ be a quadratic $\mathbb{Z}_{2}$-graded vector
space equipped with a bilinear form $B$ and $I$ be an element in
$\mathscr{E}^{(3,\overline{0})}(\mathfrak{g})$. Define
$d=-\operatorname{ad}_{\tt P}(I)$ then
$d\in\mathscr{D}^{1}_{\scriptscriptstyle{\overline{0}}}(\mathscr{E}(\mathfrak{g}))$.
Therefore, $d^{2}=0$ if and only if $\\{I,I\\}=0$. Let $F$ be the structure in
$\mathfrak{g}$ corresponding to $d$ by the isomorphism $D$ in Lemma 1.9, one
has
###### Proposition 1.12.
$F$ becomes a Lie superalgebra structure if and only if $\\{I,I\\}=0$. In this
case, with the notation $[X,Y]:=F(X,Y)$ one has:
$I(X,Y,Z)=B([X,Y],Z),\ \forall\ X,Y,Z\in\mathfrak{g}.$
Moreover, the bilinear form $B$ is invariant.
###### Proof.
We need to prove that if $F$ is a Lie superalgebra structure then
$I(X,Y,Z)=B([X,Y],Z)$, for all $X,Y,Z\in\mathfrak{g}$. Indeed, let
$\\{X_{\scriptscriptstyle{\overline{0}}}^{1},\dots,X_{\scriptscriptstyle{\overline{0}}}^{m}\\}$
be an orthonormal basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$
and
$\\{X_{\scriptscriptstyle{\overline{1}}}^{1},\dots,X_{\scriptscriptstyle{\overline{1}}}^{n},Y_{\scriptscriptstyle{\overline{1}}}^{1},\dots,Y_{\scriptscriptstyle{\overline{1}}}^{n}\\}$
be a Darboux basis of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ then
one has
$d=-\operatorname{ad}_{\tt
P}(I)=-\sum^{m}_{j=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}+\sum^{n}_{k=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}-\sum^{n}_{k=1}\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\wedge\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}.$
It implies that
$F=\sum^{m}_{j=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{j}}(I)\otimes{X_{\scriptscriptstyle{\overline{0}}}^{j}}+\sum^{n}_{k=1}\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\otimes{Y_{\scriptscriptstyle{\overline{1}}}^{k}}-\sum^{n}_{k=1}\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{k}}(I)\otimes{X_{\scriptscriptstyle{\overline{1}}}^{k}}.$
Therefore, for all $i$ we obtain
$B([X,Y],{X_{\scriptscriptstyle{\overline{0}}}^{i}})=\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{0}}}^{i}}(I)(X,Y)=I(X_{\scriptscriptstyle{\overline{0}}}^{i},X,Y)=I(X,Y,X_{\scriptscriptstyle{\overline{0}}}^{i}),$
$B([X,Y],{X_{\scriptscriptstyle{\overline{1}}}^{i}})=-\operatorname{\iota}_{X_{\scriptscriptstyle{\overline{1}}}^{i}}(I)(X,Y)=-I(X_{\scriptscriptstyle{\overline{1}}}^{i},X,Y)=I(X,Y,X_{\scriptscriptstyle{\overline{1}}}^{i}),$
$\text{ and
}B([X,Y],{Y_{\scriptscriptstyle{\overline{1}}}^{i}})=-\operatorname{\iota}_{Y_{\scriptscriptstyle{\overline{1}}}^{i}}(I)(X,Y)=-I(Y_{\scriptscriptstyle{\overline{1}}}^{i},X,Y)=I(X,Y,Y_{\scriptscriptstyle{\overline{1}}}^{i}).$
These show that $I(X,Y,Z)=B([X,Y],Z)$, for all $X,Y,Z\in\mathfrak{g}$. Since
$I$ is super-antisymmetric and $B$ is supersymmetric, then one can show that
$B$ is invariant. ∎
The two previous propositions show that on a quadratic $\mathbb{Z}_{2}$-graded
vector space $(\mathfrak{g},B)$, quadratic Lie superalgebra structures with
the same $B$ are in one to one correspondence with elements
$I\in\mathscr{E}^{(3,{\scriptscriptstyle{\overline{0}}})}(\mathfrak{g})$
satisfying $\\{I,I\\}=0$ and such that the super-exterior differential of
$\mathscr{E}(\mathfrak{g})$ is $d=-\operatorname{ad}_{\tt P}(I)$. This
correspondence provides an approach to the theory of quadratic Lie
superalgebras through $I$.
###### Definition 1.13.
Given a quadratic Lie superalgebra $(\mathfrak{g},B)$. The element $I$ defined
as above is also an invariant of $\mathfrak{g}$ since $\mathcal{L}_{X}(I)=0$,
for all $X\in\mathfrak{g}$ where
$\mathcal{L}_{X}=D(\operatorname{ad}_{\mathfrak{g}}(X))$ is the Lie super-
derivation of $\mathfrak{g}$. Therefore, $I$ is called the associated
invariant of $\mathfrak{g}$.
The following Lemma is a simple, yet interesting result.
###### Lemma 1.14.
Let $(\mathfrak{g},B)$ be a quadratic Lie superalgebra and $I$ be its
associated invariant. Then $\operatorname{\iota}_{X}(I)=0$ if and only if
$X\in\mathscr{Z}(\mathfrak{g})$.
###### Proof.
Since
$\operatorname{\iota}_{X}(I)(\mathfrak{g},\mathfrak{g})=B(X,[\mathfrak{g},\mathfrak{g}])$
and $\mathscr{Z}(\mathfrak{g})=[\mathfrak{g},\mathfrak{g}]^{\bot}$ where
$[\mathfrak{g},\mathfrak{g}]^{\bot}$ denotes the orthogonal subspace of
$[\mathfrak{g},\mathfrak{g}]$. We have then $\operatorname{\iota}_{X}(I)=0$ if
and only if $X\in\mathscr{Z}(\mathfrak{g})$. ∎
###### Definition 1.15.
Let $(\mathfrak{g},B)$ and $(\mathfrak{g}^{\prime},B^{\prime})$ be two
quadratic Lie superalgebras. We say that $(\mathfrak{g},B)$ and
$(\mathfrak{g}^{\prime},B^{\prime})$ are isometrically isomorphic (or
i-isomorphic) if there exists a Lie superalgebra isomorphism $A$ from
$\mathfrak{g}$ onto $\mathfrak{g}^{\prime}$ satisfying
$B^{\prime}(A(X),A(Y))=B(X,Y),\ \forall\ X,Y\in\mathfrak{g}.$
In other words, $A$ is an i-isomorphism if it is a (necessarily even) Lie
superalgebra isomorphism and an isometry. We write
$\mathfrak{g}\overset{\mathrm{i}}{\simeq}\mathfrak{g}^{\prime}$.
Note that two isomorphic quadratic Lie superalgebras $(\mathfrak{g},B)$ and
$(\mathfrak{g}^{\prime},B^{\prime})$ are not necessarily i-isomorphic by the
example below:
###### Example 1.16.
Let $\mathfrak{g}=\mathfrak{osp}(1,2)$ and $B$ its Killing form. Recall that
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\mathfrak{o}(3)$. Consider
another bilinear form $B^{\prime}=\lambda B$, $\lambda\in\mathbb{C}$,
$\lambda\neq 0$. In this case, $(\mathfrak{g},B)$ and $(\mathfrak{g},\lambda
B)$ can not be i-isomorphic if $\lambda\neq 1$ since
$(\mathfrak{g}_{\overline{0}},B)$ and $(\mathfrak{g}_{\overline{0}},\lambda
B)$ are not i-isomorphic.
## 2\. The dup-number of quadratic Lie superalgebras
Let $(\mathfrak{g},B)$ be a quadratic Lie superalgebra and $I$ be its
associated invariant. Then by Proposition 1.11 we have a decomposition
$I=I_{0}+I_{1}$
where
$I_{0}\in\operatorname{Alt}^{3}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$
and
$I_{1}\in\operatorname{Alt}^{1}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$.
Since $\\{I,I\\}=0$, then $\\{I_{0},I_{0}\\}=0$. It means that
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is a quadratic Lie algebra
with the associated 3-form $I_{0}$, a rather obvious result. It is easy to see
that $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is Abelian (resp.
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]=\\{0\\}$)
if and only if $I_{0}=0$ (resp. $I_{1}=0$). These cases will be fully studied
in the sequel. Define the following subspaces of $\mathfrak{g}^{*}$:
$\displaystyle\mathscr{V}_{I}$ $\displaystyle=$
$\displaystyle\\{\alpha\in\mathfrak{g}^{*}\mid\ \alpha\wedge I=0\\},$
$\displaystyle\mathscr{V}_{I_{0}}$ $\displaystyle=$
$\displaystyle\\{\alpha\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}\mid\
\alpha\wedge I_{0}=0\\},$ $\displaystyle\mathscr{V}_{I_{1}}$ $\displaystyle=$
$\displaystyle\\{\alpha\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}\mid\
\alpha\wedge I_{1}=0\\}.$
###### Lemma 2.1.
Let $\mathfrak{g}$ be a non-Abelian quadratic Lie superalgebra then one has
1. (1)
$\dim(\mathscr{V}_{I})\in\\{0,1,3\\}$,
2. (2)
$\dim(\mathscr{V}_{I})=3$ if and only if $I_{1}=0$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is non-Abelian and $I_{0}$
is decomposable in
$\operatorname{Alt}^{3}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$.
###### Proof.
Let
$\alpha=\alpha_{0}+\alpha_{1}\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}\oplus\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}^{*}$
then one has
$\alpha\wedge I=\alpha_{0}\wedge I_{0}+\alpha_{0}\wedge I_{1}+\alpha_{1}\wedge
I_{0}+\alpha_{1}\wedge I_{1},$
where $\alpha_{0}\wedge
I_{0}\in\operatorname{Alt}^{4}(\mathfrak{g}_{\overline{0}})$,
$\alpha_{0}\wedge
I_{1}\in\operatorname{Alt}^{2}(\mathfrak{g}_{\overline{0}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$,
$\alpha_{1}\wedge
I_{0}\in\operatorname{Alt}^{3}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
and $\alpha_{1}\wedge
I_{1}\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{3}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
Hence, $\alpha\wedge I=0$ if and only if $\alpha_{1}=0$ and $\alpha_{0}\wedge
I_{0}=\alpha_{0}\wedge I_{1}=0$. It means that
$\mathscr{V}_{I}=\mathscr{V}_{I_{0}}\cap\mathscr{V}_{I_{1}}$. If $I_{0}\neq 0$
then $\dim(\mathscr{V}_{I_{0}})\in\\{0,1,3\\}$ and if $I_{1}\neq 0$ then
$\dim(\mathscr{V}_{I_{1}})\in\\{0,1\\}$. Therefore,
$\dim(\mathscr{V}_{I})\in\\{0,1,3\\}$ and $\dim(\mathscr{V}_{I})=3$ if and
only if $I_{1}=0$ and $\dim(\mathscr{V}_{I_{0}})=3$. ∎
The previous Lemma allows us to introduce the notion of dup-number for
quadratic Lie superalgebras as we did for quadratic Lie algebras.
###### Definition 2.2.
Let $(\mathfrak{g},B)$ be a non-Abelian quadratic Lie superalgebra and $I$ be
its associated invariant. The $\operatorname{dup}$-number
$\operatorname{dup}(\mathfrak{g})$ is defined by
$\operatorname{dup}(\mathfrak{g})=\dim(\mathscr{V}_{I}).$
Given a subspace $W$ of $\mathfrak{g}$, if $W$ is non-degenerate (with respect
to the bilinear form $B$), that is, if the restriction of $B$ on $W\times W$
is non-degenerate, then the orthogonal subspace $W^{\bot}$ of $W$ is also non-
degenerate. In this case, we use the notation
$\mathfrak{g}=W{\ \overset{\perp}{\mathop{\oplus}}\ }W^{\bot}.$
The decomposition result below is a generalization of the quadratic Lie
algebra case. Its proof can be found in [PU07] and [DPU10].
###### Proposition 2.3.
Let $(\mathfrak{g},B)$ be a non-Abelian quadratic Lie superalgebra. Then there
are a central ideal $\mathfrak{z}$ and an ideal $\mathfrak{l}\neq\\{0\\}$ such
that:
1. (1)
$\mathfrak{g}=\mathfrak{z}{\ \overset{\perp}{\mathop{\oplus}}\ }\mathfrak{l}$
where $\left(\mathfrak{z},B|_{\mathfrak{z}\times\mathfrak{z}}\right)$ and
$\left(\mathfrak{l},B|_{\mathfrak{l}\times\mathfrak{l}}\right)$ are quadratic
Lie superalgebras. Moreover, $\mathfrak{l}$ is non-Abelian.
2. (2)
The center $\mathscr{Z}(\mathfrak{l})$ is totally isotropic, i.e.
$\mathscr{Z}(\mathfrak{l})\subset[\mathfrak{l},\mathfrak{l}]$.
3. (3)
Let $\mathfrak{g}^{\prime}$ be a quadratic Lie superalgebra and
$A:\mathfrak{g}\to\mathfrak{g}^{\prime}$ be a Lie superalgebra isomorphism.
Then
$\mathfrak{g}^{\prime}=\mathfrak{z}^{\prime}{\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{l}^{\prime}$
where $\mathfrak{z}^{\prime}=A(\mathfrak{z})$ is central,
$\mathfrak{l}^{\prime}=A(\mathfrak{z})^{\perp}$,
$\mathscr{Z}(\mathfrak{l}^{\prime})$ is totally isotropic and $\mathfrak{l}$
and $\mathfrak{l}^{\prime}$ are isomorphic. Moreover if $A$ is an
i-isomorphism, then $\mathfrak{l}$ and $\mathfrak{l}^{\prime}$ are
i-isomorphic.
The Lemma below shows that the previous decomposition has a good behavior with
respect to the $\operatorname{dup}$-number.
###### Lemma 2.4.
Let $\mathfrak{g}$ be a non-Abelian quadratic Lie superalgebra. Write
$\mathfrak{g}=\mathfrak{z}{\ \overset{\perp}{\mathop{\oplus}}\ }\mathfrak{l}$
as in Proposition 2.3 then
$\operatorname{dup}(\mathfrak{g})=\operatorname{dup}(\mathfrak{l})$.
###### Proof.
Since $[\mathfrak{z},\mathfrak{g}]=\\{0\\}$ then
$I\in\mathscr{E}^{(3,\overline{0})}(\mathfrak{l})$. Let
$\alpha\in\mathfrak{g}^{*}$ such that $\alpha\wedge I=0$, we show that
$\alpha\in\mathfrak{l}^{*}$. Assume that $\alpha=\alpha_{1}+\alpha_{2}$, where
$\alpha_{1}\in\mathfrak{z}^{*}$ and $\alpha_{2}\in\mathfrak{l}^{*}$. Since
$\alpha\wedge I=0$, $\alpha_{1}\wedge
I\in\mathscr{E}(\mathfrak{z})\otimes\mathscr{E}(\mathfrak{l})$ and
$\alpha_{2}\wedge I\in\mathscr{E}(\mathfrak{l})$ then one has
$\alpha_{1}\wedge I=0$. Therefore, $\alpha_{1}=0$ since $I$ is nonzero in
$\mathscr{E}^{(3,\overline{0})}(\mathfrak{l})$. That means
$\alpha\in\mathfrak{l}^{*}$ and then
$\operatorname{dup}(\mathfrak{g})=\operatorname{dup}(\mathfrak{l})$. ∎
Clearly, $\mathfrak{z}=\\{0\\}$ if and only if $\mathscr{Z}(\mathfrak{g})$ is
totally isotropic. By the above Lemma, it is enough to restrict our study on
the $\operatorname{dup}$-number of non- Abelian quadratic Lie superalgebras
with totally isotropic center.
###### Definition 2.5.
A quadratic Lie superalgebra $\mathfrak{g}$ is reduced if it satisfies:
1. (1)
$\mathfrak{g}\neq\\{0\\}$
2. (2)
$\mathscr{Z}(\mathfrak{g})$ is totally isotropic.
Notice that a reduced quadratic Lie superalgebra is necessarily non-Abelian.
###### Definition 2.6.
Let $\mathfrak{g}$ be a non-Abelian quadratic Lie superalgebra. We say that:
1. (1)
$\mathfrak{g}$ is an ordinary quadratic Lie superalgebra if
$\operatorname{dup}(\mathfrak{g})=0$.
2. (2)
$\mathfrak{g}$ is a singular quadratic Lie superalgebra if
$\operatorname{dup}(\mathfrak{g})\geq 1$.
* (i)
$\mathfrak{g}$ is a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$ if $\operatorname{dup}(\mathfrak{g})=1$.
* (ii)
$\mathfrak{g}$ is a singular quadratic Lie superalgebra of type
$\mathsf{S}_{3}$ if $\operatorname{dup}(\mathfrak{g})=3$.
By Lemma 2.1, if $\mathfrak{g}$ is a singular quadratic Lie superalgebra of
type $\mathsf{S}_{3}$ then $I=I_{0}$ is decomposable in
$\operatorname{Alt}^{3}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$.
One has
$I(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=B([\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}],\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$.
It implies
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}]=\\{0\\}$ since
$B$ is non-degenerate. Hence in this case,
$\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}$ is a central ideal,
$\mathfrak{g}_{\overline{0}}$ is a singular quadratic Lie algebra of type
$\mathsf{S}_{3}$ and then the classification is known in [PU07]. Therefore, we
are mainly interested in singular quadratic Lie superalgebras of type
$\mathsf{S}_{1}$.
Before proceeding, we give other simple properties of singular quadratic Lie
superalgebras:
###### Proposition 2.7.
Let $(\mathfrak{g},B)$ be a singular quadratic Lie superalgebra. If
$\mathfrak{g}_{\overline{0}}$ is non-Abelian then
$\mathfrak{g}_{\overline{0}}$ is a singular quadratic Lie algebra.
###### Proof.
By the proof of Lemma 2.1, one has
$\mathscr{V}_{I}=\mathscr{V}_{I_{0}}\cap\mathscr{V}_{I_{1}}$. Therefore,
$\dim(\mathscr{V}_{I_{0}})\geq 1$. It means that $\mathfrak{g}_{\overline{0}}$
is a singular quadratic Lie algebra. ∎
Given $(\mathfrak{g},B)$ a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$. Fix $\alpha\in\mathscr{V}_{I}$ and choose
$\Omega_{0}\in\operatorname{Alt}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$\Omega_{1}\in\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
such that $I=\alpha\wedge\Omega_{0}+\alpha\otimes\Omega_{1}$. Then one has
$\\{I,I\\}=\\{\alpha\wedge\Omega_{0},\alpha\wedge\Omega_{0}\\}+2\\{\alpha\wedge\Omega_{0},\alpha\\}\otimes\Omega_{1}+\\{\alpha,\alpha\\}\otimes\Omega_{1}\Omega_{1}.$
By the equality $\\{I,I\\}=0$, one has
$\\{\alpha\wedge\Omega_{0},\alpha\wedge\Omega_{0}\\}=0$,
$\\{\alpha,\alpha\\}=0$ and $\\{\alpha,\alpha\wedge\Omega_{0}\\}=0$. These
imply that $\\{\alpha,I\\}=0$. Hence, if we set $X_{0}=\phi^{-1}(\alpha)$ then
$X_{0}\in\mathscr{Z}(\mathfrak{g})$ and $B(X_{0},X_{0})=0$ (Corollary 1.7 and
Lemma 1.14).
###### Proposition 2.8.
Let $(\mathfrak{g},B)$ be a singular quadratic Lie superalgebra. If
$\mathfrak{g}$ is reduced then
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is reduced.
###### Proof.
As above, if $\mathfrak{g}$ is a singular quadratic Lie superalgebra of type
$\mathsf{S}_{3}$ then $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ is
central. By $\mathfrak{g}$ reduced and
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ non-degenerate,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ must be zero and then the
result follows.
If $\mathfrak{g}$ is a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$. Assume that
$\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}$ is not reduced, i.e.
$\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}=\mathfrak{z}{\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{l}$ where $\mathfrak{z}$ is a
non-trivial central ideal of
$\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}$, there is
$X\in\mathfrak{z}$ such that $B(X,X)=1$. Since $\mathfrak{g}$ is singular of
type $\mathsf{S}_{1}$ then $\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}$
is also singular. Hence, the element $X_{0}$ defined as above must be in
$\mathfrak{l}$ and
$I_{0}=\alpha\wedge\Omega_{0}\in\operatorname{Alt}^{3}(\mathfrak{l})$ (see
[DPU10] for details). We also have $B(X,X_{0})=0$.
Let $\beta=\phi(X)$ so
$\operatorname{\iota}_{X}(I)=\\{\beta,I\\}=\\{\beta,\alpha\wedge\Omega_{0}+\alpha\otimes\Omega_{1}\\}=0$.
That means $X\in\mathscr{Z}(\mathfrak{g})$. This is a contradiction since
$\mathfrak{g}$ is reduced. Hence
$\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}$ must be reduced. ∎
###### Lemma 2.9.
Let $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ be non-Abelian quadratic Lie
superalgebras. Then $\mathfrak{g}_{1}{\ \overset{\perp}{\mathop{\oplus}}\
}\mathfrak{g}_{2}$ is an ordinary quadratic Lie algebra.
###### Proof.
Set $\mathfrak{g}=\mathfrak{g}_{1}{\ \overset{\perp}{\mathop{\oplus}}\
}\mathfrak{g}_{2}$. Denote by $I$, $I_{1}$ and $I_{2}$ their non-trivial
associated invariants, respectively. One has
$\mathscr{E}(\mathfrak{g})=\mathscr{E}(\mathfrak{g}_{1})\otimes\mathscr{E}(\mathfrak{g}_{2})$,
$\mathscr{E}^{k}(\mathfrak{g})=\bigoplus_{r+s=k}\limits\mathscr{E}^{r}(\mathfrak{g}_{1})\otimes\mathscr{E}^{s}(\mathfrak{g}_{2})$
and $I=I_{1}+I_{2}$ where $I_{1}\in\mathscr{E}^{3}(\mathfrak{g}_{1})$,
$I_{2}\in\mathscr{E}^{3}(\mathfrak{g}_{2})$. Therefore, if
$\alpha=\alpha_{1}+\alpha_{2}\in\mathfrak{g}_{1}^{*}\oplus\mathfrak{g}_{2}^{*}$
such that $\alpha\wedge I=0$ then $\alpha_{1}=\alpha_{2}=0$. ∎
###### Definition 2.10.
A quadratic Lie superalgebra $\mathfrak{g}$ is indecomposable if
$\mathfrak{g}=\mathfrak{g}_{1}{\ \overset{\perp}{\mathop{\oplus}}\
}\mathfrak{g}_{2}$, with $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ ideals of
$\mathfrak{g}$, then $\mathfrak{g}_{1}$ or $\mathfrak{g}_{2}=\\{0\\}$.
The following result shows that indecomposable and reduced notions are
equivalent for singular quadratic Lie superalgebras.
###### Proposition 2.11.
Let $\mathfrak{g}$ be a singular quadratic Lie superalgebra. Then
$\mathfrak{g}$ is reduced if and only if $\mathfrak{g}$ is indecomposable.
###### Proof.
If $\mathfrak{g}$ is indecomposable then it is obvious that $\mathfrak{g}$ is
reduced. If $\mathfrak{g}$ is reduced, assume that
$\mathfrak{g}=\mathfrak{g}_{1}{\ \overset{\perp}{\mathop{\oplus}}\
}\mathfrak{g}_{2}$, with $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ ideals of
$\mathfrak{g}$, then
$\mathscr{Z}(\mathfrak{g}_{i})\subset[\mathfrak{g}_{i},\mathfrak{g}_{i}]$ for
$i=1,2$. Therefore, $\mathfrak{g}_{i}$ is reduced or
$\mathfrak{g}_{i}=\\{0\\}$. If $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are
both reduced, by Lemma 2.9, then $\mathfrak{g}$ is ordinary. Hence
$\mathfrak{g}_{1}$ or $\mathfrak{g}_{2}=\\{0\\}$. ∎
## 3\. Elementary quadratic Lie superalgebras
In this section, we consider the first non-trivial case of singular quadratic
Lie superalgebras: elementary quadratic Lie superalgebras. We begin with the
following definition.
###### Definition 3.1.
Let $\mathfrak{g}$ be a quadratic Lie superalgebra and $I$ be its associated
invariant. We say that $\mathfrak{g}$ is an elementary quadratic Lie
superalgebra if $I$ is decomposable.
Keep notations as in Section 2. If $I=I_{0}+I_{1}$ is decomposable, where
$I_{0}\in\operatorname{Alt}^{3}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})$
and
$I_{1}\in\operatorname{Alt}{}^{1}(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
then it is obvious that $I_{0}$ or $I_{1}$ is zero. The case $I_{1}=0$, i.e.
$I$ decomposable in
$\operatorname{Alt}^{3}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
corresponds to singular quadratic Lie superalgebras of type $\mathsf{S}_{3}$
and then there is nothing to do. Now we assume $I$ is a nonzero decomposable
element in
$\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
then $I$ can be written by:
$I=\alpha\otimes pq$
where $\alpha\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$ and
$p,q\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}^{*}$. It is clear that
$\mathfrak{g}$ is a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$.
###### Lemma 3.2.
Let $\mathfrak{g}$ be a reduced elementary quadratic Lie superalgebra having
$I=\alpha\otimes pq$ where
$\alpha\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$ and
$p,q\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}^{*}$. Set
$X_{\scriptscriptstyle{\overline{0}}}=\phi^{-1}(\alpha)$ then one has:
1. (1)
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})=2$ and
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\cap\mathscr{Z}(\mathfrak{g})=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$.
2. (2)
Let $X_{\scriptscriptstyle{\overline{1}}}=\phi^{-1}(p)$,
$Y_{\scriptscriptstyle{\overline{1}}}=\phi^{-1}(q)$ and
$U=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\}$
then
* (i)
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=2$ if $\dim(U)=1$ or
$U$ is non-degenerate.
* (ii)
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=4$ if $U$ is totally
isotropic.
###### Proof.
1. (1)
Let $\beta$ be an element in
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$. It is easy to see that
$\\{\beta,\alpha\\}=0$ if and only if $\\{\beta,I\\}=0$, equivalently
$\phi^{-1}(\beta)\in\mathscr{Z}(\mathfrak{g})$. Therefore,
$(\phi^{-1}(\alpha))^{\bot}\cap\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\subset\mathscr{Z}(\mathfrak{g})$.
It means that $\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\leq 2$
since $\mathfrak{g}$ is reduced (see [Bou59]). Moreover,
$X_{\scriptscriptstyle{\overline{0}}}=\phi^{-1}(\alpha)$ is isotropic then
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})=2$. If
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\cap\mathscr{Z}(\mathfrak{g}))=2$
then
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\subset\mathscr{Z}(\mathfrak{g})$.
Since $B$ is invariant we obtain $\mathfrak{g}$ Abelian (a contradiction).
Therefore,
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\cap\mathscr{Z}(\mathfrak{g})=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$.
2. (2)
It is obvious that $\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})\geq
2$. If $\dim(U)=1$ then $U$ is a totally isotropic subspace of
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ so there exists a one-
dimensional subspace $V$ of $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
such that $B$ is non-degenerate on $U\oplus V$ (see [Bou59]). Let
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=(U\oplus V){\
\overset{\perp}{\mathop{\oplus}}\ }W$ where $W=(U\oplus V)^{\bot}$ then for
all $f\in\phi(W)$ one has:
$\\{f,I\\}=\\{f,\alpha\otimes pq\\}=-\alpha\otimes(\\{f,p\\}q+p\\{f,q\\})=0.$
Therefore, $W\subset\mathscr{Z}(\mathfrak{g})$. Since $B$ is non-degenerate on
$W$ and $\mathfrak{g}$ is reduced then $W=\\{0\\}$.
If $\dim(U)=2$ then $U$ is non-degenerate or totally isotropic. If $U$ is non-
degenerate, let $\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}=U{\
\overset{\perp}{\mathop{\oplus}}\ }W$ where $W=U^{\bot}$. If $U$ is totally
isotropic, let $\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}=(U\oplus
V){\ \overset{\perp}{\mathop{\oplus}}\ }W$ where $W=(U\oplus V)^{\bot}$ in
$\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}$ and $B$ is non-degenerate
on $U\oplus V$. In the both cases, similarly as above, one has $W$ a non-
degenerate central ideal so $W=\\{0\\}$. Therefore,
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=\dim(U)=2$ if $U$ is
non-degenerate and
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=\dim(U\oplus V)=4$ if
$U$ is totally isotropic.
∎
In the sequel, we obtain the classification result.
###### Proposition 3.3.
Let $\mathfrak{g}$ be a reduced elementary quadratic Lie superalgebra then
$\mathfrak{g}$ is i-isomorphic to one of the following Lie superalgebras:
1. (1)
$\mathfrak{g}_{i}$ $(3\leq i\leq 6)$ the reduced singular quadratic Lie
algebras of type $\mathsf{S}_{3}$ given in [PU07].
2. (2)
$\mathfrak{g}_{4,1}^{s}=(\mathbb{C}X_{\overline{0}}\oplus\mathbb{C}Y_{\overline{0}})\oplus(\mathbb{C}X_{{\scriptscriptstyle{\overline{1}}}}\oplus\mathbb{C}Z_{{\scriptscriptstyle{\overline{1}}}})$
where
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}}\\}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Z_{\scriptscriptstyle{\overline{1}}}\\}$,
the bilinear form $B$ is defined by
$B(X_{\overline{0}},Y_{\overline{0}})=B(X_{{\scriptscriptstyle{\overline{1}}}},Z_{{\scriptscriptstyle{\overline{1}}}})=1,$
the other are zero and the Lie super-bracket is given by
$[Z_{{\scriptscriptstyle{\overline{1}}}},Z_{{\scriptscriptstyle{\overline{1}}}}]=-2X_{\overline{0}},\
\
[Y_{\overline{0}},Z_{{\scriptscriptstyle{\overline{1}}}}]=-2X_{{\scriptscriptstyle{\overline{1}}}},$
the other are trivial.
3. (3)
$\mathfrak{g}_{4,2}^{s}=(\mathbb{C}X_{\overline{0}}\oplus\mathbb{C}Y_{\overline{0}})\oplus(\mathbb{C}X_{{\scriptscriptstyle{\overline{1}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{1}}}})$
where
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}}\\}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\}$,
the bilinear form $B$ is defined by
$B(X_{\overline{0}},Y_{\overline{0}})=B(X_{{\scriptscriptstyle{\overline{1}}}},Y_{{\scriptscriptstyle{\overline{1}}}})=1,$
the other are zero and the Lie super-bracket is given by
$[X_{{\scriptscriptstyle{\overline{1}}}},Y_{{\scriptscriptstyle{\overline{1}}}}]=X_{\overline{0}},\
\
[Y_{\overline{0}},X_{{\scriptscriptstyle{\overline{1}}}}]=X_{{\scriptscriptstyle{\overline{1}}}},\
\
[Y_{\overline{0}},Y_{{\scriptscriptstyle{\overline{1}}}}]=-Y_{{\scriptscriptstyle{\overline{1}}}},$
the other are trivial.
4. (4)
$\mathfrak{g}_{6}^{s}=(\mathbb{C}X_{\overline{0}}\oplus\mathbb{C}Y_{\overline{0}})\oplus(\mathbb{C}X_{{\scriptscriptstyle{\overline{1}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{1}}}}\oplus\mathbb{C}Z_{{\scriptscriptstyle{\overline{1}}}}\oplus\mathbb{C}T_{{\scriptscriptstyle{\overline{1}}}})$
where
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}}\\}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}},Z_{\scriptscriptstyle{\overline{1}}},T_{\scriptscriptstyle{\overline{1}}}\\}$,
the bilinear form $B$ is defined by
$B(X_{\overline{0}},Y_{\overline{0}})=B(X_{{\scriptscriptstyle{\overline{1}}}},Z_{{\scriptscriptstyle{\overline{1}}}})=B(Y_{{\scriptscriptstyle{\overline{1}}}},T_{{\scriptscriptstyle{\overline{1}}}})=1,$
the other are zero and the Lie super-bracket is given by
$[Z_{{\scriptscriptstyle{\overline{1}}}},T_{{\scriptscriptstyle{\overline{1}}}}]=-X_{\overline{0}},\
\
[Y_{\overline{0}},Z_{{\scriptscriptstyle{\overline{1}}}}]=-Y_{{\scriptscriptstyle{\overline{1}}}},\
\
[Y_{\overline{0}},T_{{\scriptscriptstyle{\overline{1}}}}]=-X_{{\scriptscriptstyle{\overline{1}}}},$
the other are trivial.
###### Proof.
1. (1)
This statement corresponds to the case where $I$ is a decomposable 3-form in
$\operatorname{Alt}^{3}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$.
Therefore, the result is obvious.
Assume that $I=\alpha\otimes
pq\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
By the previous lemma, we can write
$\mathfrak{g}_{\overline{0}}=\mathbb{C}X_{\overline{0}}\oplus\mathbb{C}Y_{\overline{0}}$
where $X_{\overline{0}}=\phi^{-1}(\alpha)$,
$B(X_{\overline{0}},X_{\overline{0}})=B(Y_{\overline{0}},Y_{\overline{0}})=0$,
$B(X_{\overline{0}},Y_{\overline{0}})=1$. Let
$X_{\scriptscriptstyle{\overline{1}}}=\phi^{-1}(p),\
Y_{\scriptscriptstyle{\overline{1}}}=\phi^{-1}(q)$ and
$U=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\}$.
2. (2)
If $\dim(U)=1$ then
$Y_{\scriptscriptstyle{\overline{1}}}=kX_{\scriptscriptstyle{\overline{1}}}$
with some nonzero $k\in\mathbb{C}$. Therefore, $q=kp$ and $I=k\alpha\otimes
p^{2}$. Replacing $X_{\overline{0}}$ by $kX_{\overline{0}}$ and
$Y_{\scriptscriptstyle{\overline{0}}}$ by
$\frac{1}{k}Y_{\scriptscriptstyle{\overline{0}}}$, we can assume that $k=1$.
Let $Z_{\scriptscriptstyle{\overline{1}}}$ be an element in
$\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}$ such that
$B(X_{{\scriptscriptstyle{\overline{1}}}},Z_{{\scriptscriptstyle{\overline{1}}}})=1$.
Now, let
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
By using (1.7) and (1.8) of [BP89], one has:
$B(X,[Y,Z])=-2\alpha(X)p(Y)p(Z)=-2B(X_{\overline{0}},X)B(X_{{\scriptscriptstyle{\overline{1}}}},Y)B(X_{{\scriptscriptstyle{\overline{1}}}},Z).$
Since
$B|_{\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\times\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}}$
is non-degenerate then:
$[Y,Z]=-2B(X_{{\scriptscriptstyle{\overline{1}}}},Y)B(X_{{\scriptscriptstyle{\overline{1}}}},Z)X_{\overline{0}},\
\forall\ Y,Z\in\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}.$
Similarly and by the invariance of $B$, we also obtain:
$[X,Y]=-2B(X_{\overline{0}},X)B(X_{{\scriptscriptstyle{\overline{1}}}},Y)X_{{\scriptscriptstyle{\overline{1}}}},\
\forall\ X\in\mathfrak{g}_{\overline{0}},\
Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
and (2) follows.
3. (3)
If $\dim(U)=2$ and $U$ is non-degenerate then
$B(X_{{\scriptscriptstyle{\overline{1}}}},Y_{{\scriptscriptstyle{\overline{1}}}})=a\neq
0$. Replacing $X_{{\scriptscriptstyle{\overline{1}}}}$ by
$\frac{1}{a}X_{{\scriptscriptstyle{\overline{1}}}}$, $\ X_{\overline{0}}$ by
$aX_{\overline{0}}$ and $Y_{\overline{0}}$ by $\frac{1}{a}Y_{\overline{0}}$,
we can assume that $a=1$. Then one has
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\operatorname{span}\\{X_{\overline{0}},Y_{\overline{0}}\\}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\operatorname{span}\\{X_{{\scriptscriptstyle{\overline{1}}}},Y_{{\scriptscriptstyle{\overline{1}}}}\\}$,
$B(X_{\overline{0}},Y_{\overline{0}})=B(X_{{\scriptscriptstyle{\overline{1}}}},Y_{{\scriptscriptstyle{\overline{1}}}})=1$
and $I=\alpha\otimes pq$. Let
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\
Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$, we have:
$B(X,[Y,Z])=I(X,Y,Z)=-\alpha(X)(p(Y)q(Z)+p(Z)q(Y)).$
Therefore, the Lie super-bracket is defined:
$[Y,Z]=-(B(X_{{\scriptscriptstyle{\overline{1}}}},Y)B(Y_{{\scriptscriptstyle{\overline{1}}}},Z)+B(X_{{\scriptscriptstyle{\overline{1}}}},Z)B(Y_{{\scriptscriptstyle{\overline{1}}}},Y))X_{\overline{0}},\
\forall\ Y,Z\in\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}},$
$[X,Y]=-B(X_{\overline{0}},X)(B(X_{{\scriptscriptstyle{\overline{1}}}},Y)Y_{{\scriptscriptstyle{\overline{1}}}}+B(Y_{{\scriptscriptstyle{\overline{1}}}},Y)X_{{\scriptscriptstyle{\overline{1}}}}),\
\forall\ X\in\mathfrak{g}_{\overline{0}},\
Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
and (3) follows.
4. (4)
If $\dim(U)=2$ and $U$ is totally isotropic: let
$V=\operatorname{span}\\{Z_{{\scriptscriptstyle{\overline{1}}}},T_{{\scriptscriptstyle{\overline{1}}}}\\}$
be a 2-dimensional totally isotropic subspace of
$\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}$ such that
$\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}=U{\
\overset{\perp}{\mathop{\oplus}}\ }V$ and
$B(X_{{\scriptscriptstyle{\overline{1}}}},Z_{{\scriptscriptstyle{\overline{1}}}})=B(Y_{{\scriptscriptstyle{\overline{1}}}},T_{{\scriptscriptstyle{\overline{1}}}})=1$.
If
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
then:
$B(X,[Y,Z])=I(X,Y,Z)=-\alpha(X)(p(Y)q(Z)+p(Z)q(Y)).$
We obtain the Lie super-bracket as follows:
$[Y,Z]=-(B(X_{{\scriptscriptstyle{\overline{1}}}},Y)B(Y_{{\scriptscriptstyle{\overline{1}}}},Z)+B(X_{{\scriptscriptstyle{\overline{1}}}},Z)B(Y_{{\scriptscriptstyle{\overline{1}}}},Y))X_{\overline{0}},\
\forall\ Y,Z\in\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}},$
$[X,Y]=-B(X_{\overline{0}},X)(B(X_{{\scriptscriptstyle{\overline{1}}}},Y)Y_{{\scriptscriptstyle{\overline{1}}}}+B(Y_{{\scriptscriptstyle{\overline{1}}}},Y)X_{{\scriptscriptstyle{\overline{1}}}}),\
\forall\ X\in\mathfrak{g}_{\overline{0}},\
Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}.$
Thus,
$[Z_{{\scriptscriptstyle{\overline{1}}}},T_{{\scriptscriptstyle{\overline{1}}}}]=-X_{\overline{0}}$,
$[Y_{\overline{0}},Z_{{\scriptscriptstyle{\overline{1}}}}]=-Y_{{\scriptscriptstyle{\overline{1}}}}$,
$[Y_{\overline{0}},T_{{\scriptscriptstyle{\overline{1}}}}]=-X_{{\scriptscriptstyle{\overline{1}}}}$.
∎
## 4\. Quadratic Lie superalgebras with 2-dimensional even part
This section is devoted to study another particular case of singular quadratic
Lie superalgebras: quadratic Lie superalgebras with 2-dimensional even part.
As we shall see, they are can be seen as a symplectic version of solvable
singular quadratic Lie algebras. The first result classifies these algebras
with respect to the $\operatorname{dup}$-number.
###### Proposition 4.1.
Let $\mathfrak{g}$ be a non-Abelian quadratic Lie superalgebra with
$\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})=2$. Then $\mathfrak{g}$
is a singular quadratic Lie superalgebra of type $\mathsf{S}_{1}$.
###### Proof.
Let $I$ be the associated invariant of $\mathfrak{g}$. By a remark in [PU07],
every non-Abelian quadratic Lie algebra must have the dimension more than 2 so
$\mathfrak{g}_{\overline{0}}$ is Abelian and as a consequence,
$I\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
We choose a basis $\\{X_{\overline{0}},Y_{\overline{0}}\\}$ of
$\mathfrak{g}_{\overline{0}}$ such that
$B(X_{\overline{0}},X_{\overline{0}})=B(Y_{\overline{0}},Y_{\overline{0}})=0$
and $B(X_{\overline{0}},Y_{\overline{0}})=1$. Let
$\alpha=\phi(X_{\overline{0}})$, $\beta=\phi(Y_{\overline{0}})$ and we can
assume that
$I=\alpha\otimes\Omega_{1}+\beta\otimes\Omega_{2}$
where
$\Omega_{1},\Omega_{2}\in\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
Then one has:
$\\{I,I\\}=2(\Omega_{1}\Omega_{2}+\alpha\wedge\beta\otimes\\{\Omega_{1},\Omega_{2}\\}).$
Therefore, $\\{I,I\\}=0$ implies that $\Omega_{1}\Omega_{2}=0$. So
$\Omega_{1}=0$ or $\Omega_{2}=0$. It means that $\mathfrak{g}$ is a singular
quadratic Lie superalgebra of type $\mathsf{S}_{1}$. ∎
###### Proposition 4.2.
Let $\mathfrak{g}$ be a singular quadratic Lie superalgebra with Abelian even
part. If $\mathfrak{g}$ is reduced then $\dim(\mathfrak{g}_{\overline{0}})=2$.
###### Proof.
Let $I$ be the associated invariant of $\mathfrak{g}$. Since $\mathfrak{g}$
has the Abelian even part one has
$I\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
Moreover $\mathfrak{g}$ is singular then
$I=\alpha\otimes\Omega$
where $\alpha\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*},\
\Omega\in\operatorname{Sym}^{2}(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})$.
The proof follows exactly Lemma 3.2. Let
$\beta\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$ then
$\\{\beta,\alpha\\}=0$ if and only if $\\{\beta,I\\}=0$, equivalently
$\phi^{-1}(\beta)\in\mathscr{Z}(\mathfrak{g})$. Therefore,
$(\phi^{-1}(\alpha))^{\bot}\cap\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\subset\mathscr{Z}(\mathfrak{g})$.
It means that $\dim(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})=2$ since
$\mathfrak{g}$ is reduced and $\phi^{-1}(\alpha)$ is isotropic in
$\mathscr{Z}(\mathfrak{g})$. ∎
Now, let $\mathfrak{g}$ be a non-Abelian quadratic Lie superalgebra with
2-dimensional even part. By Proposition 4.1, $\mathfrak{g}$ is singular of
type $\mathsf{S}_{1}$. Fix $\alpha\in\mathscr{V}_{I}$ and choose
$\Omega\in\operatorname{Sym}^{2}(\mathfrak{g})$ such that
$I=\alpha\otimes\Omega.$
We define
$C:\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}\to\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
by $B(C(X),Y)=\Omega(X,Y)$, for all
$X,Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ and let
$X_{\overline{0}}=\phi^{-1}(\alpha)$.
###### Lemma 4.3.
The following assertions are equivalent:
1. (1)
$\\{I,I\\}=0$,
2. (2)
$\\{\alpha,\alpha\\}=0$,
3. (3)
$B(X_{\overline{0}},X_{\overline{0}})=0$.
In this case, one has $X_{\overline{0}}\in\mathscr{Z}(\mathfrak{g})$.
###### Proof.
It is easy to see that:
$\\{I,I\\}=0\Leftrightarrow\\{\alpha,\alpha\\}\otimes\Omega^{2}=0.$
Therefore the assertions are equivalent. Moreover, since $\\{\alpha,I\\}=0$
one has $X_{\overline{0}}\in\mathscr{Z}(\mathfrak{g})$. ∎
We keep the notations as in the previous sections. Then there exists an
isotropic element $Y_{\overline{0}}\in\mathfrak{g}_{\overline{0}}$ such that
$B(X_{\overline{0}},Y_{\overline{0}})=1$ and one has the following
proposition:
###### Proposition 4.4.
1. (1)
The map $C$ is skew-symmetric (with respect to $B$), that is
$B(C(X),Y)=-B(X,C(Y))$
for all $X$, $Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
2. (2)
$[X,Y]=B(C(X),Y)X_{\scriptscriptstyle{\overline{0}}}$, for all
$X,Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ and
$C=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})|_{\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}}$.
3. (3)
$\mathscr{Z}(\mathfrak{g})=\ker(C)\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$
and
$[\mathfrak{g},\mathfrak{g}]=\operatorname{Im}(C)\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$.
Therefore, $\mathfrak{g}$ is reduced if and only if
$\ker(C)\subset\operatorname{Im}(C)$.
4. (4)
The Lie superalgebra $\mathfrak{g}$ is solvable. Moreover, $\mathfrak{g}$ is
nilpotent if and only if $C$ is nilpotent.
###### Proof.
1. (1)
For all $X$, $Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$, one has
$B(C(X),Y)=\Omega(X,Y)=\Omega(Y,X)=B(C(Y),X)=-B(X,C(Y)).$
2. (2)
Let
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
then
$B(X,[Y,Z])=(\alpha\otimes\Omega)(X,Y,Z)=\alpha(X)\Omega(Y,Z).$
Since $\alpha(X)=B(X_{\overline{0}},X)$ and $\Omega(Y,Z)=B(C(Y),Z)$ so one has
$B(X,[Y,Z])=B(X_{\overline{0}},X)B(C(Y),Z).$
The non-degeneracy of $B$ implies
$[Y,Z]=B(C(Y),Z)X_{\scriptscriptstyle{\overline{0}}}$. Set
$X=Y_{\scriptscriptstyle{\overline{0}}}$ then
$B(Y_{\scriptscriptstyle{\overline{0}}},[Y,Z])=B(C(Y),Z)$. By the invariance
of $B$, we obtain $[Y_{\scriptscriptstyle{\overline{0}}},Y]=C(Y)$.
3. (3)
It follows from the assertion (2).
4. (4)
$\mathfrak{g}$ is solvable since
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is solvable, or since
$[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]\subset\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$.
If $\mathfrak{g}$ is nilpotent then
$C=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})$ is nilpotent
obviously. Conversely, if $C$ is nilpotent then it is easy to see that
$\mathfrak{g}$ is nilpotent since
$(\operatorname{ad}(X))^{k}(\mathfrak{g})\subset\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\operatorname{Im}(C^{k})$
for all $X\in\mathfrak{g}$.
∎
###### Remark 4.5.
The choice of $C$ is unique up to a nonzero scalar. Indeed, assume that
$I=\alpha^{\prime}\otimes\Omega^{\prime}$ and $C^{\prime}$ is the map
associated to $\Omega^{\prime}$. Since
$\mathscr{Z}(\mathfrak{g})\cap\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$
and $\phi^{-1}(\alpha^{\prime})\in\mathscr{Z}(\mathfrak{g})$ one has
$\alpha^{\prime}=\lambda\alpha$ for some nonzero $\lambda\in\mathbb{C}$.
Therefore, $\alpha\otimes(\Omega-\lambda\Omega^{\prime})=0$. It means that
$\Omega=\lambda\Omega^{\prime}$ and then we get $C=\lambda C^{\prime}$.
### 4.1. Double extension of a symplectic vector space
Double extensions are a very useful method initiated by V. Kac to construct
quadratic Lie algebras (see [Kac85] and [MR85]). They are generalized to many
algebras endowed with a non-degenerate invariant bilinear form, for example
quadratic Lie superalgebras (see [BB99] and [BBB]). In [DPU10], we consider a
particular case that is the double extension of a quadratic vector space by a
skew-symmetric map. From this we obtain the class of solvable singular
quadratic Lie algebras. Here, we use this notion in yet another context,
replacing the quadratic vector space by a symplectic vector space.
###### Definition 4.6.
1. (1)
Let $(\mathfrak{q},B_{\mathfrak{q}})$ be a symplectic vector space equipped
with a symplectic bilinear form $B_{\mathfrak{q}}$ and
${\overline{C}}:\mathfrak{q}\to\mathfrak{q}$ be a skew-symmetric map, that is,
$B_{\mathfrak{q}}({\overline{C}}(X),Y)=-B_{\mathfrak{q}}(X,{\overline{C}}(Y)),\
\forall\ X,Y\in\mathfrak{q}.$
Let
$(\mathfrak{t}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}}\\},B_{\mathfrak{t}})$
be a 2-dimensional quadratic vector space with the symmetric bilinear form
$B_{\mathfrak{t}}$ defined by
$B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{0}}},X_{\scriptscriptstyle{\overline{0}}})=B_{\mathfrak{t}}(Y_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}})=0,\
B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}})=1.$
Consider the vector space
$\mathfrak{g}=\mathfrak{t}{\ \overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$
equipped with the bilinear form $B=B_{\mathfrak{t}}+B_{\mathfrak{q}}$ and
define a bracket on $\mathfrak{g}$ by
$[\lambda X_{\scriptscriptstyle{\overline{0}}}+\mu
Y_{\scriptscriptstyle{\overline{0}}}+X,\lambda^{\prime}X_{\scriptscriptstyle{\overline{0}}}+\mu^{\prime}Y_{\scriptscriptstyle{\overline{0}}}+Y]=\mu{\overline{C}}(Y)-\mu^{\prime}{\overline{C}}(X)+B({\overline{C}}(X),Y)X_{\scriptscriptstyle{\overline{0}}},$
for all
$X,Y\in\mathfrak{q},\lambda,\mu,\lambda^{\prime},\mu^{\prime}\in\mathbb{C}$.
Then $(\mathfrak{g},B)$ is a quadratic solvable Lie superalgebra with
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\mathfrak{t}$ and
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\mathfrak{q}$. We say that
$\mathfrak{g}$ is the double extension of $\mathfrak{q}$ by ${\overline{C}}$.
2. (2)
Let $\mathfrak{g}_{i}$ be double extensions of symplectic vector spaces
$(\mathfrak{q}_{i},B_{i})$ by skew-symmetric maps
${\overline{C}}_{i}\in\mathcal{L}(\mathfrak{q}_{i})$, for $1\leq i\leq k$. The
amalgamated product
$\mathfrak{g}=\mathfrak{g}_{1}{\ \mathop{\times}\limits_{\mathrm{a}}\
}\mathfrak{g}_{2}{\ \mathop{\times}\limits_{\mathrm{a}}\ }\dots\
\mathop{\times}\limits_{\mathrm{a}}\ \mathfrak{g}_{k}$
is defined as follows:
* •
consider $(\mathfrak{q},B)$ be the symplectic vector space with
$\mathfrak{q}=\mathfrak{q}_{1}\oplus\mathfrak{q}_{2}\oplus\dots\oplus\mathfrak{q}_{k}$
and the bilinear form $B$ such that
$B(\sum_{i=I}^{k}X_{i},\sum_{i=I}^{k}Y_{i})=\sum_{i=I}^{k}B_{i}(X_{i},Y_{i})$,
for $X_{i},Y_{i}\in\mathfrak{q}_{i}$, $1\leq i\leq k$.
* •
the skew-symmetric map ${\overline{C}}\in\mathcal{L}(\mathfrak{q})$ is defined
by
${\overline{C}}(\sum_{i=I}^{k}X_{i})=\sum_{i=I}^{k}{\overline{C}}_{i}(X_{i})$,
for $X_{i}\in\mathfrak{q}_{i}$, $1\leq i\leq k$.
Then $\mathfrak{g}$ is the double extension of $\mathfrak{q}$ by
${\overline{C}}$.
###### Lemma 4.7.
We keep the notation above.
1. (1)
Let $\mathfrak{g}$ be the double extension of $\mathfrak{q}$ by
${\overline{C}}$. Then
$[X,Y]=B(X_{{\scriptscriptstyle{\overline{0}}}},X)C(Y)-B(X_{{\scriptscriptstyle{\overline{0}}}},Y)C(X)+B(C(X),Y)X_{\scriptscriptstyle{\overline{0}}},\
\forall\ X,Y\in\mathfrak{g},$
where $C=\operatorname{ad}(Y_{{\scriptscriptstyle{\overline{0}}}})$. Moreover,
$X_{\scriptscriptstyle{\overline{0}}}\in\mathscr{Z}(\mathfrak{g})$ and
$C|_{\mathfrak{q}}={\overline{C}}$.
2. (2)
Let $\mathfrak{g}^{\prime}$ be the double extension of $\mathfrak{q}$ by
${\overline{C^{\prime}}}=\lambda{\overline{C}}$, $\lambda\in\mathbb{C}$,
$\lambda\neq 0$. Then $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are
i-isomorphic.
###### Proof.
1. (1)
This is a straightforward computation by Definition 4.6.
2. (2)
Write $\mathfrak{g}=\mathfrak{t}{\ \overset{\perp}{\mathop{\oplus}}\
}\mathfrak{q}=\mathfrak{g}^{\prime}$. Denote by $[\cdot,\cdot]^{\prime}$ the
Lie super-bracket on $\mathfrak{g}^{\prime}$. Define
$A:\mathfrak{g}\to\mathfrak{g}^{\prime}$ by
$A(X_{\scriptscriptstyle{\overline{0}}})=\lambda
X_{\scriptscriptstyle{\overline{0}}}$,
$A(Y_{\scriptscriptstyle{\overline{0}}})=\displaystyle\frac{1}{\lambda}Y_{\scriptscriptstyle{\overline{0}}}$
and $A|_{\mathfrak{q}}=\operatorname{Id}_{\mathfrak{q}}$. Then
$A([Y_{\scriptscriptstyle{\overline{0}}},X])=C(X)=[A(Y_{\scriptscriptstyle{\overline{0}}}),A(X)]^{\prime}$
and $A([X,Y])=[A(X),A(Y)]^{\prime}$, for all $X,Y\in\mathfrak{q}$. So $A$ is
an i-isomorphism.
∎
###### Proposition 4.8.
1. (1)
Let $\mathfrak{g}$ be a non-Abelian quadratic Lie superalgebra with
2-dimensional even part. Keep the notations as in Proposition 4.4. Then
$\mathfrak{g}$ is the double extension of
$\mathfrak{q}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}})^{\perp}=\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
by
${\overline{C}}=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})|_{\mathfrak{q}}$.
2. (2)
Let $\mathfrak{g}$ be the double extension of a symplectic vector space
$\mathfrak{q}$ by a map ${\overline{C}}\neq 0$. Then $\mathfrak{g}$ is a
singular solvable quadratic Lie superalgebra with 2-dimensional even part.
Moreover:
* (i)
$\mathfrak{g}$ is reduced if and only if
$\ker({\overline{C}})\subset\operatorname{Im}({\overline{C}})$.
* (ii)
$\mathfrak{g}$ is nilpotent if and only if ${\overline{C}}$ is nilpotent.
3. (3)
Let $(\mathfrak{g},B)$ be a quadratic Lie superalgebra. Let
$\mathfrak{g}^{\prime}$ be the double extension of a symplectic vector space
$(\mathfrak{q}^{\prime},B^{\prime})$ by a map ${\overline{C^{\prime}}}$. Let
$A$ be an i-isomorphism of $\mathfrak{g}^{\prime}$ onto $\mathfrak{g}$ and
write $\mathfrak{q}=A(\mathfrak{q}^{\prime})$. Then $\mathfrak{g}$ is the
double extension of $(\mathfrak{q},B|_{\mathfrak{q}\times\mathfrak{q}})$ by
the map ${\overline{C}}=\overline{A}\ {\overline{C^{\prime}}}\
\overline{A}^{-1}$ where $\overline{A}=A|_{\mathfrak{q}^{\prime}}$.
###### Proof.
The assertions (1) and (2) follow Proposition 4.4 and Lemma 4.7. For (3),
since $A$ is i-isomorphic then $\mathfrak{g}$ has also 2-dimensional even
part. Write
$\mathfrak{g}^{\prime}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}^{\prime}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}^{\prime}$. Let
$X_{\scriptscriptstyle{\overline{0}}}=A(X_{\scriptscriptstyle{\overline{0}}}^{\prime})$
and
$Y_{\scriptscriptstyle{\overline{0}}}=A(Y_{\scriptscriptstyle{\overline{0}}}^{\prime})$.
Then
$\mathfrak{g}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and one has:
$[Y_{\scriptscriptstyle{\overline{0}}},X]=(A{\overline{C^{\prime}}}A^{-1})(X),\
\forall\ X\in\mathfrak{q},\text{ and }$
$[X,Y]=B((A{\overline{C^{\prime}}}A^{-1})(X),Y)X_{\scriptscriptstyle{\overline{0}}},\
\forall\ X,Y\in\mathfrak{q}.$
This proves the result. ∎
###### Example 4.9.
From the point of view of double extensions, for reduced elementary quadratic
Lie superalgebras with 2-dimensional even part in Section 3 one has
1. (1)
$\mathfrak{g}_{4,1}^{s}$ is the double extension of the 2-dimensional
symplectic vector space $\mathfrak{q}=\mathbb{C}^{2}$ by the map having
matrix:
${\overline{C}}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$
in a Darboux basis $\\{E_{1},E_{2}\\}$ of $\mathfrak{q}$ where
$B(E_{1},E_{2})=1$.
2. (2)
$\mathfrak{g}_{4,2}^{s}$ the double extension of the 2-dimensional symplectic
vector space $\mathfrak{q}=\mathbb{C}^{2}$ by the map having matrix:
${\overline{C}}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$
in a Darboux basis $\\{E_{1},E_{2}\\}$ of $\mathfrak{q}$ where
$B(E_{1},E_{2})=1$.
3. (3)
$\mathfrak{g}_{6}^{s}$ is the double extension of the 4-dimensional symplectic
vector space $\mathfrak{q}=\mathbb{C}^{4}$ by the map having matrix:
${\overline{C}}=\begin{pmatrix}0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&-1&0\end{pmatrix}$
in a Darboux basis $\\{E_{1},E_{2},E_{3},E_{4}\\}$ of $\mathfrak{q}$ where
$B(E_{1},E_{3})=B(E_{2},E_{4})=1$, the other are zero.
Let $(\mathfrak{q},B)$ be a symplectic vector space. We denote by
$\operatorname{Sp}(\mathfrak{q})$ the isometry group of the symplectic form
$B$ and by $\mathfrak{sp}(\mathfrak{q})$ its Lie algebra, i.e. the Lie algebra
of skew-symmetric maps with respects to $B$. The adjoint action is the action
of $\operatorname{Sp}(\mathfrak{q})$ on $\mathfrak{sp}(\mathfrak{q})$ by the
conjugation (see Appendix). Also, we denote by
$\mathbb{P}^{1}(\mathfrak{sp}(2n))$ the projective space of
$\mathfrak{sp}(2n)$ with the action induced by $\operatorname{Sp}(2n)$-adjoint
action on $\mathfrak{sp}(2n)$.
###### Proposition 4.10.
Let $(\mathfrak{q},B)$ be a symplectic vector space. Let
$\mathfrak{g}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
$\mathfrak{g}^{\prime}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ be double extensions of
$\mathfrak{q}$, by skew-symmetric maps ${\overline{C}}$ and
${\overline{C^{\prime}}}$ respectively. Then:
1. (1)
there exists a Lie superalgebra isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exist an invertible map
$P\in\mathcal{L}(\mathfrak{q})$ and a nonzero $\lambda\in\mathbb{C}$ such that
${\overline{C^{\prime}}}=\lambda\ P{\overline{C}}P^{-1}$ and
$P^{*}P{\overline{C}}={\overline{C}}$ where $P^{*}$ is the adjoint map of $P$
with respect to $B$.
2. (2)
there exists an i-isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if ${\overline{C^{\prime}}}$ is in the
$\operatorname{Sp}(\mathfrak{q})$-adjoint orbit through
$\lambda{\overline{C}}$ for some nonzero $\lambda\in\mathbb{C}$.
###### Proof.
The assertions are obvious if ${\overline{C}}=0$. We assume
${\overline{C}}\neq 0$.
1. (1)
Let $A:\mathfrak{g}\to\mathfrak{g}^{\prime}$ be a Lie superalgebra isomorphism
then
$A(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}})=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}^{\prime}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}^{\prime}$
and $A(\mathfrak{q})=\mathfrak{q}$. It is obvious that
${\overline{C^{\prime}}}\neq 0$. It is easy to see that
$\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}=\mathscr{Z}(\mathfrak{g})\cap\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$
and
$\mathbb{C}X^{\prime}_{\scriptscriptstyle{\overline{0}}}=\mathscr{Z}(\mathfrak{g}^{\prime})\cap\mathfrak{g}^{\prime}_{\scriptscriptstyle{\overline{0}}}$
then one has
$A(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}})=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}^{\prime}$.
It means $A(X_{\scriptscriptstyle{\overline{0}}})=\mu
X_{\scriptscriptstyle{\overline{0}}}^{\prime}$ for some nonzero
$\mu\in\mathbb{C}$. Let $A|_{\mathfrak{q}}=Q$ and assume
$A(Y_{\scriptscriptstyle{\overline{0}}})=\beta
Y_{\scriptscriptstyle{\overline{0}}}^{\prime}+\gamma
X_{\scriptscriptstyle{\overline{0}}}^{\prime}$. For all $X$,
$Y\in\mathfrak{q}$, we have $A([X,Y])=\mu
B({\overline{C}}(X),Y)X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$. Also,
$A([X,Y])=[Q(X),Q(Y)]^{\prime}=B({\overline{C^{\prime}}}Q(X),Q(Y))X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}.$
It results that $Q^{*}{\overline{C^{\prime}}}Q=\mu{\overline{C}}$.
Moreover,
$A([Y_{{\scriptscriptstyle{\overline{0}}}},X])=Q({\overline{C}}(X))=[\beta
Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}+\gamma
X_{{\scriptscriptstyle{\overline{0}}}}^{\prime},Q(X)]^{\prime}=\beta{\overline{C^{\prime}}}Q(X)$,
for all $X\in\mathfrak{q}$. We conclude that $Q\ {\overline{C}}\
Q^{-1}=\beta{\overline{C^{\prime}}}$ and since
$Q^{*}{\overline{C^{\prime}}}Q=\mu{\overline{C}}$, then
$Q^{*}Q{\overline{C}}=\beta\mu{\overline{C}}$.
Set $P=\displaystyle\frac{1}{(\mu\beta)^{\frac{1}{2}}}Q$ and
$\lambda=\displaystyle\frac{1}{\beta}$. It follows that
${\overline{C^{\prime}}}=\lambda P{\overline{C}}P^{-1}$ and
$P^{*}P{\overline{C}}={\overline{C}}$.
Conversely, assume that
$\mathfrak{g}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
$\mathfrak{g}^{\prime}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ be double extensions of
$\mathfrak{q}$, by skew-symmetric maps ${\overline{C}}$ and
${\overline{C^{\prime}}}$ respectively such that
${\overline{C^{\prime}}}=\lambda P{\overline{C}}P^{-1}$ and
$P^{*}P{\overline{C}}={\overline{C}}$ with an invertible map
$P\in\mathcal{L}(\mathfrak{q})$ and a nonzero $\lambda\in\mathbb{C}$. Define
$A:\mathfrak{g}\to\mathfrak{g}^{\prime}$ by
$A(X_{{\scriptscriptstyle{\overline{0}}}})=\lambda
X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$,
$A(Y_{{\scriptscriptstyle{\overline{0}}}})=\displaystyle\frac{1}{\lambda}Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$
and $A(X)=P(X)$, for all $X\in\mathfrak{q}$ then it is easy to check that $A$
is a Lie superalgebra isomorphism.
2. (2)
If $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are i-isomorphic, then the
isomorphism $A$ in the proof of (1) is an isometry. Hence
$P\in\operatorname{Sp}(\mathfrak{q})$ and ${\overline{C^{\prime}}}=\lambda
P{\overline{C}}P^{-1}$ gives the result.
Conversely, define $A$ as above (the sufficiency of (1)). Then $A$ is an
isometry and it is easy to check that $A$ is an i-isomorphism.
∎
###### Corollary 4.11.
Let $(\mathfrak{g},B)$ and $(\mathfrak{g}^{\prime},B^{\prime})$ be double
extensions of $(\mathfrak{q},\overline{B})$ and
$(\mathfrak{q}^{\prime},\overline{B^{\prime}})$ respectively where
$\overline{B}=B|_{\mathfrak{q}\times\mathfrak{q}}$ and
$\overline{B^{\prime}}=B^{\prime}|_{\mathfrak{q}^{\prime}\times\mathfrak{q}^{\prime}}$.
Write
$\mathfrak{g}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
$\mathfrak{g}^{\prime}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}^{\prime}$. Then:
1. (1)
there exists an i-isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exists an isometry
$\overline{A}:\mathfrak{q}\to\mathfrak{q}^{\prime}$ such that
${\overline{C^{\prime}}}=\lambda\ \overline{A}\ {\overline{C}}\
\overline{A}{}^{-1}$, for some nonzero $\lambda\in\mathbb{C}$.
2. (2)
there exists a Lie superalgebra isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exist invertible maps
$\overline{Q}:\mathfrak{q}\to\mathfrak{q}^{\prime}$ and
$\overline{P}\in\mathcal{L}(\mathfrak{q})$ such that
* (i)
${\overline{C^{\prime}}}=\lambda\ \overline{Q}\ {\overline{C}}\
\overline{Q}^{-1}$ for some nonzero $\lambda\in\mathbb{C}$,
* (ii)
$\overline{P}^{*}\ \overline{P}\ {\overline{C}}={\overline{C}}$ and
* (iii)
$\overline{Q}\ \overline{P}^{-1}$ is an isometry from $\mathfrak{q}$ onto
$\mathfrak{q}^{\prime}$.
###### Proof.
The proof is completely similar to Corollary 4.6 in [DPU10]. It is sketched as
follows. First we can assume $\dim(\mathfrak{g})=\dim(\mathfrak{g}^{\prime})$
and define then a map $F:\mathfrak{g}^{\prime}\rightarrow\mathfrak{g}$ by
$F(X_{\scriptscriptstyle{\overline{0}}}^{\prime})=X_{\scriptscriptstyle{\overline{0}}}$,
$F(Y_{\scriptscriptstyle{\overline{0}}}^{\prime})=Y_{\scriptscriptstyle{\overline{0}}}$
and $\bar{F}=F|_{\mathfrak{q}^{\prime}}$ is an isometry from
$\mathfrak{q}^{\prime}$ onto $\mathfrak{q}$. We define a new Lie bracket on
$\mathfrak{g}$ by
$[X,Y]^{\prime\prime}=F\left([F^{-1}(X),F^{-1}(Y)]^{\prime}\right),\ \forall
X,Y\in\mathfrak{g}.$
and denote by $(\mathfrak{g}^{\prime\prime},[\cdot,\cdot]^{\prime\prime})$
this new Lie superalgebra. So $F$ is an i-isomorphism from
$\mathfrak{g}^{\prime}$ onto $\mathfrak{g}^{\prime\prime}$ and by Proposition
4.8 (3) $\mathfrak{g}^{\prime\prime}=(\mathbb{C}X_{1}\oplus\mathbb{C}Y_{1}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ is the double extension of
$\mathfrak{q}$ by $\overline{C^{\prime\prime}}$ with
$\overline{C^{\prime\prime}}=\overline{F}\ \overline{C^{\prime}}\
\overline{F}^{-1}$. We have that $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$
are isomorphic (resp. i-isomorphic) if and only if $\mathfrak{g}$ and
$\mathfrak{g}^{\prime\prime}$ are isomorphic (resp. i-isomorphic). Finally, by
applying Proposition 4.11 to the Lie superalgebras $\mathfrak{g}$ and
$\mathfrak{g}^{\prime\prime}$ we obtain the corollary. ∎
It results that quadratic Lie superalgebra structures on the quadratic
$\mathbb{Z}_{2}$-graded vector space
$\mathbb{C}^{2}\underset{\mathbb{Z}_{2}}{\oplus}\mathbb{C}^{2n}$ can be
classified up to i-isomorphism in terms of $\operatorname{Sp}(2n)$-orbits in
$\mathbb{P}^{1}(\mathfrak{sp}(2n))$. This work is like what have been done in
[DPU10]. We need the following lemma:
###### Lemma 4.12.
Let $V$ be a quadratic $\mathbb{Z}_{2}$-graded vector space such that its even
part is 2-dimensional. We write
$V=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}^{\prime}$ with
$X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$,
$Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$ isotropic elements in
$V_{{\scriptscriptstyle{\overline{0}}}}$ and
$B(X_{{\scriptscriptstyle{\overline{0}}}}^{\prime},Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime})=1$.
Let $\mathfrak{g}$ be a quadratic Lie superalgebra with
$\dim(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})=\dim(V_{{\scriptscriptstyle{\overline{0}}}})$
and $\dim(\mathfrak{g})=\dim(V)$. Then, there exists a skew-symmetric map
${\overline{C^{\prime}}}:\mathfrak{q}^{\prime}\to\mathfrak{q}^{\prime}$ such
that $V$ is considered as the double extension of $\mathfrak{q}^{\prime}$ by
${\overline{C^{\prime}}}$ that is i-isomorphic to $\mathfrak{g}$.
###### Proof.
By Proposition 4.8, $\mathfrak{g}$ is a double extension. Let us write
$\mathfrak{g}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
${\overline{C}}=\operatorname{ad}(Y_{{\scriptscriptstyle{\overline{0}}}})|_{\mathfrak{q}}$.
Define $A:\mathfrak{g}\to V$ by
$A(X_{{\scriptscriptstyle{\overline{0}}}})=X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$,
$A(Y_{{\scriptscriptstyle{\overline{0}}}})=Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}$
and $\overline{A}=A|_{\mathfrak{q}}$ any isometry from
$\mathfrak{q}\to\mathfrak{q}^{\prime}$. It is clear that $A$ is an isometry
from $\mathfrak{g}$ to $V$. Now, define the Lie super-bracket on $V$ by:
$[X,Y]=A\left([A^{-1}(X),A^{-1}(Y)]\right),\ \forall\ X,Y\in V.$
Then $V$ is a quadratic Lie superalgebra, that is i-isomorphic to
$\mathfrak{g}$. Moreover, $V$ is obviously the double extension of
$\mathfrak{q}^{\prime}$ by ${\overline{C^{\prime}}}=\overline{A}\
{\overline{C}}\ \overline{A}{}^{-1}$. ∎
Proposition 4.8, Proposition 4.10, Corollary 4.11 and Lemma 4.12 are enough
for us to apply the classification method in [DPU10] for the set
$\mathscr{S}(2+2n)$ of quadratic Lie superalgebra structures on the quadratic
$\mathbb{Z}_{2}$-graded vector space
$\mathbb{C}^{2}\underset{\mathbb{Z}_{2}}{\oplus}\mathbb{C}^{2n}$ by only
replacing the isometry group $\operatorname{O}(m)$ by $\operatorname{Sp}(2n)$
and $\mathfrak{o}(m)$ by $\mathfrak{sp}(2n)$ to obtain completely similar
results. One has the first characterization of the set $\mathscr{S}(2+2n)$:
###### Proposition 4.13.
Let $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ be elements in
$\mathscr{S}(2+2n)$. Then $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are
i-isomorphic if and only if they are isomorphic.
By using the notion of double extension, we call the Lie superalgebra
$\mathfrak{g}\in\mathscr{S}(2+2n)$ diagonalizable (resp. invertible) if it is
a double extension by a diagonalizable (resp. invertible) map. Denote the
subsets of nilpotent elements, diagonalizable elements and invertible elements
in $\mathscr{S}(2+2n)$, respectively by $\mathscr{N}(2+2n)$,
$\mathscr{D}(2+2n)$ and by $\mathscr{S}_{\mathrm{inv}}(2+2n)$. Denote by
$\widehat{{\mathscr{N}}}(2+2n)$, $\widehat{{\mathscr{D}}}(2+2n)$,
$\widehat{\mathscr{S}_{\mathrm{inv}}}(2+2n)$ the sets of isomorphism classes
in $\mathscr{N}(2+2n)$, $\mathscr{D}(2+2n)$,
$\mathscr{S}_{\mathrm{inv}}(2+2n)$, respectively and
$\widehat{{\mathscr{D}}}_{\mathrm{red}}(2+2n)$ the subset of
$\widehat{{\mathscr{D}}}(2+2n)$ including reduced ones. Applying Appendix, we
have the classification result of these sets as follows:
###### Proposition 4.14.
1. (1)
There is a bijection between $\ \widehat{{\mathscr{N}}}(2+2n)$ and the set of
nilpotent $\ \operatorname{Sp}(2n)$-adjoint orbits of $\ \mathfrak{sp}(2n)$
that induces a bijection between $\ \widehat{{\mathscr{N}}}(2+2n)$ and the set
of partitions $\ \mathcal{P}_{-1}(2n)$.
2. (2)
There is a bijection between $\ \widehat{{\mathscr{D}}}(2+2n)$ and the set of
semisimple $\ \operatorname{Sp}(2n)$-orbits of $\
\mathbb{P}^{1}(\mathfrak{sp}(2n))$ that induces a bijection between $\
\widehat{{\mathscr{D}}}(2+2n)$ and $\Lambda_{n}/H_{n}$ where $H_{n}$ is the
group obtained from the group $G_{n}$ by adding maps
$(\lambda_{1},\dots,\lambda_{n})\mapsto\lambda(\lambda_{1},\dots,\lambda_{n}),\
\forall\lambda\in\mathbb{C},\ \lambda\neq 0$. In the reduced case,
$\widehat{{\mathscr{D}}}_{\mathrm{red}}(2+2n)$ is bijective to
$\Lambda_{n}^{+}/H_{n}$ with
$\Lambda_{n}^{+}=\\{(\lambda_{1},\dots,\lambda_{n})\mid\
\lambda_{1},\dots,\lambda_{n}\in\mathbb{C},\ \lambda_{i}\neq 0,\ \forall
i\\}$.
3. (3)
There is a bijection between $\widehat{\mathscr{S}_{\mathrm{inv}}}(2+2n)$ and
the set of invertible $\ \operatorname{Sp}(2n)$-orbits of $\
\mathbb{P}^{1}(\mathfrak{sp}(2n))$ that induces a bijection between $\
\widehat{\mathscr{S}_{\mathrm{inv}}}(2+2n)$ and $\
\mathcal{J}_{n}/\mathbb{C}^{*}$.
4. (4)
There is a bijection between $\ \widehat{\mathscr{S}}(2+2n)$ and the set of $\
\operatorname{Sp}(2n)$-orbits of $\ \mathbb{P}^{1}(\mathfrak{sp}(2n))$ that
induces a bijection between $\ \widehat{\mathscr{S}}(2+2n)$ and $\
\mathcal{D}(2n)/\mathbb{C}^{*}$.
Next, we will describe the sets $\mathscr{N}(2+2n)$,
$\mathscr{D}_{\mathrm{red}}(2+2n)$ the subset of $\mathscr{D}(2+2n)$ including
reduced ones, and $\mathscr{S}_{\mathrm{inv}}(2+2n)$ in term of amalgamated
product in Definition 4.6. Remark that except for the nilpotent case, the
amalgamated product may have a bad behavior with respect to isomorphisms.
###### Definition 4.15.
Keep the notation $J_{p}$ for the Jordan block of size $p$ as in Appendix and
define two types of double extension as follows:
* •
for $p\geq 2$, we consider the symplectic vector space
$\mathfrak{q}=\mathbb{C}^{2p}$ equipped with its canonical bilinear form
$\overline{B}$ and the map ${\overline{C}}_{2p}^{J}$ having matrix
$\begin{pmatrix}J_{p}&0\\\ 0&-{}^{t}J_{p}\end{pmatrix}$
in a Darboux basis. Then ${\overline{C}}_{2p}^{J}\in\mathfrak{sp}(2p)$ and we
denote by $\mathfrak{j}_{2p}$ the double extension of $\mathfrak{q}$ by
${\overline{C}}_{2p}^{J}$. So $\mathfrak{j}_{2p}\in\mathscr{N}(2+2p)$.
* •
for $p\geq 1$, we consider the symplectic vector space
$\mathfrak{q}=\mathbb{C}^{2p}$ equipped with its canonical bilinear form
$\overline{B}$ and the map ${\overline{C}}_{p+p}^{J}$ with matrix
$\begin{pmatrix}J_{p}&M\\\ 0&-{}^{t}J_{p}\end{pmatrix}$
in a Darboux basis where $M=(m_{ij})$ denotes the $p\times p$-matrix with
$m_{p,p}=1$ and $m_{ij}=0$ otherwise. Then
${\overline{C}}_{p+p}^{J}\in\mathfrak{sp}(2p)$ and we denote by
$\mathfrak{j}_{p+p}$ the double extension of $\mathfrak{q}$ by
${\overline{C}}_{p+p}^{J}$. So $\mathfrak{j}_{p+p}\in\mathscr{N}(2+2p)$.
The Lie superalgebras $\mathfrak{j}_{2p}$ or $\mathfrak{j}_{p+p}$ will be
called nilpotent Jordan-type Lie superalgebras.
Keep the notations as in Appendix. For $n\in\mathbb{N}$, $n\neq 0$, each
$[d]\in\mathcal{P}_{-1}(2n)$ can be written as
$[d]=(p_{1},p_{1},p_{2},p_{2},\dots,p_{k},p_{k},2q_{1},\dots 2q_{\ell}),$
with all $p_{i}$ odd, $p_{1}\geq p_{2}\geq\dots\geq p_{k}$ and $q_{1}\geq
q_{2}\geq\dots\geq q_{\ell}$.
We associate the partition $[d]$ with the map
${\overline{C}}_{[d]}\in\mathfrak{sp}(2n)$ having matrix
$\operatorname{diag}_{k+\ell}({\overline{C}}^{J}_{2p_{1}},{\overline{C}}^{J}_{2p_{2}},\dots,{\overline{C}}^{J}_{2p_{k}},{\overline{C}}^{J}_{q_{1}+q_{1}},\dots,{\overline{C}}^{J}_{q_{\ell}+q_{\ell}})$
in a Darboux basis of $\mathbb{C}^{2n}$ and denote by $\mathfrak{g}_{[d]}$ the
double extension of $\mathbb{C}^{2n}$ by ${\overline{C}}_{[d]}$. Then
$\mathfrak{g}_{[d]}\in\mathscr{N}(2+2n)$ and $\mathfrak{g}_{[d]}$ is an
amalgamated product of nilpotent Jordan-type Lie superalgebras. More
precisely,
$\mathfrak{g}_{[d]}=\mathfrak{j}_{2p_{1}}{\
\mathop{\times}\limits_{\mathrm{a}}\ }\mathfrak{j}_{2p_{2}}{\
\mathop{\times}\limits_{\mathrm{a}}\ }\dots\
\mathop{\times}\limits_{\mathrm{a}}\ \mathfrak{j}_{2p_{k}}{\
\mathop{\times}\limits_{\mathrm{a}}\ }\mathfrak{j}_{q_{1}+q_{1}}{\
\mathop{\times}\limits_{\mathrm{a}}\ }\dots\
\mathop{\times}\limits_{\mathrm{a}}\ \mathfrak{j}_{q_{\ell}+q_{\ell}}.$
###### Proposition 4.16.
Each $\mathfrak{g}\in\mathscr{N}(2+2n)$ is i-isomorphic to a unique
amalgamated product $\mathfrak{g}_{[d]},\ [d]\in\mathcal{P}_{-1}(2n)$, of
nilpotent Jordan-type Lie superalgebras.
For the reduced diagonalizable case, let $\mathfrak{g}_{4}^{s}(\lambda)$ be
the double extension of $\mathfrak{q}=\mathbb{C}^{2}$ by
${\overline{C}}=\begin{pmatrix}\lambda&0\\\ 0&-\lambda\end{pmatrix}$,
$\lambda\neq 0$. By Lemma 4.7, $\mathfrak{g}_{4}^{s}(\lambda)$ is i-isomorphic
to $\mathfrak{g}_{4}^{s}(1)=\mathfrak{g}_{4,2}^{s}$.
###### Proposition 4.17.
Let $\mathfrak{g}\in\mathscr{D}_{\mathrm{red}}(2+2n)$ then $\mathfrak{g}$ is
an amalgamated product of quadratic Lie superalgebras all i-isomorphic to
$\mathfrak{g}_{4,2}^{s}$.
Finally, for the invertible case, we recall the matrix
$J_{p}(\lambda)=\operatorname{diag}_{p}(\lambda,\dots,\lambda)+J_{p}$, $p\geq
1,\ \lambda\in\mathbb{C}$ and set
${\overline{C}}_{2p}^{J}(\lambda)=\begin{pmatrix}J_{p}(\lambda)&0\\\
0&-{}^{t}J_{p}(\lambda)\end{pmatrix}$
in a Darboux basis of $\mathbb{C}^{2p}$ then
${\overline{C}}_{2p}^{J}(\lambda)\in\mathfrak{sp}(2p)$. Let
$\mathfrak{j}_{2p}(\lambda)$ be the double extension of $\mathbb{C}^{2p}$ by
${\overline{C}}_{2p}^{J}(\lambda)$ then it is called a Jordan-type quadratic
Lie superalgebra.
When $\lambda=0$ and $p\geq 2$, we recover the nilpotent Jordan-type Lie
superalgebras $\mathfrak{j}_{2p}$. If $\lambda\neq 0$,
$\mathfrak{j}_{2p}(\lambda)$ becomes an invertible singular quadratic Lie
superalgebra and
$\mathfrak{j}_{2p}(-\lambda)\simeq\mathfrak{j}_{2p}(\lambda).$
###### Proposition 4.18.
Let $\mathfrak{g}\in\mathscr{S}_{\mathrm{inv}}(2+2n)$ then $\mathfrak{g}$ is
an amalgamated product of quadratic Lie superalgebras all i-isomorphic to
Jordan-type quadratic Lie superalgebras $\mathfrak{j}_{2p}(\lambda)$, with
$\lambda\neq 0$.
### 4.2. Quadratic dimension of reduced quadratic Lie superalgebras having
the 2-dimensional even part
Let $(\mathfrak{g},B)$ be a quadratic Lie superalgebra. To any bilinear form
$B^{\prime}$ on $\mathfrak{g}$, there is an associated map
$D:\mathfrak{g}\to\mathfrak{g}$ satisfying
$B^{\prime}(X,Y)=B(D(X),Y),\ \forall\ X,Y\in\mathfrak{g}.$
###### Lemma 4.19.
If $B^{\prime}$ is even then $D$ is even.
###### Proof.
Let $X$ be an element in $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
assume that $D(X)=Y+Z$ with
$Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$. Since $B^{\prime}$ is
even then $B^{\prime}(X,\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$.
It implies that
$B(D(X),\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=B(Z,\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$.
By the non-degeneracy of $B$ on
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$, we obtain $Z=0$ and then
$D(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\subset\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$.
Similarly to the case $X\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$,
it concludes that
$D(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})\subset\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
Thus, $D$ is even. ∎
###### Lemma 4.20.
Let $(\mathfrak{g},B)$ be a quadratic Lie superalgebra, $B^{\prime}$ be an
even bilinear form on $\mathfrak{g}$ and $D\in\mathcal{L}(\mathfrak{g})$ be
its associated map. Then:
1. (1)
$B^{\prime}$ is invariant if and only if $D$ satisfies
$D([X,Y])=[D(X),Y]=[X,D(Y)],\ \forall\ X,Y\in\mathfrak{g}.$
2. (2)
$B^{\prime}$ is supersymmetric if and only if $D$ satisfies
$B(D(X),Y)=B(X,D(Y)),\ \forall\ X,Y\in\mathfrak{g}.$
In this case, $D$ is called symmetric.
3. (3)
$B^{\prime}$ is non-degenerate if and only if $D$ is invertible.
###### Proof.
Let $X,Y$ and $Z$ be homogeneous elements in $\mathfrak{g}$ of degrees $x$,
$y$ and $z$, respectively.
1. (1)
If $B^{\prime}$ is invariant then
$B^{\prime}([X,Y],Z)=B^{\prime}(X,[Y,Z]).$
That means $B(D([X,Y]),Z=B(D(X),[Y,Z])=B([D(X),Y],Z)$. Since $B$ is non-
degenerate, one has $D([X,Y])=[D(X),Y]$. As a consequence,
$D([X,Y])=-(-1)^{xy}D([Y,X])=-(-1)^{xy}[D(Y),X]=[X,D(Y)]$ by $D$ even.
Conversely, if $D$ satisfies $D([X,Y])=[D(X),Y]=[X,D(Y)]$, for all
$X,Y\in\mathfrak{g}$, it is easy to check that $B^{\prime}$ is invariant.
2. (2)
$B^{\prime}$ is supersymmetric if and only if
$B^{\prime}(X,Y)=(-1)^{xy}B^{\prime}(Y,X)$. Therefore,
$B(D(X),Y)=(-1)^{xy}B(D(Y),X)=B(X,D(Y))$ by $B$ supersymmetric.
3. (3)
It is obvious since $B$ is non-degenerate.
∎
###### Definition 4.21.
Let $\mathfrak{g}$ be a quadratic Lie superalgebra. An even and symmetric map
$D\in\mathcal{L}(\mathfrak{g})$ satisfying Lemma 4.20 (1) is called a
centromorphism of $\mathfrak{g}$.
By [Ben03], given a quadratic Lie superalgebra $\mathfrak{g}$, the space of
centromorphisms of $\mathfrak{g}$ and the space generated by invertible ones
are the same, denote it by $\mathscr{C}(\mathfrak{g})$. As a consequence, the
space of even invariant supersymmetric bilinear forms on $\mathfrak{g}$
coincides with its subspace generated by non-degenerate ones. Moreover, all
those spaces have the same dimension called the quadratic dimension of
$\mathfrak{g}$ and denoted by $d_{q}(\mathfrak{g})$. The following proposition
gives the formula of $d_{q}(\mathfrak{g})$ for reduced quadratic Lie
superalgebras with 2-dimensional even part.
###### Proposition 4.22.
Let $\mathfrak{g}$ be a reduced quadratic Lie superalgebra with 2-dimensional
even part and $D\in\mathcal{L}(\mathfrak{g})$ be an even symmetric map. Then:
1. (1)
$D$ is a centromorphism if and only if there exist $\mu\in\mathbb{C}$ and an
even symmetric map $\mathsf{Z}:\mathfrak{g}\to\mathscr{Z}(\mathfrak{g})$ such
that $\mathsf{Z}|_{[\mathfrak{g},\mathfrak{g}]}=0$ and
$D=\mu\operatorname{Id}+\mathsf{Z}$. Moreover $D$ is invertible if and only if
$\mu\neq 0$.
2. (2)
$d_{q}(\mathfrak{g})=2+\displaystyle\frac{(\dim(\mathscr{Z}(\mathfrak{g})-1))(\dim(\mathscr{Z}(\mathfrak{g})-2)}{2}.$
###### Proof.
The proof goes exactly as Proposition 7.2 given in [DPU10], the reader may
refer to it.
∎
## 5\. Singular quadratic Lie superalgebras of type $\mathsf{S}_{1}$
Let $\mathfrak{g}$ be a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$ such that $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$
is non-Abelian. If
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]=\\{0\\}$
then $[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}]=\\{0\\}$
and therefore $\mathfrak{g}$ is an orthogonal direct sum of a singular
quadratic Lie algebra of type $\mathsf{S}_{1}$ and a vector space. There is
nothing to do. We can assume that
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq\\{0\\}$.
Fix $\alpha\in\mathscr{V}_{I}$ and choose
$\Omega_{0}\in\operatorname{Alt}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$,
$\Omega_{1}\in\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
such that
$I=\alpha\wedge\Omega_{0}+\alpha\otimes\Omega_{1}.$
Let $X_{\scriptscriptstyle{\overline{0}}}=\phi^{-1}(\alpha)$ then
$X_{\scriptscriptstyle{\overline{0}}}\in\mathscr{Z}(\mathfrak{g})$ and
$B(X_{\scriptscriptstyle{\overline{0}}},X_{\scriptscriptstyle{\overline{0}}})=0$.
We define linear maps
$C_{0}:\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\rightarrow\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$,
$C_{1}:\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}\rightarrow\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
by $\Omega_{0}(X,Y)=B(C_{0}(X),Y)$ if
$X,Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$\Omega_{1}(X,Y)=B(C_{1}(X),Y)$ if
$X,Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$. Let
$C:\mathfrak{g}\rightarrow\mathfrak{g}$ defined by $C(X+Y)=C_{0}(X)+C_{1}(Y)$,
for all $X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\
Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
###### Proposition 5.1.
For all $X,Y\in\mathfrak{g}$, the Lie super-bracket of $\mathfrak{g}$ is
defined by:
$[X,Y]=B(X_{\scriptscriptstyle{\overline{0}}},X)C(Y)-B(X_{\scriptscriptstyle{\overline{0}}},Y)C(X)+B(C(X),Y)X_{\scriptscriptstyle{\overline{0}}}.$
In particular, if $X,Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$,
$Z,T\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ then
1. (1)
$[X,Y]=B(X_{\scriptscriptstyle{\overline{0}}},X)C_{0}(Y)-B(X_{\scriptscriptstyle{\overline{0}}},Y)C_{0}(X)+B(C_{0}(X),Y)X_{\scriptscriptstyle{\overline{0}}}$,
2. (2)
$[X,Z]=B(X_{\scriptscriptstyle{\overline{0}}},X)C_{1}(Z)$,
3. (3)
$[Z,T]=B(C_{1}(Z),T)X_{\scriptscriptstyle{\overline{0}}}$
###### Proof.
By Proposition 2.7, $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is a
singular quadratic Lie algebra so the assertion (1) follows [DPU10]. Given
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$,
$Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$, one has
$B([X,Y],Z)=\alpha\otimes\Omega_{1}(X,Y,Z)=\alpha(X)\Omega_{1}(Y,Z)=B(X_{\scriptscriptstyle{\overline{0}}},X)B(C_{1}(Y),Z).$
Hence we obtain (2) and (3). ∎
Now, we show that $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is
solvable. Consider the quadratic Lie algebra
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ with 3-form
$I_{0}=\alpha\wedge\Omega_{0}$. Write
$\Omega_{0}=\sum_{i<j}{a_{ij}\alpha_{i}\wedge\alpha_{j}}$, with
$a_{ij}\in\mathbb{C}$. Set $X_{i}=\phi^{-1}(\alpha_{i})$ then
$C_{0}=\sum_{i<j}{a_{ij}(\alpha_{i}\otimes X_{j}-\alpha_{j}\otimes X_{i})}.$
Define the space
$W_{I_{0}}\subset\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$ by:
$\mathscr{W}_{I_{0}}=\\{\iota_{X\wedge Y}(I_{0})\ |\
X,Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}\\}.$
Then
$\mathscr{W}_{I_{0}}=\phi([\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}])$
and that implies
$\operatorname{Im}(C_{0})\subset[\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}]$.
In Section 2, it is known that $\\{\alpha,I_{0}\\}=0$ and then
$[X_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}]=0$.
As a sequence,
$B(X_{\scriptscriptstyle{\overline{0}}},[\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}])=0$.
That deduces
$B(X_{\scriptscriptstyle{\overline{0}}},\operatorname{Im}(C_{0}))=0$.
Therefore
$[[\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}],[\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}]]=[\operatorname{Im}(C_{0}),\operatorname{Im}(C_{0})]\subset\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\subset\mathscr{Z}(\mathfrak{g})$
and we conclude that $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is
solvable.
By $B$ non-degenerate there is an element
$Y_{\scriptscriptstyle{\overline{0}}}\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$
isotropic such that
$B(X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}})=1$.
Moreover, combined with $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ a
solvable singular quadratic Lie algebra, we can choose
$Y_{\scriptscriptstyle{\overline{0}}}$ satisfying
$C_{0}(Y_{\scriptscriptstyle{\overline{0}}})=0$ and we obtain then a
straightforward consequence as follows:
###### Corollary 5.2.
1. (1)
$C=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})$,
$\ker(C)=\mathscr{Z}(\mathfrak{g})\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}$
and
$[\mathfrak{g},\mathfrak{g}]=\operatorname{Im}(C)\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$.
2. (2)
The Lie superalgebra $\mathfrak{g}$ is solvable. Moreover, $\mathfrak{g}$ is
nilpotent if and only if C is nilpotent.
### 5.1. Singular quadratic Lie superalgebras of type $\mathsf{S}_{1}$ and
double extensions
The description of the Lie super-bracket in Proposition 5.1 allows us to
propose a definition of double extension of a quadratic
$\mathbb{Z}_{2}$-graded vector space as follows:
###### Definition 5.3.
Let
$(\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}},B_{\mathfrak{q}})$
be a quadratic $\mathbb{Z}_{2}$-graded vector space and ${\overline{C}}$ be an
even endomorphism of $\mathfrak{q}$. Assume that ${\overline{C}}$ is skew-
supersymmetric, that is, $B({\overline{C}}(X),Y)=-B(X,{\overline{C}}(Y))$, for
all $X,Y\in\mathfrak{q}$. Let
$(\mathfrak{t}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}}\\},B_{\mathfrak{t}})$
be a 2-dimensional quadratic vector space with the symmetric bilinear form
$B_{\mathfrak{t}}$ defined by:
$B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{0}}},X_{\scriptscriptstyle{\overline{0}}})=B_{\mathfrak{t}}(Y_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}})=0\
\text{and
}B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}})=1.$
Consider the vector space $\mathfrak{g}=\mathfrak{t}{\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ equipped with the bilinear
form $B=B_{\mathfrak{t}}+B_{\mathfrak{q}}$ and define on $\mathfrak{g}$ the
following bracket:
$[\lambda X_{{\scriptscriptstyle{\overline{0}}}}+\mu
Y_{{\scriptscriptstyle{\overline{0}}}}+X,\lambda^{\prime}X_{{\scriptscriptstyle{\overline{0}}}}+\mu^{\prime}Y_{{\scriptscriptstyle{\overline{0}}}}+Y]=\mu{\overline{C}}(Y)-\mu^{\prime}{\overline{C}}(X)+B({\overline{C}}(X),Y)X_{\scriptscriptstyle{\overline{0}}},$
for all
$X,Y\in\mathfrak{q},\lambda,\mu,\lambda^{\prime},\mu^{\prime}\in\mathbb{C}$.
Then $(\mathfrak{g},B)$ is a quadratic solvable Lie superalgebra with
$\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}=\mathfrak{t}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$
and
$\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$.
We say that $\mathfrak{g}$ is the double extension of $\mathfrak{q}$ by
${\overline{C}}$.
Note that an even skew-supersymmetric endomorphism ${\overline{C}}$ on
$\mathfrak{q}$ can be written by
${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}$ where
${\overline{C}}_{0}\in\mathfrak{o}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$
and
${\overline{C}}_{1}\in\mathfrak{sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$.
###### Corollary 5.4.
Let $\mathfrak{g}$ be the double extension of $\mathfrak{q}$ by
${\overline{C}}$. Denote by
$C=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})$ then one has
1. (1)
$[X,Y]=B(X_{\scriptscriptstyle{\overline{0}}},X)C(Y)-B(X_{\scriptscriptstyle{\overline{0}}},Y)C(X)+B(C(X),Y)X_{\scriptscriptstyle{\overline{0}}}$,
for all $X,Y\in\mathfrak{g}.$
2. (2)
$\mathfrak{g}$ is a singular quadratic Lie superalgebra. If
${\overline{C}}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$ is nonzero
then $\mathfrak{g}$ is of type $\mathsf{S}_{1}$.
###### Proof.
The assertion (1) is direct from the above definition. Let
$\alpha=\phi(X_{\scriptscriptstyle{\overline{0}}})$ and define the bilinear
form $\Omega:\mathfrak{g}\rightarrow\mathfrak{g}$ by $\Omega(X,Y)=B(C(X),Y)$
for all $X,Y\in\mathfrak{g}$. By $B$ even and supersymmetric, $C$ even and
skew-supersymmetric (with respect to $B$) then
$\Omega=\Omega_{0}+\Omega_{1}\in\operatorname{Alt}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\oplus\operatorname{Sym}^{2}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
The formula in (1) can be replaced by
$I=\alpha\wedge\Omega_{0}+\alpha\otimes\Omega_{1}=\alpha\wedge\Omega$.
Therefore, $\operatorname{dup}(\mathfrak{g})\geq 1$ and $\mathfrak{g}$ is
singular. If
${\overline{C}}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$ is nonzero
then $\Omega_{1}\neq 0$. In this case,
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq\\{0\\}$
and thus $\operatorname{dup}(\mathfrak{g})=1$. ∎
As a consequence of Proposition 5.1 and Definition 5.3, one has
###### Lemma 5.5.
Let $(\mathfrak{g},B)$ be a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$. Keep the notations as in Proposition 5.1 and Corollary 5.2.
Then $(\mathfrak{g},B)$ is the double extension of
$\mathfrak{q}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}})^{\bot}$
by ${\overline{C}}=C|_{\mathfrak{q}}$.
###### Remark 5.6.
The above definition is a generalization of the definition of double extension
of a quadratic vector space by a skew-symmetric map in [DPU10] and Definition
4.6. Moreover, if let
$\mathfrak{g}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\
}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$
be the double extension of
$\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
by ${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}$ then
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ is the double extension of
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$ by ${\overline{C}}_{0}$ and
the subalgebra
$(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\
}\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$ is the double extension of
$\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$ by ${\overline{C}}_{1}$.
The proof of the proposition below is completely analogous to the proof of
Proposition 4.10, so we omit it.
###### Proposition 5.7.
Let
$\mathfrak{g}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\
}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$
and
$\mathfrak{g}^{\prime}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}^{\prime}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\
}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$
be two double extensions of
$\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
by ${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}$ and
$\overline{C^{\prime}}=\overline{C_{0}^{\prime}}+\overline{C_{1}^{\prime}}$,
respectively. Assume that ${\overline{C}}_{1}$ is nonzero. Then
1. (1)
there exists a Lie superalgebra isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exist invertible maps
$P\in\mathcal{L}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$,
$Q\in\mathcal{L}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$ and a
nonzero $\lambda\in\mathbb{C}$ such that
* (i)
$\overline{C_{0}^{\prime}}=\lambda P{\overline{C}}_{0}P^{-1}$ and
$P^{*}P{\overline{C}}_{0}={\overline{C}}_{0}$.
* (ii)
$\overline{C_{1}^{\prime}}=\lambda Q{\overline{C}}_{1}Q^{-1}$ and
$Q^{*}Q{\overline{C}}_{1}={\overline{C}}_{1}$.
where $P^{*}$ and $Q^{*}$ are the adjoint maps of $P$ and $Q$ with respect to
$B|_{\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\times\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}}$
and
$B|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}\times\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$.
2. (2)
there exists an i-isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there is a nonzero
$\lambda\in\mathbb{C}$ such that $\overline{C_{0}^{\prime}}$ is in the
$\operatorname{O}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$-adjoint
orbit through $\lambda{\overline{C}}_{0}$ and $\overline{C_{1}^{\prime}}$ is
in the
$\operatorname{Sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$-adjoint
orbit through $\lambda{\overline{C}}_{1}$.
###### Remark 5.8.
If let $M=P+Q$ then $M^{-1}=P^{-1}+Q^{-1}$ and $M^{*}=P^{*}+Q^{*}$. The
formulas in Proposition 5.7 (1) can be written:
$\overline{C^{\prime}}=\lambda M{\overline{C}}M^{-1}\ \text{and
}M^{*}M{\overline{C}}={\overline{C}}.$
Hence, the classification problem of singular quadratic Lie superalgebras of
type $\mathsf{S}_{1}$ (up to i-isomorphism) can be reduced to the
classification of
$\operatorname{O}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})\times\operatorname{Sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$\-
orbits of
$\mathfrak{o}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})\oplus\mathfrak{sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$,
where
$\operatorname{O}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})\times\operatorname{Sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$
denotes the direct product of two groups
$\operatorname{O}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$ and
$\operatorname{Sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$.
###### Definition 5.9.
Let
$\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
be a quadratic $\mathbb{Z}_{2}$-graded vector space. An even isomorphism
$F\in\mathcal{L}(\mathfrak{q})$ is called an isometry of $\mathfrak{q}$ if
$F|_{\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}}$ and
$F|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$ are isometries.
To prove the following Corollary, it is enough to follow exactly the same
steps as in Corollary 4.11.
###### Corollary 5.10.
Let $(\mathfrak{g},B)$ and $(\mathfrak{g}^{\prime},B^{\prime})$ be double
extensions of $(\mathfrak{q},\overline{B})$ and
$(\mathfrak{q}^{\prime},\overline{B^{\prime}})$ by ${\overline{C}}$ and
$\overline{C^{\prime}}$ respectively where
$\overline{B}=B|_{\mathfrak{q}\times\mathfrak{q}}$ and
$\overline{B^{\prime}}=B^{\prime}|_{\mathfrak{q}^{\prime}\times\mathfrak{q}^{\prime}}$.
Write
$\mathfrak{g}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
$\mathfrak{g}^{\prime}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}^{\prime}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}^{\prime}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}^{\prime}$. Then:
1. (1)
there exists an i-isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exists an isometry
$\overline{A}:\mathfrak{q}\to\mathfrak{q}^{\prime}$ such that
${\overline{C^{\prime}}}=\lambda\ \overline{A}\ {\overline{C}}\
\overline{A}{}^{-1}$, for some nonzero $\lambda\in\mathbb{C}$.
2. (2)
there exists a Lie superalgebra isomorphism between $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ if and only if there exist even invertible maps
$\overline{Q}:\mathfrak{q}\to\mathfrak{q}^{\prime}$ and
$\overline{P}\in\mathcal{L}(\mathfrak{q})$ such that
* (i)
${\overline{C^{\prime}}}=\lambda\ \overline{Q}\ {\overline{C}}\
\overline{Q}^{-1}$ for some nonzero $\lambda\in\mathbb{C}$,
* (ii)
$\overline{P}^{*}\ \overline{P}\ {\overline{C}}={\overline{C}}$ and
* (iii)
$\overline{Q}\ \overline{P}^{-1}$ is an isometry from $\mathfrak{q}$ onto
$\mathfrak{q}^{\prime}$.
### 5.2. Fitting decomposition of a skew-supersymmetric map
We recall the following useful result (see for instance [DPU10]):
###### Lemma 5.11.
Let ${\overline{C}}$ and $\overline{C^{\prime}}$ be nilpotent elements in
$\mathfrak{o}(n)$. Then ${\overline{C}}$ is conjugate to
$\lambda\overline{C^{\prime}}$ modulo $\operatorname{O}(n)$ for some nonzero
$\lambda\in\mathbb{C}$ if and only if ${\overline{C}}$ is conjugate to
$\overline{C^{\prime}}$.
Remark that the lemma remains valid if we replace $\mathfrak{o}(n)$ by
$\mathfrak{sp}(2n)$ and $\operatorname{O}(n)$ by $\operatorname{Sp}(2n)$.
###### Proposition 5.12.
Let $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ be two nilpotent singular
quadratic Lie superalgebras. Then $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$
are isomorphic if and only if they are i-isomorphic.
###### Proof.
Singular quadratic Lie superalgebras $\mathfrak{g}$ and
$\mathfrak{g}^{\prime}$ are regarded as double extensions
$\mathfrak{g}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
$(\mathbb{C}X^{\prime}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y^{\prime}_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}^{\prime}$ by ${\overline{C}}$
and $\overline{C^{\prime}}$ where
$\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
and
$\mathfrak{q}^{\prime}=\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{1}}}$.
By Corollary 5.2, ${\overline{C}}$ and $\overline{C^{\prime}}$ are nilpotent.
Rewrite ${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}$ and
$\overline{C^{\prime}}=\overline{C_{0}^{\prime}}+\overline{C_{1}^{\prime}}$,
where
${\overline{C}}_{0}\in\mathfrak{o}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$,
$\overline{C_{0}^{\prime}}\in\mathfrak{o}(\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{0}}})$,
${\overline{C}}_{1}\in\mathfrak{sp}(\mathfrak{q})$ and
$\overline{C_{1}^{\prime}}\in\mathfrak{sp}(\mathfrak{q}^{\prime})$.
If $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are isomorphic then
$\dim(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})=\dim(\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{0}}})$
and
$\dim(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})=\dim(\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{1}}})$.
Thus, there exist isometries
$\overline{F}_{0}:\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{0}}}\rightarrow\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$
and
$\overline{F}_{1}:\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{1}}}\rightarrow\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
and then we define an isometry
$\overline{F}:\mathfrak{q}^{\prime}\rightarrow\mathfrak{q}$ by
$\overline{F}(X^{\prime}+Y^{\prime})=\overline{F}_{0}(X^{\prime})+\overline{F}^{\prime}_{0}(Y^{\prime})$
for all
$X^{\prime}\in\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{0}}}$ and
$Y^{\prime}\in\mathfrak{q}^{\prime}_{\scriptscriptstyle{\overline{1}}}$. We
now set $F:\mathfrak{g}^{\prime}\rightarrow\mathfrak{g}$ by
$F(X^{\prime}_{\scriptscriptstyle{\overline{0}}})=X_{\scriptscriptstyle{\overline{0}}}$,
$F(Y^{\prime}_{\scriptscriptstyle{\overline{0}}})=Y_{\scriptscriptstyle{\overline{0}}}$,
$F|_{\mathfrak{q}^{\prime}}=\overline{F}$ and a new Lie super-bracket on
$\mathfrak{g}$ by:
$[X,Y]^{\prime\prime}=F\left([F^{-1}(X),F^{-1}(Y)]^{\prime}\right),\ \forall
X,Y\in\mathfrak{g}.$
Denote by $\mathfrak{g}^{\prime\prime}$ this new quadratic Lie superalgebras.
It is easy to see that
$\mathfrak{g}^{\prime\prime}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ is the double extension of
$\mathfrak{q}$ by $\overline{C^{\prime\prime}}=\overline{F}\
\overline{C^{\prime}}\ \overline{F}^{-1}$ and $\mathfrak{g}^{\prime\prime}$ is
i-isomorphic to $\mathfrak{g}^{\prime}$. It need to prove that
$\mathfrak{g}^{\prime\prime}$ is i-isomorphic to $\mathfrak{g}$. Write
$\overline{C^{\prime\prime}}=\overline{C^{\prime\prime}_{0}}+\overline{C^{\prime\prime}_{1}}\in\mathfrak{o}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})\oplus\mathfrak{sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$.
Since $\mathfrak{g}$ and $\mathfrak{g}^{\prime\prime}$ are isomorphic then
there exist invertible maps
$P:\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\rightarrow\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$
and
$Q:\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}\rightarrow\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}$
such that $\overline{C^{\prime\prime}_{0}}=\lambda\overline{P}\
{\overline{C}}_{0}\ \overline{P}^{-1}$ and
$\overline{C^{\prime\prime}_{1}}=\lambda\overline{Q}\ {\overline{C}}_{1}\
\overline{Q}^{-1}$ for some nonzero $\lambda\in\mathbb{C}$. By Lemma 5.11,
${\overline{C}}_{0}$ and $\overline{C^{\prime\prime}_{0}}$ are conjugate under
$\operatorname{O}(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}})$,
${\overline{C}}_{1}$ and $\overline{C^{\prime\prime}_{1}}$ are conjugate under
$\operatorname{Sp}(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}})$ and we
can assume that $\lambda=1$. Therefore $\mathfrak{g}$ and
$\mathfrak{g}^{\prime\prime}$ are i-isomorphic. The proposition is proved. ∎
Let now $\mathfrak{g}$ be a singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$. Write $\mathfrak{g}$ as a double extension of
$(\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}},\overline{B})$
by ${\overline{C}}={\overline{C}}_{0}+{\overline{C}}_{1}$ where
${\overline{C}}=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})|_{\mathfrak{q}}$,
${\overline{C}}_{0}={\overline{C}}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}}$
and
${\overline{C}}_{1}={\overline{C}}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$.
We consider the Fitting decomposition of ${\overline{C}}_{0}$ on
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$ and ${\overline{C}}_{1}$ on
${\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}}$ by:
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{N}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{I}\
\text{ and }\
\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{N}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{I}$
where $\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{N}$ and
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{I}$ (resp.
$\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{N}$ and
$\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{I}$) are
${\overline{C}}_{0}$-stable (resp. ${\overline{C}}_{1}$-stable),
${\overline{C}}_{0}^{N}={\overline{C}}_{0}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{N}}$
and
${\overline{C}}_{1}^{N}={\overline{C}}_{1}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{N}}$
are nilpotent,
${\overline{C}}_{0}^{I}={\overline{C}}_{0}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{I}}$
and
${\overline{C}}_{1}^{I}={\overline{C}}_{1}|_{\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{I}}$
are invertible. Recall that ${\overline{C}}$ is skew-supersymmetric then
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{I}=(\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{N})^{\bot}$
in $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{I}=(\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{N})^{\bot}$
in $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
Next, we set
$\mathfrak{q}_{N}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{N}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{N}\
\text{ and }\
\mathfrak{q}_{I}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}^{I}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}^{I}$
As a consequence, ${\overline{C}}_{N}={\overline{C}}|_{\mathfrak{q}_{N}}$ is
nilpotent, ${\overline{C}}_{I}={\overline{C}}|_{\mathfrak{q}_{I}}$ is
invertible, $[\mathfrak{q}_{N},\mathfrak{q}_{I}]=\\{0\\}$, the restrictions
$\overline{B}_{N}=\overline{B}|_{\mathfrak{q}_{N}\times\mathfrak{q}_{N}}$ and
$\overline{B}_{I}=\overline{B}|_{\mathfrak{q}_{I}\times\mathfrak{q}_{I}}$ are
non-degenerate and supersymmetric. It is easy to check that
${\overline{C}}_{N}={\overline{C}}_{0}^{N}+{\overline{C}}_{1}^{N}$,
${\overline{C}}_{I}={\overline{C}}_{0}^{I}+{\overline{C}}_{1}^{I}$. Moreover,
${\overline{C}}_{N}$, ${\overline{C}}_{I}$ are skew-supersymmetric and they
are Fitting components of ${\overline{C}}$ in $\mathfrak{q}$. Let
$\mathfrak{g}_{N}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}_{N}$ and
$\mathfrak{g}_{I}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}_{I}$. Then $\mathfrak{g}_{N}$
and $\mathfrak{g}_{I}$ are subalgebras of $\mathfrak{g}$, $\mathfrak{g}_{N}$
is the double extension of $\mathfrak{q}_{N}$ by ${\overline{C}}_{N}$,
$\mathfrak{g}_{I}$ is the double extension of $\mathfrak{q}_{I}$ by
${\overline{C}}_{I}$ and $\mathfrak{g}_{N}$ is a nilpotent singular quadratic
Lie superalgebra.
###### Definition 5.13.
The subalgebras $\mathfrak{g}_{N}$ and $\mathfrak{g}_{I}$ as above are
respectively the nilpotent and invertible Fitting components of
$\mathfrak{g}$.
###### Definition 5.14.
A double extension is called an invertible quadratic Lie superalgebra if the
corresponding skew-supersymmetric map is invertible.
It is easy to check that the dimension of an invertible quadratic Lie
superalgebra must be even. Moreover, following Corollary 5.10, two invertible
quadratic Lie superalgebras are isomorphic if and only if they are
i-isomorphic. This property is still right for singular quadratic Lie
superalgebras of type $\mathsf{S}_{1}$.
###### Proposition 5.15.
Let $\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ be singular quadratic Lie
superalgebras of type $\mathsf{S}_{1}$ and $\mathfrak{g}_{N}$,
$\mathfrak{g}_{I}$, $\mathfrak{g}^{\prime}_{N}$, $\mathfrak{g}^{\prime}_{I}$
be their Fitting components, respectively. Then
1. (1)
$\mathfrak{g}\overset{\mathrm{i}}{\simeq}\mathfrak{g}^{\prime}$ if and only if
$\mathfrak{g}_{N}\overset{\mathrm{i}}{\simeq}\mathfrak{g}^{\prime}_{N}$ and
$\mathfrak{g}_{I}\overset{\mathrm{i}}{\simeq}\mathfrak{g}_{I}^{\prime}$. The
result remains valid if we replace $\overset{\mathrm{i}}{\simeq}$ by $\simeq$.
2. (2)
$\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ are isomorphic if and only if they
are i-isomorphic.
###### Proof.
The proposition is proved as Proposition 6.4 in [DPU10]. It is sketched as
follows. We assume that $\mathfrak{g}\simeq\mathfrak{g}^{\prime}$. They are
regarded as double extensions
$\mathfrak{g}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ and
$(\mathbb{C}X^{\prime}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y^{\prime}_{\scriptscriptstyle{\overline{0}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}^{\prime}$ by ${\overline{C}}$
and $\overline{C^{\prime}}$. By Corollary 5.10, there are an even invertible
map $\overline{P}:\mathfrak{q}\to\mathfrak{q}^{\prime}$ and a nonzero
$\lambda\in\mathbb{C}$ such that $\overline{C^{\prime}}=\lambda\ \overline{P}\
{\overline{C}}\ \overline{P}^{-1}$, so
$\mathfrak{q}_{N}^{\prime}=\overline{P}(\mathfrak{q}_{N})$ and
$\mathfrak{q}_{I}^{\prime}=\overline{P}(\mathfrak{q}_{I})$, then
$\dim(\mathfrak{q}^{\prime}_{N})=\dim(\mathfrak{q}_{N})$ and
$\dim(\mathfrak{q}^{\prime}_{I})=\dim(\mathfrak{q}_{I})$. Thus, there exist
isometries $F_{N}:\mathfrak{q}^{\prime}_{N}\to\mathfrak{q}_{N}$ and
$F_{I}:\mathfrak{q}^{\prime}_{I}\to\mathfrak{q}_{I}$ and we can define an
isometry $\overline{F}:\mathfrak{q}^{\prime}\to\mathfrak{q}$ by
$\overline{F}(X_{N}^{\prime}+X_{I}^{\prime})=F_{N}(X_{N}^{\prime})+F_{I}(X_{I}^{\prime})$,
$\forall X_{N}^{\prime}\in\mathfrak{q}_{N}^{\prime}$ and
$X_{I}^{\prime}\in\mathfrak{q}_{I}^{\prime}$. We now define
$F:\mathfrak{g}^{\prime}\to\mathfrak{g}$ by $F(X_{1}^{\prime})=X_{1}$,
$F(Y_{1}^{\prime})=Y_{1}$, $F|_{\mathfrak{q}^{\prime}}=\overline{F}$ and a new
Lie super-bracket on $\mathfrak{g}$:
$[X,Y]^{\prime\prime}=F\left([F^{-1}(X),F^{-1}(Y)]^{\prime}\right),\ \forall
X,Y\in\mathfrak{g}.$
Denote by $\mathfrak{g}^{\prime\prime}$ this new quadratic Lie superalgebra.
It is obvious that
$\mathfrak{g}^{\prime}\overset{\mathrm{i}}{\simeq}\mathfrak{g}^{\prime\prime}$.
It remains to prove the assertions for two quadratic Lie superalgebras
$\mathfrak{g}$ and $\mathfrak{g}^{\prime\prime}$. Those follow Corollary 5.10,
Lemma 5.11 and Proposition 5.12.
∎
###### Proposition 5.16.
The $\operatorname{dup}$-number is invariant under Lie superalgebra
isomorphisms, i.e. if $(\mathfrak{g},B)$ and
$(\mathfrak{g}^{\prime},B^{\prime})$ are quadratic Lie superalgebras with
$\mathfrak{g}\simeq\mathfrak{g}^{\prime}$, then
$\operatorname{dup}(\mathfrak{g})=\operatorname{dup}(\mathfrak{g}^{\prime})$.
###### Proof.
By Lemma 2.4 we can assume that $\mathfrak{g}$ is reduced. By Proposition 2.3,
$\mathfrak{g}^{\prime}$ is also reduced. Since
$\mathfrak{g}\simeq\mathfrak{g}^{\prime}$ then we can identify
$\mathfrak{g}=\mathfrak{g}^{\prime}$ as a Lie superalgebra equipped with the
bilinear forms $B$, $B^{\prime}$ and we have two $\operatorname{dup}$-numbers:
$\operatorname{dup}_{B}(\mathfrak{g})$ and
$\operatorname{dup}_{B^{\prime}}(\mathfrak{g})$.
We start with the case $\operatorname{dup}_{B}(\mathfrak{g})=3$. Since
$\mathfrak{g}$ is reduced then
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\\{0\\}$ and $\mathfrak{g}$
is a reduced singular quadratic Lie algebra of type $\mathsf{S}_{3}$. By
[PU07], $\dim([\mathfrak{g},\mathfrak{g}])=3$ and then
$\operatorname{dup}_{B^{\prime}}(\mathfrak{g})=3$.
If $\operatorname{dup}_{B}(\mathfrak{g})=1$, then $\mathfrak{g}$ is of type
$\mathsf{S}_{1}$ with respect to $B$. There are two cases:
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]=\\{0\\}$
and
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq\\{0\\}$.
If
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]=0$
then $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\\{0\\}$ by
$\mathfrak{g}$ reduced. In this case, $\mathfrak{g}$ is a reduced singular
quadratic Lie algebra of type $\mathsf{S}_{1}$. By [DPU10], $\mathfrak{g}$ is
also a reduced singular quadratic Lie algebra of type $\mathsf{S}_{1}$ with
the bilinear form $B^{\prime}$, i.e.
$\operatorname{dup}_{B^{\prime}}(\mathfrak{g})=1$.
Assume that
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq\\{0\\}$,
we need the following lemma:
###### Lemma 5.17.
Let $\mathfrak{g}$ be a reduced quadratic Lie superalgebras of type
$\mathsf{S}_{1}$ such that
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq
0$ and $D\in\mathcal{L}(\mathfrak{g})$ be an even symmetric map. Then $D$ is a
centromorphism if and only if there exist $\mu\in\mathbb{C}$ and an even
symmetric map $\mathsf{Z}:\mathfrak{g}\to\mathscr{Z}(\mathfrak{g})$ such that
$\mathsf{Z}|_{[\mathfrak{g},\mathfrak{g}]}=0$ and
$D=\mu\operatorname{Id}+\mathsf{Z}$. Moreover $D$ is invertible if and only if
$\mu\neq 0$.
###### Proof.
First, $\mathfrak{g}$ can be realized as the double extension
$\mathfrak{g}=(\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}}){\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{q}$ by
$C=\operatorname{ad}(Y_{\scriptscriptstyle{\overline{0}}})$ and let
${\overline{C}}=C|_{\mathfrak{q}}$.
Assume that $D$ is an invertible centromorphism. The condition (1) of Lemma
4.20 implies that $D\circ\operatorname{ad}(X)=\operatorname{ad}(X)\circ D$,
for all $X\in\mathfrak{g}$ and then $DC=CD$. Using formula (1) of Corollary
5.4 and $CD=DC$, from
$[D(X),Y_{{\scriptscriptstyle{\overline{0}}}}]=[X,D(Y_{\scriptscriptstyle{\overline{0}}})]$
we find
$D(C(X))=\mu C(X),\ \forall\ X\in\mathfrak{g},\ \text{where
}\mu=B(D(X_{\scriptscriptstyle{\overline{0}}}),Y_{\scriptscriptstyle{\overline{0}}}).$
Since $D$ is invertible, one has $\mu\neq 0$ and
$C(D-\mu\operatorname{Id})=0$. Recall that
$\ker(C)=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\ker({\overline{C}})\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}=\mathscr{Z}(\mathfrak{g})\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}$,
there exist a map $\mathsf{Z}:\mathfrak{g}\to\mathscr{Z}(\mathfrak{g})$ and
$\varphi\in\mathfrak{g}^{*}$ such that
$D-\mu\operatorname{Id}=\mathsf{Z}+\varphi\otimes
Y_{\scriptscriptstyle{\overline{0}}}.$
It needs to show that $\varphi=0$. Indeed, $D$ maps
$[\mathfrak{g},\mathfrak{g}]$ into itself and
$Y_{\scriptscriptstyle{\overline{0}}}\notin[\mathfrak{g},\mathfrak{g}]$, so
$\varphi|_{[\mathfrak{g},\mathfrak{g}]}=0$. One has
$[\mathfrak{g},\mathfrak{g}]=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\operatorname{Im}({\overline{C}})$.
If $X\in\operatorname{Im}({\overline{C}})$, let $X=C(Y)$. Then
$D(X)=D(C(Y))=\mu C(Y)$, so $D(X)=\mu X$. For
$Y_{\scriptscriptstyle{\overline{0}}}$,
$D([Y_{\scriptscriptstyle{\overline{0}}},X])=DC(X)=\mu C(X)$ for all
$X\in\mathfrak{g}$. But also,
$D([Y_{\scriptscriptstyle{\overline{0}}},X])=[D(Y_{\scriptscriptstyle{\overline{0}}}),X]=\mu
C(X)+\varphi(Y_{\scriptscriptstyle{\overline{0}}})C(X)$, hence
$\varphi(Y_{\scriptscriptstyle{\overline{0}}})=0$. As a consequence,
$D(Y_{\scriptscriptstyle{\overline{0}}})=\mu
Y_{\scriptscriptstyle{\overline{0}}}+\mathsf{Z}(Y_{\scriptscriptstyle{\overline{0}}})$.
Now, we prove that $D(X_{\scriptscriptstyle{\overline{0}}})=\mu
X_{\scriptscriptstyle{\overline{0}}}$. Indeed, since $D$ is even and
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$
then one has
$D(X_{\scriptscriptstyle{\overline{0}}})\subset
D([\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}])=[D(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}),\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\subset[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}.$
It implies that,
$D(X_{\scriptscriptstyle{\overline{0}}})=aX_{\scriptscriptstyle{\overline{0}}}$.
Combined with
$B(D(Y_{\scriptscriptstyle{\overline{0}}}),X_{\scriptscriptstyle{\overline{0}}})=B(Y_{\scriptscriptstyle{\overline{0}}},D(X_{\scriptscriptstyle{\overline{0}}}))$,
we obtain $\mu=a$.
Let $X\in\mathfrak{q}$, $B(D(X_{\scriptscriptstyle{\overline{0}}}),X)=\mu
B(X_{\scriptscriptstyle{\overline{0}}},X)=0$. Moreover,
$B(D(X_{\scriptscriptstyle{\overline{0}}}),X)=B(X_{\scriptscriptstyle{\overline{0}}},D(X))$,
so $\varphi(X)=0$.
Since $\mathscr{C}(\mathfrak{g})$ is generated by invertible centromorphisms
then the necessary condition of Lemma is finished. The sufficiency is obvious.
∎
Let us return now to the proposition. By the previous lemma, the bilinear form
$B^{\prime}$ defines an associated invertible centromorphism
$D=\mu\operatorname{Id}+\mathsf{Z}$ for some nonzero $\mu\in\mathbb{C}$ and
$\mathsf{Z}:\mathfrak{g}\to\mathscr{Z}(\mathfrak{g})$ satisfying
$\mathsf{Z}|_{[\mathfrak{g},\mathfrak{g}]}=0$. For all $X,Y,Z\in\mathfrak{g}$,
one has:
$I^{\prime}(X,Y,Z)=B^{\prime}([X,Y],Z)=B(D([X,Y]),Z)=B([D(X),Y],Z)=\mu
B([X,Y],Z).$
That means $I^{\prime}=\mu I$ and then
$\operatorname{dup}_{B^{\prime}}(\mathfrak{g})=\operatorname{dup}_{B}(\mathfrak{g})=1$.
Finally, if $\operatorname{dup}_{B}(\mathfrak{g})=0$ then $\mathfrak{g}$
cannot be of type $\mathsf{S}_{3}$ or $\mathsf{S}_{1}$ with respect to
$B^{\prime}$, so $\operatorname{dup}_{B^{\prime}}(\mathfrak{g})=0$. ∎
Let $\mathfrak{g}$ be a reduced singular quadratic Lie superalgebra of type
$\mathsf{S}_{1}$ such that
$[\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}]\neq
0$. Keep the notation as in Lemma 5.17. We set
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}}=\mathscr{Z}(\mathfrak{g})\cap\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$,
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}}=\mathscr{Z}(\mathfrak{g})\cap\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$,
$[\mathfrak{g},\mathfrak{g}]_{\scriptscriptstyle{\overline{0}}}=[\mathfrak{g},\mathfrak{g}]\cap\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$
and
$[\mathfrak{g},\mathfrak{g}]_{\scriptscriptstyle{\overline{1}}}=[\mathfrak{g},\mathfrak{g}]\cap\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$.
It is obvious that
$X_{\scriptscriptstyle{\overline{0}}}\in\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}}\subset[\mathfrak{g},\mathfrak{g}]_{\scriptscriptstyle{\overline{0}}}$
and
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}}\subset[\mathfrak{g},\mathfrak{g}]_{\scriptscriptstyle{\overline{1}}}$.
In other words, $\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}}$
and $\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}}$ are totally
isotropic subspaces of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$, respectively. Rewrite
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}}=\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{l}_{\scriptscriptstyle{\overline{0}}}$.
Then there exist totally isotropic subspaces
$\mathfrak{u}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}$
of $\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$ and
$\mathfrak{u}_{\scriptscriptstyle{\overline{1}}}$ of
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ such that
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=[\mathfrak{g},\mathfrak{g}]_{\scriptscriptstyle{\overline{0}}}\oplus(\mathfrak{u}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}})$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=[\mathfrak{g},\mathfrak{g}]_{\scriptscriptstyle{\overline{1}}}\oplus\mathfrak{u}_{\scriptscriptstyle{\overline{1}}}$,
the subspaces
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}}\oplus(\mathfrak{u}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}})$
and
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}}\oplus\mathfrak{u}_{\scriptscriptstyle{\overline{1}}}$
are non-degenerate. Let us define
$\mathsf{Z}:\mathfrak{u}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{u}_{\scriptscriptstyle{\overline{1}}}\rightarrow\mathfrak{l}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}\oplus\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}}$
by: set bases
$\\{X_{1}=X_{\scriptscriptstyle{\overline{0}}},X_{2},...,X_{r}\\}$ of
$\mathfrak{l}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{0}}}$,
$\\{Y_{1},...,Y_{t}\\}$ of
$\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}}$,
$\\{X_{1}^{\prime}=Y_{\scriptscriptstyle{\overline{0}}},X_{2}^{\prime},...,X_{r}^{\prime}\\}$
of
$\mathfrak{u}_{\scriptscriptstyle{\overline{0}}}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{0}}}$
and $\\{Y^{\prime}_{1},...,Y^{\prime}_{t}\\}$ of
$\mathfrak{u}_{\scriptscriptstyle{\overline{1}}}$ such that
$B(X_{i},X^{\prime}_{j})=\delta_{ij}$, $B(Y_{k},Y^{\prime}_{l})=\delta_{kl}$.
Then the map $\mathsf{Z}$ is completely defined by
$\mathsf{Z}\left(\sum_{j=1}^{r}x_{j}X_{j}^{\prime}\right)=\sum_{i=1}^{r}\left(\sum_{j=1}^{r}\mu_{ij}x_{j}\right)X_{i},$
$\mathsf{Z}\left(\sum_{j=1}^{t}y_{j}Y_{j}^{\prime}\right)=\sum_{i=1}^{t}\left(\sum_{j=1}^{t}\nu_{ij}y_{j}\right)Y_{i}$
with $\mu_{ij}=\mu_{ji}=B(X_{i}^{\prime},\mathsf{Z}(X_{j}^{\prime}))$ and
$\nu_{ij}=-\nu_{ji}=B(Y_{i}^{\prime},\mathsf{Z}(Y_{j}^{\prime}))$.
It results that the quadratic dimension of $\mathfrak{g}$ can be calculated as
follows:
$d_{q}(\mathfrak{g})=1+\displaystyle\frac{\dim(\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}})(1+\dim(\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{0}}}))}{2}+\displaystyle\frac{\dim(\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}})(\dim(\mathscr{Z}(\mathfrak{g})_{\scriptscriptstyle{\overline{1}}})-1)}{2}.$
## 6\. Quasi-singular quadratic Lie algebras
By Definition 5.3, it is natural to question: let
$(\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}},B_{\mathfrak{q}})$
be a quadratic $\mathbb{Z}_{2}$-graded vector space and ${\overline{C}}$ be an
endomorphism of $\mathfrak{q}$. Let
$(\mathfrak{t}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\},B_{\mathfrak{t}})$
be a 2-dimensional symplectic vector space with
$B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}})=1$.
Is there an extension $\mathfrak{g}=\mathfrak{q}{\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{t}$ such that $\mathfrak{g}$
equipped with the bilinear form $B=B_{\mathfrak{q}}+B_{\mathfrak{t}}$ becomes
a quadratic Lie superalgebra such that
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}\oplus\mathfrak{t}$
and the Lie super-bracket is represented by ${\overline{C}}$? In this section,
we will give an affirmative answer to this question.
The dup-number and the form of the associated invariant $I$ in the previous
sections suggest that it would be also interesting to study a quadratic Lie
superalgebra $\mathfrak{g}$ whose associated invariant $I$ has the form
$I=J\wedge p$
where $p\in\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}^{*}$ is nonzero,
$J\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
is indecomposable. We obtain the first result as follows:
###### Proposition 6.1.
$\\{J,J\\}=\\{p,J\\}=0.$
###### Proof.
Apply Proposition 1.2 to obtain
$\\{I,I\\}=\\{J\wedge p,J\wedge p\\}=\\{J\wedge p,J\\}\wedge
p+J\wedge\\{J\wedge p,p\\}$ $=-\\{J,J\\}\wedge p\wedge
p+2J\wedge\\{p,J\\}\wedge p-J\wedge J\wedge\\{p,p\\}.$
Since the super-exterior product is commutative then one has $J\wedge J=0$.
Moreover, $\\{I,I\\}=0$ implies that:
$\\{J,J\\}\wedge p\wedge p=2J\wedge\\{p,J\\}\wedge p.$
That means $\\{J,J\\}\wedge p=2J\wedge\\{p,J\\}$.
If $\\{J,J\\}\neq 0$ then $\\{J,J\\}\wedge p\neq 0$, so $J\wedge\\{p,J\\}\neq
0$. Note that
$\\{p,J\\}\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$
so $J$ must contain the factor $p$, i.e. $J=\alpha\otimes p$ where
$\alpha\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}^{*}$. But
$\\{p,J\\}=\\{p,\alpha\otimes p\\}=-\alpha\otimes\\{p,p\\}=0$ since
$\\{p,p\\}=0$. This is a contradiction and therefore $\\{J,J\\}=0$.
As a consequence, $J\wedge\\{p,J\\}=0$. Set
$\alpha=\\{p,J\\}\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})$
then we have $J\wedge\alpha=0$. If $\alpha\neq 0$ then $J$ must have the form
$J=\alpha\otimes q$ where
$q\in\operatorname{Sym}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
That is a contradiction since $J$ is indecomposable. ∎
###### Definition 6.2.
We continue to keep the condition $I=J\wedge p$ with
$p\in\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}}^{*}$ nonzero and
$J\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}})\otimes\operatorname{Sym}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$
indecomposable. We can assume that
$J=\sum_{i=1}^{n}\alpha_{i}\otimes p_{i}$
where
$\alpha_{i}\in\operatorname{Alt}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{0}}}}),\
i=1,\dots,n$ are linearly independent and
$p_{i}\in\operatorname{Sym}^{1}(\mathfrak{g}_{{\scriptscriptstyle{\overline{1}}}})$.
A quadratic Lie superalgebra having such associated invariant $I$ is called a
quasi- singular quadratic Lie superalgebra.
Let $U=\operatorname{span}\\{\alpha_{1},\dots,\alpha_{n}\\}$ and
$V=\operatorname{span}\\{p_{1},\dots,p_{n}\\}$, one has $\dim(U)$ and
$\dim(V)$ more than $1$ by if there is a contrary then $J$ is decomposable.
Using Definition 1.1, we have:
$\\{J,J\\}=\left\\{\sum_{i=1}^{n}\alpha_{i}\otimes
p_{i},\sum_{i=1}^{n}\alpha_{i}\otimes
p_{i}\right\\}=-\sum_{i,j=1}^{n}\left(\\{\alpha_{i},\alpha_{j}\\}\otimes
p_{i}p_{j}+(\alpha_{i}\wedge\alpha_{j})\otimes\\{p_{i},p_{j}\\}\right).$
Since $\\{J,J\\}=0$ and $\alpha_{i},\ i=1,\dots,n$ are linearly independent
then $\\{p_{i},p_{j}\\}=0$, for all $i,j$. It implies that $\\{p_{i},J\\}=0$,
for all $i$.
Moreover, since $\\{p,J\\}=0$ we obtain $\\{p,p_{i}\\}=0$, consequently
$\\{p_{i},I\\}=0$, for all $i$ and $\\{p,I\\}=0$. By Corollary 1.7 (2) and
Lemma 1.14 we conclude that $\phi^{-1}(V+\mathbb{C}p)$ is a subspace of
$\mathscr{Z}(\mathfrak{g})$ and totally isotropic.
Now, let $\\{q_{1},\dots,q_{m}\\}$ be a basis of $V$ then $J$ can be rewritten
by
$J=\sum_{j=1}^{m}\beta_{j}\otimes q_{j}$
where $\beta_{j}\in U$, for all $j$. One has:
$\\{J,J\\}=\left\\{\sum_{j=1}^{m}\beta_{j}\otimes
q_{j},\sum_{j=1}^{m}\beta_{j}\otimes
q_{j}\right\\}=-\sum_{i,j=1}^{m}\left(\\{\beta_{i},\beta_{j}\\}\otimes
q_{i}q_{j}+(\beta_{i}\wedge\beta_{j})\otimes\\{q_{i},q_{j}\\}\right).$
By the linear independence of the system $\\{q_{i}q_{j}\\}$, we obtain
$\\{\beta_{i},\beta_{j}\\}=0$, for all $i,j$. It implies that
$\\{\beta_{j},I\\}=0$, equivalently
$\phi^{-1}(\beta_{j})\in\mathscr{Z}(\mathfrak{g})$, for all $j$. Therefore, we
always can begin with $J=\sum_{i=1}^{n}\limits\alpha_{i}\otimes p_{i}$
satisfying the following conditions:
* (i)
$\alpha_{i},\ i=1,\dots,n$ are linearly independent,
* (ii)
$\phi^{-1}(U)$ and $\phi^{-1}(V+\mathbb{C}p)$ are totally isotropic subspaces
of $\mathscr{Z}(\mathfrak{g})$ where
$U=\operatorname{span}\\{\alpha_{1},\dots,\alpha_{n}\\}$ and
$V=\operatorname{span}\\{p_{1},\dots,p_{n}\\}$.
Let $X_{\scriptscriptstyle{\overline{0}}}^{i}=\phi^{-1}(\alpha_{i})$,
$X_{\scriptscriptstyle{\overline{1}}}^{i}=\phi^{-1}(p_{i})$, for all $i$ and
$C:\mathfrak{g}\rightarrow\mathfrak{g}$ defined by
$J(X,Y)=B(C(X),Y),\ \forall\ X,Y\in\mathfrak{g}.$
###### Lemma 6.3.
The mapping $C$ is a skew-supersymmetric homogeneous endomorphism of odd
degree and $\operatorname{Im}(C)\subset\mathscr{Z}(\mathfrak{g})$. Recall that
if $C$ is a homogeneous endomorphism of degree $c$ of $\mathfrak{g}$
satisfying
$B(C(X),Y)=-(-1)^{cx}B(X,C(Y)),\ \forall\ X\in\mathfrak{g}_{x},\
Y\in\mathfrak{g}$
then we say $C$ skew-supersymmetric (with respect to $B$).
###### Proof.
Since
$J(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})=J(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})=0$
and $B$ is even then
$C(\mathfrak{g}_{\scriptscriptstyle{\overline{0}}})\subset\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$
and
$C(\mathfrak{g}_{\scriptscriptstyle{\overline{1}}})\subset\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}$.
That means $C$ is of odd degree. For all
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\
Y\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ one has:
$B(C(X),Y)=J(X,Y)=\sum_{i=1}^{n}\alpha_{i}\otimes
p_{i}(X,Y)=\sum_{i=1}^{n}\alpha_{i}(X)p_{i}(Y)=\sum_{i=1}^{n}B(X_{\scriptscriptstyle{\overline{0}}}^{i},X)B(X_{\scriptscriptstyle{\overline{1}}}^{i},Y).$
By the non-degeneracy of $B$ and $J(X,Y)=-J(Y,X)$, we obtain:
$C(X)=\sum_{i=1}^{n}B(X_{\scriptscriptstyle{\overline{0}}}^{i},X)X_{\scriptscriptstyle{\overline{1}}}^{i}\
\ \text{ and }\ \
C(Y)=-\sum_{i=1}^{n}B(X_{\scriptscriptstyle{\overline{1}}}^{i},Y)X_{\scriptscriptstyle{\overline{0}}}^{i}.$
Combined with $B$ supersymmetric, one has:
$-B(Y,C(X))=B(C(X),Y)=-B(C(Y),X)=-B(X,C(Y)).$
It shows that $C$ is skew-supersymmetric. Finally,
$\operatorname{Im}(C)\subset\mathscr{Z}(\mathfrak{g})$ since
$X_{\scriptscriptstyle{\overline{0}}}^{i},\
X_{\scriptscriptstyle{\overline{1}}}^{i}\in\mathscr{Z}(\mathfrak{g})$, for all
$i$. ∎
###### Proposition 6.4.
Let $X_{\scriptscriptstyle{\overline{1}}}=\phi^{-1}(p)$ then for all
$X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\
Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ one has:
1. (1)
$[X,Y]=-B(C(X),Y)X_{\scriptscriptstyle{\overline{1}}}-B(X_{\scriptscriptstyle{\overline{1}}},Y)C(X)$,
2. (2)
$[Y,Z]=B(X_{\scriptscriptstyle{\overline{1}}},Y)C(Z)+B(X_{\scriptscriptstyle{\overline{1}}},Z)C(Y)$,
3. (3)
$X_{\scriptscriptstyle{\overline{1}}}\in\mathscr{Z}(\mathfrak{g})$ and
$C(X_{\scriptscriptstyle{\overline{1}}})=0$.
###### Proof.
Let $X\in\mathfrak{g}_{\scriptscriptstyle{\overline{0}}},\
Y,Z\in\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ then
$\displaystyle B([X,Y],Z)=J\wedge p(X,Y,Z)=-J(X,Y)p(Z)-J(X,Z)p(Y)$
$\displaystyle=-B(C(X),Y)B(X_{\scriptscriptstyle{\overline{1}}},Z)-B(C(X),Z)B(X_{\scriptscriptstyle{\overline{1}}},Y).$
By the non-degeneracy of $B$ on
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}\times\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$,
it shows that:
$[X,Y]=-B(C(X),Y)X_{\scriptscriptstyle{\overline{1}}}-B(X_{\scriptscriptstyle{\overline{1}}},Y)C(X).$
Combined with $B$ invariant and $C$ skew-supersymmetric, one has:
$[Y,Z]=B(X_{\scriptscriptstyle{\overline{1}}},Y)C(Z)+B(X_{\scriptscriptstyle{\overline{1}}},Z)C(Y).$
Since $\\{p,I\\}=0$ then
$X_{\scriptscriptstyle{\overline{1}}}\in\mathscr{Z}(\mathfrak{g})$. Moreover,
$\\{p,p_{i}\\}=0$ imply
$B(X_{\scriptscriptstyle{\overline{1}}},X_{\scriptscriptstyle{\overline{1}}}^{i})=0$,
for all $i$. It means
$B(X_{\scriptscriptstyle{\overline{1}}},\operatorname{Im}(C))=0$. And since
$B(C(X_{\scriptscriptstyle{\overline{1}}}),X)=B(X_{\scriptscriptstyle{\overline{1}}},C(X))=0$,
for all $X\in\mathfrak{g}$ then $C(X_{\scriptscriptstyle{\overline{1}}})=0$. ∎
Let $W$ be a complementary subspace of
$\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}}^{1},\dots,X_{\scriptscriptstyle{\overline{1}}}^{n},X_{\scriptscriptstyle{\overline{1}}}\\}$
in $\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}$ and
$Y_{\scriptscriptstyle{\overline{1}}}$ be an element in $W$ such that
$B(X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}})=1$.
Let
$X_{\scriptscriptstyle{\overline{0}}}=C(Y_{\scriptscriptstyle{\overline{1}}})$,
$\mathfrak{q}=(\mathbb{C}X_{1}\oplus\mathbb{C}Y_{\scriptscriptstyle{\overline{1}}})^{\bot}$
and $B_{\mathfrak{q}}=B|_{\mathfrak{q}\times\mathfrak{q}}$ then we have the
following corollary:
###### Corollary 6.5.
1. (1)
$[Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}]=2X_{\scriptscriptstyle{\overline{0}}}$,
$[Y_{\scriptscriptstyle{\overline{1}}},X]=C(X)-B(X,X_{\scriptscriptstyle{\overline{0}}})X_{\scriptscriptstyle{\overline{1}}}$
and $[X,Y]=-B(C(X),Y)X_{\scriptscriptstyle{\overline{1}}}$, for all
$X,Y\in\mathfrak{q}\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}$.
2. (2)
$[\mathfrak{g},\mathfrak{g}]\subset\operatorname{Im}(C)+\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}\subset\mathscr{Z}(\mathfrak{g})$
so $\mathfrak{g}$ is 2-step nilpotent. If $\mathfrak{g}$ is reduced then
$[\mathfrak{g},\mathfrak{g}]=\operatorname{Im}(C)+\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}=\mathscr{Z}(\mathfrak{g})$.
3. (3)
$C^{2}=0$.
###### Proof.
1. (1)
The assertion (1) is obvious by Proposition 6.4.
2. (2)
Note that $X_{\scriptscriptstyle{\overline{0}}}\in\operatorname{Im}(C)$ so
$[\mathfrak{g},\mathfrak{g}]\subset\operatorname{Im}(C)+\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}$.
By Lemma 6.3 and Proposition 6.4,
$\operatorname{Im}(C)+\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}\subset\mathscr{Z}(\mathfrak{g})$.
If $\mathfrak{g}$ is reduced then
$\mathscr{Z}(\mathfrak{g})\subset[\mathfrak{g},\mathfrak{g}]$ and therefore
$[\mathfrak{g},\mathfrak{g}]=\operatorname{Im}(C)+\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}=\mathscr{Z}(\mathfrak{g})$.
3. (3)
Since $\mathfrak{g}$ is 2-step nilpotent then
$0=[Y_{\scriptscriptstyle{\overline{1}}},[Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}]]=[Y_{\scriptscriptstyle{\overline{1}}},2X_{\scriptscriptstyle{\overline{0}}}]=2C(X_{\scriptscriptstyle{\overline{0}}})-2B(X_{\scriptscriptstyle{\overline{0}}},X_{\scriptscriptstyle{\overline{0}}})X_{\scriptscriptstyle{\overline{1}}}.$
Since
$X_{\scriptscriptstyle{\overline{0}}}=C(Y_{\scriptscriptstyle{\overline{1}}})$
and $\operatorname{Im}(C)$ is totally isotropic then
$B(X_{\scriptscriptstyle{\overline{0}}},X_{\scriptscriptstyle{\overline{0}}})=0$
and therefore
$C(X_{\scriptscriptstyle{\overline{0}}})=C^{2}(Y_{\scriptscriptstyle{\overline{1}}})=0$.
If $X\in\mathfrak{q}\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}$ then
$0=[Y_{\scriptscriptstyle{\overline{1}}},[Y_{\scriptscriptstyle{\overline{1}}},X]]=[Y_{\scriptscriptstyle{\overline{1}}},C(X)]$.
By the choice of $Y_{\scriptscriptstyle{\overline{1}}}$, it is sure that
$C(X)\in\mathfrak{q}\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}$.
Therefore, one has:
$0=[Y_{\scriptscriptstyle{\overline{1}}},C(X)]=C^{2}(X)-B(C(X),X_{\scriptscriptstyle{\overline{0}}})X_{\scriptscriptstyle{\overline{1}}}=C^{2}(X)-B(C(X),C(Y_{\scriptscriptstyle{\overline{1}}}))X_{\scriptscriptstyle{\overline{1}}}.$
By $\operatorname{Im}(C)$ totally isotropic, one has $C^{2}(X)=0$.
∎
Now, we consider a special case: $X_{\scriptscriptstyle{\overline{0}}}=0$. As
a consequence,
$[Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}]=0$,
$[Y_{\scriptscriptstyle{\overline{1}}},X]=C(X)$ and
$[X,Y]=-B(C(X),Y)X_{\scriptscriptstyle{\overline{1}}}$, for all
$X,Y\in\mathfrak{q}$. Let $X\in\mathfrak{q}$ and assume that
$C(X)=C_{1}(X)+aX_{1}$ where $C_{1}(X)\in\mathfrak{q}$ then
$0=B([Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}],X)=B(Y_{\scriptscriptstyle{\overline{1}}},[Y_{\scriptscriptstyle{\overline{1}}},X])=B(Y_{\scriptscriptstyle{\overline{1}}},C_{1}(X)+aX_{1})=a.$
It shows that $C(X)\in\mathfrak{q}$, for all $X\in\mathfrak{q}$ and therefore
we have an affirmative answer of the above question as follows:
###### Proposition 6.6.
Let
$(\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}},B_{\mathfrak{q}})$
be a quadratic $\mathbb{Z}_{2}$-graded vector space and ${\overline{C}}$ be an
odd endomorphism of $\mathfrak{q}$ such that ${\overline{C}}$ is skew-
supersymmetric and ${\overline{C}}^{2}=0$. Let
$(\mathfrak{t}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\},B_{\mathfrak{t}})$
be a 2-dimensional symplectic vector space with
$B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}})=1$.
Consider the space $\mathfrak{g}=\mathfrak{q}{\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{t}$ and define the product on
$\mathfrak{g}$ by:
$[Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}]=[X_{\scriptscriptstyle{\overline{1}}},\mathfrak{g}]=0,\
[Y_{\scriptscriptstyle{\overline{1}}},X]={\overline{C}}(X)\ \text{ and
}[X,Y]=-B_{\mathfrak{q}}({\overline{C}}(X),Y)X_{\scriptscriptstyle{\overline{1}}}$
for all $X\in\mathfrak{q}$. Then $\mathfrak{g}$ becomes a 2-nilpotent
quadratic Lie superalgebra with the bilinear form
$B=B_{\mathfrak{q}}+B_{\mathfrak{t}}$. Moreover, one has
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}\oplus\mathfrak{t}$.
###### Remark 6.7.
The method above remains valid for the elementary quadratic Lie superalgebra
$\mathfrak{g}_{6}^{s}$ with $I$ decomposable (see Section 3) as follows: let
$\mathfrak{q}=(\mathbb{C}X_{{\scriptscriptstyle{\overline{0}}}}\oplus\mathbb{C}Y_{{\scriptscriptstyle{\overline{0}}}})\oplus(\mathbb{C}Z_{{\scriptscriptstyle{\overline{1}}}}\oplus\mathbb{C}T_{{\scriptscriptstyle{\overline{1}}}})$
where
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{0}}},Y_{\scriptscriptstyle{\overline{0}}}\\}$,
$\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}=\operatorname{span}\\{Z_{\scriptscriptstyle{\overline{1}}},T_{\scriptscriptstyle{\overline{1}}}\\}$
and the bilinear form $B_{\mathfrak{q}}$ is defined by
$B(X_{{\scriptscriptstyle{\overline{0}}}},Y_{{\scriptscriptstyle{\overline{0}}}})=B(Z_{{\scriptscriptstyle{\overline{1}}}},T_{{\scriptscriptstyle{\overline{1}}}})=1$,
the other are zero. Let $C:\mathfrak{q}\ \rightarrow\ \mathfrak{q}$ be a
linear map defined by:
${\overline{C}}=\begin{pmatrix}0&0&0&-1\\\ 0&0&0&0\\\ 0&1&0&0\\\
0&0&0&0\end{pmatrix}.$
Then $C$ is odd and $C^{2}=0$. Set the vector space
$\mathfrak{g}=\mathfrak{q}{\ \overset{\perp}{\mathop{\oplus}}\ }\mathfrak{t}$,
where
$(\mathfrak{t}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\},B_{\mathfrak{t}})$
is a 2-dimensional symplectic vector space with
$B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}})=1$.
Then $\mathfrak{g}=\mathfrak{g}_{6}^{s}$ with the Lie super-bracket defined as
in Proposition 6.6.
It remains to consider $X_{\scriptscriptstyle{\overline{0}}}\neq 0$. The fact
is that $C$ may be not stable on $\mathfrak{q}$, that is,
$C(X)\in\mathfrak{q}\oplus\mathbb{C}X_{\scriptscriptstyle{\overline{1}}}$ if
$X\in\mathfrak{q}$ but that we need here is an action stable on
$\mathfrak{q}$. Therefore, we decompose $C$ by
$C(X)={\overline{C}}(X)+\varphi(X)X_{\scriptscriptstyle{\overline{1}}}$, for
all $X\in\mathfrak{q}$ where
${\overline{C}}:\mathfrak{q}\rightarrow\mathfrak{q}$ and
$\varphi:\mathfrak{q}\rightarrow\mathbb{C}$. Since
$B(C(Y_{\scriptscriptstyle{\overline{1}}}),X)=B(Y_{\scriptscriptstyle{\overline{1}}},C(X)$
then
$\varphi(X)=-B(X_{\scriptscriptstyle{\overline{0}}},X)=-B(X,X_{\scriptscriptstyle{\overline{0}}})$,
for all $X\in\mathfrak{q}$. Moreover, $C$ is odd degree on $\mathfrak{g}$ and
skew-supersymmetric (with respect to $B$) implies that ${\overline{C}}$ is
also odd on $\mathfrak{q}$ and skew-supersymmetric (with respect to
$B_{\mathfrak{q}}$). It is easy to see that ${\overline{C}}^{2}=0$,
${\overline{C}}(X_{\scriptscriptstyle{\overline{0}}})=0$ and we have the
following result:
###### Corollary 6.8.
Keep the notations as in Corollary 6.5 and replace
$2X_{\scriptscriptstyle{\overline{0}}}$ by
$X_{\scriptscriptstyle{\overline{0}}}$ then for all $X,Y\in\mathfrak{q}$, one
has:
* •
$[Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}]=X_{\scriptscriptstyle{\overline{0}}},$
* •
$[Y_{\scriptscriptstyle{\overline{1}}},X]={\overline{C}}(X)-B(X,X_{\scriptscriptstyle{\overline{0}}})X_{\scriptscriptstyle{\overline{1}}},$
* •
$[X,Y]=-B({\overline{C}}(X),Y)X_{\scriptscriptstyle{\overline{1}}}$.
Hence, we have a more general result of Proposition 6.6:
###### Proposition 6.9.
Let
$(\mathfrak{q}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}\oplus\mathfrak{q}_{\scriptscriptstyle{\overline{1}}},B_{\mathfrak{q}})$
be a quadratic $\mathbb{Z}_{2}$-graded vector space and ${\overline{C}}$ an
odd endomorphism of $\mathfrak{q}$ such that ${\overline{C}}$ is skew-
supersymmetric and ${\overline{C}}^{2}=0$. Let
$X_{\scriptscriptstyle{\overline{0}}}$ be an isotropic element of
$\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$,
$X_{\scriptscriptstyle{\overline{0}}}\in\ker({\overline{C}})$ and
$(\mathfrak{t}=\operatorname{span}\\{X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}\\},B_{\mathfrak{t}})$
be a 2-dimensional symplectic vector space with
$B_{\mathfrak{t}}(X_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}})=1$.
Consider the space $\mathfrak{g}=\mathfrak{q}{\
\overset{\perp}{\mathop{\oplus}}\ }\mathfrak{t}$ and define the product on
$\mathfrak{g}$ by:
$[Y_{\scriptscriptstyle{\overline{1}}},Y_{\scriptscriptstyle{\overline{1}}}]=X_{\scriptscriptstyle{\overline{0}}},\
[Y_{\scriptscriptstyle{\overline{1}}},X]={\overline{C}}(X)-B_{\mathfrak{q}}(X,X_{\scriptscriptstyle{\overline{0}}})X_{\scriptscriptstyle{\overline{1}}}\text{
and
}[X,Y]=-B_{\mathfrak{q}}({\overline{C}}(X),Y)X_{\scriptscriptstyle{\overline{1}}}$
for all $X\in\mathfrak{q}$. Then $\mathfrak{g}$ becomes a 2-nilpotent
quadratic Lie superalgebra with the bilinear form
$B=B_{\mathfrak{q}}+B_{\mathfrak{t}}$. Moreover, one has
$\mathfrak{g}_{\scriptscriptstyle{\overline{0}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{0}}}$,
$\mathfrak{g}_{\scriptscriptstyle{\overline{1}}}=\mathfrak{q}_{\scriptscriptstyle{\overline{1}}}\oplus\mathfrak{t}$.
A quadratic Lie superalgebra obtained in the above proposition is a special
case of the generalized double extensions given in [BBB] where the authors
consider the generalized double extension of a quadratic
$\mathbb{Z}_{2}$-graded vector space (regarded as an Abelian superalgebra) by
a one-dimensional Lie superalgebra.
## 7\. Appendix: Adjoint orbits of $\mathfrak{sp}(2n)$ and $\mathfrak{o}(m)$
This appendix recalls a fundamental and really interesting problem in Lie
theory that is necessary for the paper: the classification of adjoint orbits
of classical Lie algebras $\mathfrak{sp}(2n)$ and $\mathfrak{o}(m)$ where $m$,
$n\in\mathbb{N}^{*}$. A brief overview can be found in [Hum95] with
interesting discussions. Many results with detailed proofs can be found in
[CM93].
A different point here is to use the Fitting decomposition to review this
problem. In particular, we parametrize the invertible component in the Fitting
decomposition of a skew-symmetric map and from this, we give an explicit
classification for $\operatorname{Sp}(2n)$-adjoint orbits of
$\mathfrak{sp}(2n)$ and $\operatorname{O}(m)$-adjoint orbits of
$\mathfrak{o}(m)$ in the general case. In other words, we establish a one-to-
one correspondence between the set of orbits and some set of indices. This is
an rather obvious and classical result but in our knowledge there is not a
reference for that mentioned before.
Let $V$ be a $m$-dimensional complex vector space endowed with a non-
degenerate bilinear form $B_{\epsilon}$ where $\epsilon=\pm 1$ such that
$B_{\epsilon}(X,Y)=\epsilon B_{\epsilon}(Y,X)$, for all $X,Y\in V$. If
$\epsilon=1$ then the form $B_{1}$ is symmetric and we say $V$ a quadratic
vector space. If $\epsilon=-1$ then $m$ must be even and we say $V$ a
symplectic vector space with symplectic form $B_{-1}$. We denote by
$\mathcal{L}(V)$ the algebra of linear operators of $V$ and by
$\operatorname{GL}(V)$ the group of invertible operators in $\mathcal{L}(V)$.
A map $C\in\mathcal{L}(V)$ is called skew-symmetric (with respect to
$B_{\epsilon}$) if it satisfies the following condition:
$B_{\epsilon}(C(X),Y)=-B_{\epsilon}(X,C(Y)),\ \forall\ X,Y\in V.$
We define
$I_{\epsilon}(V)=\\{A\in\operatorname{GL}(V)\mid\
B_{\epsilon}(A(X),A(Y))=B_{\epsilon}(X,Y),\ \forall\ X,Y\in V\\}$ $\text{and
}\ \mathfrak{g}_{\epsilon}(V)=\\{C\in\mathcal{L}(V)\mid\ C\text{ is skew-
symmetric}\\}.$
Then $I_{\epsilon}(V)$ is the isometry group of the bilinear form
$B_{\epsilon}$ and $\mathfrak{g}_{\epsilon}(V)$ is its Lie algebra. Denote by
$A^{*}\in\mathcal{L}(V)$ the adjoint map of an element $A\in\mathcal{L}(V)$
with respect to $B_{\epsilon}$, then $A\in I_{\epsilon}(V)$ if and only if
$A^{-1}=A^{*}$ and $C\in\mathfrak{g}_{\epsilon}(V)$ if and only if $C^{*}=-C$.
If $\epsilon=1$ then $I_{\epsilon}(V)$ is denoted by $\operatorname{O}(V)$ and
$\mathfrak{g}_{\epsilon}(V)$ is denoted by $\mathfrak{o}(V)$. If $\epsilon=-1$
then $\operatorname{Sp}(V)$ stands for $I_{\epsilon}(V)$ and
$\mathfrak{sp}(V)$ stands for $\mathfrak{g}_{\epsilon}(V)$.
Recall that the adjoint action $\operatorname{Ad}$ of $I_{\epsilon}(V)$ on
$\mathfrak{g}_{\epsilon}(V)$ is given by
$\operatorname{Ad}_{U}(C)=UCU^{-1},\ \forall\ U\in I_{\epsilon}(V),\
C\in\mathfrak{g}_{\epsilon}(V).$
We denote by $\mathcal{O}_{C}=\operatorname{Ad}_{I_{\epsilon}(V)}(C)$, the
adjoint orbit of an element $C\in\mathfrak{g}_{\epsilon}(V)$ by this action.
If $V=\mathbb{C}^{m}$, we call $B_{\epsilon}$ a canonical bilinear form of
$\mathbb{C}^{m}$. And with respect to $B_{\epsilon}$, we define a canonical
basis $\mathcal{B}=\\{E_{1},\dots,E_{m}\\}$ of $\mathbb{C}^{m}$ as follows. If
$m$ even, $m=2n$, write
$\mathcal{B}=\\{E_{1},\dots,E_{n},F_{1},\dots,F_{n}\\}$, if $m$ is odd,
$m=2n+1$, write $\mathcal{B}=\\{E_{1},\dots,E_{n},G,F_{1},\dots,F_{n}\\}$ and
one has:
* •
if $m=2n$ then
$B_{1}(E_{i},F_{j})=B_{1}(F_{j},E_{i})=\delta_{ij},\
B_{1}(E_{i},E_{j})=B_{1}(F_{i},F_{j})=0,$
$B_{-1}(E_{i},F_{j})=-B_{-1}(F_{j},E_{i})=\delta_{ij},\
B_{-1}(E_{i},E_{j})=B_{-1}(F_{i},F_{j})=0,$
where $1\leq i,j\leq n$.
In the case of $\epsilon=-1$, $\mathcal{B}$ is also called a Darboux basis of
$\mathbb{C}^{2n}$.
* •
if $m=2n+1$ then $\epsilon=1$ and
$\begin{cases}B_{1}(E_{i},F_{j})=\delta_{ij},\
B_{1}(E_{i},E_{j})=B_{1}(F_{i},F_{j})=0,\\\
B_{1}(E_{i},G)=B_{1}(F_{j},G)=0,\\\ B_{1}(G,G)=1\end{cases}$
where $1\leq i,j\leq n$.
Also, in the case $V=\mathbb{C}^{m}$, we denote by $\operatorname{GL}(m)$
instead of $\operatorname{GL}(V)$, $\operatorname{O}(m)$ stands for
$\operatorname{O}(V)$ and $\mathfrak{o}(m)$ stands for $\mathfrak{o}(V)$. If
$m=2n$ then $\operatorname{Sp}(2n)$ stands for $\operatorname{Sp}(V)$ and
$\mathfrak{sp}(2n)$ stands for $\mathfrak{sp}(V)$. We will also write
$I_{\epsilon}=I_{\epsilon}(\mathbb{C}^{m})$ and
$\mathfrak{g}_{\epsilon}=\mathfrak{g}_{\epsilon}(\mathbb{C}^{m})$. Our goal is
classifying all of $I_{\epsilon}$-adjoint orbits of $\mathfrak{g}_{\epsilon}$.
Finally, let $V$ is an m-dimensional vector space. If $V$ is quadratic then
$V$ is isometrically isomorphic to the quadratic space
$(\mathbb{C}^{m},B_{1})$ and if $V$ is symplectic then $V$ is isometrically
isomorphic to the symplectic space $(\mathbb{C}^{m},B_{-1})$ [Bou59].
### 7.1. Nilpotent orbits
Let $n\in\mathbb{N}^{*}$, a partition $[d]$ of $n$ is a tuple
$[d_{1},...,d_{k}]$ of positive integers satisfying
$d_{1}\geq...\geq d_{k}\text{ and }d_{1}+...+d_{k}=n.$
Occasionally, we use the notation $[t_{1}^{i_{1}},\dots,t_{r}^{i_{r}}]$ to
replace the partition $[d_{1},...,d_{k}]$ where
$d_{j}=\left\\{\begin{matrix}t_{1}&1\leq j\leq i_{1}\\\ t_{2}&i_{1}+1\leq
j\leq i_{1}+i_{2}\\\ t_{3}&i_{1}+i_{2}+1\leq j\leq i_{1}+i_{2}+i_{3}\\\
\dots\end{matrix}\right.$
Each $i_{j}$ is called the multiplicity of $t_{j}$. Denote by $\mathcal{P}(n)$
the set of partitions of $n$.
Let $p\in\mathbb{N}^{*}$. We denote the Jordan block of size $p$ by
$J_{1}=(0)$ and for $p\geq 2$,
$J_{p}:=\begin{pmatrix}0&1&0&\dots&0\\\ 0&0&1&\dots&0\\\
\vdots&\vdots&\dots&\ddots&\vdots\\\ 0&0&\dots&0&1\\\
0&0&0&\dots&0\end{pmatrix}.$
Then $J_{p}$ is a nilpotent endomorphism of $\mathbb{C}^{p}$. Given a
partition $[d]=[d_{1},...,d_{k}]\in\mathcal{P}(n)$ there is a nilpotent
endomorphism of $\mathbb{C}^{n}$ defined by
$X_{[d]}:=\operatorname{diag}_{k}(J_{d_{1}},...,J_{d_{k}}).$
Moreover, $X_{[d]}$ is also a nilpotent element of $\mathfrak{sl}(n)$ since
its trace is zero. Conversely, if $C$ is a nilpotent element in
$\mathfrak{sl}(n)$ then $C$ is $\operatorname{GL}(n)$-conjugate to its Jordan
normal form $X_{[d]}$ for some partition $[d]\in\mathcal{P}(n)$.
Given two different partitions $[d]=[d_{1},...,d_{k}]$ and
$[d^{\prime}]=[d^{\prime}_{1},...,d^{\prime}_{l}]$ of $n$ then the
$\operatorname{GL}(n)$-adjoint orbits through $X_{[d]}$ and $X_{[d^{\prime}]}$
respectively are disjoint by the unicity of Jordan normal form. Therefore, one
has the following proposition:
###### Proposition 7.1.
There is a one-to-one correspondence between the set of nilpotent
$\operatorname{GL}(n)$-adjoint orbits of $\mathfrak{sl}(n)$ and the set
$\mathcal{P}(n)$.
Define the set
$\mathcal{P}_{\epsilon}(m)=\\{[d_{1},...,d_{m}]\in\mathcal{P}(m)|\
\sharp\\{j\mid\ d_{j}=i\\}\text{ is even for all i such that
}(-1)^{i}=\epsilon\\}.$
In particular, $\mathcal{P}_{1}(m)$ is the set of partitions of $m$ in which
even parts occur with even multiplicity and $\mathcal{P}_{-1}(m)$ is the set
of partitions of $m$ in which odd parts occur with even multiplicity.
###### Proposition 7.2 (Gerstenhaber).
Nilpotent $I_{\epsilon}$-adjoint orbits in $\mathfrak{g}_{\epsilon}$ are in
one-to-one correspondence with the set of partitions in
$\mathcal{P}_{\epsilon}(m)$.
Here, we give a construction of a nilpotent element in
$\mathfrak{g}_{\epsilon}$ from a partition $[d]$ of $m$ that is useful for
this paper. Define maps in $\mathfrak{g}_{\epsilon}$ as follows:
* •
For $p\geq 2$, we equip the vector space $\mathbb{C}^{2p}$ with its canonical
bilinear form $B_{\epsilon}$ and the map $C_{2p}^{J}$ having the matrix
$C_{2p}^{J}=\begin{pmatrix}J_{p}&0\\\ 0&-{}^{t}J_{p}\end{pmatrix}$
in a canonical basis where ${}^{t}J_{p}$ denotes the transpose matrix of the
Jordan block $J_{p}$. Then
$C_{2p}^{J}\in\mathfrak{g}_{\epsilon}(\mathbb{C}^{2p})$.
* •
For $p\geq 1$ we equip the vector space $\mathbb{C}^{2p+1}$ with its canonical
bilinear form $B_{1}$ and the map $C_{2p+1}^{J}$ having the matrix
$C_{2p+1}^{J}=\begin{pmatrix}J_{p+1}&M\\\ 0&-{}^{t}J_{p}\end{pmatrix}$
in a canonical basis where $M=(m_{ij})$ denotes the $(p+1)\times p$-matrix
with $m_{p+1,p}=-1$ and $m_{ij}=0$ otherwise. Then
$C_{2p+1}^{J}\in\mathfrak{o}(2p+1)$
* •
For $p\geq 1$, we consider the vector space $\mathbb{C}^{2p}$ equipped with
its canonical bilinear form $B_{-1}$ and the map $C_{p+p}^{J}$ with matrix
$\begin{pmatrix}J_{p}&M\\\ 0&-{}^{t}J_{p}\end{pmatrix}$
in a canonical basis where $M=(m_{ij})$ denotes the $p\times p$-matrix with
$m_{p,p}=1$ and $m_{ij}=0$ otherwise. Then $C_{p+p}^{J}\in\mathfrak{sp}(2p)$.
For each partition $[d]\in\mathcal{P}_{-1}(2n)$, $[d]$ can be written as
$(p_{1},p_{1},p_{2},p_{2},\dots,p_{k},p_{k},2q_{1},\dots,2q_{\ell})$
with all $p_{i}$ odd, $p_{1}\geq p_{2}\geq\dots\geq p_{k}$ and $q_{1}\geq
q_{2}\geq\dots\geq q_{\ell}$. We associate $[d]$ to the map $C_{[d]}$ with
matrix:
$\operatorname{diag}_{k+\ell}(C^{J}_{2p_{1}},C^{J}_{2p_{2}},\dots,C^{J}_{2p_{k}},C^{J}_{q_{1}+q_{1}},\dots,C^{J}_{q_{\ell}+q_{\ell}})$
in a canonical basis of $\mathbb{C}^{2n}$ then $C_{[d]}\in\mathfrak{sp}(2n)$.
Similarly, let $[d]\in\mathcal{P}_{1}(m)$, $[d]$ can be written as
$(p_{1},p_{1},p_{2},p_{2},\dots,p_{k},p_{k},2q_{1}+1,\dots,2q_{\ell}+1)$
with all $p_{i}$ even, $p_{1}\geq p_{2}\geq\dots\geq p_{k}$ and $q_{1}\geq
q_{2}\geq\dots\geq q_{\ell}$. We associate $[d]$ to the map $C_{[d]}$ with
matrix:
$\operatorname{diag}_{k+\ell}(C^{J}_{2p_{1}},C^{J}_{2p_{2}},\dots,C^{J}_{2p_{k}},C^{J}_{2q_{1}+1},\dots,C^{J}_{2q_{\ell}+1}).$
in a canonical basis of $\mathbb{C}^{m}$ then $C_{[d]}\in\mathfrak{o}(m)$.
By Proposition 7.2, it is sure that our construction is a bijection between
the set $\mathcal{P}_{\epsilon}(m)$ and the set of nilpotent
$I_{\epsilon}$-adjoint orbits in $\mathfrak{g}_{\epsilon}$.
### 7.2. Semisimple orbits
We recall a well-known result [CM93]:
###### Proposition 7.3.
Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{h}$ be a Cartan
subalgebra of $\mathfrak{g}$ and $W$ be the associated Weyl group. Then there
is a bijection between the set of semisimple orbits of $\mathfrak{g}$ and
$\mathfrak{h}/W$.
For each $\mathfrak{g}_{\epsilon}$, we choose the Cartan subalgebra
$\mathfrak{h}$ given by the vector space of diagonal matrices of type
$\operatorname{diag}_{2n}(\lambda_{1},\dots,\lambda_{n},-\lambda_{1},\dots,-\lambda_{n})$
if $\mathfrak{g}_{\epsilon}=\mathfrak{o}(2n)$ or
$\mathfrak{g}_{\epsilon}=\mathfrak{sp}(2n)$ and of type
$\operatorname{diag}_{2n+1}(\lambda_{1},\dots,\lambda_{n},0,-\lambda_{1},\dots,-\lambda_{n})$
if $\mathfrak{g}_{\epsilon}=\mathfrak{o}(2n+1)$.
Any diagonalizable (equivalently semisimple) $C\in\mathfrak{g}_{\epsilon}$ is
conjugate to an element of $\mathfrak{h}$.
If $\mathfrak{g}_{\epsilon}=\mathfrak{sp}(2n)$ then any two eigenvectors
$v,w\in\mathbb{C}^{2n}$ of $C\in\mathfrak{g}_{\epsilon}$ with eigenvalues
$\lambda,\lambda^{\prime}\in\mathbb{C}$ such that
$\lambda+\lambda^{\prime}\neq 0$ are orthogonal. Moreover, each eigenvalue
pair $\lambda,-\lambda$ is corresponding to an eigenvector pair $(v,w)$
satisfying $B_{\epsilon}(v,w)=1$ and we can easily arrange for vectors
$v,v^{\prime}$ lying in a distinct pair $(v,w),(v^{\prime},w^{\prime})$ to be
orthogonal, regardless of the eigenvalues involved. That means the associated
Weyl group is of all coordinate permutations and sign changes of
$(\lambda_{1},\dots,\lambda_{n})$. We denote it by $G_{n}$.
If $\mathfrak{g}_{\epsilon}=\mathfrak{o}(2n)$, the associated Weyl group, when
considered in the action of the group $\operatorname{SO}(2n)$, consists all
coordinate permutations and even sign changes of
$(\lambda_{1},\dots,\lambda_{n})$. However, we only focus on
$\operatorname{O}(2n)$-adjoint orbits of $\mathfrak{o}(2n)$ obtained by the
action of the full orthogonal group, then similarly to preceding analysis any
sign change effects. The corresponding group is still $G_{n}$. If
$\mathfrak{g}_{\epsilon}=\mathfrak{o}(2n+1)$, the Weyl group is $G_{n}$ and
there is nothing to add.
Now, let $\Lambda_{n}=\\{(\lambda_{1},\dots,\lambda_{n})\mid\
\lambda_{1},\dots,\lambda_{n}\in\mathbb{C},\ \lambda_{i}\neq 0\ \text{ for
some }\ i\\}$.
###### Corollary 7.4.
There is a bijection between nonzero semisimple $I_{\epsilon}$-adjoint orbits
of $\mathfrak{g}_{\epsilon}$ and $\Lambda_{n}/G_{n}$.
### 7.3. Invertible orbits
###### Definition 7.5.
We say that the $I_{\epsilon}$-orbit $\mathcal{O}_{X}$ is invertible if $X$ is
an invertible element in $\mathfrak{g}_{\epsilon}$.
Keep the above notations. We say an element $X\in V$ isotropic if
$B_{\epsilon}(X,X)=0$ and a subset $W\subset V$ totally isotropic if
$B_{\epsilon}(X,Y)=0$ for all $X,Y\in W$.
We recall the classification method given in [DPU10] as follows. First, we
need the lemma:
###### Lemma 7.6.
Let $V$ be an even-dimensional vector space with a non-degenerate bilinear
form $B_{\epsilon}$. Assume that $V=V_{+}\oplus V_{-}$ where $V_{\pm}$ are
totally isotropic vector subspaces.
1. (1)
Let $N\in\mathcal{L}(V)$ such that $N(V_{\pm})\subset V_{\pm}$. We define maps
$N_{\pm}$ by $N_{+}|_{V_{+}}=N|_{V_{+}}$, $N_{+}|_{V_{-}}=0$,
$N_{-}|_{V_{-}}=N|_{V_{-}}$ and $N_{-}|_{V_{+}}=0$. Then
$N\in\mathfrak{g}_{\epsilon}(V)$ if and only if $N_{-}=-N_{+}^{*}$ and, in
this case, $N=N_{+}-N_{+}^{*}$.
2. (2)
Let $U_{+}\in\mathcal{L}(V)$ such that $U_{+}$ is invertible,
$U_{+}(V_{+})=V_{+}$ and $U_{+}|_{V_{-}}=\operatorname{Id}_{V_{-}}$. We define
$U\in\mathcal{L}(V)$ by $U|_{V_{+}}=U_{+}|_{V_{+}}$ and
$U|_{V_{-}}=\left(U_{+}^{-1}\right)^{*}|_{V_{-}}$. Then $U\in
I_{\epsilon}(V)$.
3. (3)
Let $N^{\prime}\in\mathfrak{g}_{\epsilon}(V)$ such that $N^{\prime}$ satisfies
the assumptions of (1). Define $N_{\pm}$ as in (1). Moreover, we assume that
there exists $U_{+}\in\mathcal{L}(V_{+})$, $U_{+}$ invertible such that
$N_{+}^{\prime}|_{V_{+}}=\left(U_{+}\ N_{+}\ U_{+}^{-1}\right)|_{V_{+}}.$
We extend $U_{+}$ to $V$ by $U_{+}|_{V_{-}}=\operatorname{Id}_{V_{-}}$ and
define $U\in I_{\epsilon}(V)$ as in (2). Then
$N^{\prime}=U\ N\ U^{-1}.$
###### Proof.
1. (1)
It is obvious that $N=N_{+}+N_{-}$. Recall that
$N\in\mathfrak{g}_{\epsilon}(V)$ if and only if $N^{*}=-N$ so
$N_{+}^{*}+N_{-}^{*}=-N_{+}-N_{-}$. Since
$B_{\epsilon}(N_{+}^{*}(V_{+}),V)=B_{\epsilon}(V_{+},N_{+}(V))=0$ then
$N_{+}^{*}(V_{+})=0$. Similarly, $N_{-}^{*}(V_{-})=0$. Hence,
$N_{-}=-N_{+}^{*}.$
2. (2)
We shows that $B_{\epsilon}(U(X),U(Y))=B_{\epsilon}(X,Y)$, for all $X,Y\in V$.
Indeed, let $X=X_{+}+X_{-},Y=Y_{+}+Y_{-}\in V_{+}\oplus V_{-}$, one has
$B_{\epsilon}(U(X_{+}+X_{-}),U(Y_{+}+Y_{-}))=B_{\epsilon}(U_{+}(X_{+})+\left(U_{+}^{-1}\right)^{*}(X_{-}),U_{+}(Y_{+})+\left(U_{+}^{-1}\right)^{*}(Y_{-}))$
$=B_{\epsilon}(U_{+}(X_{+}),\left(U_{+}^{-1}\right)^{*}(Y_{-}))+B_{\epsilon}(\left(U_{+}^{-1}\right)^{*}(X_{-}),U_{+}(Y_{+}))$
$=B_{\epsilon}(X_{+},Y_{-})+B_{\epsilon}(X_{-},Y_{+})=B_{\epsilon}(X,Y).$
3. (3)
Since $B_{\epsilon}(U^{-1}(V_{+}),V_{+})=B_{\epsilon}(V_{+},U(V_{+}))=0$, one
has $U^{-1}(V_{+})=V_{+}$ and $U^{-1}(V_{-})=V_{-}$. Consequently, $(U\ N\
U^{-1})(V_{+})\subset V_{+}$ and $(U\ N\ U^{-1})(V_{-})\subset V_{-}$.
Clearly, $U\ N\ U^{-1}\in\mathfrak{g}_{\epsilon}(V)$. By (1), we only show
that
$(U\ N\ U^{-1})|_{V_{+}}=N_{+}^{\prime}$
This is obvious since $U^{-1}|_{V_{+}}=U_{+}^{-1}$.
∎
Let us now consider $C\in\mathfrak{g}_{\epsilon}$, $C$ invertible. Then, $m$
must be even (obviously, it happened if $\epsilon=-1$), $m=2n$. Indeed, we
assume that $\epsilon=1$ then the skew-symmetric form $\Delta_{C}$ on
$\mathbb{C}^{m}$ defined by $\Delta_{C}(v_{1},v_{2})=B_{1}(v_{1},C(v_{2}))$ is
non-degenerate. and the assertion follows. We decompose $C=S+N$ into
semisimple and nilpotent parts, $S$, $N\in\mathfrak{g}_{\epsilon}$ by its
Jordan decomposition. It is clear that $S$ is invertible. We have
$\lambda\in\Lambda$ if and only if $-\lambda\in\Lambda$ where $\Lambda$ is the
spectrum of $S$. Also, $m(\lambda)=m(-\lambda)$, for all $\lambda\in\Lambda$
with the multiplicity $m(\lambda)$. Since $N$ and $S$ commute, we have
$N(V_{\pm\lambda})\subset V_{\pm\lambda}$ where $V_{\lambda}$ is the
eigenspace of $S$ corresponding to $\lambda\in\Lambda$. Denote by $W(\lambda)$
the direct sum
$W(\lambda)=V_{\lambda}\oplus V_{-\lambda}.$
Define the equivalence relation $\mathscr{R}$ on $\Lambda$ by:
$\lambda\mathscr{R}\mu\ \text{ if and only if }\ \lambda=\pm\mu.$
Then
$\mathbb{C}^{2n}=\bigoplus^{\bot}_{\lambda\in\Lambda/\mathscr{R}}W(\lambda),$
and each $(W(\lambda),B_{\lambda})$ is a vector space with the non-degenerate
form $B_{\lambda}$ given by:
$B_{\lambda}=B_{\epsilon}|_{W(\lambda)\times W(\lambda)}.$
Fix $\lambda\in\Lambda$. We write $W(\lambda)=V_{+}\oplus V_{-}$ with
$V_{\pm}=V_{\pm\lambda}$. Then, according to the notation in Lemma 7.6, define
$N_{\pm\lambda}=N_{\pm}$. Since $N|_{V_{-}}=-N_{\lambda}^{*}$, it is easy to
verify that the matrices of $N|_{V_{+}}$ and $N|_{V_{-}}$ have the same Jordan
form. Let $(d_{1}(\lambda),\dots,d_{r_{\lambda}}(\lambda))$ be the size of the
Jordan blocks in the Jordan decomposition of $N|_{V_{+}}$. This does not
depend on a possible choice between $N|_{V_{+}}$ or $N|_{V_{-}}$ since both
maps have the same Jordan type.
Next, we consider
$\mathcal{D}=\bigcup_{r\in\mathbb{N}^{*}}\\{(d_{1},\dots,d_{r})\in\mathbb{N}^{r}\mid\
d_{1}\geq d_{2}\geq\dots\geq d_{r}\geq 1\\}.$
Define $d:\Lambda\to\mathcal{D}$ by
$d(\lambda)=(d_{1}(\lambda),\dots,d_{r_{\lambda}}(\lambda))$. It is clear that
$\Phi\circ d=m$ where $\Phi:\mathcal{D}\to\mathbb{N}$ is the map defined by
$\Phi(d_{1},\dots,d_{r})=\sum_{i=1}^{r}d_{i}$.
Finally, we can associate to $C\in\mathfrak{g}_{\epsilon}$ a triple
$(\Lambda,m,d)$ defined as above.
###### Definition 7.7.
Let $\mathcal{J}_{n}$ be the set of all triples $(\Lambda,m,d)$ such that:
1. (1)
$\Lambda$ is a subset of $\mathbb{C}\setminus\\{0\\}$ with $\sharp\Lambda\leq
2n$ and $\lambda\in\Lambda$ if and only if $-\lambda\in\Lambda$.
2. (2)
$m:\Lambda\to\mathbb{N}^{*}$ satisfies $m(\lambda)=m(-\lambda)$, for all
$\lambda\in\Lambda$ and $\underset{\lambda\in\Lambda}{\sum}m(\lambda)=2n$.
3. (3)
$d:\Lambda\to\mathcal{D}$ satisfies $d(\lambda)=d(-\lambda)$, for all
$\lambda\in\Lambda$ and $\Phi\circ d=m$.
Let $\mathcal{I}(2n)$ be the set of invertible elements in
$\mathfrak{g}_{\epsilon}$ and $\widetilde{{\mathcal{I}}}(2n)$ be the set of
$I_{\epsilon}$-adjoint orbits of elements in $\mathcal{I}(2n)$. By the
preceding remarks, there is a map $i:\mathcal{I}(2n)\to\mathcal{J}_{n}$. Then
we have a parametrization of the set $\widetilde{{\mathcal{I}}}(2n)$ as
follows:
###### Proposition 7.8.
The map $i:\mathcal{I}(2n)\to\mathcal{J}_{n}$ induces a bijection
$\widetilde{i}:\widetilde{{\mathcal{I}}}(2n)\to\mathcal{J}_{n}$.
###### Proof.
Let $C$ and $C^{\prime}\in\mathcal{I}(2n)$ such that $C^{\prime}=U\ C\ U^{-1}$
with $U\in I_{\epsilon}$. Let $S$, $S^{\prime}$, $N$, $N^{\prime}$ be
respectively the semisimple and nilpotent parts of $C$ and $C^{\prime}$. Write
$i(C)=(\Lambda,m,d)$ and
$i(C^{\prime})=(\Lambda^{\prime},m^{\prime},d^{\prime})$. One has
$S^{\prime}+N^{\prime}=U\ (S+N)\ U^{-1}=U\ S\ U^{-1}+U\ N\ U^{-1}.$
By the unicity of Jordan decomposition, $S^{\prime}=U\ S\ U^{-1}$ and
$N^{\prime}=U\ N\ U^{-1}$. So $\Lambda^{\prime}=\Lambda$ and $m^{\prime}=m$.
Also, since $U\ S=S^{\prime}\ U$ one has $U\ S(V_{\lambda})=S^{\prime}\
U(V_{\lambda})$. It implies that
$S^{\prime}\ (U(V_{\lambda}))=\lambda U(V_{\lambda}).$
That means $U(V_{\lambda})=V^{\prime}_{\lambda}$, for all $\lambda\in\Lambda$.
Since $N^{\prime}=U\ N\ U^{-1}$ then $N|_{V_{\lambda}}$ and
$N^{\prime}|_{V^{\prime}_{\lambda}}$ have the same Jordan decomposition, so
$d=d^{\prime}$ and $\widetilde{i}$ is well defined.
To prove that $\widetilde{i}$ is onto, we start with
$\Lambda=\\{\lambda_{1},-\lambda_{1},\dots,\lambda_{k},-\lambda_{k}\\}$, $m$
and $d$ as in Definition 7.7. Define on the canonical basis:
$S=\operatorname{diag}_{2n}(\overbrace{\lambda_{1},\dots,\lambda_{1}}^{m(\lambda_{1})},\dots,\overbrace{\lambda_{k},\dots,\lambda_{k}}^{m(\lambda_{k})},\overbrace{-\lambda_{1},\dots,-\lambda_{1}}^{m(\lambda_{1})},\dots,\overbrace{-\lambda_{k},\dots,-\lambda_{k}}^{m(\lambda_{k})}).$
For all $1\leq i\leq k$, let $d(\lambda_{i})=(d_{1}(\lambda_{i})\geq\dots\geq
d_{r_{\lambda_{i}}}(\lambda_{i})\geq 1)$ and define
$N_{+}(\lambda_{i})=\operatorname{diag}_{d(\lambda_{i})}\left(J_{d_{1}(\lambda_{i})},J_{d_{2}(\lambda_{i})},\dots,J_{d_{r_{\lambda_{i}}}(\lambda_{i})}\right)$
on the eigenspace $V_{\lambda_{i}}$ and $0$ on the eigenspace
$V_{-\lambda_{i}}$ where $J_{d}$ is the Jordan block of size $d$.
By Lemma 7.6, $N(\lambda_{i})=N_{+}(\lambda_{i})-N_{+}^{*}(\lambda_{i})$ is
skew-symmetric on $V_{\lambda_{i}}\oplus V_{-\lambda_{i}}$. Finally,
$\mathbb{C}^{2n}=\bigoplus^{\bot}_{1\leq i\leq k}\left(V_{\lambda_{i}}\oplus
V_{-\lambda_{i}}\right).$
Define $N\in\mathfrak{g}_{\epsilon}$ by
$N\left(\sum_{i=1}^{k}v_{i}\right)=\sum_{i=1}^{k}N(\lambda_{i})(v_{i})$,
$v_{i}\in V_{\lambda_{i}}\oplus V_{-\lambda_{i}}$ and
$C=S+N\in\mathfrak{g}_{\epsilon}$. By construction, $i(C)=(\Lambda,m,d)$, so
$\widetilde{i}$ is onto.
To prove that $\widetilde{i}$ is one-to-one, assume that $C$,
$C^{\prime}\in\mathcal{I}(2n)$ and that $i(C)=i(C^{\prime})=(\Lambda,m,d)$.
Using the previous notation, since their respective semisimple parts $S$ and
$S^{\prime}$ have the same spectrum and same multiplicities, there exist $U\in
I_{\epsilon}$ such that $S^{\prime}=USU^{-1}$. For $\lambda\in\Lambda$, we
have $U(V_{\lambda})=V^{\prime}_{\lambda}$ for eigenspaces $V_{\lambda}$ and
$V^{\prime}_{\lambda}$ of $S$ and $S^{\prime}$ respectively.
Also, for $\lambda\in\Lambda$, if $N$ and $N^{\prime}$ are the nilpotent parts
of $C$ and $C^{\prime}$, then $N^{\prime\prime}(V_{\lambda})\subset
V_{\lambda}$, with $N^{\prime\prime}=U^{-1}N^{\prime}U$. Since
$i(C)=i(C^{\prime})$, then $N|_{V_{\lambda}}$ and
$N^{\prime}|_{V^{\prime}_{\lambda}}$ have the same Jordan type. Since
$N^{\prime\prime}=U^{-1}N^{\prime}U$, then $N^{\prime\prime}|_{V_{\lambda}}$
and $N^{\prime}|_{V^{\prime}_{\lambda}}$ have the same Jordan type. So
$N|_{V_{\lambda}}$ and $N^{\prime\prime}|_{V_{\lambda}}$ have the same Jordan
type. Therefore, there exists $D_{+}\in\mathcal{L}(V_{\lambda})$ such that
$N^{\prime\prime}|_{V_{\lambda}}=D_{+}N|_{V_{\lambda}}D_{+}^{-1}$. By Lemma
7.6, there exists $D(\lambda)\in I_{\epsilon}(V_{\lambda}\oplus V_{-\lambda})$
such that
$N^{\prime\prime}|_{V_{\lambda}\oplus
V_{-\lambda}}=D(\lambda)N|_{V_{\lambda}\oplus V_{-\lambda}}D(\lambda)^{-1}.$
We define $D\in I_{\epsilon}$ by $D|_{V_{\lambda}\oplus
V_{-\lambda}}=D(\lambda)$, for all $\lambda\in\Lambda$. Then
$N^{\prime\prime}=DND^{-1}$ and $D$ commutes with $S$ since
$S|_{V_{\pm\lambda}}$ is scalar. Then $S^{\prime}=(UD)S(UD)^{-1}$ and
$N^{\prime}=(UD)N(UD)^{-1}$ and we conclude that $C^{\prime}=(UD)C(UD)^{-1}$.
∎
### 7.4. Adjoint orbits in the general case
Let us now classify $I_{\epsilon}$-adjoint orbits of $\mathfrak{g}_{\epsilon}$
in the general case as follows. Let $C$ be an element in
$\mathfrak{g}_{\epsilon}$ and consider the Fitting decomposition of $C$
$\mathbb{C}^{m}=V_{N}\oplus V_{I},$
where $V_{N}$ and $V_{I}$ are stable by $C$, $C_{N}=C|_{V_{N}}$ is nilpotent
and $C_{I}=C|_{V_{I}}$ is invertible. Since $C$ is skew-symmetric,
$B_{\epsilon}(C^{k}(V_{N}),V_{I})=(-1)^{k}B_{\epsilon}(V_{N},C^{k}(V_{I}))$
for any $k$ then one has $V_{I}=(V_{N})^{\perp}$. Also, the restrictions
$B_{\epsilon}^{N}=B_{\epsilon}|_{V_{N}\times V_{N}}$ and
$B_{\epsilon}^{I}=B_{\epsilon}|_{V_{I}\times V_{I}}$ are non-degenerate.
Clearly, $C_{N}\in\mathfrak{g}_{\epsilon}(V_{N})$ and
$C_{I}\in\mathfrak{g}_{\epsilon}(V_{I})$. By Subsection 7.1 and Subsection
7.3, $C_{N}$ is attached with a partition $[d]\in\mathcal{P}_{\epsilon}(n)$
and $C_{I}$ corresponds to a triple $T\in\mathcal{J}_{\ell}$ where
$n=\dim(V_{N})$, $2\ell=\dim(V_{I})$. Let $\mathcal{D}(m)$ be the set of all
pairs $([d],T)$ such that $[d]\in\mathcal{P}_{\epsilon}(n)$ and
$T\in\mathcal{J}_{\ell}$ satisfying $n+2\ell=m$. By the preceding remarks,
there exists a map $p:\mathfrak{g}_{\epsilon}\rightarrow\mathcal{D}(m)$ .
Denote by $\mathcal{O}(\mathfrak{g}_{\epsilon})$ the set of
$I_{\epsilon}$-adjoint orbits of $\mathfrak{g}_{\epsilon}$ then we obtain the
classification of $\mathcal{O}(\mathfrak{g}_{\epsilon})$ as follows:
###### Proposition 7.9.
The map $p:\mathfrak{g}_{\epsilon}\rightarrow\mathcal{D}(m)$ induces a
bijection
$\widetilde{p}:\mathcal{O}(\mathfrak{g}_{\epsilon})\rightarrow\mathcal{D}(m)$.
###### Proof.
Let $C$ and $C^{\prime}$ be two elements in $\mathfrak{g}_{\epsilon}$. Assume
that $C$ and $C^{\prime}$ lie in the same $I_{\epsilon}$-adjoint orbit. It
means that there exists an isometry $P$ such that $C^{\prime}=PCP^{-1}$. So
$C^{\prime k}\ P=P\ C^{k}$ for any $k$ in $\mathbb{N}$. As a consequence,
$P(V_{N})\subset V^{\prime}_{N}$ and $P(V_{I})\subset V^{\prime}_{I}$.
However, $P$ is an isometry then $V^{\prime}_{N}=P(V_{N})$ and
$V^{\prime}_{I}=P(V_{I})$. Therefore, one has
$C^{\prime}_{N}=P_{N}\ C_{N}P^{-1}_{N}\text{ and }C^{\prime}_{I}=P_{I}\
C_{I}P_{I}^{-1},$
where $P_{N}=P:V_{N}\rightarrow V^{\prime}_{N}$ and $P_{I}=P:V_{I}\rightarrow
V^{\prime}_{I}$ are isometries. It implies that $C_{N}$, $C^{\prime}_{N}$ have
the same partition and $C_{I}$, $C^{\prime}_{I}$ have the same triple. Hence,
the map $\widetilde{p}$ is well defined.
For a pair $([d],T)\in\mathcal{D}(m)$ with $[d]\in\mathcal{P}_{\epsilon}(n)$
and $T\in\mathcal{J}_{\ell}$, we set a nilpotent map
$C_{N}\in\mathfrak{g}_{\epsilon}(V_{N})$ corresponding to $[d]$ as in Section
7.1 and an invertible map $C_{I}\in\mathfrak{g}_{\epsilon}(V_{I})$ as in
Proposition 7.8 where $\dim(V_{N})=n$ and $\dim(V_{I})=2\ell$. Define
$C\in\mathfrak{g}_{\epsilon}$ by $C(X_{N}+X_{I})=C_{N}(X_{N})+C_{I}(X_{I})$,
for all $X_{N}\in V_{N},\ X_{I}\in V_{I}$. By construction, $p(C)=([d],T)$ and
$\widetilde{p}$ is onto.
To prove $\widetilde{p}$ is one-to-one, let
$C,C^{\prime}\in\mathfrak{g}_{\epsilon}$ such that
$p(C)=p(C^{\prime})=([d],T)$. Keep the above notations, since $C_{N}$ and
$C^{\prime}_{N}$ have the same partition then there exists an isometry
$P_{N}:V_{N}\rightarrow V^{\prime}_{N}$ such that $C^{\prime}_{N}=P_{N}\
C_{N}\ P_{N}^{-1}$. Similarly $C_{I}$ and $C^{\prime}_{I}$ have the same
triple and then there exists an isometry $P_{I}:V_{I}\rightarrow
V^{\prime}_{I}$ such that $C^{\prime}_{I}=P_{I}\ C_{I}\ P_{I}^{-1}$. Define
$P:V\rightarrow V$ by $P(X_{N}+X_{I})=P_{N}(X_{N})+P_{I}(X_{I})$, for all
$X_{N}\in V_{N},X_{I}\in V_{I}$ then $P$ is an isometry and $C^{\prime}=P\ C\
P^{-1}$. Therefore, $\widetilde{p}$ is one-to-one.
∎
## References
|
arxiv-papers
| 2012-06-24T14:43:20 |
2024-09-04T02:49:32.158598
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Minh Thanh Duong and Rosane Ushirobira",
"submitter": "Thanh Duong Minh",
"url": "https://arxiv.org/abs/1206.5504"
}
|
1206.5535
|
* # The Nature of The Light Variations of Unique Binary DK CVn
Hasan Ali DAL 1,2, Esin SİPAHİ 1, Orkun ÖZDARCAN 1 1 Department of Astronomy
and Space Sciences, University of Ege, Bornova, 35100 İzmir, Turkey
2 Corresponding Author, Email: ali.dal@ege.edu.tr
Abstract: In this study, we present the BVR observations of DK CVn in 2007 and
2008. We analysed the BVR light curves of the system and obtained the system’s
parameters. Using the ”q-search” method, we estimated the mass ratio of the
system ($q$) as 0.55. Taking the temperature of the primary component as 4040
K, the temperature of the secondary was found to be 3123 K in the analyses.
Several flares were detected in this study, and the distributions of flare
equivalent duration versus flare total duration are modelled using the One
Phase Exponential Association function for these flares. The parameters of the
model demonstrated that the flares are the same with analogous detected from
UV Ceti type stars. Apart from flare activity, we reached some results about
the sinusoidal-like variation at out-of-eclipse. The clues demonstrate that
the variation at out-of-eclipse must be caused by some cool spot(s) on one of
the components. The star is found to show two active longitudes in which the
spots are mainly formed. Consequently, this study reveals that DK CVn should
be a chromospherically active binary star.
Keywords: binaries: close, stars: activity, stars: spots, stars: flare, stars:
individual: DK CVn.
## 1 Introduction
DK Canum Venaticorum (= GSC 03018 01509) is classified as an eclipsing binary
star in the SIMBAD database. The system was discovered by the Robotic Optical
Transient Search Experiment for the first time (Akerlof et al., 2000).
Examining the light variation of the system, Diethelm, (2001) noted that the
reflection effect is dominant in the light variation. DK CVn identified as a
variable star by Kazarovets et al., (2003) was listed as an Algol type
variable (EA) in the Catalogue of Eclipsing Variables by Malkov et al.,
(2006). Terrell et al., (2005) noticed from $UBVR_{C}I_{C}$ bands
observations of the season 2002 that the amplitudes of the humps around
$0^{P}.25$ increase from $I_{C}$ band to U band. A flare was detected by
Terrell et al., (2005) in the observations of the season 2003. Using the low
resolution spectroscopy, they determined that the primary component is from K7
spectral type. Considering the light curves obtained in $I_{C}$ band, they
indicate that the secondary component must be a star from the late M spectral
types.
In this study, BVR photometry of DK CVn was made in the seasons 2007 and 2008.
All the details of the observations and reduction procedures are given in
Section 2.1. The new light elements are given in Section 2.2. Using the PHOEBE
V.0.31a software (Prša & Zwitter, 2005), the BVR light curves were analysed,
and some physical parameters were obtained. The details of the analysis are
described in Section 2.3 and 2.4. Extracting the obtained synthetic light
curve from the observed light curves, it was examined whether there is any
variation out-of-eclipses. All the procedures are described in Section 2.5.
Several flares were detected. Some parameters were computed for each flare.
The One Phase Exponential Association (hearafter OPEA) model was derived for
the flares. The analyses and the OPEA model of the flares are described in
Section 2.6. All the results are given and discussed in Section 3.
## 2 Observations and Analyses
### 2.1 Observations
Observations were acquired with a thermoelectrically cooled ALTA U$+$42
2048$\times$2048 pixel CCD camera attached to a 40 cm - Schmidt - Cassegrains
- type MEADE telescope at Ege University Observatory. The observations made in
BVR bands were continued two nights in the season 2007 and five nights in the
season 2008. Some basic parameters of program stars are listed in Table 1.
Although the program and comparison stars are very close on the sky,
differential atmospheric extinction corrections were applied. The atmospheric
extinction coefficients were obtained from observations of the comparison
stars on each night. Moreover, the comparison stars were observed with the
standard stars in their vicinity and the reduced differential magnitudes, in
the sense variable minus comparison, were transformed to the standard system
using procedures outlined by Hardie, (1962). The standard stars are listed in
catalogues of Landolt, (1983, 1992). Furthermore, the dereddened colours of
the system were computed. Heliocentric corrections were also applied to the
times of the observations.
The mean averages of the standard deviations are $0^{m}.021$, $0^{m}.013$, and
$0^{m}.019$ for observations acquired in the BVR bands, respectively. To
compute the standard deviations of observations, we used the standard
deviations of the reduced differential magnitudes in the sense comparisons
minus check stars for each night. There was no variation observed in the
standard brightnesses of the comparison stars.
In Figure 1, the light and colour curves of DK CVn are shown. Comparing the
light curves of the seasons 2007 and 2008, it is obviously seen that there are
some differences between two light curves. The decreases in the levels of both
maxima and minima of 2007 and 2008 light curves are seen, and the shapes of
the light curves also changed from 2007 to 2008. The parts of the light curves
between $0^{P}.70$ and $1^{P}.20$ are especially different from each other. On
the other hand, there is no variation in the colour curves except the flare
moments. Moreover, there are no variations in the mean colour, while there are
some variations in the mean brightness. The secondary minimum was not obtained
in 2007 due to the absence of the observation in those phases. In Figure 2,
there are two V band light curves, whose data were taken from the database of
The Northern Sky Variability Survey (hereafter NSVS) (Woźniak et al., 2004).
As seen from Figure 2, the shape of DK CVn’s light curve is rapidly varying,
even in one observing season.
### 2.2 Times of Minima and Orbital Period
We could find 54 times of minima (included both primary and secondary) from
the literature. The data cover an interval from 2001 to the current time and
listed in Table 2. All the minimum times used in the analysis were obtained in
photoelectric observations. Using the linear least-squares method, an update
linear ephemeris is given by Equation (1):
$HJD~{}=~{}24~{}53422.9838(2)~{}+~{}0^{d}.494964(1)~{}\times~{}E$ (1)
The linear correction $(O-C)_{I}$ is shown in Figure 3. The phases in all the
figures are calculated with these new light elements.
### 2.3 Light Curve Analysis
Although the secondary minimum was not obtained in 2007, whole the light curve
was obtained in 2008. Because of this, we analysed the BVR light curves
obtained in 2008. The analyses were carried out with using the PHOEBE V.0.31a
software (Prša & Zwitter, 2005). The method used in the PHOEBE V.0.31a
software depends on the method used in the version 2003 of the Wilson-Devinney
Code (Wilson & Devinney, 1971; Wilson, 1990). The BVR light curves were
analysed with the ”detached system”, ”semi-detached system with the primary
component filling its Roche-Lobe” and ”semi-detached system with the secondary
component filling its Roche-Lobe” modes. An acceptable result was only
obtained in ”detached system” mode, while no acceptable results were obtained
in all the others modes. Terrell et al., (2005), who examined the low
resolution spectrum of the system, demonstrated that the primary component is
a K7V star. Besides, we took JHK brightness of the system ($J=10^{m}.489$,
$H=9^{m}.839$, $K=9^{m}.664$) from the NOMAD Catalogue (Zacharias et al.,
2005). Using these brightnesses, we derived dereddened colours as a
$(J-H)_{\circ}$=0m.561 and $(H-K)_{\circ}$=0m.140 for the system. Using the
calibrations given by Tokunaga, (2000), we derived the temperature of the
primary component as 4040 K depending on these dereddened colours. The derived
temperature is in agreement with the spectral type given by Terrell et al.,
(2005). In the analyses, the temperature of the primary component was fixed to
4040 K, and the temperature of the secondary was taken as a free parameter.
Considering the spectral types, the albedos ($A_{1}$ and $A_{2}$) and the
gravity darkening coefficients ($g_{1}$ and $g_{2}$) of the components were
adopted for the stars with the convective envelopes (Lucy, 1967; Rucinski,
1969). The non-linear limb-darkening coefficients ($x_{1}$ and $x_{2}$) of the
components were taken from van Hamme, (1993). In the analyses, their
dimensionless potentials ($\Omega_{1}$ and $\Omega_{2}$), the fractional
luminosity ($L_{1}$) of the primary component and the inclination ($i$) of the
system were taken as the adjustable free parameters.
In order to find the best photometric mass ratio of the components, we used
the ”q-search” method with using a step of 0.05 due to the absence of any
spectroscopic mass ratios. As seen from Figure 4, the minimum sum of weighted
squared residuals ($\Sigma res^{2}$) is found for the mass ratio value of
$q=0.55$. According to this result, we assume that a possible mass ratio of
the system is $q=0.55$.
As it is clearly seen from Figure 1, there is the clear asymmetry in the light
curves of both 2007 and 2008. Moreover, the shape of the asymmetric light
curve changed from 2007 to 2008. In order to remove the asymmetry, we assumed
that the primary component has two cool spots on its surface. The synthetic
light curves obtained from the best light curve solution are seen in Figure 5,
and the result parameters of the analysis are also listed in Table 3. The 3D
model of Roche geometry is shown in Figure 6.
### 2.4 Estimated Absolute Parameters
Although there is not any radial velocity curve of the system, we tried to
estimate the absolute parameters of the components. Considering its spectral
type, we took the mass of the primary component from Tokunaga, (2000), and
the mass of the secondary component was calculated from the estimated mass
ratio of the system. Using Kepler’s third law, we calculated possible the
semi-major axis ($a$), and then the mean radii of the components were
calculated. The mass of the primary component was found to be 0.44
$M_{\odot}$, and it was found to be 0.24 $M_{\odot}$ for the secondary
component. Considering estimated $a$ value, the radius of the primary
component was computed as 0.58 $R_{\odot}$, while it was computed as 0.59
$R_{\odot}$. Using the estimated radii and the obtained temperatures of the
components, the luminosity of the primary component was estimated to be 0.08
$L_{\odot}$, and it was found as 0.03 $L_{\odot}$ for the secondary component.
The absolute parameters seem to be an acceptable as an astrophysical. However,
the radii of the both components are larger than the expected values in
respect to the theoretical models. In Figure 7, we plotted the distribution of
the radii versus the masses for some stars. In the figure, the filled circles
represent the known active stars, which were taken from the catalogue of
Gershberg et al., (1999). Some of these stars exhibit the spot activity,
while some of them exhibit the flare activity. Some stars exhibit both spot
and flare activities. In the figure, the asterisk represents the secondary
component, while the open triangle represents the primary component. The line
represents the ZAMS theoretical model developed by Siess et al., (2000).
### 2.5 The Variations Out-of-Eclipses
In order to be sure of the reasons of the variations out-of-eclipses, we
investigated the pre-whitened light curves in R band. For this aim, using the
physical parameters, we derived the synthetic light curve of the system for
unspotted case for the R band. Then, the synthetic light curve of R band was
extracted from both R light curves obtained in 2007 and 2008. In the second
step, the R band light curves, which were obtained in 2002, 2003, 2004 and
2005, presented by Terrell et al., (2005) were scanned, and the observational
data were obtained from these light curves in order to compare our data with
the data existing in the literature. Then, the synthetic light curve derived
for unspotted case was extracted from Terrell’s light curves.
The pre-whitened light curves of the season 2002, 2003, 2004, 2005, 2007 and
2008 are shown in Panels a, b, c, d, e and f in Figure 8, respectively. If the
pre-whitened light curves are carefully examined, three points will be seen in
the general nature of the light curves. One of them is a sinusoidal-like
variation. The sinusoidal-like variations are seen in all the pre-whitened
light curves as a dominant feature. The second one is that the sinusoidal-like
variations have an asymmetric shape. Moreover, the shapes of the sinusoidal-
like variations (the minima and maxima phases, their amplitudes and etc.) are
varying from a year to the next one. Finally, there are some sudden and short-
duration flares in the this curves.
In the pre-whitened light curves, one asymmetric minimum is seen in the
sinusoidal-like variations, generally. However, there are two minima in the
pre-whitened light curve of the season 2003. Examining these sinusoidal-like
variations, some parameters were computed. These parameters are listed in
Table 4. In the table, we listed the observing seasons, $\theta_{min}$, the
amplitude of the pre-whitened R band light curves and references,
respectively. The variation of the amplitude of the pre-whitened R band light
curves is shown in Figure 9. As seen from the figure, although the amplitude
of the pre-whitened light curves are decreasing from 2002 to 2008 in the
general view, a cyclic behaviour in sinusoidal form is also seen in the
amplitude variation. Figure 10 is a plot of the phase of light minimum against
the observing year. Apart from the amplitude, the minima phases
($\theta_{min}$) also exhibit a variation. The minima phases are separating in
two mean longitudes. The $\theta_{min}$ determined from such light curves
would give the effective longitude of the spot or spots group. There is an
indication of two effective longitudes in which the spots are generally
formed. One of them is around 0P.80, while the other is around 0P.20.
### 2.6 Flare Activity
Apart from both eclipses and the sinusoidal-like variations out-of-eclipses,
DK CVn also exhibits the flare activity. In this study, we detected one flare
in the observations of 2007 and two flares in 2008. The flare shown in Figure
11 was detected in 2007 and exhibits itself in each band, while the flares
detected in 2008 exhibits themselves in just B and V bands. They are shown in
Figure 12. These two flares can not be detected in R band due to their lower
powers. In addition to the flares detected in this study, the flare detected
in B band by Terrell et al., (2005) was scanned and we got its observational
data as well. Terrell’s flare is shown in Figure 13. The flare parameters were
calculated for each flare, and we list them in Table 5. In the table, the
observing season, observing band, HJD of the flare maxima, flare rise time
(s), flare decay time (s), flare total duration (s), flare equivalent duration
(s), flare amplitude (mag) are listed in each column, respectively. In the
last column, we noted which study the flare data are belonged.
To calculate the flare parameters, we used the method described by Dal &
Evren, (2010). However, a different way was followed to determine the
quiescent level of the brightness due to the eclipsing binary nature of DK
CVn. Using the synthetic light curve; we obtained the quiescent level of the
brightness for each phase. In order to test whether the method is correctly
working or not, we compared the light curves observed in the consecutive-close
nights and the synthetic light curve. Some examples for these comparisons are
shown in Figures 11 and 12. As seen from the figures, DK CVn was observed two
or more times in each phase intervals in close dates. Sometimes a flare was
detected in one observing night, while no flare was detected in the same phase
interval in another-close observing night. Comparing these observations with
the synthetic light curve aids to determine the actual flare light variation.
As seen from Figure 11, a flare was detected around the primary minima on
April 18, 2007. The system was observed in the same phase interval on March 8,
2007, but no flare was detected. The flare detected on April 18, 2007
distorted almost the shape of the primary minimum. The similar case is seen in
the observations of 2008. Two flares were detected on March 16, 2008, while no
flare was detected on March 2, 2008. The flare taken from Terrell et al.,
(2005) is shown in Figure 13. We compared Terrell’s flare light curve with
only the synthetic light curve due to absence of another observation in a
consecutive-close night.
Using the Equations (2) and (3) described in the method developed by
Gershberg, (1972), the flare equivalent durations and flare energies can be
calculated. In Equation (2), $I_{0}$ is the intensity of the star in the
quiescent level and $I_{flare}$ is the intensity during flare.
$P=\int[(I_{flare}-I_{0})/I_{0}]dt$ (2)
$E=P\times L$ (3)
where $E$ is the flare energy, $P$ is the flare equivalent duration given by
Equation (2), and $L$ is the luminosity of the stars in the quiescent level.
To understand whether the flares observed from DK CVn, which is an eclipsing
binary system, are similar to the flares occurring on the surface of UV Ceti
type stars, DK CVn’s flares were compared with B band flares of five UV Ceti
type stars presented by Dal & Evren, (2011). To be able to compare them,
first of all, following the method developed by Dal & Evren, (2011), the
distribution of the flare equivalent durations versus flare total durations
were derived for B band flares of DK CVn. Using SPSS V17.0 software (Green et
al., 1999) and GrahpPad Prism V5.02 software (Motulsky, 2007; Dawson &
Trapp, 2004), the best model function was determined. Using the least-squares
method, regression calculations showed that the best model function of
distribution is the OPEA function (Motulsky, 2007; Spanier & Oldham, 1987)
given by Equation (4):
$y~{}=~{}y_{0}~{}+~{}(Plateau~{}-~{}y_{0})~{}\times~{}(1~{}-~{}e^{-k~{}\times~{}x})$
(4)
The derived OPEA model of DK CVn’s flares is shown in Figure 14. Using this
model, some parameters of the flare equivalent duration, such as $y_{0}$,
$Plateau$, $K$, $Span$, $Half-Life$ values, were computed. In the OPEA model
function given by Equation (4), the $y$ values were taken as flare equivalent
duration in logarithmic scale, while $x$ values were taken as flare total
durations. The parameter $y_{0}$ is the lowest flare equivalent duration
obtained in logarithmic scale, while the parameter $Plateau$ is the upper
limit the flare equivalent durations can reach. According to Equation (3),
$Plateau$ value depends only on flare energy, while $y_{0}$ value depends on
the brightness of the target and sensitivity of the optical system, as well as
flare power. The parameter $K$ is a constant value depending on $x$ values.
The $Span$ value is a difference between the $Plateau$ and $y_{0}$ values. The
$Half-Life$ value is half of the first $x$ values, where the model reaches the
$Plateau$ value. In other words, it is half of the flare total duration, where
flares with the highest energy start to be seen. The $Half-Life$ value is an
indicator of the duration the flare process occurring on the surface of a star
needs to reach the saturation. As seen from the distribution of flare
equivalent durations, the flare equivalent durations increase with the flare
total duration until a specific total duration value, and then the flare
equivalent durations became constant, no matter how long the flare total
duration is. The $Half-Life$ value is half of this specific total duration
value. All the parameters computed from the OPEA model are listed in Table 6.
In order to test whether the $Plateau$ value is statistically acceptable for
this distribution of the flare equivalent durations, using the Independent
Sample t-Test (hereafter t-Test, Motulsky, 2007; Dawson & Trapp, 2004; Green
et al., 1999), we compute the mean average value of the flares, which are
located in the $Plateau$ phase of the model. The found mean average value is
listed in the last row of Table 6.
All the parameters listed in Table 6 were compared with the parameters given
by Dal & Evren, (2011) for five UV Ceti type stars. In the comparison, we
assume that the flares are occurring on the surface of the cool component of
the system. The light curve analysis of the system indicates that the
temperature of the cool component is 3123 K. According to Tokunaga, (2000),
this temperature is corresponding to B-V = $1^{m}.630$ as the colour index.
All the comparisons are shown in Figures 15 and 16.
As seen from Figures 15 and 16, in fact, the flares detected in the
observations of DK CVn seem to have a same nature with the flares detected
from UV Ceti type stars. Moreover, our assumption also seems to be correct,
because the parameters are in agreement with its analogue according to assumed
temperature.
## 3 Results and Discussion
In this study, we obtained the light curves of an eclipsing binary system, DK
CVn, in two observing seasons, and we analysed the BVR light curves obtained
in 2008 to find the physical properties of DK CVn. In addition, using Kepler’s
third law under some assumptions, the possible absolute parameters were found.
The observations demonstrate that the radii are generally larger than the
expected values. We compared the radii of DK CVn’s components with the known
active stars and a model. The radii of the active stars, which exhibit spot or
flare activity or both of them, are dramatically larger than the values given
by the model developed for the stars with $Z=0.02$ by Siess et al., (2000).
The components of DK CVn are also in agreement with the other active stars,
listed in the catalogue of Gershberg et al., (1999). According to several
theoretical models and observational studies (Ribas, 2006; Chabrier et al.,
2007; Morales et al., 2008, 2010), the case seen in Figure 7 is a well known
phenomenon for low-mass active stars. For instance, YY Gem (Torres & Ribas,
2002), CU Cam (Ribas, 2003) and CM Dra (Morales et al., 2009) are the most
popular system for this case. There are several similarities between these
three systems and DK CVn. Firstly, all of them exhibit the spot and flare
activities, and they consist of the low mass components.
The observations in B and V bands demonstrate that the system exhibits flare
activity. Considering the effective temperature, we assumed that the flares
occur on the secondary component. The derived parameters demonstrate that DK
CVn’s flares seem to behave in the same way with the flares of UV Ceti stars.
The maximum energy level of the flares seen in the DK CVn system is in
agreement with the analogues of UV Ceti stars from the late spectral types.
Consequently, the flares detected from DK Cvn must be produced by the same
process occurring on the surface of an UV Ceti star.
Considering the flare activity exhibited by DK CVn system, both components
could be chromospherically active stars. In addition, the spectral types of
both components are also supported to the chromospherical activity. Another
support comes from the light curve analysis. The observed light curve can be
modelled with two cool spots on the one of the components due to the asymmetry
seen in the shape of the curve. We assumed that the spots are located on the
primary component, and we analysed the light curves with this assumption.
According to its effective temperature, the secondary component is so close
the border of the full-convective area among the M dwarfs. Although the full-
convective M dwarfs exhibit very strong flare activity, a few of them just
exhibit spot activity. However, the K dwarfs are generally potential stars,
which are possible to exhibit spot activity. This is why we assumed the
spotted star is the primary component.
Considering the sum of weighted squared residuals, we found from the BVR light
curve analyses that there are a large spot and a small one on the primary
component. The larger spot with a temperature factor of 0.95 is located in
longitude of 188∘, while the small one with a temperature factor of 0.90 is
located in longitude of 290∘. It is well known that the longitudes of the
spotted areas can be found exactly from the light curves obtained with
photometric observations. However, the co-latitude, radius and temperature
factor of a spot are not very well. A similar synthetic curves fitting the
observations can be derived with some different values of these parameters. In
this study, when we considered the sum of weighted squared residuals, the
radii, co-latitudes and temperature factors derived for the spots give one of
the best synthetic curves, which are seen in Figure 5. They are statistically
acceptable, and the physical parameters of the components are also acceptable
in the astrophysical sense with these spot parameters.
However, the asymmetry and variation on the pre-whitened light curves reveal
that cool spots vary with time, as well. Although the spots can be sometimes
changing in few months as it is seen from NSVS data, generally the spots do
not seem to change on the short time-scale. The spots occur at two longitudes,
i.e., in phases of 0.80-0.90 and 0.00-0.20. The phenomenon reveals that spots
concentrate on two active longitudes. The amplitude of the pre-whitened light
curves demonstrated very well variation. In general, the amplitudes of the
variations at out-of-eclipses have been decreasing since 2002, while a cyclic
variation in sinusoidal shape is also seen, combining with the general
decreasing.
The phases of the minima are dramatically changing in this system. The
variations of the pre-whitened light curves indicate that the spotted areas
are not stable on the component. In the pre-whitened light curves of the
season 2003, two minima are seen separately from each other. In the season
2005, the pre-whitened light curve has a very strong asymmetry. The variations
seen out-of-eclipses are similar to the variations exhibiting by the young-
fast-rotating stars, such as YY Gem, ER Vul, SV Cam, CU Cam and CM Dra
(Strassmeier, 2009; Torres & Ribas, 2002; Ribas, 2003; Morales et al.,
2009). Therefore, the spotted component (and the system) could be a young
star. On the other hand, it must be noted that there are many systems exhibit
an unexpected cases in contrast to this approach (Rocha-Pinto et al., 2002).
In brief, the variation seen at out-of-eclipses should be due to the cool
spots gathering into two separated longitudes on the surface of the primary
component. Thus, this variation is caused by the rotational modulation due to
the chromospherical activity. Consequently, DK CVn seems to be an analogue of
RS CVn type stars. Besides, both spot and flare activities indicate that the
system has high level chromospherical activity.
## Acknowledgments
The authors acknowledge generous allotments of observing time at the Ege
University Observatory. We thank Dr. Dirk Terrell for reviewing the manuscript
and his useful comments that have contributed to the improvement of the paper.
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Figure 1: The V band light curve with both B-V and V-R colour curves of DK
CVn. a) The observations of the season 2007. b) The observations of the season
2008. Figure 2: The V band light curves of the observing season 1999. The data
were taken from the NSVS database (Woźniak et al., 2004). Figure 3: DK CVn’s
$O-C$ diagram (The dashed line represents the linear fit). Figure 4: The
variation of the sum of weighted squared residuals versus mass ratio in the ”q
search”. Figure 5: DK CVn’s BVR band light curves (filled circles) observed in
2008 and the synthetic curves (lines) derived from the light curve solution.
Figure 6: The geometric configurations at the (a) phase 0.25 and (b) 0.40,
illustrated for DK CVn. Figure 7: The places of the components of DK CVn among
UV Ceti type stars in the Mass-Radius distribution. In the figure, the filled
circles represent the active stars listed in the catalogue of Gershberg et
al., (1999). The asterisk represents the secondary component, while the open
triangle represents the primary component of DK CVn. The line represents the
ZAMS theoretical model developed by Siess et al., (2000). Figure 8: All the
pre-whitened light curves in R band. (All the light curves are shown as double
cycle for better visibility of light variations). Figure 9: The variation of
the amplitude of the variation at out-of-eclipses throughout the years. The
dashed line represents the fits of the variations in the figure. Figure 10:
The minimum phases for the variation at out-of-eclipses throughout the years.
Figure 11: The fast flare sample detected around primary minimum on April 18,
2007. In figure, open circles represent the observations on April 18, 2007,
while filled circles represent the observations on March 8, 2007. The line
represents the synthetic light curve. Figure 12: Two fast flare samples
detected on March 16, 2008. In figure, open circles represent the observations
on March 16, 2008, while filled circles represent the observations on March 2,
2008. The line represents the synthetic light curve. Figure 13: The fast flare
sample detected by Terrell et al., (2005) in B band. Figure 14: The
distribution of the flare equivalent durations versus the flare total
durations for DK CVn flares (filled circles). The OPEA model (dashed line)
derived for this distribution. Figure 15: The variations of (a) the $Plateau$
values derived from the OPEA models and (b) the mean equivalent durations
computed by t-Test analyses versus B-V colour index. In the figures, the
filled circles represent the DK CVn, while open circles represent five UV Ceti
type stars taken from Dal & Evren, (2011). The dashed lines are just used to
show the variation trend. Figure 16: The variations of (a) the maximum flare
rise times and (b) the $Half-Life$ values versus B-V colour index. In the
figures, all the symbols are the same with Figure 15.
Table 1: Basic parameters for the observed stars. V band brightness and B-V indexes were obtained in this study. Considering B-V indexes, the spectral types were taken from Tokunaga, (2000). Star | Alpha / Delta (J2000) | V (mag) | B-V (mag) | Spectral Type
---|---|---|---|---
DK CVn | 12h 33m 09s.34 / +37∘ 58′ 20′′.28 | 12.967 | 0.890 | K2
GSC 3018 2499 | 12h 33m 11s.14 / +37∘ 45′ 12′′.90 | 12.004 | 0.527 | F8
GSC 3018 2425 | 12h 32m 58s.36 / +37∘ 54′ 20′′.30 | 12.702 | 0.628 | G2
Table 2: The minima times and $O-C$ residuals.
HJD (+24 00000) | E | $(O-C)_{I}$ | $(O-C)_{II}$ | Filter | Ref.
---|---|---|---|---|---
52001.4487 | -2872.0 | -0.0019 | 0.0010 | CCD | 1
52361.7820 | -2144.0 | -0.0017 | 0.0006 | R | 2
52363.7604 | -2140.0 | -0.0031 | -0.0008 | I | 2
52363.7618 | -2140.0 | -0.0017 | 0.0006 | R | 2
52363.7623 | -2140.0 | -0.0012 | 0.0011 | V | 2
52408.8049 | -2049.0 | -0.0003 | 0.0020 | U | 2
52693.9027 | -1473.0 | -0.0012 | 0.0006 | CCD | 2
52712.7104 | -1435.0 | -0.0021 | -0.0003 | R | 2
52713.7008 | -1433.0 | -0.0016 | 0.0002 | R | 2
53082.4493 | -688.0 | -0.0005 | 0.0006 | CCD | 2
53083.9362 | -685.0 | 0.0015 | 0.0026 | R | 2
53085.9135 | -681.0 | -0.0011 | 0.0000 | V | 2
53094.8234 | -663.0 | -0.0005 | 0.0006 | V | 2
53108.6837 | -635.0 | 0.0008 | 0.0019 | I | 2
53109.6731 | -633.0 | 0.0003 | 0.0014 | B | 2
53383.8786 | -79.0 | -0.0037 | -0.0031 | V | 2
53383.8788 | -79.0 | -0.0035 | -0.0029 | V | 2
53383.8798 | -79.0 | -0.0025 | -0.0019 | R | 2
53385.8600 | -75.0 | -0.0022 | -0.0015 | V | 2
53388.8299 | -69.0 | -0.0020 | -0.0014 | R | 2
53389.8191 | -67.0 | -0.0028 | -0.0022 | R | 2
53390.8090 | -65.0 | -0.0028 | -0.0022 | R | 2
53420.0139 | -6.0 | -0.0007 | -0.0001 | B | 2
53421.9926 | -2.0 | -0.0019 | -0.0013 | B | 2
53422.9834 | 0.0 | -0.0010 | -0.0004 | B | 2
53426.9420 | 8.0 | -0.0021 | -0.0015 | B | 2
53427.9322 | 10.0 | -0.0018 | -0.0013 | B | 2
53430.9040 | 16.0 | 0.0002 | 0.0007 | V | 2
53430.9043 | 16.0 | 0.0005 | 0.0010 | B | 2
53432.8825 | 20.0 | -0.0012 | -0.0006 | B | 2
53432.8840 | 20.0 | 0.0003 | 0.0009 | V | 2
53433.8726 | 22.0 | -0.0010 | -0.0004 | B | 2
53443.7712 | 42.0 | -0.0016 | -0.0011 | V | 2
53445.7507 | 46.0 | -0.0020 | -0.0015 | B | 2
53448.7210 | 52.0 | -0.0015 | -0.0010 | B | 2
53451.6915 | 58.0 | -0.0007 | -0.0002 | V | 2
53456.6409 | 68.0 | -0.0010 | -0.0005 | R | 2
53457.6302 | 70.0 | -0.0016 | -0.0011 | B | 2
53478.9140 | 113.0 | -0.0012 | -0.0007 | R | 2
53479.9051 | 115.0 | 0.0000 | 0.0004 | B | 2
53496.7330 | 149.0 | -0.0009 | -0.0004 | R | 2
54168.4007 | 1506.0 | 0.0020 | 0.0013 | BVR | 3
54217.4049 | 1605.0 | 0.0049 | 0.0041 | VR | 3
54220.3719 | 1611.0 | 0.0021 | 0.0013 | R | 4
54220.3725 | 1611.0 | 0.0027 | 0.0019 | VR | 3
54531.4535 | 2239.5 | -0.0006 | -0.0019 | V | 5
54531.4553 | 2239.5 | 0.0012 | -0.0001 | I | 5
54531.4557 | 2239.5 | 0.0016 | 0.0003 | R | 5
54564.3733 | 2306.0 | 0.0042 | 0.0029 | BVR | 3
54619.3154 | 2417.0 | 0.0054 | 0.0040 | VR | 3
54929.4066 | 3043.5 | 0.0023 | 0.0003 | R | 4
54936.3379 | 3057.5 | 0.0041 | 0.0021 | CCD | 4
55691.4004 | 4583.0 | 0.0005 | -0.0028 | VR | 6
55697.3413 | 4595.0 | 0.0018 | -0.0014 | RI | 6
1 Brát et al., (2007)
2 Terrell et al., (2005)
3 Sipahi et al., (2009)
4 Brát et al., (2009)
5 Brát et al., (2008)
6 This Study
Table 3: The parameters obtained from the light curve analysis. Parameter | Value
---|---
$q$ | 0.55 (Fixed)
$i$ (∘) | 71.12$\pm$0.04
$T_{1}$ (K) | 4040
$T_{2}$ (K) | 3123$\pm$16
$\Omega_{1}$ | 4.542$\pm$0.003
$\Omega_{2}$ | 3.424$\pm$0.001
L1/LT $(B)$ | 0.848$\pm$0.004
L1/LT $(V)$ | 0.909$\pm$0.002
L1/LT $(R)$ | 0.883$\pm$0.003
$g_{1}$ | 0.32 (Fixed)
$g_{2}$ | 0.32 (Fixed)
$A_{1}$ | 0.50 (Fixed)
$A_{2}$ | 0.50 (Fixed)
$x_{1,bol}$ | 0.563 (Fixed)
$x_{1,B}$ | 0.826 (Fixed)
$x_{1,V}$ | 0.799 (Fixed)
$x_{1,R}$ | 0.747 (Fixed)
$x_{2,bol}$ | 0.468 (Fixed)
$x_{2,B}$ | 0.868 (Fixed)
$x_{2,V}$ | 0.839 (Fixed)
$x_{2,R}$ | 0.748 (Fixed)
$<r_{1}>$ | 0.254$\pm$0.005
$<r_{2}>$ | 0.258$\pm$0.004
$Co-Lat_{Spot~{}I}$ (∘) | 110 (Fixed)
$Long_{Spot~{}I}$ (∘) | 188 (Fixed)
$R_{Spot~{}I}$ (∘) | 26 (Fixed)
$T_{eff,Spot~{}I}$ | 0.95 (Fixed)
$Co-Lat_{Spot~{}II}$ (∘) | 50 (Fixed)
$Long_{Spot~{}II}$ (∘) | 290 (Fixed)
$R_{Spot~{}II}$ (∘) | 15 (Fixed)
$T_{eff,~{}Spot~{}II}$ | 0.90 (Fixed)
Table 4: The minimum phases and the amplitudes of the sinusoidal-like
variations seen in the pre-whitened light curves in R band.
Year | $\theta_{min}$ | Amplitude in R (mag) | Ref
---|---|---|---
2002 | 0.80 | 0.139 | 1
2003 | 0.90 | 0.125 | 1
2004 | 1.00 | 0.079 | 1
2005 | 0.80 | 0.099 | 1
2007 | 0.20 | 0.115 | 2
2008 | 0.18 | 0.062 | 2
1 Terrell et al., (2005)
2 This Study
Table 5: The flare parameters obtained from the flares detected in this study
and the flare detected by Terrell et al., (2005).
Year | Filter | HJD of Maxima | Rise | Decay | Total | Equivalent | Amplitude | Ref.
---|---|---|---|---|---|---|---|---
| | (+24 00000) | Time (s) | Time (s) | Duration (s) | Duration (s) | (mag) |
2005 | B | 53394.90620 | 568 | 4315 | 4884 | 929.1551 | 0.668 | 1
2007 | B | 54209.48337 | 363 | 1476 | 1839 | 429.6051 | 0.516 | 2
2008 | B | 54542.31363 | 626 | 940 | 1566 | 595.2623 | 0.901 | 2
2008 | B | 54542.33176 | 313 | 626 | 939 | 264.5768 | 0.488 | 2
2007 | V | 54209.48084 | 363 | 1476 | 1839 | 214.9586 | 0.244 | 2
2008 | V | 54542.31137 | 313 | 627 | 940 | 68.9973 | 0.180 | 2
2007 | R | 54209.48168 | 363 | 1476 | 1839 | 189.9065 | 0.220 | 2
1 Terrell et al., (2005)
2 This study.
Table 6: The parameters computed from the OPEA model of DK CVn’s flares. Parametre | Value
---|---
Max. Rise Time (s) | 626
Max. Tot. Time (s) | 4884
$Plateau$ (s) | 2.965 $\pm$ 0.07374
$y_{0}$ (s) | 1.167 $\pm$ 0.1555
$K$ (s) | 0.001289 $\pm$ 0.000495
$Span$ (s) | 1.798 $\pm$ 0.1243
$Half-Life$ (s) | 537.7
Mean (s) | 2.731
|
arxiv-papers
| 2012-06-24T19:45:12 |
2024-09-04T02:49:32.183131
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, E. Sipahi, O. \\\"Ozdarcan",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.5535"
}
|
1206.5536
|
* # High-Level Magnetic Activity on Low Mass Close Binary: GSC 2038-0293
DAL, H.A. 1,2, SİPAHİ, E. 1, ÖZDARCAN, O. 1 1 Department of Astronomy and
Space Sciences, University of Ege, Bornova, 35100 İzmir, Turkey
2 Corresponding Author, Email: ali.dal@ege.edu.tr
Abstract: Taking into account results obtained from light curve analysis and
from analyses of out-of-eclipses, we discussed the nature of GSC 02038-00293
and also its magnetic activity behaviour. We obtained the light curves of the
system in observing seasons 2007, 2008 and 2011. We obtained its secondary
minimum clearly in I band observations in 2008 for the first time. Analysing
this light curve, we found physical parameters of the components. The light
curve analysis indicates that possible mass ratio of the system is 0.35. We
obtained the remained V band light curves, extracting the eclipses. We
modelled these remained curves by using the SPOTMODEL program, and we found
possible spot configurations of magnetically active component for each
observing season. The models demonstrated that there are two active longitudes
on the active component. The models reveal that both active longitudes migrate
toward the decreasing longitudes. We also examined the light curves at out-of-
eclipses in respect to its minimum and maximum brightness and the amplitude,
and etc. The amplitude of the curves at out-of-eclipses is varying in a
sinusoidal way with a period of $\sim$8.9 years; the mean brightness of the
system is dramatically decreasing. The phases of the deeper minimum at out-of-
eclipses exhibit a migration toward the decreasing phases.
Keywords: (stars:) binaries: eclipsing, stars: low-mass, stars: activity,
(stars:) starspots, stars: individual: GSC 2038-0293.
## 1 Introduction
Many stars such as BY Dra and RS CVn type stars on the main sequence toward
the late spectral types exhibit stellar spot activity called as magnetic
activity. In the literature, the BY Dra variables were found by Kron, (1952)
for the first time, who demonstrated some sinusoidal-like variations at out-
of-eclipses of the eclipsing binary star YY Gem. He explained the variation at
out-of-eclipses as a heterogeneous temperature on star surface. Then, it was
called BY Dra stars by Kunkel, (1975). Based on some rigorous arguments
obtained from observations, the variability seen in BY Dra type stars was
confirmed in terms of dark regions on the surface of the stars by later works
such as Torres & Ferraz Mello, (1973); Bopp & Evans, (1973); Vogt, (1975);
Friedemann & Gürtler, (1975); Bopp & Fekel, (1977). RS Cvn type stars
exhibit the same variability with BY Dra stars. However, one of the components
of RS CVn stars is evolved and it is generally a giant or subgiant star, while
the other component is a main sequence star (Thomas & Weiss, 2008). The
incidence of late type stars in our Galaxy is about 65 $\%$. Seventy-five
percent of them show magnetic activity such as the spot and flare activities
(Rodonó, 1986). The researches such as López-Morales, (2007); Morales et
al., (2008, 2010) demonstrate that magnetic activity dramatically affects the
stellar structure of the late type stars and also their evolutions. Morales et
al., (2008); Casagrande et al., (2008); Fernandez et al., (2009); Morales
et al., (2010); Torres et al., (2010); Kraus et al., (2011) have revealed
that the radii found from the analyses of the observations are generally
larger than the radii theoretically expected for several magnetically active
low mass binaries, while the effective temperatures found from the
observations are usually lower than those theoretically expected. They claimed
that the reason is most likely magnetic activity. In this respect,
magnetically active low mass components of the binaries take very important
places to understand their evolution.
In this study, we introduce GSC 02038-00293 as a new candidate for the
magnetically active low mass stars. Using the ROTSE 1 database (Woźniak et
al., 2004), the system was discovered by Bernhard & Frank, (2006) in the
optical identification program of X-ray sources listed in the ROSAT All-Sky
Survey Bright Source Catalogue (Voges et al., 1999). The identification
reveals that the system is actually the uncatalogued variable NSVS object ID
7869362 as the optic counterpart of the X-ray source 1RXS J160248.3+252031.
Combining their 2005 and 2006 data with the available data taken from the
ROTSE 1 and ASAS 3 databases, Bernhard & Frank, (2006) determined light
elements as follows: $JD(Hel.)_{MinI}$ = 53560.491(3)+$0^{d}.49541(1)$ $\times
E$. Norton et al., (2007) gave the same period analysing the Super-WASP
observations of the system. In addition, Frank & Bernhard, (2007) also
confirmed the period with 2007 observations. Bernhard & Frank, (2006)
indicted a 6 - 8 year long activity cycle. However, consecutive observations
demonstrated that the light curve shapes can change even in one week. Finally,
Bernhard & Frank, (2006) identified the system as a RS CVn type binary. Using
low resolution spectra, Dragomir et al. (2007) indicated that the spectral
type of the system likely to be a K type. A detailed spectral study was done
by Korhonen et al., (2010). Considering some features seen in the low
resolution spectra of the system such as neutral metals ($Mg$, $Na$), weak
Balmer lines and also absence of molecular bands, they confirmed that the
system is from K spectral types. Adopting $log~{}g=4.5$, the $T_{eff}$ was
found to be $4750\pm 250$ $K$ and the $v\sin i$ was found to be $90\pm 10$
$kms^{-1}$. However, using their the v sin i measured together with the
rotation period of 0.495410 days found by Bernhard & Frank, (2006), Korhonen
et al., (2010) estimated the system’s radius as $R\times\sin i=0.88\pm 0.10$
$R_{\odot}$. The value of $R\times\sin i$ indicated a late-G spectral type or
later.
The observations in the literature demonstrated that GSC 02038-00293 exhibits
magnetic activity. In this study we analysed the light curves of the system,
and also we examined the variations at out-of-eclipses. We finally compared
the system with its analogue in respect to the theoretical models.
## 2 Observations and Data
Observations of GSC 02038-00293 were carried on with two telescopes in BVRI
bands at Ege University Observatory. The first part of observations was
acquired with a High-Speed Three Channel Photometer attached to the 48 cm
Cassegrain type telescope in observing seasons 2007 and 2008. The second part
of observations was acquired with a thermoelectrically cooled ALTA U+42
2048$\times$2048 pixel CCD camera attached to a 40 cm - Schmidt - Cassegrains
- type MEADE telescope in the observing seasons 2008 and 2011. The comparison
and check stars used in all observations are the same stars used in the
literature. Some basic parameters of program stars are listed in Table 1. The
names of the stars are listed in first column, while J2000 coordinates are
listed in second column. The V magnitudes are in third column, and B-V colours
are listed in the last column.
Although the program and comparison stars are very close on the sky,
differential atmospheric extinction corrections were applied. The atmospheric
extinction coefficients were obtained from observations of the comparison
stars on each night. Moreover, the comparison stars were observed with the
standard stars in their vicinity and the reduced differential magnitudes, in
the sense variable minus comparison, were transformed to the standard system
using procedures outlined by Hardie, (1962). The standard stars are listed in
catalogues of Landolt, (1983, 1992). Furthermore, the dereddened colours of
the system were computed. Heliocentric corrections were also applied to the
times of the observations.
In BVR bands, the first part of observations was continued for 6 nights
between April 22 and July 19 in 2007, and it was carried on for 4 nights
between April 7 and July 23 in 2008 with 48 cm Cassegrain type telescope. In
addition, the second part of observations was continued for 9 nights between
May 20, 2008 and August 20, 2008 in BVRI bands, and it was carried on for 2
nights on April 23 and May 9 in 2011 with 40 cm - Schmidt - Cassegrains - type
MEADE telescope. The mean averages of the standard deviations were found to be
$0^{m}.009$, $0^{m}.007$ and $0^{m}.007$ from the observations of 48 cm
Cassegrain type telescope for BVR bands, respectively. They were found to be
$0^{m}.023$, $0^{m}.011$, $0^{m}.010$ and $0^{m}.013$ for observations
acquired with 40 cm - Schmidt - Cassegrains - type MEADE telescope in the BVRI
bands, respectively. To compute the standard deviations of observations, we
used the standard deviations of the reduced differential magnitudes in the
sense comparison minus check stars for each night. There was no variation
observed in the standard brightness of comparison stars.
The minima times obtained in this study are listed in Table 2. In the table,
the first 9 minimum times have already been published by Sipahi et al.,
(2009), while the last one is unpublished. Using all available minima times in
the literature, we adjusted the light elements of the system, as follows:
$JD~{}(Hel.)~{}=~{}24~{}53560.4925(9)~{}+~{}0^{d}.4954115(5)~{}\times~{}E.$
(1)
Using the light elements given by Equation (1), we phased all our observations
and also all data taken from the literature. In Figure 1, the light and colour
curves of the system are shown for three observing seasons 2007, 2008 and
2011. As it is seen from the figures, there is a remarkable variation in the
shape of the light curves from a season to next one. As it is well known from
the literature, the system exhibits magnetic activity. Figure 1 demonstrates
that available magnetic activity causes dramatic distortion on the light curve
shape. Although the secondary minimum can show itself in observing season
2008, it can not clearly reveal itself in general. In addition, the B-V colour
curves exhibits some variations, while there is no clear variation over the
standard deviation in the V-R colour curves. However, we have a chance to
observe the system in I band in the program of the 40 cm - Schmidt -
Cassegrains - type MEADE telescope for both seasons 2008 and 2011. I band
observations show the secondary minimum better than all other bands (see the
Section 3). The secondary minimum exhibits itself better toward long
wavelengths.
In order to analyse, we collected all available data from the literature. For
this aim, we got the ROTSE 1’s V band data from the Northern Sky Variability
Survey (hereafter NSVS) database (Woźniak et al., 2004), and also we got the
available data in the ASAS Database (Pojmánski, 1997). In addition, we took
the observations published by Bernhard & Frank, (2006) and Frank & Bernhard,
(2007). The standard V band data of ROTSE 1 cover the observing seasons of
1999 and 2000, while the standard V band data of ASAS cover the seasons from
2003 to 2006. The observations in this study were started in 2007. Although
the data taken from Bernhard & Frank, (2006) and Frank & Bernhard, (2007)
were not standard, using the comparison and check stars given by them, we
transformed to the standard system. After transforming their data, we compared
all the available data whether the data taken from different sources are
suitable to use together. For this aim, we compared the ASAS data of 2005 and
2006 with the data taken from Bernhard & Frank, (2006), and also we compared
the data taken from Frank & Bernhard, (2007) with our 2007 data. As seen from
the comparisons, the levels of the data taken from different sources are
statistically the same in 3$\sigma$ value. The light curves obtained from the
available data are shown in Figure 2. As seen from the figure, the light
variation shapes of seasons 1999, 2000, and 2005 are similar to the variation
observed in season 2011. The light curves obtained in 2011 are similar to the
light curves obtained previous studies. Fortunately, I band observations of
2008 gave a chance to us; we can do the light curve analyses. In the case of
lack of the secondary minimum, the light curve analyse does not give reliable
results.
## 3 Light Curve Analysis
The light curve analysis of such a magnetically active star is generally quite
difficult due to the absence of the secondary minima. In fact, no secondary
minimum is seen in our observations of the season 2007 and also generally all
the other observations published in the literature. However, I band
observations in 2008 clearly show the secondary minimum. This gave us a chance
for the light curve analysis. This is why we analysed only I light curve
obtained in 2008 with using the PHOEBE V.0.31a software (Prša & Zwitter,
2005), whose method depends on the method used in the version 2003 of the
Wilson-Devinney Code (Wilson & Devinney, 1971; Wilson, 1990). We tried to
analyse I band curve with three different modes, such as the ”detached
system”, ”semi-detached system with the primary component filling its Roche-
Lobe” and ”semi-detached system with the secondary component filling its
Roche-Lobe” modes. The initial analyses demonstrated that an astrophysical
acceptable result can be obtained if the analysis is carried out in the
”detached system” mode. The initial experience revealed that no acceptable
results in the astrophysical sense could be obtained in all the others modes.
Using low resolution spectra, Korhonen et al., (2010) found that the system
is from K spectral types, and the $T_{eff}$ was found to be $4750\pm 250$ $K$.
Considering the case, the temperature of the primary component was fixed to
4750 K, and the temperature of the secondary was taken as a free parameter in
the analyses. Considering the spectral type, the albedos ($A_{1}$ and $A_{2}$)
and the gravity darkening coefficients ($g_{1}$ and $g_{2}$) of the components
were adopted for the stars with the convective envelopes (Lucy, 1967;
Rucinski, 1969). The non-linear limb-darkening coefficients ($x_{1}$ and
$x_{2}$) of the components were taken from van Hamme, (1993). In the
analyses, their dimensionless potentials ($\Omega_{1}$ and $\Omega_{2}$), the
fractional luminosity ($L_{1}$) of the primary component and the inclination
($i$) of the system were taken as the adjustable free parameters.
There is no obtained spectroscopic mass ratio for the system. Because of this,
we used the ”q-search” method with using a step of 0.05 to find the best
photometric mass ratio of the components. The general result of the q-search
is shown in Figure 3. As seen from the figure, the minimum sum of weighted
squared residuals ($\Sigma res^{2}$) is found for the mass ratio value of
$q=0.35$. According to this result, we assume that a possible mass ratio of
the system is $q=0.35$.
As it is clearly seen from Figures 1, 2, there is a dramatic asymmetry in the
light curves due to the magnetic activity. In the light curve analysis, we
assumed that the primary component has two cool spots on its surface to remove
this asymmetry. According to the results obtained from the first iterations,
the secondary component seems to be a M dwarf, which is close to the full-
convective boundary (Browning, 2011). The full-convective M dwarfs exhibit
very strong flare activity, while a few of them just exhibit spot activity
(Dal & Evren, 2011). In addition, if the secondary component is the spotted
one, to remove the asymmetries seen in the light curves, the analysis
demonstrated that the spots should be as large as that cover all the surface
of the star or their effective temperatures must be half of the surface
temperature due to the secondary component’s light rate in the total light of
the system. Although there are a few stars, which are close to the full-
convective boundary and have some large spots on their surface. But, this is
not a common phenomena. However, the K dwarfs are generally potential stars,
which are possible to exhibit spot activity. This is why we assumed the
spotted star is the primary component. Moreover, the light curve analyses with
this assumption gave more acceptable results in the astrophysical sense. On
the other hand, it must be noted here that it is well known that spot solution
suffers from non-uniqueness. The synthetic light curve obtained from the light
curve solution is seen in Figure 4, and the resulting parameters of the
analysis are also listed in Table 3.
Although there is not any available radial velocity curve, we tried to
estimate the absolute parameters of the components. According to Tokunaga,
(2000), the mass of the primary component must be 0.73$\pm$0.05 $M_{\odot}$
corresponding to its surface temperature. Considering possible mass ratio of
the system, the mass of the secondary component was found to be 0.25$\pm$0.04
$M_{\odot}$.
Using Kepler’s third law, we calculated possible the semi-major axis as a 2.62
$R_{\odot}$. Considering this estimated semi-major axis, the radius of the
primary component was computed as 0.87$\pm$0.05 $R_{\odot}$, while it was
computed as 0.27$\pm$0.04 $R_{\odot}$ for the secondary component. As it is
seen from these results, the primary component’s radius derived with the
assumption of spotted primary component is in agreement with the value of
$R\times\sin i$ found by Korhonen et al., (2010). Using the estimated radii
and the obtained temperatures of the components, the luminosity of the primary
component was estimated to be 0.35 $L_{\odot}$, and it was found as 0.01
$L_{\odot}$ for the secondary component. The absolute parameters are generally
acceptable in the astrophysical sense. However, the radius of the primary
component is larger than the expected values in respect to the theoretical
models. We plotted the distribution of the radii versus the masses for some
stars in Figure 5. The filled circles in the figure represent well-known
active stars listed in the catalogue of Gershberg et al., (1999). Some of
these stars exhibit the spot activity, while some of them exhibit the flare
activity. Some stars exhibit both spot and flare activities. In the figure,
the asterisk represents the secondary component, while the open triangle
represents the primary component. The line represents the ZAMS theoretical
model developed for the stars with $Z=0.02$ by Siess et al., (2000).
## 4 Variations at Out-of-Eclipses
The light curve analysis demonstrated that the light variation of the system
is caused by eclipses combined with the effect of magnetic activity existing
on the primary component. Considering the contact times derived from the
theoretical synthetic light curve, we removed the eclipses from all the V band
light curves, and we obtained the remaining curve for each season. Then, in
order to reveal the magnetic activity behaviour along the years, we
investigated the data at out-of-eclipses firstly for the short-term and
secondly for the long-term variations.
To reveal the spot configuration on the primary component’s surface
(especially the longitudinal distributions of the spotted areas) in each
short-term interval, we modelled the remaining curves under some assumptions
with using the SPOTMODEL program (Ribárik et al., 2003; Ribárik, 2002).
Although the data obtained in this study contain multi-band observations, the
available data in the literature do not. All the data taken from both the NSVS
and ASAS databases and also from both Bernhard & Frank, (2006) and Frank &
Bernhard, (2007) contain just the V band observations. However, modelling
progress in the SPOTMODEL program requires two band observations or any spot
temperature factor at least. Although the large part of the using data is
mono-chromatic, it is likely that we can get a temperature factors for the
spotted areas from light curve analysis of I band observations in the season
2008. According to surface temperature of the primary component, the derived
temperature factors are in agreement with the temperature factors found from
other analogue stars (Thomas & Weiss, 2008). Considering the spot temperature
factor derived from the light curve analysis, we assumed the temperature
factors of the spotted areas are 0.80 in the SPOTMODEL program, and we took it
as constant parameters for the models in order to just determine surface
distributions of the spotted areas for each short-term interval. The
longitudes, latitudes and radii of the spots were taken as adjustable
parameters in the program for the model of each season.
The derived models are shown in Figure 6. The models of the data taken from
the NSVS database are shown in panels a and b of Figure 6, while the ASAS data
are in panels c, d. For the seasons 2005 and 2006, there are some data in both
the ASAS database and Bernhard & Frank, (2006) and Frank & Bernhard, (2007).
These data were combined for each year. The models of these years are shown in
panels e and f of Figure 6. For the season 2007, some data were obtained in
this study, and also some data were taken from Frank & Bernhard, (2007). The
combined data were modelled, and the result is shown in panel g of Figure 6.
In panels h and i, the models of light curves obtained in the seasons 2008 and
2011. The parameters of the models are listed in Table 4. In the table, the
data set, mean times of observations, mean years of observations are listed in
first three columns, while the spot parameters such as longitude ($I$),
latitude ($b$) and radius ($g$) are listed in the following columns for two
spots, respectively.
As expected, the spot distribution on the surface is rapidly changing. The
distributions are varying from a year to next one. As seen from Figure 6,
there is always a spotted area on the surface of the primary star, while there
are two spotted areas on its surface. Figure 7 demonstrates that the spotted
areas are always separated from each other. Although there are usually about
180∘ longitudinal differences between them, they are some times getting closer
to each other. As seen from Figure 7, the longitudes of the one of the spotted
areas exhibit a quasi-sinusoidal variation along the years, while both of them
migrate toward the decreasing longitudes. Using GraphPad Prism V5.02 software
(Motulsky, 2007), the quasi-sinusoidal variation was fitted by a polynomial
function. To test whether the polynomial fit is statistically acceptable, we
computed the probability value (hereafter p-value). The value of $\alpha$ was
taken as 0.005 for the p-value, which allowed us to test whether the p-value
are statistically acceptable or not (Dawson et al., 2004). The p-value was
found to be p-value $<$ 0.00167. Considering the $\alpha$ value, this means
that the result is statistically acceptable.
Apart from the short-term variations, considering the remained V band light
curves without any eclipses, we examined the variations of the mean
brightness, the amplitude and the deeper minimum phases of light curves
($\theta_{min}$). All of them are listed in Table 5. The variations of three
parameters are shown versus the years in Figure 8. The variation of the
amplitudes for the remaining curves is shown in upper panel (a). The mean
brightness variation is shown in middle panel (b). As it is seen from the
figures, the amplitude is varying in a sinusoidal way with a period of
$\sim$8.9 years. Using GraphPad Prism V5.02 software, the variation was fitted
by a polynomial fit. According to the statistical analysis, just two point,
which are shown by open circles in the figure, diverged from the general
trend. Apart from these point, the polynomial function is fitted the general
trend with the correlation coefficient of 0.91. Moreover, The p-value was
found to be p-value $<$ 0.00093. Considering the $\alpha$ value, p-value the
result is statistically acceptable. However, the mean brightness is
dramatically decreasing through the years from 1999 to 2011. In these years,
it was decreased from $10^{m}.40$ to $10^{m}.60$. Both figures indicate that
the primary star has high level the magnetic activity. The distribution of the
deeper minimum phases of the remaining curves is shown in the bottom panel of
Figure 8.
As seen from Figure 8, the deeper minimum was located in the phase interval
between $0^{P}.35$ and $0^{P}.55$ from 1999 to 2005. However, it was suddenly
shifted to the phase interval between $0^{P}.95$ and $0^{P}.05$ from 2006 to
2008. It was again seen in the phases between $0^{P}.35$ and $0^{P}.355$ in
the season 2011. The phenomenon is generally ”flip-flop” in the literature
(Berdyugina, 2006; Oláh et al., 2006; Korhonen & Elstner, 2005; Korhonen &
Järvinen, 2007). Considering the minima seen in the phase interval between
$0^{P}.35$ and $0^{P}.55$ reveal that the minima slowly migrated toward the
decreasing phases.
## 5 Results and Discussion
In this study, we tried to reveal the nature of close binary system GSC
02038-00293. The light curve analysis indicated that the mass ratio of the
system is 0.35. We estimated that the mean radii are 0.87 $R_{\odot}$ for the
primary and 0.27 $R_{\odot}$ for the secondary component. Korhonen et al.,
(2010) had indicated that the effective temperature is 4750 $K$. They found to
be $R\times\sin i=0.88$ $R_{\odot}$. In this study, the inclination ($i$) of
the system was found to be 77.91∘. In this case, the primary component’s
radius estimated in this study is in agreement with one given by Korhonen et
al., (2010). On the other hand, the radius of the primary star is actually
larger than an expected value. As it is seen from Figure 5, the secondary
component is located almost on the ZAMS. This indicates that this component
should be very young. However, it seems that the primary component has
departed from the ZAMS. It is more likely that, considering their masses, the
primary component should be evolved more rapidly than the secondary component.
On the other hand, as it was discussed by López-Morales, (2007); Morales et
al., (2008); Casagrande et al., (2008); Fernandez et al., (2009); Morales
et al., (2010); Torres et al., (2010) and Kraus et al., (2011), the case of
larger radius is a common phenomena for the active stars. López-Morales,
(2007) demonstrated that the case is very common especially for the
magnetically active stars in the mass range from 0.35 $M_{\odot}$ to 0.70
$M_{\odot}$. The mass of the primary star of the system is close to this
interval. Consequently, it is more possible that its radius was found larger
due to the magnetic activity. The radius of the secondary component is in
agreement with the expected value for a star with 0.25 $M_{\odot}$.
The light curve analysis of I band observations of the season 2008 and the
spot models of the light curves at out-of-eclipses confirmed that there are
two spotted areas on this component. The models and the examining of
variations at out-of-eclipses generally reveal some properties for the primary
component’s magnetic activity behaviour. (1) There are two active longitudes
on the primary component. (2) One of them is usually active, while second one
can be sometime less active. (3) Although there are two stable active
longitudes on the star, the locations of the spotted areas on this star can
rapidly change. Both active longitudes migrate toward the decreasing
longitude. In addition, one of them exhibits a quasi-sinusoidal variation
during the migration. Here, it should be noted that we assumed that the
temperature factors of the spots are stable and constant along the years.
Because of this assumption, the radii and latitudes of the spotted areas
obtained from the spot models can clearly change with taking another
temperature factor for the spots. However, the longitudes will not change.
Although the temperature factor, radius and latitude depend on each other, but
the longitude does not depend directly on them.
Examinations of the long-term amplitude variation of the light curves at out-
of-eclipses indicate that the amplitude is varying in a sinusoidal way, while
the mean brightness of the system is dramatically decreasing. In this case, it
is possible that the spots should cover more part of the surface of the active
component, while the spotted areas sometimes gather toward an active
longitude. On the other hand, it is also possible that there can be another
scenario. The decreasing of the mean brightness can be due to the any spotted
polar-cap areas. The increasing total area covered by spots on the polar can
cause the same effect on the mean brightness. However, the phases of the
deeper minima at out-of-eclipses migrate toward the decreasing phases. The
other point is that the second active longitude is more active than the first
one between the seasons 2006 and 2008. This is a small clue for the ”flip-
flop” behaviour, which is general property seen in many active stars
(Berdyugina, 2006; Oláh et al., 2006; Korhonen & Elstner, 2005; Korhonen &
Järvinen, 2007).
In this study, the nature of the system is a bit cleared. As a result of the
analyses, GSC 02038-00293 is a close binary, whose primary component exhibits
high level magnetic activity. The long-term photometric observations together
with the spectral study will reveal the nature in a better view, as well.
## Acknowledgments
The authors acknowledge generous allotments of observing time at the Ege
University Observatory (EUO). We wish to thank Dr. Bernhard, who shared all
his data with us, and braced us with delivering all the ROTSE 1 and ASAS 3
data of GSC 02038-00293. We also thank the referee for useful comments that
have contributed to the improvement of the paper.
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Figure 1: The light and colour curves of GSC 02038-00293 for the observing
seasons 2007 (a), 2008 (b) and 2011 (c). Figure 2: The available observations
of GSC 02038-00293 in the literature. In panels (a) and (b), the light curves
of the ROTSE 1 V band data are shown for the seasons 1999 and 2000. The V band
light curves of 2003 (c), 2004 (d), 2005 (e), 2006 (f) and 2007 (g), whose
data were taken from the ASAS database (Pojmánski, 1997) and from Bernhard &
Frank, (2006) and Frank & Bernhard, (2007). For the seasons 2005 and 2006,
there are some data in both the ASAS database and Bernhard & Frank, (2006)
and Frank & Bernhard, (2007). To easy compare, all the data set are shown
together in panel h. Figure 3: The variation of the sum of weighted squared
residuals versus mass ratio in the ”q search”. Figure 4: The synthetic light
curve obtained from the light curve analysis of I band. Figure 5: The places
of the components of GSC 02038-00293 among well-known active stars in the
Mass-Radius distribution. In the figure, the filled circles represent the
active stars listed in the catalogue of Gershberg et al., (1999). The
asterisk represents the secondary component, while the open triangle
represents the primary component of GSC 02038-00293. The line represents the
ZAMS theoretical model developed by Siess et al., (2000). Figure 6: The
synthetic light curves at out-of-eclipses and 3D surface models. In each panel
on left side, the filled circles represent the observations, while the lines
represent theoretical fit derived by the SPOTMODEL program. The 3D surface
model for two phases (especially the phases the spots seen) is seen just on
right side of the light curve of each model. Figure 7: The variation of the
longitudes of the spotted areas. In the figure, filled circles represent the
Spot I, while the open circles represent Spot II. The dashed lines represent
the linear fit for the Spot I and the polynomial one for the Spot II. Figure
8: The variations of some parameters determined from the remaining curves. a)
The amplitude variation. b) The mean brightness variation. c) The spot minimum
phase distribution. The filled circles represent the determined parameters in
the figure, while the dashed lines represent the polynomial fits in panel a
and b, while the dashed line represents the linear fit for the theta min of
one spotted area. The open circles in panel a represent the points diverged
from the general trend.
Table 1: Basic parameters for the observed stars. V band brightness and B-V index were obtained in this study. Star | RA / DE (J2000) | V | B-V
---|---|---|---
Name | (h m s) / (∘ ′ ′′) | (mag) | (mag)
GSC 02038-00293 | 16 02 48.54 +25 20 38.9 | 10.540 | 0.972
GSC 02038-00867 (Comparison) | 16 05 35.85 +25 16 59.6 | 9.185 | 1.243
GSC 02038-00565 (Check) | 16 02 53.98 +25 10 43.2 | 11.811 | 0.394
GSC 02038-00663 (Check) | 16 03 13.37 +25 12 10.9 | 11.513 | 0.738
Table 2: The minima times of GSC 02038-00293. HJD (+24 00000) | Error | Type | Filter
---|---|---|---
54213.4434 | 0.0014 | I | BVR
54219.3903 | 0.0032 | I | BVR
54226.3264 | 0.0014 | I | BVR
54526.5460 | 0.0008 | I | BVR
54587.4837 | 0.0007 | I | BVR
54621.4215 | 0.0019 | II | I
54651.3956 | 0.0067 | I | BVRI
54659.3202 | 0.0038 | I | BVRI
54671.4488 | 0.0097 | II | BVRI
55675.4011 | 0.0005 | I | BVRI
Table 3: The parameters obtained from I band light curve analysis. Parameter | Value
---|---
$q$ | 0.35
$i$ (∘) | 77.91$\pm$0.87
$T_{1}$ (K) | 4750 (Fixed)
$T_{2}$ (K) | 3515$\pm$61
$\Omega_{1}$ | 3.42$\pm$0.05
$\Omega_{2}$ | 4.72$\pm$0.14
$L_{1}/L_{T}$ (I) | 0.977$\pm$0.067
$g_{1}$, $g_{2}$ | 0.32, 0.32
$A_{1}$, $A_{2}$ | 0.5, 0.5
$x_{1,bol}$, $x_{2,bol}$ | 0.625, 0.625
$x_{1,I}$, $x_{2,I}$ | 0.681, 0.681
$<r_{1}>$ | 0.334$\pm$0.006
$<r_{2}>$ | 0.103$\pm$0.004
$Co-Lat_{Spot~{}I}$ (∘) | 90.00 (fixed)
$Long_{Spot~{}I}$ (∘) | 0.00 (fixed)
$R_{Spot~{}I}$ (∘) | 54.43 (fixed)
$T_{eff,~{}Spot~{}I}$ | 0.97 (fixed)
$Co-Lat_{Spot~{}II}$ (∘) | 90.00 (fixed)
$Long_{Spot~{}II}$ (∘) | 263.56 (fixed)
$R_{Spot~{}II}$ (∘) | 54.43 (fixed)
$T_{eff,~{}Spot~{}II}$ | 0.98 (fixed)
Table 4: The spot parameters derived by the SPOTMODEL program are listed for
each data set. In the table, the subscripts 1 and 2 represent Spot I and Spot
II.
Data | Mean HJD | Mean | $l_{1}$ | $l_{2}$ | $b_{1}$ | $b_{2}$ | $g_{1}$ | $g_{2}$ | Data
---|---|---|---|---|---|---|---|---|---
Set | (+24 50000) | Year | (∘) | (∘) | (∘) | (∘) | (∘) | (∘) | Source
A | 1350.2176 | 1999.46 | 10 | - | 144 | - | 53 | - | 1
B | 1607.0397 | 2000.17 | 39 | - | 358 | - | 43 | - | 1
B | 1607.0397 | 2000.17 | - | 181 | | 358 | - | 43 | 1
C | 2784.3294 | 2003.39 | 354 | - | 357 | - | 38 | - | 2
C | 2784.3294 | 2003.39 | - | 186 | | 351 | - | 37 | 2
D | 3131.0668 | 2004.34 | 333 | - | 168 | - | 31 | - | 2
E | 3553.9587 | 2005.50 | 330 | - | 342 | - | 28 | - | 2, 3
E | 3553.9587 | 2005.50 | - | 178 | | 357 | - | 55 | 2, 3
F | 3881.8078 | 2006.39 | 349 | - | 357 | - | 32 | - | 2, 3
F | 3881.8078 | 2006.39 | - | 175 | | 355 | - | 28 | 2, 3
G | 4255.8110 | 2007.42 | 352 | - | 30 | - | 51 | - | 3, 4
G | 4255.8110 | 2007.42 | - | 127 | | 24 | - | 20 | 3, 4
H | 4624.8493 | 2008.43 | 20 | - | 40 | - | 47 | - | 4
H | 4624.8493 | 2008.43 | - | 166 | | 3 | - | 26 | 4
I | 5683.9083 | 2011.33 | 331 | - | 332 | - | 18 | - | 4
I | 5683.9083 | 2011.33 | - | 129 | | 24 | - | 41 | 4
1 The NSVS Database (Woźniak et al., 2004)
2 The ASAS Database (Pojmánski, 1997)
3 Bernhard & Frank, (2006) and Frank & Bernhard, (2007)
4 This Study
Table 5: Some parameters determined from the remaining V light curves.
Data | Mean HJD | Mean | $\theta_{min}$ | $V_{min}$ | $V_{max}$ | Amplitude | $V_{mean}$ | Data
---|---|---|---|---|---|---|---|---
Set | (+24 50000) | Year | | (mag) | (mag) | (mag) | (mag) | Source
A | 1350.2176 | 1999.46 | 0.530 | 10.519 | 10.353 | 0.166 | 10.436 | 1
B | 1607.0397 | 2000.17 | 0.520 | 10.457 | 10.340 | 0.117 | 10.399 | 1
C | 2784.3294 | 2003.39 | 0.520 | 10.534 | 10.441 | 0.093 | 10.488 | 2
D | 3131.0668 | 2004.34 | 0.435 | 10.571 | 10.496 | 0.075 | 10.534 | 2
E | 3553.9587 | 2005.50 | 0.495 | 10.648 | 10.479 | 0.169 | 10.564 | 2, 3
F | 3881.8078 | 2006.39 | 0.983 | 10.602 | 10.521 | 0.081 | 10.562 | 2, 3
G | 4255.8110 | 2007.42 | 0.980 | 10.676 | 10.536 | 0.140 | 10.606 | 3, 4
H | 4624.8493 | 2008.43 | 1.050 | 10.702 | 10.539 | 0.163 | 10.621 | 4
I | 5683.9083 | 2011.33 | 0.370 | 10.662 | 10.547 | 0.115 | 10.605 | 4
1 The NSVS Database (Woźniak et al., 2004)
2 The ASAS Database (Pojmánski, 1997)
3 Bernhard & Frank, (2006) and Frank & Bernhard, (2007)
4 This Study
|
arxiv-papers
| 2012-06-24T19:54:33 |
2024-09-04T02:49:32.192679
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, E. Sipahi, O. \\\"Ozdarcan",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.5536"
}
|
1206.5791
|
# A New Method To Classify Flares Of UV Ceti Type Stars:
Differences Between Slow And Fast Flares
H. A. Dal and S. Evren Department Of Astronomy and Space Sciences, University
of Ege,
Bornova, 35100 İzmir, Turkey ali.dal@ege.edu.tr
###### Abstract
In this study, a new method is presented to classify flares derived from the
photoelectric photometry of UV Ceti type stars. Using Independent Samples
t-Test, the method is based on statistical analysis. The data used in the
analyses were obtained from four flare stars observed between the years 2004
and 2007. Total number of flares obtained in the observations of AD Leo
(catalog ), EV Lac (catalog ), EQ Peg (catalog ) and V1054 Oph is 321 in the
standard Johnson U band. As a result, flare can be separated into two types as
slow and fast depending on the ratio of flare decay time to flare rise time.
The ratio is below the value 3.5 for all slow flares, while it is above 3.5
for all fast flares. Also, according to the Independent Samples t-Test, there
are about 157 seconds difference between equivalent durations of slow and fast
flares. In addition, there are significant differences between amplitudes and
rise times of slow and fast flares.
methods: data analysis — methods: statistical — stars: flare — stars:
individual(AD Leo, EV Lac, EQ Peg, V1054 Oph)
††slugcomment: Not to appear in Nonlearned J., 45.
## 1 Introduction
Flares and flare processes are very hard worked subtitles of astrophysics.
Lots of studies on flares have been carried out since the first flare was
detected on the Sun by R.C. Carrington and R. Hodgson in 1 September 1959.
Flare processes have not been perfectly understood yet (Benz & Güdel, 2010).
However, the researches indicate that the incidence of red dwarfs in our
galaxy is 65$\%$. Seventy-five per cent of them show flare activity, these
stars are known as UV Ceti type stars (Rodonó, 1986). In this respect, it will
be easier to understand the evolution of red dwarfs if the flare processes are
well known. This is because flare activity dramatically affects the evolutions
of the red dwarfs. In this respect, an attempt to classify flares by
considering shapes of flare light variations observed in the UV Ceti type
stars, flares have been tried to classify. It is believed that classification
of flares makes the flares and flare processes more intelligible.
Flares of the UV Ceti type stars were first detected in 1939 (Van Maanen,
1940). Discovering the flare stars with high flare frequency such as UV Cet,
YZ CMi, EV Lac (catalog ), AD Leo (catalog ) and EQ Peg (catalog ), the
detected flare numbers and the variety of flare light variations increased.
The light variation of each flare is almost different from each other. In the
first place, it is seen that there are lots of shapes for flare light
variations (Moffett, 1974; Gershberg, 2005). On the other hand, when large
numbers of flares are examined, it is seen that there are only two main shapes
for flare light variations. One of them is called the fast flare. Fast flares
have higher energy and the shapes of their light variations are similar to the
shapes of solar hard X-ray flares. The second type flares are called slow
flares. Unlike fast flares, slow flares exhibit lower energy. The rise times
of slow flares are almost equal to their decay times.
The terms of fast and slow flares were used for the first time in 1960’s in
astrophysics. If the rise time of a flare is below 30 minutes, Haro &
Parsamian (1969) called that flare a fast flare. If its rise time is above 30
minutes, they called it a slow flare. Like Haro & Parsamian (1969),
considering the shapes of flare light variations, Osawa et al. (1968)
described two types of flares. However, Oskanian (1969) separated flares into
four classes. Like Oskanian (1969), considering only light variation shapes of
the flares, Moffett (1974) directly classified flares such as classical,
complex, slow, and flare event. On the other hand, Kunkel had asserted another
idea in his PhD thesis in 1967 (Gershberg, 2005). According to Kunkel, the
observed flare light variations must be some combinations of some slow and
fast flares. According to this idea, there are two main flare types. The
complex flares mentioned by Moffett (1974) are actually a combination of some
fast and slow flares. And also, both slow flares and flare events mentioned by
Moffett (1974) can be classified as the same type flares.
Gurzadyan (1988) described two flare types to model the flare light curves.
Gurzadyan (1988) indicated that thermal processes are dominated in the
processes of slow flares. And these flares are ninety-five per cent of all
flares observed in UV Ceti type stars. The non-thermal processes are dominated
in the processes of fast flares, which are all the other flares. According to
Gurzadyan (1988), there is a large energy difference between these two types
of flares.
In this study, we introduce a new statistical method for classifying flares.
Using statistical Independent Samples t-Test (hereafter t-Test) analysis, this
method is based on the distribution of flare equivalent durations versus flare
rise times. Considering the studies in Osawa et al. (1968), Kunkel s PhD
thesis (Gershberg, 2005) and Gurzadyan (1988), we assume that there are two
flare types such as fast and slow flares types. We classify flares into two
types and demonstrate the similarities and differences between these two types
of flares. In respect of these analyses, we discuss the results obtained from
analyses of 321 flares detected in U band observations of flare stars AD Leo
(catalog ), EV Lac (catalog ), EQ Peg (catalog ) and V1054 Oph (catalog )
between 2004 and 2007. The programme stars were selected for this study due to
their high flare frequencies (Moffett, 1974). The flare data obtained in this
study are useful for such analysis. This is because the data were obtained
with systematical observations, using the same method.
The flare activity of AD Leo (catalog ) was discovered by Gordon & Kron (1949)
for the first time. The star is a red dwarf and a member of The Castor Moving
Group, whose age is about 200 million years (Montes et al., 2001). Crespo-Chac
et al. (2006) found that the flare frequency of AD Leo (catalog ) was 0.71
$h^{-1}$. The variation of the flare frequency was investigated by Ishida et
al. (1991). They mentioned that there is no variation in the flare frequency
of AD Leo (catalog ). The other star in this study is EV Lac (catalog ), which
is one of the well known UV Ceti stars. According to the spatial velocities,
EV Lac (catalog ) seems to be a member of 300 million years old Ursa Major
Group (Montes et al., 2001). It has been known since 1950 that EV Lac (catalog
) shows flares (Lippincott, 1952; Van de Kamp, 1953). The largest observed
flare amplitude is 6.4 mag in U band observations of EV Lac (catalog ).
Andrews (1982) detected 50 flares in U and B band observations of EV Lac
(catalog ). The author indicated that about 42 flares of 50 flares were found
in some groups, which were occurred in every 5-6 days. The seasonal flare
frequencies were computed from 1972 to 1981 and these frequencies were
compared with the seasonal averages of B band magnitudes. According to this
comparison, it was found that the activity cycle is about 5 years for EV Lac
(catalog ) (Mavridis & Avgoloupis, 1986). On the other hand, Ishida et al.
(1991) indicated that there was no flare frequency variation from 1971 to
1988. In another study, Leto et al. (1997) showed that flare frequencies of EV
Lac (catalog ) increased from 1968 to 1977. EQ Peg (catalog ) is another
active flare star, whose flare activity was discovered by Roques (1954). EQ
Peg (catalog ) is classified as a metal-rich star and it is a member of the
young disk population in the galaxy (Veeder, 1974; Fleming et al., 1995). EQ
Peg (catalog ) is a visual binary (Wilson, 1954). Both of the components are a
flare star (Pettersen et al., 1983). Angular distance between components is
given as a value between 3$\arcsec$.5 and 5$\arcsec$.2 (Haisch et al., 1987;
Robrade et al., 2004). One of the components is 10.4 mag and the other is 12.6
mag in V band (Kukarin, 1969). Observations show that flares of EQ Peg
(catalog ) generally come from the fainter component (Fossi et al., 1995).
Rodonó (1978) proved that 65$\%$ of the flares come from faint component and
about 35$\%$ from the brighter component. The fourth star in this study is
V1054 Oph, whose flare activity was discovered by Eggen (1965). V1054 Oph (=
Wolf 630ABab, Gliese 644ABab) is a member of Wolf star group (Joy, 1947; Joy &
Abt, 1974). Wolf 630ABab, Wolf 629AB (= Gliese 643AB) and VB8 (= Gliese 644C),
are the members of the main triplet system, whose scheme is demonstrated in
Fig.1 given in the paper of Pettersen et al. (1984). Wolf 630 and Wolf 629 are
visual binary and they are separated 72$\arcsec$ from each other. Wolf 630AB
is a close visual binary in itself. Wolf 629AB is a spectroscopic binary. B
component of Wolf 629AB seems to be a spectroscopic binary. VB8 is
220$\arcsec$ far away from the other components. There is an angular distance
about 0$\arcsec$.218 between A and B components of Wolf 630 (Joy, 1947; Joy &
Abt, 1974). The masses were derived for each components of Wolf 630ABab by
Mazeh et al. (2001). The author showed that the masses are 0.41 $M_{\odot}$
for Wolf 629A, 0.336 $M_{\odot}$ for Wolf 630Ba and 0.304 $M_{\odot}$ for Wolf
630Bb. In addition, Mazeh et al. (2001) demonstrated that the age of the
system is about 5 Gyr.
## 2 Observations and Analyses
### 2.1 Observations
The observations were acquired with a High-Speed Three Channel Photometer
attached to a 48 cm Cassegrain type telescope at Ege University Observatory.
Using a tracking star set in a second channel of photometer, the observations
were continued in standard Johnson U band with the exposure time between 2 and
10 seconds. The basic parameters of all program stars and their comparisons
are given in Table 1. The parameters given in the table are star name,
magnitude in V band, B-V colour index, spectral type, distance (pc) and
bolometric luminosity ($LogL_{bol}$, ergs $s^{-1}$). The magnitudes and colour
indexes were obtained in this study. Considering B-V colour indexes, the
spectral types were taken from Tokunaga (2000). The distances and bolometric
luminosities were taken from Fossi et al. (1995) and Gershberg et al. (1999).
Although the program and comparison stars are so close on the sky,
differential extinction corrections were applied. The extinction coefficients
were obtained from the observations of the comparison stars on each night.
Moreover, the comparison stars were observed with the standard stars in their
vicinity and the reduced differential magnitudes, in the sense variable minus
comparison, were transformed to the standard system using procedures outlined
in Hardie (1962). The standard stars are listed in the catalogues of Landolt
(1983, 1992). Heliocentric corrections were applied to the times of the
observations. The standard deviation of observation points acquired in
standard Johnson U band is about 0m.015 on each night. Observational reports
of all program stars are given in the Table 2. It is seen that there is no
variation of differential magnitudes in the sense comparison minus check
stars.
Gershberg (1972) developed a method for calculating flare energies. Flare
equivalent durations and energies were calculated with using Equation (1) and
(2) of this method.
$P~{}=~{}\int[(I_{flare}~{}-~{}I_{0})~{}/~{}I_{0}]~{}dt$ (1)
In Equation (1), $I_{0}$ is the intensity of the star in quiescent level.
$I_{flare}$ is the intensity during flare.
$E~{}=~{}P~{}\times~{}L$ (2)
where $E$ is the flare energy; $P$ is the flare equivalent duration; $L$ is
the luminosity of the stars in quiescent level in the Johnson U band.
HJD of flare maximum times, flare rise and decay times, amplitudes of flares
and flare equivalent durations were calculated for each flare. Brightness of a
star without a flare was taken as a quiescent level of brightness of this star
on each night. Considering this level, all flare parameters were calculated
for each night. It was seen that some flares have a few peaks. In this case,
flare maximum time and amplitudes were calculated from the first highest peak.
Instead of flare energies, flare equivalent durations were used for all
statistical analyses. This is because of the luminosity term in Equation (2).
The luminosities of stars from different spectral types have great
differences. Although the equivalent durations of two flares obtained from two
stars in different spectral types are the same, calculated energies of these
flares are different due to different luminosities of these spectral types.
Therefore, we could not use these flare energies in the same analyses. On the
other hand, flare equivalent duration depends just on flare power. Another
reason of using equivalent duration is that the given distances of the same
star in different studies are quite different. Therefore, the calculated
luminosities become different because of these different distances.
All the calculated parameters of flares are given in Table 3. The given
parameters in columns are star name, observation date, HJD of flare maximum,
flare total time (s), decay time (s), equivalent duration (s), energy (ergs),
flare amplitude (mag) and flare types. In the last column, it is given whether
the flare was used in the analyses, or not.
When the observed flares are examined, it is seen that almost each flare has a
distinctive light variation shape (Figures 1, 2, 3, and 4). In these figures,
horizontal dashed lines represent the level of quiescent brightness. The flare
seen in Figure 1 is a fast flare. This flare type occurs frequently in UV Ceti
type stars. On the other hand, the first flare seen in Figure 2 is a compact
flare. This flare type among the others is the hardest flare type to classify.
This flare type must be a combination of two flares. The observed flares,
whose light variations are similar to the first flare in Figure 2, were not
used in the analyses. The second flare seen in Figure 2 is a fast flare. The
flare seen in Figures 3 is a powerful flare, but its light variation was not
completed because we did not carry on observation until the flare completely
decreased to quiescent phase due to the Sun rising. If the light variation of
a flare was not completed like this one, the flare was not used in the
analyses. The flare seen in Figure 4 seems to be very different from previous
flares. Moffett (1974) called flares like this one as the flare events. They
are called slow flares in some other studies. In this study, we called them as
the slow flares.
### 2.2 Analyses
The impulsive phase of a flare is the time interval where sudden-high energy
occurs. On the other hand, the mean phase is other part of the flare, where
the energy is emitted to all space (Gurzadyan, 1988; Benz & Güdel, 2010).
Moreover, rate of brightness increase for fast flares is clearly higher than
that of slow flares (Gurzadyan, 1988; Gershberg, 2005). Moreover, Gurzadyan
(1988) stated that, the energies of fast flares are always higher than slow
flare energies.
According to this approach, we examined all flare data and we saw that the
equivalent durations of some flares are different, while their rise times are
the same. For example, the rise time of 22 flares is 15 second. The equivalent
durations of eight flares among them are very high, while the equivalent
durations of other 14 flares are dramatically low. It was seen in 30 different
rise times. It means that there are least two flares in 30 different rise
times and their equivalent durations are different from each other. 140 flares
were chosen in total. These 140 flares used in the analyses are specified in
the last column of Table 3. Considering also their light variation shapes, we
separated flares into two groups as flares with high energy and low energy. It
was found that 61 flares have high energy, while 79 flares have low energy.
Considering light variations, it was seen that 61 flares are fast flare, and
the others are slow flare. For each one of 30 rise times, the averages of the
equivalent durations were computed separately for the flares with high energy
and low energy.
The most suitable statistical test for these data is the t-Test to determine
the difference between the equivalent durations of two groups. This is because
the t-Test examines whether there is any statistical difference between
independent variable of two groups, or not (Wall & Jenkins, 2003; Dawson &
Trapp, 2004). In this study, the flare rise times were taken as a dependent
variable, while flare equivalent durations were taken as an independent
variable. In the analyses, the SPSS V17.0 software was used (Green et al.,
1999). The average of equivalent durations in the logarithmic scale for 79
slow flares was calculated and found to be 1.348 $\pm$ 0.092. And it was found
to be 2.255 $\pm$ 0.126 for 61 fast flares. This shows that there is a
difference about 0.907 between average equivalent durations in the logarithmic
scale. The p-probability value (hereafter p-value) was computed to test the
results of the t-Test and it was found as p-value $<$ 0.0001. It means that
the result is statistically acceptable. All the results of the analyses are
given in Table 4.
In the next step, we compared their distributions, the distribution of the
equivalent durations versus flare rise times. The best fits for distributions
were searched. Using GrahpPad Prism V5.02 software (Motulsky, 2007),
regression calculations showed that the best fits of distributions seen in
Figure 5 were linear functions given by Equations (3) and (4).
$Log(P_{u})~{}=~{}1.109~{}\times~{}Log(T_{r})~{}-~{}0.581$ (3)
$Log(P_{u})~{}=~{}1.227~{}\times~{}Log(T_{r})~{}+~{}0.122$ (4)
It was tested whether these linear functions belong to two independent
distributions, or not. In this point, the slopes of linear functions were
principally examined. As it can be seen in Table 4, the slope of the linear
function is 1.109 $\pm$ 0.127 for slow flares, while it is 1.227 $\pm$ 0.243
for fast flares. This shows that the increasing of equivalent durations versus
flare rise times for both fast and slow flares are parallel. When the
probability, p-value, was calculated to say whether it is statistically
significant, it was found that p-value = 0.670. This value indicates that
there is no significant difference between the slopes of fits.
Finally, the y-intercept values were calculated and compared for two linear
fits. While this value is -0.581 for the slow flares, it is 0.122 for the fast
flares in the logarithmic scale. It means that there is a difference about
0.703 between these values in the logarithmic scale. When the probability
value was calculated for y-intercept values to say whether there is a
statistically significant difference, it was found that p-value $<$ 0.0001.
This result indicates that the difference between two y-intercept values is
clearly important.
Some other differences like ones demonstrated by t-Test are directly seen in
the graphics. For example, the lengths of flare rise times for both type
flares can be compared in Figure 6. While the lengths of rise times for slow
flares can reach to 1400 seconds, they are not longer than 400 seconds for
fast flares.
The comparison of another parameter is given in Figure 7. The flare amplitudes
are seen in this figure. As it can be seen, while the amplitudes of fast
flares can reach to 4.0 mag, the amplitudes of slow flares can exceed 1.0 mag.
61 fast and 79 slow flares were chosen among 321 flares observed in this
study. The ratios of flare decay time to flare rise time were computed for
both 61 fast and 79 slow flares. As a result, it is seen that the ratio is
below the value of 3.5 for each one of 79 slow flares. On the other hand, the
ratio is above the value of 3.5 for each fast flares. The value of 3.5 is
considered as a limit for these two type flares. Considering the ratio of 3.5,
other 181 flares of 321 flares were separated as slow and fast flares. When
the results obtained from analyses of 140 flares were rechecked for 321
flares, it was seen that the results are the same with the previous ones.
## 3 Results and Discussion
We observed 321 flares in U band observations of AD Leo (catalog ), EV Lac
(catalog ), EQ Peg (catalog ) and V1054 Oph. Examining 321 flares, 61 fast and
79 slow flares were identified for analyses. The t-Test was used as an
analysis method. Flare rise times were accepted as dependent variables, while
flare equivalent durations were taken as independent variables. The results
obtained from the t-Test analyses of the data show that there are distinctive
differences between two flare types. These differences are important
properties because the models of white light flares observed in photoelectric
photometry must support these properties to explain both flare types.
The distributions of the equivalent durations were represented by linear fits
given by Equations (3) and (4) for these flare types. The slope of linear fit
is 1.109 for slow flares, which are low energy flares. And, it is 1.227 for
fast flares, which are high energy flares. The values are almost close to each
other. It seems that the equivalent durations versus rise times increase in
similar ways.
In the case of UV Ceti stars, when flare models are considered, it is seen
that there are two main energy sources for flares (Gurzadyan, 1988; Benz &
Güdel, 2010). These depend on the thermal and non-thermal processes
(Gurzadyan, 1988). Flares with small amplitude are generally the flares with
low energy. The thermal processes are commonly dominant for these type flares.
On the other hand, the flares, which have sudden rapid increases, are more
energetic events. Non-thermal processes are dominant for this type. And thus,
there is an energy difference between these two types of flares (Gurzadyan,
1988). When the averages of equivalent durations for two type flares were
computed in logarithmic scale, it was found that the average of equivalent
durations is 1.348 for slow flares and it is 2.255 for fast flares. The
difference of 0.907 between these values in logarithmic scale is equal to
157.603 second difference between the equivalent durations. As it can be seen
from Equation (2), this difference between average equivalent durations
affects the energies in the same way. Therefore, there is 157.603 times
difference between energies of these two type flare. This difference must be
the difference mentioned by Gurzadyan (1988).
The slopes of linear fits are almost close. On the other hand, if the
y-intercept values of the linear fits are compared, it is seen that there is
0.703 times difference in logarithmic scale, while there is 0.907 times
difference between general averages. Considering also Figure 5, it is seen
that equivalent durations of fast flares can increase more than slow flare
equivalent durations towards the long rise times. Some other effects should be
involved in the fast flare process for long rise times. These effects can make
fast flares more powerful than they are.
When the lengths of rise times for both flare types are compared, it is seen
that there is a difference between them. The lengths of rise times can reach
to 1400 seconds for slow flares, but are not longer than 400 seconds for fast
flares. In addition, when the flare amplitudes are examined for both type
flares, an adverse difference is seen according to rise times. While the
amplitudes of slow flares reach to 1.0 mag at most, the amplitudes of fast
flares can exceed 4.0 mag.
Finally, when the ratios of flare decay times to flare rise times are computed
for two flares types, the ratios never exceed the value of 3.5 for all slow
flares. On the other hand, the ratios are always above the value of 3.5 for
fast flares. It means that if decay time of a flare is 3.5 times longer than
its rise time at least, the flare is a fast flare. If not, the flare is a slow
flare. Therefore, the type of an observed flare can be determined by
considering this value of the ratio. In the studies like Osawa et al. (1968),
Oskanian (1969), Haro & Parsamian (1969) and Moffett (1974), considering
directly the shapes of flare light variations, the flares have been classified
into two types as fast and slow flares. For instance, according to Haro &
Parsamian (1969), if the rise time of a flare is above 30 minutes, the flare
is slow flare. If not, it is a fast flare. However it is shown in this study
that there are some fast flares, whose rise times are longer than the rise
times of some slow flares. This is clearly seen from Table 3. This case
indicates that a classification by considering only the rise time may not be
right. Nevertheless, Moffett (1974) separated flares into more than two groups
such as classic, complex, spike and flare events. On the other hand, according
to our results of t-Test analyses, neither only one parameter nor the shape of
the light variation was enough to classify a flare. The flare equivalent
durations and also one more parameter should be taken into consideration in
order to make such a classification.
The values 3.5, the ratio of flare decay times to flare rise times, can give
an idea about the rate of energy emitting in a flare process. The rise times
of flares are some limits for each type. Maximum flare rise time is about 400
seconds for fast flares, while it can reach the values over 1400 seconds for
slow flares. However, the decay times can take any duration without any
limited values. Consequently, the ratio of flare decay times to flare rise
times depends on rise time more than decay times. In the case of rise time,
the difference between two type flares must be caused by whether the flare
processes are thermal or non-thermal. We computed the duration as a rise time
from the phase in which the brightness increases. Increasing of the brightness
is caused by increasing the temperature of some region on the surface of the
star. The flare rise time is an indicator of heating this region on the
surface. Therefore, the ratio of flare decay times to flare rise times, so the
values of 3.5, must be a critical value between thermal or non-thermal
processes.
As it is seen from the models of Gurzadyan (1988), the differences between
flare durations and flare amplitudes are seen between two flare types derived
from observed flares in this study. The difference between amplitudes of slow
and fast flares was given by Equation (22) in the paper of Gurzadyan (1988).
In the case of flare amplitude, the result obtained in this study is in
agreement with this equation.
Providing that the value 3.5 is a limit ratio for flare types, fast flare rate
is 63$\%$ of all 321 flares observed in this study, while slow flare rate is
37$\%$. It means that one of every three flares is a fast flare, while two of
them are slow flares. This result diverges from what Gurzadyan (1988) stated.
According to Gurzadyan (1988), slow flares with low energies and low
amplitudes are 95$\%$ of all flares. The remainder are fast flares. When
looking individually over each star, the rate of flare types is changing from
star to star. Detected flare number of AD Leo (catalog ) is 110 as it can be
seen from Table 2. Slow flare rate of AD Leo (catalog ) flares is 78$\%$,
while the rate of fast flares is 22$\%$. Detected flare number is 98 in
observation of EV Lac (catalog ) and 40 in observation of V1054 Oph. Slow
flare rates of both stars are 75$\%$, while fast flare rates are 25$\%$. EQ
Peg (catalog ) flare number is 78. Slow flare rate of them is 63$\%$, and fast
flare rate is 37$\%$.
In this study, one of the remarkable points is the correlation coefficients of
linear fits. As it is seen in Table 4, the correlation coefficient is 0.732
for linear fit of slow flare type and 0.476 for fast flares. Although the
correlation coefficient of the linear fit to the distribution of equivalent
durations versus rise times is in an acceptable level for slow flares, it is
relatively lower for fast flares. Regression calculations show that the best
fits are linear for the distribution of equivalent durations versus rise times
in logarithmic scales. The correlation coefficients of other fits are not
higher than linear correlation coefficients. Especially, the correlation
coefficient is lower for the fast flares due to the distribution of their
equivalent durations. As it is seen from Figure 5, the equivalent durations of
fast flares can take values in a wide range towards the longer riser times.
This must be owing to the same reason of differences between y-intercept
values and the mean averages of equivalent durations of two flare types. As it
is discussed above, while the slopes of the fits are nearly close to each
other, there is a considerable difference between y-intercept values and mean
average of equivalent duration for two flare types. Consequently, all these
deviations are seen in fast flares. The magnetic reconnection is dominant in
this type of flares. A parameter in magnetic reconnection process causes some
fast flares to be more powerful than the expected values. Eventually, some
fast flares are more powerful than they are, while some of them are at
expected energy levels. On the other hand, this parameter in magnetic
reconnection process is not dominated in slow flare processes. And so,
distribution of their equivalent durations is not scattered. This must be why
the correlation coefficient of the fit is relatively higher for slow flares.
In this classification method, the complex flares are an exceptional case.
These flares must be composed of some different flares. The complex flare
should be separated into component flares before classification. If the fast
and slow flares can be modelled, using these models, the complex flares can be
decomposed into component flares.
In conclusion, some parameters can be computed from flares observed in
photoelectric photometry. And, if the behaviours between these parameters can
be analysed by suitable methods, the flare types can be determined. In this
study, we analysed the distributions of equivalent durations versus flare rise
time by the statistical analysis method, t-Test. Finally, it is seen that
using the ratios of flare decay times to flare rise times, flares can be
classified. Thus, flares are classified into two types as fast and slow
flares. It is seen that there are considerable differences between these two
types of the flares. The differences and the similarities between flare types
are important to understand the flare processes. This gives new ideas to model
white light flares of UV Ceti stars.
## Acknowledgments
The authors acknowledge generous allotments of observing time at Ege
University Observatory. We wish to thank both Dr. Hayal Boyacıoǧlu, who gave
us important suggestions about statistical analyses, and Prof. Dr. M. Can
Akan, who gave us valuable suggestions, which improved the language of the
manuscript. We also thank the referee for useful comments which have
contributed to improve the manuscript. We finally wish to thank the Ege
University Research Found Council for supporting this work through grant Nr.
2005/FEN/051.
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Figure 1: A flare light curve sample for fast flares obtained from U band
observation of V1054 Oph in 14 June 2004. Figure 2: A flare light curve sample
for fast flares obtained from U band observation of EV Lac in 10 August 2004.
Figure 3: A flare light curve sample for more powerful flares obtained from U
band observation of AD Leo in 6 February 2007. Figure 4: A flare light curve
sample for slow flares obtained from U band observation of EQ Peg in 16
September 2004. Figure 5: The distributions for the mean averages of the
equivalent durations ($Log(P_{u})$) versus flare rise times ($Log(T_{r})$) in
logarithmic scale. In the figure, open circles represent slow flares, while
filled circles show the fast flares. And the lines represent fits given
equations (3) and (4). Figure 6: The distributions of the equivalent durations
($Log(P_{u})$) in logarithmic scale versus flare rise times ($T_{r}$) for all
321 flares detected in observations of program stars. In the figure, open
circles represent slow flares, while filled circles show the fast flares.
Figure 7: The distributions of flare amplitudes versus flare rise times
($Log(T_{r})$) in logarithmic scale for all 321 flares detected in
observations of program stars. In the figure, open circles represent slow
flares, while filled circles show the fast flares.
Table 1: Basic parameters for the targets studied and their comparison (C1) and check (C2) stars. Stars | V (mag) | B-V (mag) | Spectral Type | Distance (pc) | $LogL_{bol}$
---|---|---|---|---|---
AD Leo | 9.388 | 1.498 | M3 | 4.90 | 31.87
C1 = HD 89772 | 8.967 | 1.246 | K6-K7 | - | -
C2 = HD 89471 | 7.778 | 1.342 | K8 | - | -
EV Lac | 10.313 | 1.554 | M3 | 5.00 | 31.72
C1 = HD 215576 | 9.227 | 1.197 | K6 | - | -
C2 = HD 215488 | 10.037 | 0.881 | K1-K2 | - | -
EQ Peg | 10.170 | 1.574 | M3-M4 | 6.58 | 31.42
C1 = SAO 108666 | 9.598 | 0.745 | G8 | - | -
C2 = SAO 91312 | 9.050 | 1.040 | K3-K4 | - | -
V1054 Oph | 8.996 | 1.552 | M3 | 5.70 | 31.93
C1 = HD 152678 | 7.976 | 1.549 | M3 | - | -
C2 = SAO 141448 | 9.978 | 0.805 | K0 | - | -
Table 2: Observational reports of the each program star for each observing season. Stars | Year | HJD | Filter | Observation | Observation | U Filter
---|---|---|---|---|---|---
| | (+2400000) | | Number | Time (hour) | Flare Number
AD Leo | 2005 | 53377 - 53514 | U | 12 | 36.35 | 39
| 2006 | 53717 - 53831 | U | 15 | 37.48 | 54
| 2007 | 54048 - 54248 | U | 8 | 20.80 | 17
EV Lac | 2004 | 53197 - 53257 | U | 17 | 47.62 | 31
| 2005 | 53554 - 53606 | U | 9 | 26.65 | 32
| 2006 | 53940 - 53996 | U | 16 | 44.66 | 35
EQ Peg | 2004 | 53236 - 53335 | U | 13 | 64.42 | 38
| 2005 | 53621 - 53686 | U | 10 | 35.84 | 35
V1054 Oph | 2004 | 53136 - 53202 | U | 19 | 42.64 | 14
| 2005 | 53502 - 53564 | U | 10 | 33.13 | 26
Table 3: All the parameters were computed from observed flares. From the first
column to the last, star name, the date of observation, HJD of flare maximum
moment, flare total time (sec), decay time (sec), equivalent duration (sec),
flare energy (erg), flare amplitude (mag) and flare type are given,
respectively. And in the last column, it is specified whether the flare was
used in the analyses, or not.
Stars | Observation | HJD For | Total | Decay | Equivalent | Flare Energy | Amplitude | Flare | Used In
---|---|---|---|---|---|---|---|---|---
| Date | Maximum of Flare | Time (sec) | Time (sec) | Duration (sec) | (ergs) | In U Band (mag) | Type | Analyses
AD Leo | 06.01.2005 | 53377.50660 | 60 | 36 | 5.1690 | 6.8480E+30 | 0.348 | Slow | Yes
AD Leo | 06.01.2005 | 53377.59771 | 684 | 660 | 94.5297 | 1.2523E+32 | 0.723 | Fast | Yes
AD Leo | 10.01.2005 | 53381.51272 | 816 | 732 | 162.6476 | 2.1548E+32 | 0.519 | Fast | Yes
AD Leo | 10.01.2005 | 53381.52939 | 288 | 216 | 31.5306 | 4.1772E+31 | 0.399 | Slow | Yes
AD Leo | 10.02.2005 | 53412.49006 | 1308 | 1008 | 491.1727 | 6.5071E+32 | 0.863 | Slow | No
AD Leo | 10.02.2005 | 53412.52798 | 4164 | 3756 | 3907.4481 | 5.1767E+33 | 1.589 | Fast | No
AD Leo | 10.02.2005 | 53412.57464 | 1224 | 948 | 845.6318 | 1.1203E+33 | 0.792 | Slow | No
AD Leo | 10.02.2005 | 53412.58687 | 1764 | 1656 | 3250.0883 | 4.3058E+33 | 2.387 | Fast | Yes
AD Leo | 11.02.2005 | 53413.46661 | 36 | 24 | 3.0958 | 4.1014E+30 | 0.200 | Slow | No
AD Leo | 11.02.2005 | 53413.55759 | 384 | 360 | 60.6736 | 8.0382E+31 | 0.363 | Fast | Yes
AD Leo | 12.03.2005 | 53442.39045 | 720 | 680 | 78.4325 | 1.0391E+32 | 0.437 | Fast | Yes
AD Leo | 14.03.2005 | 53444.39896 | 432 | 300 | 30.9237 | 4.0968E+31 | 0.183 | Slow | No
AD Leo | 14.03.2005 | 53444.41327 | 1140 | 1068 | 152.7276 | 2.0234E+32 | 0.183 | Fast | Yes
AD Leo | 14.03.2005 | 53444.42674 | 588 | 504 | 54.4535 | 7.2141E+31 | 0.155 | Fast | Yes
AD Leo | 14.03.2005 | 53444.47105 | 1488 | 1428 | 320.0239 | 4.2397E+32 | 1.168 | Fast | Yes
AD Leo | 14.03.2005 | 53444.50563 | 1428 | 1356 | 206.4180 | 2.7347E+32 | 0.542 | Fast | Yes
AD Leo | 14.03.2005 | 53444.53049 | 240 | 72 | 25.4230 | 3.3681E+31 | 0.258 | Slow | No
AD Leo | 14.03.2005 | 53444.54716 | 576 | 324 | 47.9470 | 6.3521E+31 | 0.247 | Slow | No
AD Leo | 14.03.2005 | 53444.55396 | 84 | 60 | 10.7092 | 1.4188E+31 | 0.222 | Slow | Yes
AD Leo | 14.03.2005 | 53444.55507 | 84 | 48 | 5.5891 | 7.4046E+30 | 0.243 | Slow | Yes
AD Leo | 14.03.2005 | 53444.55632 | 132 | 72 | 8.6651 | 1.1480E+31 | 0.192 | Slow | No
AD Leo | 14.03.2005 | 53444.55966 | 24 | 12 | 8.6411 | 1.1448E+31 | 0.213 | Slow | No
AD Leo | 14.03.2005 | 53444.56021 | 60 | 36 | 8.6411 | 1.1448E+31 | 0.192 | Slow | Yes
AD Leo | 14.03.2005 | 53444.56313 | 204 | 156 | 24.4442 | 3.2384E+31 | 0.207 | Slow | Yes
AD Leo | 16.03.2005 | 53446.29405 | 2472 | 1812 | 293.6454 | 3.8903E+32 | 0.221 | Slow | No
AD Leo | 16.03.2005 | 53446.34849 | 48 | 24 | 2.4130 | 3.1968E+30 | 0.206 | Slow | Yes
AD Leo | 16.03.2005 | 53446.36113 | 456 | 408 | 33.4773 | 4.4351E+31 | 0.195 | Fast | Yes
AD Leo | 16.03.2005 | 53446.39558 | 2244 | 1032 | 198.8119 | 2.6339E+32 | 0.193 | Slow | No
AD Leo | 16.03.2005 | 53446.41252 | 852 | 756 | 229.8011 | 3.0444E+32 | 0.553 | Fast | No
AD Leo | 16.03.2005 | 53446.42960 | 756 | 588 | 99.5567 | 1.3189E+32 | 0.359 | Slow | No
AD Leo | 16.03.2005 | 53446.44822 | 132 | 72 | 12.2456 | 1.6223E+31 | 0.433 | Slow | No
AD Leo | 09.04.2005 | 53470.31255 | 1176 | 924 | 677.9344 | 8.9814E+32 | 1.259 | Fast | No
AD Leo | 09.04.2005 | 53470.33269 | 2088 | 1272 | 194.9401 | 2.5826E+32 | 0.121 | Slow | No
AD Leo | 09.04.2005 | 53470.36824 | 936 | 804 | 226.5383 | 3.0012E+32 | 0.938 | Fast | No
AD Leo | 10.04.2005 | 53471.30082 | 240 | 192 | 19.8520 | 2.6300E+31 | 0.234 | Fast | Yes
AD Leo | 10.04.2005 | 53471.37263 | 756 | 588 | 124.5185 | 1.6496E+32 | 0.512 | Slow | No
AD Leo | 02.05.2005 | 53493.31977 | 1284 | 1212 | 203.5007 | 2.6960E+32 | 0.699 | Fast | Yes
AD Leo | 09.05.2005 | 53500.32863 | 264 | 156 | 26.3800 | 3.4949E+31 | 0.207 | Slow | No
AD Leo | 09.05.2005 | 53500.35640 | 852 | 300 | 47.2850 | 6.2644E+31 | 0.164 | Slow | No
AD Leo | 08.01.2006 | 53744.50793 | 105 | 45 | 11.0325 | 1.4616E+31 | 0.236 | Slow | No
AD Leo | 08.01.2006 | 53744.55585 | 45 | 15 | 3.4390 | 4.5561E+30 | 0.162 | Slow | No
AD Leo | 08.01.2006 | 53744.64085 | 525 | 255 | 44.7429 | 5.9276E+31 | 0.192 | Slow | No
AD Leo | 27.01.2006 | 53763.62811 | 168 | 120 | 14.9402 | 1.9793E+31 | 0.171 | Slow | Yes
AD Leo | 27.01.2006 | 53763.63019 | 96 | 36 | 8.4330 | 1.1172E+31 | 0.162 | Slow | No
AD Leo | 02.02.2006 | 53769.52752 | 420 | 396 | 62.8336 | 8.3243E+31 | 0.510 | Fast | Yes
AD Leo | 02.02.2006 | 53769.55037 | 108 | 60 | 5.6936 | 7.5429E+30 | 0.140 | Slow | Yes
AD Leo | 02.02.2006 | 53769.61581 | 156 | 132 | 13.7184 | 1.8174E+31 | 0.138 | Fast | Yes
AD Leo | 02.02.2006 | 53769.61803 | 48 | 24 | 2.0431 | 2.7067E+30 | 0.161 | Slow | Yes
AD Leo | 02.02.2006 | 53769.62335 | 72 | 48 | 5.7062 | 7.5597E+30 | 0.160 | Slow | Yes
AD Leo | 02.02.2006 | 53769.62765 | 96 | 48 | 8.1257 | 1.0765E+31 | 0.173 | Slow | Yes
AD Leo | 04.02.2006 | 53771.46932 | 1188 | 1032 | 798.6289 | 1.0580E+33 | 1.210 | Fast | No
AD Leo | 04.02.2006 | 53771.49590 | 168 | 84 | 17.2768 | 2.2889E+31 | 0.321 | Slow | Yes
AD Leo | 04.02.2006 | 53771.50608 | 60 | 24 | 3.8830 | 5.1443E+30 | 0.183 | Slow | Yes
AD Leo | 21.02.2006 | 53788.42126 | 210 | 198 | 56.5353 | 7.4899E+31 | 0.618 | Fast | No
AD Leo | 21.02.2006 | 53788.42609 | 12 | 6 | 1.8036 | 2.3895E+30 | 0.261 | Slow | No
AD Leo | 21.02.2006 | 53788.49379 | 58 | 44 | 22.5737 | 2.9906E+31 | 0.741 | Slow | Yes
AD Leo | 21.02.2006 | 53788.49497 | 148 | 114 | 37.6468 | 4.9875E+31 | 0.522 | Slow | Yes
AD Leo | 21.02.2006 | 53788.49696 | 10 | 6 | 1.4739 | 1.9527E+30 | 0.290 | Slow | No
AD Leo | 21.02.2006 | 53788.53818 | 374 | 358 | 186.3927 | 2.4694E+32 | 1.365 | Fast | Yes
AD Leo | 21.02.2006 | 53788.54429 | 8 | 2 | 1.2042 | 1.5953E+30 | 0.267 | Slow | No
AD Leo | 21.02.2006 | 53788.54697 | 16 | 2 | 2.2268 | 2.9501E+30 | 0.337 | Slow | Yes
AD Leo | 17.03.2006 | 53812.29784 | 585 | 555 | 52.7018 | 6.9820E+31 | 0.294 | Fast | Yes
AD Leo | 17.03.2006 | 53812.32348 | 165 | 135 | 21.5461 | 2.8545E+31 | 0.230 | Fast | Yes
AD Leo | 25.03.2006 | 53820.28531 | 12 | 4 | 3.0728 | 4.0708E+30 | 0.363 | Slow | No
AD Leo | 25.03.2006 | 53820.31442 | 138 | 62 | 23.2816 | 3.0844E+31 | 0.358 | Slow | No
AD Leo | 25.03.2006 | 53820.31970 | 14 | 2 | 2.8248 | 3.7423E+30 | 0.582 | Slow | No
AD Leo | 25.03.2006 | 53820.40075 | 256 | 222 | 237.9794 | 3.1528E+32 | 1.171 | Fast | Yes
AD Leo | 25.03.2006 | 53820.40674 | 90 | 48 | 16.2012 | 2.1464E+31 | 0.435 | Slow | No
AD Leo | 25.03.2006 | 53820.40742 | 20 | 10 | 3.1643 | 4.1921E+30 | 0.542 | Slow | No
AD Leo | 25.03.2006 | 53820.41637 | 8 | 4 | 1.7195 | 2.2781E+30 | 0.448 | Slow | No
AD Leo | 25.03.2006 | 53820.41686 | 12 | 6 | 3.0961 | 4.1018E+30 | 0.477 | Slow | No
AD Leo | 01.04.2006 | 53827.27390 | 10 | 6 | 1.3930 | 1.8455E+30 | 0.464 | Slow | No
AD Leo | 01.04.2006 | 53827.29613 | 12 | 4 | 2.5516 | 3.3804E+30 | 0.493 | Slow | No
AD Leo | 01.04.2006 | 53827.30144 | 10 | 4 | 2.0011 | 2.6511E+30 | 0.461 | Slow | No
AD Leo | 01.04.2006 | 53827.31787 | 10 | 2 | 1.8332 | 2.4286E+30 | 0.519 | Slow | No
AD Leo | 01.04.2006 | 53827.37920 | 12 | 4 | 1.8984 | 2.5150E+30 | 0.427 | Slow | No
AD Leo | 05.04.2006 | 53831.39559 | 66 | 46 | 34.7529 | 4.6041E+31 | 0.713 | Slow | Yes
AD Leo | 05.04.2006 | 53831.39755 | 10 | 6 | 1.6828 | 2.2295E+30 | 0.446 | Slow | No
AD Leo | 05.04.2006 | 53831.40919 | 76 | 62 | 38.0426 | 5.0400E+31 | 0.917 | Fast | No
AD Leo | 05.04.2006 | 53831.41058 | 16 | 8 | 2.7778 | 3.6801E+30 | 0.459 | Slow | No
AD Leo | 05.04.2006 | 53831.41937 | 14 | 2 | 3.3755 | 4.4720E+30 | 0.536 | Slow | No
AD Leo | 01.12.2006 | 54071.52083 | 78 | 39 | 8.0437 | 1.0657E+31 | 0.190 | Slow | No
AD Leo | 01.12.2006 | 54071.52940 | 156 | 104 | 12.6557 | 1.6767E+31 | 0.229 | Slow | No
AD Leo | 01.12.2006 | 54071.53617 | 208 | 130 | 23.8354 | 3.1578E+31 | 0.208 | Slow | No
AD Leo | 01.12.2006 | 54071.54174 | 52 | 26 | 7.0330 | 9.3175E+30 | 0.186 | Slow | No
AD Leo | 01.12.2006 | 54071.59966 | 2258 | 1699 | 214.7257 | 2.8447E+32 | 0.201 | Slow | No
AD Leo | 15.12.2006 | 54085.63076 | 20 | 10 | 1.8367 | 2.4332E+30 | 0.139 | Slow | No
AD Leo | 15.12.2006 | 54085.63572 | 30 | 20 | 1.9239 | 2.5488E+30 | 0.160 | Slow | No
AD Leo | 23.12.2006 | 54093.60562 | 30 | 10 | 3.2230 | 4.2698E+30 | 0.151 | Slow | Yes
AD Leo | 23.12.2006 | 54093.60712 | 40 | 10 | 3.5123 | 4.6532E+30 | 0.170 | Slow | No
AD Leo | 23.12.2006 | 54093.61349 | 150 | 110 | 10.6923 | 1.4165E+31 | 0.195 | Slow | No
AD Leo | 23.12.2006 | 54093.63486 | 650 | 400 | 58.1135 | 7.6990E+31 | 0.186 | Slow | No
AD Leo | 23.12.2006 | 54093.64631 | 1676 | 1086 | 154.4312 | 2.0459E+32 | 0.200 | Slow | No
AD Leo | 21.01.2007 | 54122.47422 | 16 | 8 | 1.8714 | 2.4792E+30 | 0.229 | Slow | No
AD Leo | 21.01.2007 | 54122.48635 | 32 | 16 | 3.9082 | 5.1777E+30 | 0.274 | Slow | Yes
AD Leo | 21.01.2007 | 54122.48876 | 16 | 8 | 2.0181 | 2.6736E+30 | 0.302 | Slow | No
AD Leo | 21.01.2007 | 54122.49107 | 32 | 16 | 3.6712 | 4.8637E+30 | 0.213 | Slow | Yes
AD Leo | 21.01.2007 | 54122.49573 | 24 | 16 | 2.9831 | 3.9520E+30 | 0.248 | Slow | No
AD Leo | 21.01.2007 | 54122.49693 | 16 | 8 | 1.7260 | 2.2867E+30 | 0.264 | Slow | No
AD Leo | 21.01.2007 | 54122.49712 | 16 | 8 | 1.7700 | 2.3449E+30 | 0.215 | Slow | No
AD Leo | 21.01.2007 | 54122.49860 | 16 | 8 | 1.8787 | 2.4889E+30 | 0.214 | Slow | No
AD Leo | 21.01.2007 | 54122.50156 | 24 | 8 | 2.9281 | 3.8792E+30 | 0.259 | Slow | Yes
AD Leo | 21.01.2007 | 54122.50582 | 40 | 24 | 3.0146 | 3.9938E+30 | 0.211 | Slow | Yes
AD Leo | 21.01.2007 | 54122.50628 | 24 | 16 | 2.5792 | 3.4170E+30 | 0.217 | Slow | No
AD Leo | 21.01.2007 | 54122.53384 | 24 | 16 | 3.2242 | 4.2715E+30 | 0.279 | Slow | No
AD Leo | 08.03.2007 | 54168.46811 | 60 | 30 | 8.2226 | 1.0893E+31 | 0.285 | Slow | No
AD Leo | 08.03.2007 | 54168.46933 | 90 | 60 | 14.2830 | 1.8922E+31 | 0.232 | Slow | No
AD Leo | 16.03.2007 | 54176.37237 | 84 | 60 | 4.8172 | 6.3820E+30 | 0.157 | Slow | Yes
AD Leo | 16.03.2007 | 54176.39611 | 192 | 120 | 18.7237 | 2.4805E+31 | 0.230 | Slow | Yes
AD Leo | 16.03.2007 | 54176.45538 | 372 | 156 | 28.2549 | 3.7433E+31 | 0.168 | Slow | No
EQ Peg | 18.08.2004 | 53236.43176 | 290 | 250 | 44.8869 | 4.4695E+31 | 0.369 | Fast | Yes
EQ Peg | 18.08.2004 | 53236.48106 | 210 | 170 | 28.0058 | 2.7886E+31 | 0.226 | Fast | Yes
EQ Peg | 18.08.2004 | 53236.50398 | 220 | 180 | 42.7730 | 4.2590E+31 | 0.473 | Fast | Yes
EQ Peg | 18.08.2004 | 53236.56139 | 1060 | 900 | 475.3540 | 4.7332E+32 | 1.279 | Fast | No
EQ Peg | 19.08.2004 | 53237.38157 | 490 | 440 | 111.7547 | 1.1128E+32 | 0.503 | Fast | Yes
EQ Peg | 19.08.2004 | 53237.40715 | 150 | 110 | 34.0142 | 3.3869E+31 | 0.258 | Slow | No
EQ Peg | 19.08.2004 | 53237.41224 | 340 | 310 | 68.8944 | 6.8600E+31 | 0.464 | Fast | Yes
EQ Peg | 19.08.2004 | 53237.48805 | 230 | 200 | 51.5306 | 5.1310E+31 | 0.734 | Fast | Yes
EQ Peg | 19.08.2004 | 53237.49291 | 180 | 160 | 30.2882 | 3.0159E+31 | 0.343 | Fast | Yes
EQ Peg | 19.08.2004 | 53237.53122 | 610 | 380 | 106.3842 | 1.0593E+32 | 0.384 | Slow | No
EQ Peg | 22.08.2004 | 53240.36934 | 570 | 510 | 260.2007 | 2.5909E+32 | 0.880 | Fast | Yes
EQ Peg | 08.09.2004 | 53257.36799 | 340 | 270 | 56.6293 | 5.6387E+31 | 0.427 | Fast | No
EQ Peg | 08.09.2004 | 53257.38223 | 430 | 140 | 72.0350 | 7.1727E+31 | 0.478 | Slow | No
EQ Peg | 08.09.2004 | 53257.45202 | 1170 | 1040 | 199.7623 | 1.9891E+32 | 0.406 | Fast | No
EQ Peg | 08.09.2004 | 53257.51000 | 120 | 110 | 14.8846 | 1.4821E+31 | 0.413 | Fast | No
EQ Peg | 08.09.2004 | 53257.51845 | 60 | 30 | 11.3037 | 1.1255E+31 | 0.520 | Slow | No
EQ Peg | 09.09.2004 | 53258.36870 | 1840 | 1390 | 409.0358 | 4.0729E+32 | 0.478 | Slow | No
EQ Peg | 09.09.2004 | 53258.40585 | 190 | 160 | 32.7460 | 3.2606E+31 | 0.574 | Fast | Yes
EQ Peg | 09.09.2004 | 53258.51094 | 200 | 160 | 33.3344 | 3.3192E+31 | 0.484 | Fast | Yes
EQ Peg | 12.09.2004 | 53261.39106 | 260 | 220 | 53.8905 | 5.3660E+31 | 0.583 | Fast | Yes
EQ Peg | 12.09.2004 | 53261.46988 | 130 | 110 | 19.8878 | 1.9803E+31 | 0.332 | Fast | Yes
EQ Peg | 12.09.2004 | 53261.55529 | 150 | 130 | 35.5859 | 3.5434E+31 | 0.756 | Fast | Yes
EQ Peg | 12.09.2004 | 53261.59048 | 210 | 180 | 74.9052 | 7.4585E+31 | 0.785 | Fast | Yes
EQ Peg | 14.09.2004 | 53263.34072 | 1070 | 950 | 110.0519 | 1.0958E+32 | 0.251 | Fast | No
EQ Peg | 14.09.2004 | 53263.36815 | 190 | 130 | 41.2794 | 4.1103E+31 | 0.532 | Slow | No
EQ Peg | 14.09.2004 | 53263.37752 | 1620 | 1430 | 625.0200 | 6.2235E+32 | 0.868 | Fast | No
EQ Peg | 14.09.2004 | 53263.40484 | 200 | 150 | 44.5257 | 4.4335E+31 | 0.414 | Slow | Yes
EQ Peg | 14.09.2004 | 53263.47371 | 50 | 40 | 4.9279 | 4.9068E+30 | 0.289 | Fast | No
EQ Peg | 14.09.2004 | 53263.48725 | 70 | 50 | 9.4754 | 9.4349E+30 | 0.357 | Slow | Yes
EQ Peg | 14.09.2004 | 53263.54697 | 60 | 30 | 5.7809 | 5.7562E+30 | 0.361 | Slow | No
EQ Peg | 15.09.2004 | 53264.46607 | 330 | 120 | 44.1024 | 4.3914E+31 | 0.246 | Slow | No
EQ Peg | 15.09.2004 | 53264.50102 | 580 | 550 | 294.4328 | 2.9317E+32 | 2.109 | Fast | Yes
EQ Peg | 15.09.2004 | 53264.51167 | 510 | 490 | 166.0980 | 1.6539E+32 | 1.420 | Fast | Yes
EQ Peg | 15.09.2004 | 53264.56595 | 1440 | 390 | 186.4048 | 1.8561E+32 | 0.284 | Slow | No
EQ Peg | 16.09.2004 | 53265.39674 | 910 | 860 | 180.5574 | 1.7979E+32 | 0.622 | Fast | Yes
EQ Peg | 16.09.2004 | 53265.51329 | 420 | 250 | 84.7088 | 8.4347E+31 | 0.203 | Slow | No
EQ Peg | 16.09.2004 | 53265.54003 | 410 | 350 | 99.4698 | 9.9044E+31 | 0.463 | Fast | Yes
EQ Peg | 16.09.2004 | 53265.56942 | 3180 | 1950 | 746.5378 | 7.4335E+32 | 0.345 | Slow | No
EQ Peg | 07.09.2005 | 53621.52654 | 150 | 105 | 34.9365 | 3.4787E+31 | 0.630 | Slow | No
EQ Peg | 08.09.2005 | 53622.45159 | 285 | 270 | 78.7774 | 7.8441E+31 | 0.711 | Fast | Yes
EQ Peg | 08.09.2005 | 53622.46218 | 540 | 510 | 156.1378 | 1.5547E+32 | 0.968 | Fast | Yes
EQ Peg | 08.09.2005 | 53622.48087 | 120 | 105 | 39.0373 | 3.8870E+31 | 0.640 | Fast | Yes
EQ Peg | 08.09.2005 | 53622.49927 | 75 | 60 | 12.4900 | 1.2437E+31 | 0.362 | Fast | Yes
EQ Peg | 12.09.2005 | 53626.33898 | 1839 | 1764 | 6818.1425 | 6.7890E+33 | 4.006 | Fast | Yes
EQ Peg | 12.09.2005 | 53626.36062 | 75 | 45 | 8.6906 | 8.6534E+30 | 0.261 | Slow | No
EQ Peg | 12.09.2005 | 53626.36218 | 105 | 75 | 15.2364 | 1.5171E+31 | 0.338 | Slow | No
EQ Peg | 12.09.2005 | 53626.36409 | 105 | 30 | 10.5085 | 1.0464E+31 | 0.337 | Slow | Yes
EQ Peg | 12.09.2005 | 53626.36582 | 922 | 802 | 352.2107 | 3.5070E+32 | 1.285 | Fast | No
EQ Peg | 12.09.2005 | 53626.40874 | 225 | 180 | 51.6810 | 5.1460E+31 | 0.567 | Fast | Yes
EQ Peg | 12.09.2005 | 53626.50223 | 90 | 45 | 24.2883 | 2.4184E+31 | 0.440 | Slow | No
EQ Peg | 12.09.2005 | 53626.56201 | 315 | 225 | 181.2224 | 1.8045E+32 | 0.964 | Slow | Yes
EQ Peg | 12.09.2005 | 53626.56478 | 135 | 120 | 83.2370 | 8.2881E+31 | 1.090 | Fast | Yes
EQ Peg | 12.09.2005 | 53626.56635 | 60 | 45 | 12.1029 | 1.2051E+31 | 0.252 | Slow | Yes
EQ Peg | 12.09.2005 | 53626.56721 | 180 | 150 | 27.7286 | 2.7610E+31 | 0.282 | Fast | Yes
EQ Peg | 12.09.2005 | 53626.56930 | 60 | 30 | 5.7164 | 5.6920E+30 | 0.248 | Slow | No
EQ Peg | 12.09.2005 | 53626.58426 | 165 | 135 | 31.8950 | 3.1759E+31 | 0.676 | Fast | Yes
EQ Peg | 27.09.2005 | 53641.34719 | 135 | 120 | 76.2420 | 7.5916E+31 | 1.111 | Fast | Yes
EQ Peg | 27.09.2005 | 53641.36437 | 90 | 45 | 9.0307 | 8.9921E+30 | 0.225 | Slow | No
EQ Peg | 27.09.2005 | 53641.39000 | 150 | 120 | 35.8623 | 3.5709E+31 | 0.535 | Fast | Yes
EQ Peg | 27.09.2005 | 53641.42135 | 1545 | 1410 | 1739.8937 | 1.7325E+33 | 1.762 | Fast | No
EQ Peg | 27.09.2005 | 53641.47316 | 977 | 585 | 203.3585 | 2.0249E+32 | 0.387 | Slow | No
EQ Peg | 03.10.2005 | 53647.34180 | 180 | 135 | 66.8966 | 6.6611E+31 | 1.076 | Slow | No
EQ Peg | 03.10.2005 | 53647.40952 | 255 | 180 | 56.0173 | 5.5778E+31 | 0.536 | Slow | Yes
EQ Peg | 28.10.2005 | 53672.27432 | 2764 | 1594 | 378.4947 | 3.7688E+32 | 0.333 | Slow | No
EQ Peg | 28.10.2005 | 53672.30797 | 210 | 120 | 52.2187 | 5.1995E+31 | 0.402 | Slow | Yes
EQ Peg | 28.10.2005 | 53672.33656 | 225 | 195 | 55.9946 | 5.5755E+31 | 0.682 | Fast | Yes
EQ Peg | 29.10.2005 | 53673.34763 | 570 | 450 | 155.7805 | 1.5511E+32 | 0.740 | Fast | No
EQ Peg | 29.10.2005 | 53673.35896 | 240 | 135 | 72.3348 | 7.2025E+31 | 0.716 | Slow | No
EQ Peg | 29.10.2005 | 53673.36104 | 330 | 285 | 96.7821 | 9.6368E+31 | 0.461 | Fast | Yes
EQ Peg | 29.10.2005 | 53673.38314 | 738 | 468 | 171.8148 | 1.7108E+32 | 0.428 | Slow | No
EQ Peg | 11.11.2005 | 53686.29364 | 540 | 405 | 151.3632 | 1.5072E+32 | 0.511 | Slow | No
EQ Peg | 11.11.2005 | 53686.30076 | 795 | 645 | 566.9639 | 5.6454E+32 | 1.019 | Fast | No
EQ Peg | 11.11.2005 | 53686.37753 | 270 | 120 | 97.9336 | 9.7515E+31 | 0.562 | Slow | No
EV Lac | 11.07.2004 | 53198.47514 | 260 | 190 | 50.2493 | 2.3691E+31 | 0.431 | Slow | No
EV Lac | 11.07.2004 | 53198.48325 | 590 | 560 | 128.0124 | 6.0353E+31 | 0.735 | Fast | Yes
EV Lac | 17.07.2004 | 53204.51795 | 1230 | 820 | 174.6712 | 8.2351E+31 | 0.313 | Slow | No
EV Lac | 20.07.2004 | 53207.49970 | 310 | 230 | 64.2165 | 3.0276E+31 | 0.588 | Slow | No
EV Lac | 20.07.2004 | 53207.51648 | 510 | 490 | 114.2672 | 5.3873E+31 | 0.905 | Fast | Yes
EV Lac | 24.07.2004 | 53211.54445 | 830 | 590 | 165.8313 | 7.8184E+31 | 0.397 | Slow | No
EV Lac | 25.07.2004 | 53212.48570 | 490 | 340 | 141.3253 | 6.6630E+31 | 0.602 | Slow | No
EV Lac | 25.07.2004 | 53212.51625 | 380 | 160 | 167.7941 | 7.9109E+31 | 0.514 | Slow | No
EV Lac | 26.07.2004 | 53213.48702 | 80 | 50 | 17.0178 | 8.0233E+30 | 0.676 | Slow | No
EV Lac | 26.07.2004 | 53213.52209 | 100 | 90 | 15.4025 | 7.2618E+30 | 0.385 | Fast | No
EV Lac | 28.07.2004 | 53215.48409 | 180 | 160 | 97.5996 | 4.6015E+31 | 1.315 | Fast | Yes
EV Lac | 28.07.2004 | 53215.49960 | 170 | 120 | 36.9820 | 1.7436E+31 | 0.436 | Slow | Yes
EV Lac | 07.08.2004 | 53225.42498 | 350 | 190 | 94.2676 | 4.4444E+31 | 0.611 | Slow | No
EV Lac | 07.08.2004 | 53225.46827 | 140 | 80 | 47.9538 | 2.2609E+31 | 0.643 | Slow | No
EV Lac | 08.08.2004 | 53226.46483 | 140 | 120 | 26.0719 | 1.2292E+31 | 0.377 | Fast | Yes
EV Lac | 08.08.2004 | 53226.50951 | 2482 | 2422 | 594.8340 | 2.8044E+32 | 0.816 | Fast | Yes
EV Lac | 09.08.2004 | 53227.38165 | 940 | 910 | 728.2759 | 3.4336E+32 | 1.763 | Fast | Yes
EV Lac | 09.08.2004 | 53227.42505 | 630 | 490 | 188.9706 | 8.9093E+31 | 0.555 | Slow | No
EV Lac | 09.08.2004 | 53227.43928 | 360 | 260 | 83.5153 | 3.9375E+31 | 0.591 | Slow | No
EV Lac | 09.08.2004 | 53227.44287 | 240 | 220 | 44.6927 | 2.1071E+31 | 0.427 | Fast | Yes
EV Lac | 09.08.2004 | 53227.46382 | 1000 | 960 | 1134.4908 | 5.3487E+32 | 1.987 | Fast | Yes
EV Lac | 09.08.2004 | 53227.49692 | 90 | 70 | 18.4229 | 8.6857E+30 | 0.511 | Slow | Yes
EV Lac | 09.08.2004 | 53227.55063 | 70 | 60 | 9.8307 | 4.6348E+30 | 0.353 | Fast | No
EV Lac | 10.08.2004 | 53228.37056 | 90 | 50 | 19.5669 | 9.2251E+30 | 0.404 | Slow | No
EV Lac | 10.08.2004 | 53228.38654 | 2200 | 2090 | 4235.9481 | 1.9971E+33 | 2.718 | Fast | No
EV Lac | 10.08.2004 | 53228.41698 | 1800 | 1730 | 849.9341 | 4.0071E+32 | 2.020 | Fast | No
EV Lac | 10.08.2004 | 53228.52716 | 50 | 20 | 8.2244 | 3.8775E+30 | 0.437 | Slow | No
EV Lac | 10.08.2004 | 53228.55471 | 120 | 100 | 24.2433 | 1.1430E+31 | 0.520 | Fast | Yes
EV Lac | 14.08.2004 | 53232.36686 | 80 | 50 | 14.2793 | 6.7322E+30 | 0.433 | Slow | No
EV Lac | 14.08.2004 | 53232.37785 | 290 | 220 | 72.8660 | 3.4354E+31 | 0.660 | Slow | No
EV Lac | 14.08.2004 | 53232.39359 | 520 | 470 | 147.9080 | 6.9733E+31 | 0.751 | Fast | Yes
EV Lac | 05.07.2005 | 53557.52397 | 168 | 116 | 71.6667 | 3.3788E+31 | 0.693 | Slow | No
EV Lac | 05.07.2005 | 53557.53656 | 4 | 2 | 1.4724 | 6.9419E+29 | 0.516 | Slow | No
EV Lac | 26.07.2005 | 53578.50257 | 600 | 585 | 184.9502 | 8.7197E+31 | 0.727 | Fast | Yes
EV Lac | 26.07.2005 | 53578.51594 | 105 | 75 | 15.5441 | 7.3285E+30 | 0.381 | Slow | No
EV Lac | 26.07.2005 | 53578.52566 | 30 | 15 | 3.7640 | 1.7746E+30 | 0.381 | Slow | Yes
EV Lac | 26.07.2005 | 53578.54667 | 120 | 45 | 12.2619 | 5.7810E+30 | 0.372 | Slow | Yes
EV Lac | 03.08.2005 | 53586.50688 | 570 | 480 | 384.1497 | 1.8111E+32 | 1.565 | Fast | Yes
EV Lac | 03.08.2005 | 53586.51348 | 30 | 15 | 2.6479 | 1.2484E+30 | 0.313 | Slow | Yes
EV Lac | 03.08.2005 | 53586.52856 | 180 | 135 | 38.1427 | 1.7983E+31 | 0.437 | Slow | No
EV Lac | 14.08.2005 | 53597.33755 | 135 | 105 | 41.7339 | 1.9676E+31 | 0.361 | Slow | No
EV Lac | 14.08.2005 | 53597.40029 | 60 | 45 | 8.4471 | 3.9825E+30 | 0.297 | Slow | Yes
EV Lac | 14.08.2005 | 53597.47199 | 105 | 60 | 23.3056 | 1.0988E+31 | 0.359 | Slow | No
EV Lac | 14.08.2005 | 53597.47615 | 210 | 120 | 44.6968 | 2.1073E+31 | 0.401 | Slow | Yes
EV Lac | 14.08.2005 | 53597.47806 | 105 | 75 | 35.7253 | 1.6843E+31 | 0.304 | Slow | No
EV Lac | 14.08.2005 | 53597.48015 | 105 | 75 | 30.4018 | 1.4333E+31 | 0.493 | Slow | No
EV Lac | 14.08.2005 | 53597.48223 | 165 | 60 | 32.2519 | 1.5206E+31 | 0.228 | Slow | No
EV Lac | 15.08.2005 | 53598.42911 | 390 | 345 | 92.8472 | 4.3774E+31 | 0.435 | Fast | Yes
EV Lac | 15.08.2005 | 53598.43380 | 195 | 165 | 33.7418 | 1.5908E+31 | 0.219 | Fast | Yes
EV Lac | 15.08.2005 | 53598.46517 | 1005 | 870 | 398.3372 | 1.8780E+32 | 0.949 | Fast | No
EV Lac | 15.08.2005 | 53598.50008 | 810 | 720 | 282.6663 | 1.3327E+32 | 1.023 | Fast | Yes
EV Lac | 15.08.2005 | 53598.52741 | 1365 | 1245 | 1264.2000 | 5.9603E+32 | 2.096 | Fast | No
EV Lac | 22.08.2005 | 53605.39060 | 60 | 45 | 21.6520 | 1.0208E+31 | 0.467 | Slow | Yes
EV Lac | 22.08.2005 | 53605.39147 | 330 | 300 | 187.0415 | 8.8183E+31 | 1.058 | Fast | Yes
EV Lac | 22.08.2005 | 53605.41460 | 195 | 105 | 61.7938 | 2.9134E+31 | 0.469 | Slow | Yes
EV Lac | 22.08.2005 | 53605.42241 | 75 | 30 | 20.6508 | 9.7361E+30 | 0.496 | Slow | No
EV Lac | 22.08.2005 | 53605.42467 | 105 | 75 | 24.9765 | 1.1776E+31 | 0.439 | Slow | No
EV Lac | 22.08.2005 | 53605.43113 | 45 | 30 | 12.4383 | 5.8642E+30 | 0.434 | Slow | Yes
EV Lac | 23.08.2005 | 53606.38421 | 45 | 30 | 8.4172 | 3.9684E+30 | 0.312 | Slow | Yes
EV Lac | 23.08.2005 | 53606.42194 | 45 | 30 | 7.6081 | 3.5869E+30 | 0.460 | Slow | Yes
EV Lac | 23.08.2005 | 53606.45404 | 120 | 105 | 30.8166 | 1.4529E+31 | 0.333 | Fast | Yes
EV Lac | 23.08.2005 | 53606.49226 | 45 | 15 | 1.6396 | 7.7299E+29 | 0.476 | Slow | No
EV Lac | 23.08.2005 | 53606.49851 | 30 | 15 | 3.5732 | 1.6846E+30 | 0.412 | Slow | Yes
EV Lac | 23.07.2006 | 53940.47444 | 60 | 30 | 7.9475 | 3.7470E+30 | 0.422 | Slow | No
EV Lac | 23.07.2006 | 53940.52270 | 480 | 375 | 113.1827 | 5.3362E+31 | 0.572 | Fast | No
EV Lac | 23.07.2006 | 53940.53726 | 120 | 90 | 16.4250 | 7.7438E+30 | 0.334 | Slow | No
EV Lac | 29.07.2006 | 53946.47599 | 1054 | 989 | 570.7213 | 2.6907E+32 | 1.991 | Fast | No
EV Lac | 29.07.2006 | 53946.54317 | 75 | 45 | 12.2062 | 5.7548E+30 | 0.394 | Slow | No
EV Lac | 03.08.2006 | 53951.53751 | 2898 | 2688 | 2579.1462 | 1.2160E+33 | 1.574 | Fast | No
EV Lac | 04.08.2006 | 53952.37326 | 125 | 65 | 12.4023 | 5.8473E+30 | 0.238 | Slow | No
EV Lac | 04.08.2006 | 53952.37413 | 30 | 20 | 3.6127 | 1.7033E+30 | 0.334 | Slow | No
EV Lac | 04.08.2006 | 53952.37911 | 20 | 10 | 2.8304 | 1.3344E+30 | 0.314 | Slow | No
EV Lac | 04.08.2006 | 53952.37934 | 20 | 10 | 2.6850 | 1.2659E+30 | 0.345 | Slow | No
EV Lac | 04.08.2006 | 53952.37980 | 30 | 20 | 4.2637 | 2.0102E+30 | 0.319 | Slow | No
EV Lac | 07.08.2006 | 53955.48708 | 90 | 60 | 14.2999 | 6.7419E+30 | 0.460 | Slow | No
EV Lac | 07.08.2006 | 53955.52215 | 90 | 60 | 7.6945 | 3.6277E+30 | 0.480 | Slow | No
EV Lac | 07.08.2006 | 53955.53095 | 120 | 90 | 28.0671 | 1.3233E+31 | 0.578 | Slow | No
EV Lac | 08.08.2006 | 53956.34904 | 60 | 45 | 13.7110 | 6.4642E+30 | 0.652 | Slow | Yes
EV Lac | 08.08.2006 | 53956.35441 | 120 | 105 | 38.2660 | 1.8041E+31 | 0.633 | Fast | Yes
EV Lac | 08.08.2006 | 53956.39257 | 1680 | 1590 | 900.4373 | 4.2452E+32 | 1.484 | Fast | Yes
EV Lac | 08.08.2006 | 53956.44677 | 30 | 15 | 8.5613 | 4.0364E+30 | 0.459 | Slow | Yes
EV Lac | 08.08.2006 | 53956.44989 | 150 | 75 | 23.5724 | 1.1114E+31 | 0.489 | Slow | Yes
EV Lac | 08.08.2006 | 53956.45215 | 75 | 45 | 16.8672 | 7.9523E+30 | 0.516 | Slow | No
EV Lac | 08.08.2006 | 53956.45285 | 30 | 15 | 8.7699 | 4.1347E+30 | 0.450 | Slow | Yes
EV Lac | 08.08.2006 | 53956.45510 | 195 | 105 | 56.8137 | 2.6786E+31 | 0.486 | Slow | Yes
EV Lac | 12.08.2006 | 53960.53098 | 2940 | 2100 | 1453.1544 | 6.8511E+32 | 0.821 | Slow | No
EV Lac | 15.08.2006 | 53963.40072 | 30 | 10 | 7.4410 | 3.5082E+30 | 0.578 | Slow | Yes
EV Lac | 25.08.2006 | 53973.49652 | 2330 | 2260 | 2288.6227 | 1.0790E+33 | 2.164 | Fast | No
EV Lac | 05.09.2006 | 53984.44139 | 360 | 315 | 253.0786 | 1.1932E+32 | 1.215 | Fast | Yes
EV Lac | 05.09.2006 | 53984.47750 | 135 | 60 | 35.1887 | 1.6590E+31 | 0.563 | Slow | Yes
EV Lac | 05.09.2006 | 53984.50997 | 30 | 15 | 8.6037 | 4.0564E+30 | 0.475 | Slow | Yes
EV Lac | 08.09.2006 | 53987.47180 | 90 | 45 | 19.8145 | 9.3418E+30 | 0.537 | Slow | No
EV Lac | 08.09.2006 | 53987.47389 | 75 | 30 | 24.6323 | 1.1613E+31 | 0.845 | Slow | No
EV Lac | 15.09.2006 | 53994.36716 | 70 | 20 | 19.0094 | 8.9622E+30 | 0.441 | Slow | Yes
EV Lac | 15.09.2006 | 53994.37087 | 320 | 150 | 59.0751 | 2.7852E+31 | 0.359 | Slow | No
EV Lac | 17.09.2006 | 53996.33563 | 435 | 390 | 82.6432 | 3.8963E+31 | 0.395 | Fast | Yes
EV Lac | 17.09.2006 | 53996.34258 | 360 | 150 | 65.9648 | 3.1100E+31 | 0.253 | Slow | No
EV Lac | 17.09.2006 | 53996.35595 | 195 | 135 | 35.6703 | 1.6817E+31 | 0.317 | Slow | No
V1054 Oph | 11.06.2004 | 53168.42510 | 140 | 80 | 13.5776 | 3.9221E+31 | 0.274 | Slow | No
V1054 Oph | 14.06.2004 | 53171.39288 | 650 | 600 | 440.3598 | 1.2720E+33 | 1.829 | Fast | Yes
V1054 Oph | 14.06.2004 | 53171.42147 | 180 | 130 | 10.7079 | 3.0931E+31 | 0.226 | Slow | Yes
V1054 Oph | 14.06.2004 | 53171.42563 | 3270 | 3190 | 1441.3629 | 4.1636E+33 | 1.239 | Fast | Yes
V1054 Oph | 14.06.2004 | 53171.46290 | 1780 | 1750 | 281.5233 | 8.1322E+32 | 0.359 | Fast | Yes
V1054 Oph | 20.06.2004 | 53177.40665 | 300 | 270 | 32.6526 | 9.4321E+31 | 0.355 | Fast | Yes
V1054 Oph | 20.06.2004 | 53177.42459 | 120 | 60 | 13.8765 | 4.0084E+31 | 0.230 | Slow | No
V1054 Oph | 20.06.2004 | 53177.43755 | 1000 | 520 | 59.0017 | 1.7043E+32 | 0.181 | Slow | No
V1054 Oph | 04.07.2004 | 53191.36033 | 340 | 90 | 43.0587 | 1.2438E+32 | 0.305 | Slow | No
V1054 Oph | 04.07.2004 | 53191.36334 | 1340 | 1280 | 329.4567 | 9.5168E+32 | 0.957 | Fast | Yes
V1054 Oph | 04.07.2004 | 53191.39505 | 1800 | 340 | 247.6494 | 7.1537E+32 | 0.194 | Slow | No
V1054 Oph | 06.07.2004 | 53193.34635 | 1070 | 960 | 429.5400 | 1.2408E+33 | 1.568 | Fast | No
V1054 Oph | 06.07.2004 | 53193.36302 | 1460 | 980 | 192.7904 | 5.5690E+32 | 0.180 | Slow | No
V1054 Oph | 10.07.2004 | 53197.39384 | 670 | 530 | 88.1072 | 2.5451E+32 | 0.502 | Fast | No
V1054 Oph | 11.05.2005 | 53502.41191 | 48 | 36 | 2.3143 | 6.6853E+30 | 0.151 | Slow | No
V1054 Oph | 11.05.2005 | 53502.41469 | 216 | 180 | 24.2944 | 7.0178E+31 | 0.365 | Fast | Yes
V1054 Oph | 04.06.2005 | 53526.42970 | 36 | 24 | 2.2093 | 6.3819E+30 | 0.122 | Slow | No
V1054 Oph | 04.06.2005 | 53526.44400 | 60 | 36 | 6.1596 | 1.7793E+31 | 0.175 | Slow | Yes
V1054 Oph | 05.06.2005 | 53527.40053 | 468 | 396 | 51.9875 | 1.5017E+32 | 0.271 | Fast | Yes
V1054 Oph | 05.06.2005 | 53527.40636 | 48 | 36 | 3.9567 | 1.1429E+31 | 0.192 | Slow | No
V1054 Oph | 05.06.2005 | 53527.45803 | 108 | 84 | 5.6538 | 1.6332E+31 | 0.152 | Slow | Yes
V1054 Oph | 06.06.2005 | 53528.38594 | 1596 | 708 | 160.4190 | 4.6339E+32 | 0.250 | Slow | No
V1054 Oph | 06.06.2005 | 53528.39942 | 1368 | 1008 | 168.3597 | 4.8633E+32 | 0.207 | Slow | No
V1054 Oph | 06.06.2005 | 53528.41775 | 288 | 276 | 31.2711 | 9.0331E+31 | 0.227 | Fast | No
V1054 Oph | 13.06.2005 | 53535.44630 | 2304 | 1260 | 280.2476 | 8.0953E+32 | 0.258 | Slow | No
V1054 Oph | 13.06.2005 | 53535.47463 | 1416 | 768 | 102.0127 | 2.9468E+32 | 0.182 | Slow | No
V1054 Oph | 13.06.2005 | 53535.48505 | 192 | 156 | 12.1108 | 3.4984E+31 | 0.216 | Fast | Yes
V1054 Oph | 13.06.2005 | 53535.49797 | 144 | 96 | 11.4981 | 3.3214E+31 | 0.314 | Slow | Yes
V1054 Oph | 13.06.2005 | 53535.50880 | 132 | 120 | 18.6571 | 5.3893E+31 | 0.395 | Fast | No
V1054 Oph | 24.06.2005 | 53546.42356 | 72 | 48 | 8.2024 | 2.3694E+31 | 0.374 | Slow | Yes
V1054 Oph | 24.06.2005 | 53546.48564 | 48 | 36 | 2.6777 | 7.7348E+30 | 0.227 | Slow | No
V1054 Oph | 24.06.2005 | 53546.48717 | 48 | 36 | 5.5471 | 1.6023E+31 | 0.295 | Slow | No
V1054 Oph | 24.06.2005 | 53546.48912 | 36 | 24 | 3.2846 | 9.4881E+30 | 0.242 | Slow | No
V1054 Oph | 24.06.2005 | 53546.49148 | 36 | 24 | 3.4479 | 9.9596E+30 | 0.266 | Slow | No
V1054 Oph | 26.06.2005 | 53548.40336 | 48 | 36 | 1.5149 | 4.3761E+30 | 0.151 | Slow | No
V1054 Oph | 26.06.2005 | 53548.47419 | 36 | 12 | 4.9893 | 1.4412E+31 | 0.421 | Slow | Yes
V1054 Oph | 26.06.2005 | 53548.48919 | 300 | 276 | 10.4920 | 3.0308E+31 | 0.313 | Fast | Yes
V1054 Oph | 02.07.2005 | 53554.36689 | 108 | 60 | 5.4493 | 1.5741E+31 | 0.212 | Slow | Yes
V1054 Oph | 03.07.2005 | 53555.33448 | 564 | 36 | 33.8135 | 9.7675E+31 | 0.209 | Slow | No
V1054 Oph | 03.07.2005 | 53555.33921 | 960 | 624 | 103.7402 | 2.9967E+32 | 0.231 | Slow | No
Table 4: For both fast and slow flares whose rise times are the same, the results obtained from both the regression calculations and the t-Test analyses performed to the mean averages of the equivalent durations ($Log(P_{u})$) versus flare rise times ($Log(T_{r})$) in logarithmic scale are listed. Flare Groups : | Slow Flare | Fast Flare
---|---|---
Data | |
Total Flare Number : | $30$ | $30$
Best Representation Values | |
Slope : | $1.109\pm 0.127$ | $1.227\pm 0.243$
$y$ intercept when $x=0.0$ : | $-0.581\pm 0.226$ | $0.122\pm 0.433$
$x$ intercept when $y=0.0$ : | $0.524$ | $-0.099$
Mean Average of All Y Values | |
Mean Average : | $1.348$ | $2.255$
Mean Average Error : | $0.092$ | $0.126$
Goodness of Fit | |
$r^{2}$ : | $0.732$ | $0.476$
Is slope significantly non-zero? | |
p-value : | $<$ 0.0001 | $<$ 0.0001
Deviation from zero? : | $Significant$ | $Significant$
|
arxiv-papers
| 2012-06-24T18:50:37 |
2024-09-04T02:49:32.206768
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, S. Evren",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.5791"
}
|
1206.5792
|
Rotation Modulations and Distributions of The Flare Occurrence Rates
On The Surface Of Five UV Ceti Type Stars
H. A. Dal and S. Evren
Department Of Astronomy and Space Sciences, University of Ege,
Bornova, 35100 İzmir, Turkey
methods: data analysis — methods: statistical — stars: activity — stars: starspots — stars: flare — stars: individual(AD Leo, EV Lac, V1005 Ori, EQ Peg, V1054 Oph)
In this study, we discuss stellar spots, stellar flares and also the relation between these two magnetic proccess that take place on UV Ceti stars. In addition, the hypothesis about slow flares described by Gurzadian , 1986 will be discussed. All these discussions are based on the results of three years of observations of the UV Ceti type stars AD Leo, EV Lac, V1005 Ori, EQ Peg and V1054 Oph. First of all, the results show that the stellar spot activity occurs on the stellar surface of EV Lac, V1005 Ori and EQ Peg, while AD Leo does not show any short-term variability and V1054 Oph does not exhibits any variability. We report new ephemerides, for EV Lac, V1005 Ori and EQ Peg, obtained from the time series analyses. The phases, computed in intervals of 0.10 phase length, where the mean flare occurence rates get maximum amplitude, and the phases of rotational modulation were compared to investigate whether there is any longitudinal relation between stellar flares and spots. Although, the results show that flare events are related with spotted areas on the stellar surfaces in some of the observing seasons, we did not find any clear correlation among them. Finally, it is tested whether slow flares are the fast flares occurring on the opposite side of the stars according to the direction of the observers as mentioned in the hypothesis developed by Gurzadian , 1986. The flare occurence rates reveal that both slow and fast flares can occur in any rotational phases. The flare occurence rates of both fast and slow flares are varying in the same way along the longitudes for all program stars. These results are not expected based on the case mentioned in the hypothesis.
§ INTRODUCTION
Many samples of UV Ceti type stars posses the stellar spot activity known as BY Dra Syndrome. The BY Dra Syndrome among the UV Ceti stars was first found by Kron , 1952. He reported the existence of sinusoidal-like variations at out-of-eclipses of the eclipsing binary star YY Gem. Kron , 1952 explained this sinusoidal-like variation at out-of-eclipse as a heterogeneous temperature on star surface, which was called BY Dra Syndrome by Kunkel , 1975. This interpretation of BY Dra Syndrome in terms of dark regions of the surface of rotating stars was confirmed, based on more rigorous arguments, by later works of Torres & Ferraz Mello , 1973, Bopp & Evans , 1973, Vogt , 1975, Friedemann & Guertler , 1975.
Since the most of solar flares occur over solar spot regions, in the stellar case it is also expected to find a correlation between the frequency of flares and the effects caused by spots in the light curve. In order to determine a similar relation among the stars, lots of studies have been made using UV Ceti type stars showing stellar spot activity such as BY Dra. One of these studies was reported by Bopp , 1974 on YY Gem. Bopp , 1974 did not find a clear correlation between the location and extent in longitude of flares and spots on YY Gem. Moreover, he notes that the longitudinal extent he derived for a flare-producing region is in good agreement with the longitudinal extents of starspots previously calculated for BY Dra and CC Eri by Bopp & Evans , 1973. In another study, Pettersen et al. , 1983 compared longitudes of the stellar spots obtained from two years observations of YZ CMi and EV Lac with longitudes of flare events and distributions of flare energy and frequencies along the longitude obtained in the same work. Direct comparisons and statistical tests are not able to reveal positive relationships between flare frequency or flare energy and the position of the spotted region. In another extended work, Leto et al. , 1997 looked for whether there is any relation between stellar spots and flares observed from 1967 to 1977 in the observations of EV Lac. The authors were able to find a relation in the year 1970. They could not find any relation in other observing seasons because of the higher threshold of the system used for flare detection. In the last years, García-Alvarez et al. , 2003 found some flares occurring in the same active area with other activity patterns with using simultaneous observations.
Since no correlation is found between stellar spots and flares, a hypothesis about fast and slow flares was put forward. The hypothesis is based on the work named as Fast Electron Hypothesis. According to this hypothesis, the shape of a flare light variation depends on the location of the event on the star surface in respect to direction of observer. If the flaring area is on the front side of the star according to the observer, the light variation shape looks as a fast flare. If the flaring area is on the opposite side of the star according to the observer, the light variation shape looks as a slow flare [Gurzadian , 1965, Gurzadian , 1986]. In addition, Gurzadian , 1988 described two types of flares to model flare light curves. Gurzadian , 1988 indicated that thermal processes are dominant in the processes of slow flares, which are 95$\%$ of all flares observed in UV Ceti type stars. Non-thermal processes are dominant in the processes of fast flares, which are classified as "other" flares. According to Gurzadian , 1988, there is a large energy difference between these two types of flares. Moreover, Dal & Evren , 2010 developed a rule to the classifying of fast and slow flares. When the ratios of flare decay times to flare rise times are computed for two types of flares, the ratios never exceed 3.5 for all slow flares. On the other hand, the ratios are always above 3.5 for fast flares. It means that if the decay time of a flare is 3.5 times longer than its rise time at least, the flare is a fast flare. If not, the flare is a slow flare.
In this paper, the results obtained from Johnson UBVR observations of AD Leo, EV Lac and V1005 Ori will be discussed. Shakhovskaya , 1974 and Bopp & Espanak , 1977 reported that V1005 Ori is a flare star that exhibits rotational modulation due to stellar spots. The authors found an amplitude variation of $0^{m}.08$ with a period of $1^{d}.96$ in V band. Besides, Bopp et al. , 1978 examined photometric data in the time series analyses and found 4 periods for rotational modulation. The period of $1^{d}.858$ is suggested as the most probable period among them. On contrary, in B band observations of 1981, a $4.56\pm0.01$ day period variation with an amplitude of $0^{m}.16$ was found [Byrne et al. , 1984]. No important light curve changes are seen in the years 1996 and 1997, while the minimum phases of rotation modulation is varied from $0^{P}.40$ to $0^{P}.55$. The amplitude of the curves is $0^{m}.10$ in the year 1996, but in the year 1997 it gets larger than the previous ones [Amado et al. , 2001].
In the case of AD Leo, it is a debate issue whether AD Leo has any stellar spot activity, or not. Chugainov , 1974 and Mullan , 1974 show that AD Leo does not exhibit any rotational modulation caused by stellar spots. Besides, Anderson , 1979 found no variations at the $0.02$ magnitude level during the period 1978, May 10 to 17. However, Spiesman & Hawley , 1986 reveals that AD Leo demonstrates BY Dra Syndrome with a period of $2.7\pm0.05$ days. In addition, Panov , 1993 confirmed this period of AD Leo for BY Dra variation.
On the other hand, EV Lac is a well known active star with both high level flare and stellar spot activities. Mahmoud & Oláh , 1981 indicated that the star has no variation caused by rotational modulation in the observations in B band from 1972 to 1976. However, Pettersen , 1980 showed a rotational modulation with a period of $4^{d}.378$ and an amplitude of $0^{m}.07$. Pettersen et al. , 1983, based on continuous observations from 1979 to 1981, renewed the ephemerides of the variation as a period of $4^{d}.375$ and an amplitude of $0^{m}.08$. This indicates that the light curves of EV Lac were almost constant for 2.5 years because the spot groups on the star are stable during these 2.5 years. Using the renewed ephemerides, Kleinman et al. , 1987 found the amplitude of light curve enlarging from $0^{m}.08$ to $0^{m}.16$ in the year 1986. On the other hand, no variation was seen in the light curve of the year 1987. Pettersen et al. , 1992 showed that the spotted area is located in the same semi-sphere on EV Lac for 10 years. Comparing the phases of the light curve minima caused by rotational modulation with the flare frequencies and the distribution of the flare equivalent durations for YZ CMi and EV Lac, Pettersen et al. , 1983 showed that there is no relation between the flare activity and stellar spot activity on these stars.
EQ Peg is classified as a metal-rich star and it is a member of the young disk population in the galaxy [Veeder , 1974, Fleming et al. , 1995]. EQ Peg is a visual binary [Wilson , 1954]. Both components are flare stars [Pettersen et al. , 1983]. Angular distance between components is given as a value between 3$\arcsec$.5 and 5$\arcsec$.2 [Haisch et al. , 1987, Robrade et al. , 2004]. One of the components is 10.4 mag and the other is 12.6 mag in V band [Kukarin , 1969]. Observations show that flares on EQ Peg generally come from the fainter component [Foster , 1995]. Rodonó , 1978 proved that 65$\%$ of the flares come from faint component and about 35$\%$ from the brighter component.
The fourth star in this study is V1054 Oph, whose flare activity was discovered by Eggen , 1965. Norton et al. , 2007 demonstrated that EQ Peg has a variability with the period of $1^{d}.0664$. V1054 Oph (= Wolf 630ABab, Gliese 644ABab) is a member of Wolf star group [Joy , 1947, Joy & Abt , 1974]. Wolf 630ABab, Wolf 629AB (= Gliese 643AB) and VB8 (= Gliese 644C), are the members of the main triplet system, whose scheme is shown in Fig.1 given by Pettersen et al. , 1984. The masses were derived for each components of Wolf 630ABab by Mazeh et al. , 2001. The author showed that the masses are 0.41 $M_{\odot}$ for Wolf 629A, 0.336 $M_{\odot}$ for Wolf 630Ba and 0.304 $M_{\odot}$ for Wolf 630Bb. In addition, Mazeh et al. , 2001 demonstrated that the age of the system is about 5 Gyr.
In this study, for each program stars, we analyse the variations at out-of-flare for each light curves obtained in Johnson UBVR observations, or not. Although all of them show high flare activity, EV Lac, V1005 Ori and EQ Peg exhibit stellar spot activity. On the other hand, the spot activity is not obvious for AD Leo. It is discussed whether AD Leo has any stellar spot activity, or not. Finally, this work do not demonstrate any variation from rotational modulations. To perform this kind of studies we would require a long term observing program. As a part of this study, the phase distributions for both fast and slow flares are examined in terms of the minimum phases of rotational modulation. Thus, hypothesis developed by Gurzadian , 1986 is tested.
§ OBSERVATIONS AND ANALYSES
§.§ Observations
The observations were acquired with a High-Speed Three Channel Photometer attached to the 48 cm Cassegrain type telescope at Ege University Observatory. Observations were grouped in two schedules. Using a tracking star in second channel of the photometer, flare observations were only continued in standard Johnson U band with exposure times between 2 and 10 seconds. The same comparison stars were used for all observations. The second observation schedule was used for determining whether there was any variation out-of-flare. Pausing flare patrol of program stars, we observed them once or twice a night, when they were close to the celestial meridian. Using a tracking star in second channel of the photometer, the observations in this schedule were made with the exposure time of 10 seconds in each band of standard Johnson UBVR system, respectively. There were any delay between the exposure in different filters due to the High-Speed Three Channel Photometer.
Although the program and comparison stars are so close on the sky, differential atmospheric extinction corrections were applied. The atmospheric extinction coefficients were obtained from the observations of the comparison stars on each night. Moreover, the comparison stars were observed with the standard stars in their vicinity and the reduced differential magnitudes, in the sense variable minus comparison, were transformed to the standard system using procedures outlined by Hardie , 1962. The standard stars are listed in the catalogues of Landolt , 1983 and Landolt , 1992. And also, the de-reddened colour of the systems were computed. Heliocentric corrections were also applied to the times of observations. The mean averages of the standard deviations are $0^{m}.015$, $0^{m}.009$, $0^{m}.007$ and $0^{m}.007$ for the observations acquired in standard Johnson UBVR bands, respectively. To compute the standard deviations of observations, we use the standard deviations of the reduced differential magnitudes in the sense comparisons (C1) minus check (C2) stars for each night. There is no variation in the standard brightness comparison stars.
Basic parameters for the stars studied and their comparison (C1) and check (C2) stars.
Stars V (mag) B-V (mag) Spectral Type
AD Leo 9.388 1.498 M3
C1 = HD 89772 8.967 1.246 K6-K7
C2 = HD 89471 7.778 1.342 K8
V1005 Ori 10.090 1.307 K7
C1 = BD+01 870 8.800 1.162 K5
C2 = HD 31452 9.990 0.920 K2
EV Lac 10.313 1.554 M3
C1 = HD 215576 9.227 1.197 K6
C2 = HD 215488 10.037 0.881 K1-K2
EQ Peg 10.170 1.574 M3-M4
C1 = SAO 108666 9.598 0.745 G8
C2 = SAO 91312 9.050 1.040 K3-K4
V1054 Oph 8.996 1.552 M3
C1 = HD 152678 7.976 1.549 M3
C2 = SAO 141448 9.978 0.805 K0
The identities of all programme stars and their comparisons are given in Table 1. All the magnitudes and colour indexes in the table were were taken from Dal & Evren , 2010 and Dal & Evren , 2011.
Observational reports of the program stars for each observing season.
Stars Observing HJD Interval Filter Number
Season (+24 00000) of Night
AD Leo 2004/2005 53377-53514 $UBVR$ 15
2005/2006 53717-53831 $UBVR$ 16
2006/2007 54071-54248 $UBVR$ 13
EV Lac 2004 53202-53240 $UBVR$ 11
2004 53260-53312 $UBVR$ 15
2005 53554-53652 $UBVR$ 18
2006 53863-54058 $UBVR$ 27
V1005 Ori 2004/2005 53353-53453 $UBVR$ 11
2005/2006 53640-53812 $UBVR$ 22
2006/2007 53984-54158 $UBVR$ 15
EQ Peg 2004 53236-53335 $UBVR$ 13
2005 53621 - 53686 $UBVR$ 9
V1054 Oph 2004 53136-53167 $UBVR$ 9
2005 53505 - 53564 $UBVR$ 9
2006 53861 - 53894 $UBVR$ 7
The observation reports of programme stars are given in Table 2. In this table, "observing seasons" are given with HJD intervals. The observing season refer the period of the year in which each star can be seen from the site of observation. In the last column of the table, the number of night refers to total number of nights dedicated to observe the corresponding star in a given observing season.
§.§ The Time Series Analyses
All data sets were analysed with the method of Discrete Fourier Transform (DFT) [Scargle , 1982]. The results obtained from DFT were tested by two other methods. One of them is CLEANest, which is another Fourier method [Foster , 1995], and the second method is the Phase Dispersion Minimization (PDM), which is a statistical method [Stellingwerf , 1978]. These methods confirmed the results obtained by DFT. Although analyses showed a variation in each observation season for EV Lac, V1005 Ori and EQ Peg, analyses do not indicate any variation for both AD Leo and V1054 Oph in each season. All data set obtained in Johnson UBVR bands are given in Table 3 for each star. The star names are given in the first column, the observing seasons are given in the second column and HJDs are in the third column. V band magnitudes, U-B, B-V and V-R colour indexes are given in the next four columns. The number between brackets are the standard deviations in the columns.
The values of U-B colour indexes listed in Table 3 were not use in time series analyses, because the U-B colour indexes are much more sensitive to the flare activity on the surface of the programme stars. When a small flare occurred to detected in respect to the higher threshold, it is not seen any sign in V light, B-V and V-R colours. On the other hand, some distinctive sign is seen in U band light and U-B colour.
The light and colour curves obtained from the observations of V1054 Oph in this study are seen for the seasons 2004 (a), 2005 (b) and 2006 (c). Dashed lines represent the mean brightness and colour level obtained from the time series analyses.
All observation data obtained in Johnson UBVR bands for all program stars. The number between brackets are the standard deviations in the columns.
Star Observing HJD V U-B B-V V-R
Season (+24 00000) (mag) (mag) (mag) (mag)
AD Leo 2004/2005 53377.4886(0.0055) 9.402(0.001) 0.957(0.009) 1.502(0.005) 1.124(0.004)
AD Leo 2004/2005 53381.4822(0.0047) 9.411(0.008) 0.936(0.015) 1.486(0.007) 1.127(0.005)
AD Leo 2004/2005 53412.4029(0.0008) 9.380(0.003) 1.004(0.019) 1.492(0.001) 1.134(0.001)
AD Leo 2004/2005 53413.4187(0.0045) 9.412(0.002) 0.869(0.015) 1.484(0.003) 1.136(0.001)
AD Leo 2004/2005 53444.3612(0.0044) 9.424(0.002) 0.997(0.006) 1.500(0.005) 1.129(0.004)
AD Leo 2004/2005 53446.2663(0.0048) 9.399(0.001) 0.983(0.011) 1.497(0.003) 1.119(0.002)
AD Leo 2004/2005 53465.3122(0.0073) 9.404(0.009) 0.960(0.039) 1.497(0.011) 1.127(0.008)
AD Leo 2004/2005 53469.2653(0.0046) 9.379(0.003) 0.954(0.018) 1.519(0.002) 1.105(0.004)
AD Leo 2004/2005 53470.2777(0.0040) 9.408(0.003) 0.938(0.024) 1.510(0.005) 1.113(0.002)
AD Leo 2004/2005 53471.2669(0.0045) 9.404(0.005) 0.947(0.007) 1.496(0.007) 1.119(0.003)
AD Leo 2004/2005 53499.2888(0.0039) 9.405(0.003) 0.918(0.004) 1.494(0.003) 1.148(0.006)
AD Leo 2004/2005 53500.2896(0.0047) 9.380(0.008) 0.968(0.044) 1.500(0.007) 1.119(0.009)
AD Leo 2004/2005 53502.2878(0.0055) 9.396(0.004) 0.957(0.008) 1.501(0.005) 1.109(0.006)
AD Leo 2004/2005 53505.2925(0.0046) 9.395(0.003) 0.928(0.016) 1.502(0.004) 1.120(0.003)
AD Leo 2004/2005 53514.2952(0.0037) 9.394(0.011) 0.961(0.035) 1.500(0.005) 1.127(0.009)
AD Leo 2005/2006 53717.5037(0.0058) 9.404(0.001) 0.998(0.010) 1.506(0.001) 1.120(0.002)
AD Leo 2005/2006 53744.4914(0.0044) 9.399(0.003) 0.988(0.011) 1.498(0.004) 1.121(0.004)
AD Leo 2005/2006 53757.4158(0.0036) 9.387(0.001) 0.964(0.006) 1.501(0.001) 1.109(0.001)
AD Leo 2005/2006 53763.5181(0.0035) 9.385(0.002) 0.969(0.006) 1.500(0.003) 1.112(0.004)
AD Leo 2005/2006 53764.5537(0.0040) 9.400(0.002) 0.936(0.009) 1.494(0.003) 1.122(0.002)
AD Leo 2005/2006 53765.5831(0.0038) 9.389(0.002) 0.920(0.009) 1.504(0.002) 1.115(0.003)
AD Leo 2005/2006 53769.4748(0.0038) 9.396(0.002) 0.926(0.007) 1.499(0.001) 1.114(0.002)
AD Leo 2005/2006 53771.4091(0.0039) 9.380(0.003) 0.952(0.010) 1.506(0.003) 1.117(0.004)
AD Leo 2005/2006 53782.4414(0.0036) 9.395(0.002) 0.971(0.009) 1.529(0.002) 1.120(0.003)
AD Leo 2005/2006 53787.4247(0.0035) 9.394(0.003) 1.008(0.008) 1.497(0.003) 1.112(0.003)
AD Leo 2005/2006 53788.3733(0.0035) 9.391(0.002) 0.968(0.006) 1.486(0.003) 1.121(0.003)
AD Leo 2005/2006 53812.2796(0.0027) 9.385(0.003) 0.959(0.011) 1.500(0.003) 1.113(0.004)
AD Leo 2005/2006 53816.2875(0.0038) 9.386(0.002) 0.870(0.009) 1.502(0.004) 1.107(0.003)
AD Leo 2005/2006 53821.2994(0.0035) 9.381(0.008) 0.985(0.012) 1.494(0.007) 1.113(0.010)
AD Leo 2005/2006 53827.2550(0.0035) 9.385(0.003) 0.934(0.024) 1.505(0.005) 1.103(0.003)
AD Leo 2005/2006 53831.2530(0.0034) 9.383(0.001) 1.014(0.007) 1.520(0.004) 1.109(0.002)
AD Leo 2006/2007 54071.5791(0.0034) 9.379(0.005) 0.995(0.054) 1.495(0.006) 1.110(0.010)
AD Leo 2006/2007 54085.6298(0.0040) 9.378(0.007) 0.987(0.029) 1.488(0.006) 1.101(0.009)
AD Leo 2006/2007 54093.4845(0.0037) 9.376(0.005) 0.938(0.012) 1.501(0.012) 1.107(0.009)
AD Leo 2006/2007 54097.5495(0.0047) 9.376(0.009) 0.932(0.050) 1.491(0.007) 1.109(0.006)
AD Leo 2006/2007 54114.5302(0.0035) 9.375(0.009) 0.976(0.045) 1.500(0.012) 1.104(0.007)
AD Leo 2006/2007 54115.5539(0.0057) 9.369(0.005) 0.982(0.044) 1.498(0.008) 1.100(0.007)
AD Leo 2006/2007 54122.5021(0.0044) 9.372(0.008) 0.970(0.030) 1.500(0.008) 1.097(0.007)
AD Leo 2006/2007 54138.4519(0.0000) 9.381(0.006) 0.941(0.066) 1.496(0.005) 1.109(0.005)
AD Leo 2006/2007 54227.2863(0.0039) 9.374(0.005) 0.968(0.046) 1.497(0.010) 1.103(0.004)
AD Leo 2006/2007 54232.3069(0.0034) 9.372(0.003) 0.936(0.032) 1.492(0.010) 1.087(0.008)
AD Leo 2006/2007 54236.3290(0.0035) 9.368(0.006) 0.977(0.037) 1.482(0.007) 1.087(0.008)
AD Leo 2006/2007 54237.3158(0.0048) 9.363(0.009) 0.862(0.056) 1.486(0.011) 1.086(0.010)
AD Leo 2006/2007 54248.3084(0.0050) 9.356(0.004) 0.904(0.046) 1.484(0.007) 1.097(0.003)
EV Lac 2004-Set 1 53202.4799(0.0000) 10.271(0.007) 1.253(0.045) 1.577(0.010) 1.163(0.010)
EV Lac 2004-Set 1 53204.4809(0.0009) 10.336(0.006) 1.231(0.019) 1.564(0.008) 1.174(0.002)
EV Lac 2004-Set 1 53207.4523(0.0058) 10.251(0.008) 1.183(0.032) 1.573(0.011) 1.159(0.010)
EV Lac 2004-Set 1 53211.3915(0.0056) 10.276(0.007) 1.228(0.067) 1.553(0.017) 1.178(0.008)
EV Lac 2004-Set 1 53212.3637(0.0014) 10.265(0.010) 1.168(0.056) 1.590(0.017) 1.197(0.014)
EV Lac 2004-Set 1 53226.3835(0.0008) 10.320(0.010) 1.182(0.046) 1.561(0.012) 1.165(0.009)
EV Lac 2004-Set 1 53227.3451(0.0011) 10.304(0.007) 1.236(0.052) 1.582(0.012) 1.166(0.011)
EV Lac 2004-Set 1 53228.3398(0.0009) 10.278(0.013) 1.300(0.062) 1.576(0.019) 1.154(0.012)
EV Lac 2004-Set 1 53232.3968(0.0058) 10.293(0.014) 1.236(0.017) 1.558(0.010) 1.171(0.012)
EV Lac 2004-Set 1 53236.3282(0.0011) 10.292(0.003) 1.210(0.018) 1.574(0.005) 1.177(0.008)
EV Lac 2004-Set 1 53240.3283(0.0011) 10.325(0.011) 1.324(0.013) 1.564(0.017) 1.178(0.009)
EV Lac 2004-Set 2 53260.2571(0.0016) 10.298(0.007) 1.301(0.046) 1.574(0.006) 1.165(0.003)
EV Lac 2004-Set 2 53263.2966(0.0012) 10.300(0.010) 1.299(0.064) 1.558(0.010) 1.176(0.009)
EV Lac 2004-Set 2 53264.2561(0.0012) 10.281(0.017) 1.404(0.062) 1.571(0.019) 1.153(0.015)
EV Lac 2004-Set 2 53265.2587(0.0011) 10.315(0.013) 1.137(0.086) 1.533(0.031) 1.164(0.011)
EV Lac 2004-Set 2 53280.2741(0.0049) 10.306(0.015) 1.259(0.046) 1.564(0.006) 1.152(0.014)
EV Lac 2004-Set 2 53282.4419(0.0058) 10.306(0.011) 1.240(0.062) 1.571(0.016) 1.155(0.008)
EV Lac 2004-Set 2 53287.3673(0.0047) 10.311(0.009) 1.184(0.053) 1.549(0.011) 1.157(0.009)
EV Lac 2004-Set 2 53288.2904(0.0043) 10.328(0.009) 1.355(0.027) 1.582(0.013) 1.161(0.011)
EV Lac 2004-Set 2 53289.2385(0.0047) 10.291(0.008) 1.176(0.052) 1.559(0.014) 1.157(0.010)
EV Lac 2004-Set 2 53302.2839(0.0074) 10.304(0.012) 1.195(0.067) 1.550(0.012) 1.160(0.007)
EV Lac 2004-Set 2 53303.2428(0.0047) 10.284(0.009) 1.226(0.076) 1.549(0.008) 1.166(0.010)
EV Lac 2004-Set 2 53304.3083(0.0076) 10.288(0.011) 0.914(0.086) 1.557(0.019) 1.162(0.010)
EV Lac 2004-Set 2 53309.2371(0.0046) 10.328(0.008) 1.341(0.089) 1.563(0.008) 1.164(0.008)
EV Lac 2004-Set 2 53310.2340(0.0043) 10.327(0.007) 1.305(0.042) 1.561(0.017) 1.156(0.008)
EV Lac 2004-Set 2 53312.2830(0.0047) 10.288(0.009) 1.103(0.076) 1.551(0.016) 1.155(0.011)
EV Lac 2005-Set 3 53554.4680(0.0045) 10.302(0.011) 1.133(0.088) 1.542(0.012) 1.149(0.005)
EV Lac 2005-Set 3 53557.4785(0.0053) 10.330(0.008) 1.198(0.048) 1.549(0.009) 1.163(0.007)
EV Lac 2005-Set 3 53564.4841(0.0040) 10.290(0.006) 1.229(0.064) 1.558(0.007) 1.141(0.005)
EV Lac 2005-Set 3 53566.3986(0.0051) 10.330(0.008) 1.227(0.055) 1.561(0.008) 1.160(0.017)
EV Lac 2005-Set 3 53578.3593(0.0042) 10.312(0.009) 1.185(0.045) 1.567(0.007) 1.137(0.005)
EV Lac 2005-Set 3 53581.5294(0.0038) 10.310(0.008) 1.085(0.014) 1.525(0.007) 1.163(0.006)
EV Lac 2005-Set 3 53584.4970(0.0040) 10.315(0.006) 1.207(0.015) 1.556(0.009) 1.154(0.005)
EV Lac 2005-Set 3 53585.5324(0.0036) 10.297(0.010) 1.293(0.074) 1.545(0.011) 1.159(0.009)
EV Lac 2005-Set 3 53586.3264(0.0008) 10.300(0.004) 1.062(0.016) 1.549(0.003) 1.133(0.010)
EV Lac 2005-Set 3 53597.4090(0.0017) 10.304(0.008) 1.259(0.050) 1.564(0.011) 1.153(0.009)
EV Lac 2005-Set 3 53602.4255(0.0044) 10.314(0.006) 0.532(0.042) 1.540(0.011) 1.167(0.014)
EV Lac 2005-Set 3 53605.3681(0.0038) 10.337(0.011) 1.217(0.018) 1.543(0.019) 1.171(0.007)
EV Lac 2005-Set 3 53606.2978(0.0035) 10.313(0.007) 1.183(0.029) 1.553(0.007) 1.151(0.008)
EV Lac 2005-Set 3 53621.3298(0.0051) 10.296(0.007) 0.859(0.030) 1.555(0.006) 1.152(0.007)
EV Lac 2005-Set 3 53626.2689(0.0037) 10.328(0.001) 1.161(0.046) 1.549(0.003) 1.164(0.002)
EV Lac 2005-Set 3 53641.2634(0.0040) 10.297(0.005) 0.900(0.013) 1.579(0.006) 1.158(0.011)
EV Lac 2005-Set 3 53647.2519(0.0038) 10.303(0.007) 1.138(0.026) 1.540(0.010) 1.161(0.013)
EV Lac 2005-Set 3 53652.2573(0.0037) 10.313(0.009) 1.092(0.042) 1.549(0.009) 1.146(0.010)
EV Lac 2006-Set 4 53863.5667(0.0029) 10.274(0.004) 1.244(0.014) 1.567(0.010) 1.154(0.006)
EV Lac 2006-Set 4 53877.5499(0.0027) 10.285(0.014) 1.092(0.078) 1.571(0.012) 1.152(0.012)
EV Lac 2006-Set 4 53883.5378(0.0038) 10.372(0.006) 1.069(0.062) 1.582(0.007) 1.158(0.006)
EV Lac 2006-Set 4 53886.4627(0.0034) 10.295(0.006) 1.130(0.040) 1.542(0.008) 1.167(0.005)
EV Lac 2006-Set 4 53894.5260(0.0028) 10.267(0.004) 1.215(0.060) 1.559(0.005) 1.147(0.004)
EV Lac 2006-Set 4 53907.5303(0.0036) 10.264(0.005) 1.163(0.034) 1.571(0.009) 1.128(0.008)
EV Lac 2006-Set 4 53908.5431(0.0031) 10.299(0.002) 1.176(0.015) 1.566(0.004) 1.146(0.003)
EV Lac 2006-Set 4 53915.5406(0.0030) 10.306(0.004) 1.125(0.033) 1.538(0.006) 1.152(0.005)
EV Lac 2006-Set 4 53938.5340(0.0038) 10.277(0.006) 1.204(0.048) 1.559(0.006) 1.145(0.007)
EV Lac 2006-Set 4 53940.4593(0.0038) 10.382(0.003) 1.121(0.073) 1.580(0.008) 1.172(0.004)
EV Lac 2006-Set 4 53946.4700(0.0008) 10.260(0.002) 1.030(0.063) 1.550(0.002) 1.135(0.005)
EV Lac 2006-Set 4 53950.4652(0.0008) 10.303(0.005) 1.231(0.043) 1.545(0.008) 1.139(0.007)
EV Lac 2006-Set 4 53951.4800(0.0026) 10.263(0.003) 1.172(0.053) 1.555(0.007) 1.133(0.005)
EV Lac 2006-Set 4 53955.3607(0.0039) 10.253(0.007) 1.258(0.087) 1.579(0.007) 1.135(0.007)
EV Lac 2006-Set 4 53956.3324(0.0034) 10.274(0.011) 0.901(0.033) 1.581(0.012) 1.137(0.009)
EV Lac 2006-Set 4 53961.4980(0.0027) 10.321(0.004) 1.117(0.011) 1.571(0.007) 1.148(0.004)
EV Lac 2006-Set 4 53963.3328(0.0032) 10.322(0.003) 0.931(0.036) 1.546(0.008) 1.160(0.006)
EV Lac 2006-Set 4 53972.3414(0.0035) 10.292(0.006) 1.199(0.012) 1.567(0.006) 1.144(0.011)
EV Lac 2006-Set 4 53973.4465(0.0034) 10.268(0.003) 1.287(0.022) 1.557(0.005) 1.149(0.004)
EV Lac 2006-Set 4 53984.2589(0.0008) 10.390(0.007) 1.146(0.083) 1.562(0.008) 1.179(0.011)
EV Lac 2006-Set 4 53988.2519(0.0027) 10.360(0.003) 1.009(0.044) 1.573(0.007) 1.172(0.007)
EV Lac 2006-Set 4 53992.4226(0.0027) 10.358(0.003) 1.074(0.036) 1.550(0.003) 1.160(0.006)
EV Lac 2006-Set 4 54009.2659(0.0027) 10.314(0.003) 1.158(0.055) 1.571(0.013) 1.157(0.003)
EV Lac 2006-Set 4 54037.2116(0.0037) 10.388(0.005) 1.145(0.056) 1.565(0.012) 1.150(0.009)
EV Lac 2006-Set 4 54047.2046(0.0027) 10.271(0.008) 1.112(0.026) 1.541(0.014) 1.142(0.010)
EV Lac 2006-Set 4 54055.2251(0.0027) 10.326(0.008) 1.159(0.043) 1.553(0.018) 1.152(0.012)
EV Lac 2006-Set 4 54058.2051(0.0000) 10.364(0.002) 1.146(0.088) 1.587(0.007) 1.145(0.007)
V1005 Ori 2004/2005 53353.4006(0.0037) 10.184(0.011) 1.152(0.053) 1.306(0.010) 0.882(0.008)
V1005 Ori 2004/2005 53355.3312(0.0047) 10.253(0.012) 1.232(0.048) 1.313(0.014) 0.892(0.011)
V1005 Ori 2004/2005 53374.3572(0.0030) 10.125(0.007) 1.151(0.044) 1.316(0.009) 0.877(0.008)
V1005 Ori 2004/2005 53376.3710(0.0035) 10.206(0.010) 1.260(0.057) 1.314(0.016) 0.886(0.012)
V1005 Ori 2004/2005 53377.2512(0.0000) 10.246(0.010) 1.207(0.038) 1.319(0.017) 0.897(0.012)
V1005 Ori 2004/2005 53381.3452(0.0038) 10.256(0.007) 1.252(0.052) 1.313(0.016) 0.903(0.011)
V1005 Ori 2004/2005 53383.2424(0.0045) 10.116(0.006) 0.997(0.031) 1.293(0.010) 0.877(0.008)
V1005 Ori 2004/2005 53412.2611(0.0040) 10.274(0.006) 1.212(0.015) 1.319(0.005) 0.917(0.014)
V1005 Ori 2004/2005 53413.2330(0.0042) 10.234(0.004) 1.257(0.075) 1.315(0.008) 0.897(0.009)
V1005 Ori 2004/2005 53414.2662(0.0007) 10.176(0.007) 1.258(0.042) 1.317(0.014) 0.898(0.006)
V1005 Ori 2004/2005 53453.2770(0.0046) 10.189(0.008) 0.830(0.051) 1.331(0.008) 0.894(0.009)
V1005 Ori 2005/2006 53640.5712(0.0039) 10.211(0.005) 1.168(0.046) 1.325(0.008) 0.884(0.006)
V1005 Ori 2005/2006 53647.5820(0.0038) 10.188(0.006) 1.232(0.089) 1.319(0.007) 0.880(0.007)
V1005 Ori 2005/2006 53664.5827(0.0062) 10.181(0.012) 1.179(0.061) 1.306(0.033) 0.883(0.006)
V1005 Ori 2005/2006 53669.4940(0.0046) 10.185(0.004) 1.100(0.052) 1.320(0.010) 0.878(0.005)
V1005 Ori 2005/2006 53672.4511(0.0039) 10.181(0.008) 1.162(0.034) 1.303(0.008) 0.878(0.006)
V1005 Ori 2005/2006 53673.4626(0.0036) 10.191(0.013) 1.132(0.014) 1.329(0.030) 0.872(0.011)
V1005 Ori 2005/2006 53686.5181(0.0040) 10.202(0.003) 1.310(0.025) 1.339(0.007) 0.892(0.007)
V1005 Ori 2005/2006 53702.3409(0.0042) 10.216(0.006) 1.173(0.052) 1.317(0.014) 0.896(0.002)
V1005 Ori 2005/2006 53717.3477(0.0065) 10.213(0.007) 1.150(0.025) 1.338(0.016) 0.910(0.013)
V1005 Ori 2005/2006 53724.3014(0.0040) 10.211(0.002) 1.259(0.045) 1.305(0.005) 0.875(0.004)
V1005 Ori 2005/2006 53725.2527(0.0035) 10.194(0.005) 1.226(0.017) 1.298(0.003) 0.874(0.005)
V1005 Ori 2005/2006 53729.2900(0.0036) 10.207(0.002) 1.209(0.055) 1.306(0.006) 0.892(0.003)
V1005 Ori 2005/2006 53737.2813(0.0046) 10.224(0.004) 1.253(0.021) 1.301(0.002) 0.893(0.004)
V1005 Ori 2005/2006 53744.3067(0.0068) 10.213(0.005) 1.361(0.045) 1.340(0.004) 0.906(0.004)
V1005 Ori 2005/2006 53757.3701(0.0028) 10.184(0.002) 1.313(0.018) 1.308(0.003) 0.884(0.002)
V1005 Ori 2005/2006 53763.2739(0.0042) 10.225(0.004) 1.143(0.013) 1.330(0.005) 0.901(0.005)
V1005 Ori 2005/2006 53764.2990(0.0036) 10.216(0.003) 1.202(0.025) 1.317(0.004) 0.887(0.003)
V1005 Ori 2005/2006 53765.4071(0.0030) 10.207(0.003) 1.195(0.008) 1.302(0.007) 0.907(0.004)
V1005 Ori 2005/2006 53771.3851(0.0039) 10.178(0.004) 1.256(0.035) 1.312(0.008) 0.883(0.002)
V1005 Ori 2005/2006 53788.2932(0.0030) 10.160(0.003) 1.217(0.035) 1.328(0.006) 0.877(0.004)
V1005 Ori 2005/2006 53796.2760(0.0035) 10.197(0.005) 1.177(0.048) 1.309(0.011) 0.887(0.006)
V1005 Ori 2005/2006 53812.2458(0.0030) 10.253(0.005) 1.231(0.052) 1.302(0.006) 0.906(0.003)
V1005 Ori 2006/2007 53984.5749(0.0037) 10.237(0.005) 1.161(0.046) 1.312(0.008) 0.868(0.006)
V1005 Ori 2006/2007 53987.5873(0.0035) 10.185(0.008) 1.341(0.049) 1.312(0.005) 0.880(0.008)
V1005 Ori 2006/2007 53988.5834(0.0038) 10.250(0.011) 1.341(0.010) 1.331(0.011) 0.884(0.013)
V1005 Ori 2006/2007 53994.5802(0.0037) 10.180(0.006) 1.237(0.070) 1.304(0.009) 0.866(0.008)
V1005 Ori 2006/2007 54033.5946(0.0038) 10.259(0.004) 1.146(0.042) 1.313(0.009) 0.895(0.006)
V1005 Ori 2006/2007 54047.4878(0.0038) 10.183(0.010) 1.154(0.084) 1.301(0.026) 0.886(0.008)
V1005 Ori 2006/2007 54049.5410(0.0035) 10.184(0.011) 1.110(0.029) 1.304(0.012) 0.871(0.016)
V1005 Ori 2006/2007 54064.5743(0.0035) 10.265(0.010) 1.225(0.057) 1.319(0.010) 0.890(0.009)
V1005 Ori 2006/2007 54066.3884(0.0035) 10.145(0.017) 1.266(0.070) 1.304(0.009) 0.882(0.010)
V1005 Ori 2006/2007 54071.3212(0.0035) 10.155(0.010) 1.267(0.049) 1.337(0.010) 0.868(0.010)
V1005 Ori 2006/2007 54085.5616(0.0049) 10.225(0.013) 1.289(0.037) 1.331(0.016) 0.868(0.013)
V1005 Ori 2006/2007 54095.4691(0.0037) 10.273(0.007) 1.237(0.017) 1.315(0.007) 0.899(0.006)
V1005 Ori 2006/2007 54109.3354(0.0035) 10.212(0.006) 1.208(0.078) 1.297(0.007) 0.893(0.006)
V1005 Ori 2006/2007 54122.3280(0.0035) 10.225(0.005) 1.221(0.077) 1.312(0.005) 0.884(0.010)
V1005 Ori 2006/2007 54158.2553(0.0038) 10.140(0.013) 1.127(0.023) 1.315(0.018) 0.863(0.014)
V1054 Oph 2004 53136.5478(0.0013) 8.996(0.012) 1.147(0.005) 1.548(0.007)
V1054 Oph 2004 53138.4673(0.0010) 8.991(0.012) 1.141(0.005) 1.542(0.006)
V1054 Oph 2004 53146.5107(0.0012) 8.989(0.008) 1.149(0.005) 1.545(0.006)
V1054 Oph 2004 53151.4962(0.0017) 8.996(0.012) 1.161(0.006) 1.543(0.006)
V1054 Oph 2004 53152.4609(0.0013) 8.996(0.011) 1.161(0.006) 1.548(0.006)
V1054 Oph 2004 53153.5004(0.0017) 8.989(0.008) 1.144(0.006) 1.545(0.007)
V1054 Oph 2004 53157.5139(0.0020) 8.988(0.010) 1.138(0.006) 1.547(0.007)
V1054 Oph 2004 53163.5248(0.0015) 8.995(0.008) 1.142(0.006) 1.545(0.006)
V1054 Oph 2004 53167.4368(0.0015) 8.991(0.008) 1.146(0.005) 1.542(0.005)
V1054 Oph 2005 53505.4608(0.0004) 9.004(0.004) 1.066(0.006) 1.555(0.005) 1.062(0.004)
V1054 Oph 2005 53526.3480(0.0005) 9.000(0.008) 1.074(0.006) 1.548(0.005) 1.064(0.004)
V1054 Oph 2005 53527.3366(0.0004) 8.998(0.009) 1.055(0.006) 1.545(0.005) 1.064(0.004)
V1054 Oph 2005 53528.3373(0.0004) 9.004(0.007) 1.111(0.006) 1.548(0.005) 1.061(0.005)
V1054 Oph 2005 53546.3539(0.0004) 9.000(0.005) 1.120(0.006) 1.551(0.005) 1.060(0.004)
V1054 Oph 2005 53547.3576(0.0004) 9.006(0.008) 1.092(0.007) 1.555(0.006) 1.060(0.005)
V1054 Oph 2005 53548.3520(0.0000) 8.996(0.009) 1.070(0.006) 1.554(0.005) 1.058(0.005)
V1054 Oph 2005 53557.3294(0.0004) 9.003(0.010) 1.056(0.006) 1.545(0.006) 1.066(0.005)
V1054 Oph 2005 53564.3437(0.0018) 8.998(0.007) 1.079(0.005) 1.550(0.004) 1.060(0.004)
V1054 Oph 2006 53861.4662(0.0003) 8.994(0.008) 1.134(0.006) 1.543(0.005) 1.064(0.004)
V1054 Oph 2006 53863.3951(0.0004) 8.989(0.007) 1.145(0.006) 1.556(0.005) 1.064(0.005)
V1054 Oph 2006 53875.3481(0.0004) 8.997(0.006) 1.165(0.005) 1.551(0.004) 1.065(0.004)
V1054 Oph 2006 53877.3435(0.0004) 8.994(0.006) 1.143(0.006) 1.542(0.005) 1.057(0.005)
V1054 Oph 2006 53882.3311(0.0004) 8.986(0.005) 1.187(0.005) 1.556(0.004) 1.061(0.004)
V1054 Oph 2006 53893.3528(0.0003) 8.995(0.005) 1.168(0.005) 1.554(0.004) 1.059(0.004)
V1054 Oph 2006 53894.3664(0.0004) 8.991(0.004) 1.127(0.005) 1.549(0.005) 1.061(0.005)
EQ Peg 2004 53236.3967(0.0011) 10.157(0.008) 0.963(0.014) 1.574(0.013) 1.183(0.012)
EQ Peg 2004 53237.3662(0.0011) 10.129(0.007) 0.933(0.016) 1.598(0.012) 1.130(0.013)
EQ Peg 2004 53259.4116(0.0011) 10.149(0.007) 0.939(0.011) 1.568(0.011) 1.166(0.016)
EQ Peg 2004 53260.4846(0.0008) 10.173(0.006) 1.053(0.015) 1.560(0.016) 1.179(0.013)
EQ Peg 2004 53261.6059(0.0008) 10.144(0.007) 0.947(0.011) 1.576(0.011) 1.160(0.013)
EQ Peg 2004 53263.3286(0.0008) 10.165(0.008) 1.011(0.010) 1.597(0.014) 1.178(0.013)
EQ Peg 2004 53264.3962(0.0023) 10.186(0.008) 0.931(0.016) 1.578(0.016) 1.165(0.013)
EQ Peg 2004 53265.3348(0.0007) 10.197(0.006) 1.079(0.016) 1.567(0.013) 1.172(0.016)
EQ Peg 2004 53280.3039(0.0050) 10.162(0.006) 1.052(0.015) 1.587(0.011) 1.174(0.012)
EQ Peg 2004 53281.2675(0.0022) 10.180(0.008) 1.107(0.017) 1.591(0.010) 1.160(0.013)
EQ Peg 2004 53287.3928(0.0022) 10.142(0.008) 1.070(0.011) 1.581(0.013) 1.157(0.013)
EQ Peg 2004 53289.3472(0.0047) 10.154(0.006) 1.125(0.013) 1.574(0.012) 1.163(0.012)
EQ Peg 2004 53335.2905(0.0010) 10.169(0.008) 0.864(0.012) 1.565(0.014) 1.148(0.013)
EQ Peg 2005 53621.3608(0.0030) 10.144(0.006) 1.021(0.015) 1.561(0.013) 1.155(0.012)
EQ Peg 2005 53622.4013(0.0029) 10.146(0.005) 0.992(0.017) 1.567(0.011) 1.150(0.011)
EQ Peg 2005 53626.2942(0.0012) 10.160(0.005) 1.127(0.013) 1.562(0.012) 1.138(0.012)
EQ Peg 2005 53641.2928(0.0030) 10.163(0.005) 1.127(0.012) 1.554(0.011) 1.158(0.011)
EQ Peg 2005 53652.2815(0.0027) 10.144(0.007) 1.138(0.015) 1.546(0.012) 1.151(0.011)
EQ Peg 2005 53664.2628(0.0034) 10.175(0.005) 1.002(0.013) 1.556(0.012) 1.175(0.011)
EQ Peg 2005 53672.2250(0.0043) 10.153(0.006) 1.094(0.014) 1.571(0.012) 1.137(0.012)
EQ Peg 2005 53673.2791(0.0030) 10.158(0.007) 1.095(0.014) 1.585(0.020) 1.157(0.013)
EQ Peg 2005 53686.2577(0.0028) 10.164(0.006) 1.067(0.015) 1.560(0.012) 1.144(0.011)
The time series analyses do not demonstrate any short-term variation in one season for V1054 Oph. However, the levels of both the brightness in V band and the colours are varying from one season to next one. This is seen from Figure 1. The light and colour curves in this figure are versus HJD instead of phase. This is because there is no known rotational period for V1054 Oph. As it is seen from the figure, there is no data for V-R colour variation in the first observing season of the star, this is because the star was observed in the UBV bands in the first season. Like V1054 Oph, the time series analyses do not reveal any regular variation for AD Leo. For other analyses, the ephemeris given in Equation (1) taken from Panov , 1993 was used in phase calculations for all UBVR observations of AD Leo.
\begin{equation}
\end{equation}
In Figure 2, standard V band light and U-B, B-V and V-R colour curves are shown for three observing seasons. When this Figure is examined, it is clearly seen that some observation points exceed above the level of the standard deviations, but the time series analysis does not give any regular variation. Although AD Leo does not show any regular variation in one observing season, as it is seen from the Figure, level of the brightness in V band is increasing about $0^{m}.01$ from the season 2004/2005 to next one and $0^{m}.02$ from the season 2005/2006 to last season. This is the same in the other bands. The colour curves in Figure 2 show that the stars do no exhibit any distinctive colour variation in a season. On the other hand, V-R colour gets bluer from a season to next one, when the star get brighter.
The light and colour curves obtained from the observations of AD Leo in this study are seen for the seasons 2004/2005 (a), 2005/2006 (b) and 2006/2007 (c). Dashed lines represent the mean brightness and colour level obtained from the time series analyses.
In case of EV Lac, the time series analyses gave almost the same periods changing unsystematically from $4^{d}.331\pm0.037$ to $4^{d}.379\pm0.017$ for each data set of three observing season. When all the data sets were analysed together, the time series analysis gave a period of $4^{d}.3517\pm0.001$ given in Equation (2) for rotational modulation of EV Lac. The periods found from the each sets and the all seasons are similar to those found by Pettersen et al. , 1983 and Mahmoud & Oláh , 1981.
It must be noted that we could not observed EV Lac for a while, about 20 days, in the middle of the seasons 2004. When we started to observe the star after 20 days later, we saw that the light curves had been partly changed. Therefore, the data set was divided in two parts as Set 1 and Set 2 to solve this problem. We think that this is because of the rapid variations of active areas on the stellar surface of EV Lac. We did not see any similar variations on the other program stars. For example, we could not observed AD Leo for almost 100 days, in the middle of some seasons, but we saw that the light curves had not been changed yet. This is why we did not divided the data sets of AD Leo into two or more parts. This is the same for other stars.
The light and colour curves obtained in this study are seen for four data sets composed from the observations of EV Lac. a) Observing season 2004 - the first part, b) Observing season 2004 - the second part, c) Observing season 2005, d) Observing season 2006. Dashed lines represent the models derived from the Discrete Fourier Transform.
\begin{equation}
\end{equation}
Using the ephemeris given in Equation (2), the phases were computed for all data sets. V band light and U-B, B-V and V-R colour curves are shown for four data sets in Figure 3. For V band of EV Lac, the variations of minimum, maximum and mean average of brightness, amplitude and periods found from each data set are given in Figure 4. The time series analyses also show that minimum phases of the curves are changing. It is at about $0^{P}.62$ for Set 1 and 2, $0^{P}.54$ for Set 3, and $0^{P}.60$ for Set 4. Comparing the colour curves with light curves for Set 1, it is seen that B-V and V-R colours are getting bluer towards the phase of minimum seen in the light curve. There is no variation in colour curves for data Set 2 and 3. On the other hand, both the colours are getting reddening towards the minimum phases of light curves in the last season.
The figure shows the results of the analyses for 4 data sets of EV Lac in V band. The variations of minimum, maximum and mean average levels of brightness are shown. In the figure, open circles show minimum and maximum levels, while filled circles show the mean average of the brightness for each data set. Dashed lines represent the fits of brightnesses to just show the variations.
V1005 Ori was observed in three observing seasons. Because of the winter weather conditions, some observation could not be carried on. This is why there are some empty phases in the light curve of the season of 2004/2005. However, the data obtained is enough for the time series analysis. The time series analyses gave almost the same periods changing unsystematically from $4^{d}.419\pm0.005$ to $4^{d}.429\pm0.014$ for each data set of three observing season. The periods found in this study are close to the period found by Byrne et al. , 1984. When all the data sets were analysed together, the time series analysis gave a period of $4^{d}.4236\pm0.001$ given in Equation (3) for rotational modulation of V1005 Ori.
The light and colour curves obtained from the observations of V1005 Ori in this study are seen for the seasons 2004/2005 (a), 2005/2006 (b) and 2006/2007 (c). Dashed lines represent the models derived from the Discrete Fourier Transform.
\begin{equation}
JD~(Hel.)~= ~24~53353.40036~+~4^{d}.4236~\times~E.
\end{equation}
Using the ephemeris given in Equation (3), the phases were computed for all data sets of V1005 Ori. Standard V band light and U-B, B-V and V-R colour curves are shown for four data sets in Figure 5. For V band of V1005 Ori, the variations of minimum, maximum and mean average of brightness, amplitude and periods for each season are given in Figure 6. When the light curves of V1005 Ori are examined, the phase of minimum was about $0^{P}.34$ for the season of 2004/2005 and it was $0^{P}.66$ for the season of 2006/2007. Although the amplitude of the curve for the season of 2005/2006 was the smallest among the previous and the latter, the analyses indicate that the phase of minimum might be $0^{P}.78$. It is seen that there is no variation in the B-V colour index for the first season, while there is reddening in the V-R colour index towards the minimum phase of the light curves. On the other hand, for the season of 2005/2006, although the light curves in Johnson UBVR bands have the smallest amplitudes, an extreme excess is seen in the B-V colour. Moreover, the B-V colour is getting bluer towards the minimum phase of the light curves. The V-R colour is reddening towards the minimum phase of the light curve in this season. In the last season, both B-V and V-R colours are reddening towards the minimum phase of the light curves.
The figure shows the results of the analyses for 3 data sets of V1005 Ori in V band. The variations of minimum, maximum and mean average levels of brightness are shown. In the figure, open circles show minimum and maximum levels, while filled circles show the mean average of the brightness for each data set. Dashed lines represent the fits of brightnesses to just show the variations.
In case of EQ Peg, the time series analyses demonstrated that EQ Peg has a variability. The analyses gave almost the same period as $1^{d}.0608\pm0.0001$ for data set combined from two observing seasons. Using the ephemeris given in Equation (4), the phases were computed for all data sets of EQ Peg.
\begin{equation}
JD~(Hel.)~= ~24~53236.39669~+~1^{d}.0608~\times~E.
\end{equation}
Standard V band light and U-B, B-V and V-R colour curves are shown for four data sets in Figure 7. In the figure, the data sets of season 2004 and season 2005 were not separated into two panel. This is because the light curves of both seasons have the same shape. The time series analyses showed that the phases of the rotational modulation is $0^{P}.32$ for the light curves of EQ Peg. Instead of light curves, the data set combined from EQ Peg observations do not exhibits any variations.
The light and colour curves obtained from the observations of EQ Peg in this study are seen together for the seasons 2004 and 2005. Dashed lines represent the models derived from the Discrete Fourier Transform.
§.§ The Distributions of Flare Occurrence Rates Versus Photometric Period
These stars, for which BY Dra Syndrome is discussed, show also high level flare activity. To investigate whether there is any relation between stellar spots and flare activities, the rates of flare occurrence versus rotational phase and the phases of the minima of the V band light curves were compared for each star. V band light curves for each star were obtained in this study. The flare data of the stars have been given by Dal & Evren , 2010 and Dal & Evren , 2011.
To carry out this investigation, first of all, the phases of flare maxima were computed for all flare types (together with fast and slow flares) with the same method used for the phases of light curves. The flare maximum times were used to compute the phases due to main energy emitting in this part of the flare light curves. In addition, the periods of stars are too long according to the average of flare total durations. In the second step, computing the ratio of flare number to monitoring time in intervals of 0.10 phase length as the same method used by Leto et al. , 1997, the mean flare occurence rate was derived for data set of each star with using Equation (5).
\begin{equation}
N~=~\Sigma n_{f}~/~\Sigma T_{t}
\end{equation}
where $N$ is the mean flare occurence rate in intervals of 0.10 phase length. $n_{f}$ is flare number in that intervals. $T_{t}$ is total monitoring time in that interval. All the parameters are listed in Table 4. In the table, stars' names are given in the first column, observing seasons are listed in the second column. The average phases, total monitoring time in intervals of 0.10 phase length, the number of flares detected in that interval and the mean flare occurence rates in that interval are listed in the next columns,respectively.
The report of the mean flare occurence rate for all flares (together with fast and slow flares) for all program stars.
Observing Observing Average Observing Flare Flare
Star Season Phase Duration (h) Number Occur. Rate ($h^{-1}$)
AD Leo 2004-2007 0.05 9.17 11 1.200
AD Leo 2004-2007 0.15 2.74 5 1.825
AD Leo 2004-2007 0.35 6.73 15 2.229
AD Leo 2004-2007 0.45 4.60 12 2.610
AD Leo 2004-2007 0.55 15.65 23 1.470
AD Leo 2004-2007 0.65 8.19 18 2.198
AD Leo 2004-2007 0.75 9.92 19 1.915
AD Leo 2004-2007 0.85 5.11 6 1.174
AD Leo 2004-2007 0.95 2.40 3 1.250
EV Lac 2004 0.05 3.88 3 0.773
EV Lac 2004 0.15 3.62 2 0.553
EV Lac 2004 0.25 8.20 4 0.488
EV Lac 2004 0.45 0.91 1 1.099
EV Lac 2004 0.55 4.19 4 0.955
EV Lac 2004 0.75 5.17 7 1.354
EV Lac 2004 0.85 3.53 3 0.850
EV Lac 2004 0.95 8.12 7 0.862
EV Lac 2005 0.25 1.58 2 1.266
EV Lac 2005 0.45 1.88 4 2.128
EV Lac 2005 0.55 4.73 8 1.690
EV Lac 2005 0.75 4.36 7 1.606
EV Lac 2005 0.85 4.63 5 1.080
EV Lac 2005 0.95 4.74 5 1.055
EV Lac 2006 0.05 3.51 3 0.855
EV Lac 2006 0.15 2.35 2 0.850
EV Lac 2006 0.25 5.01 8 1.597
EV Lac 2006 0.35 3.65 7 1.918
EV Lac 2006 0.45 2.01 3 1.493
EV Lac 2006 0.55 2.17 3 1.382
EV Lac 2006 0.65 2.10 3 1.429
EV Lac 2006 0.85 1.33 1 0.752
EV Lac 2006 0.95 9.54 5 0.524
V1005 Ori 2005-2006 0.05 3.78 7 1.850
V1005 Ori 2005-2006 0.25 2.48 4 1.613
V1005 Ori 2005-2006 0.35 7.79 6 0.770
V1005 Ori 2005-2006 0.65 1.42 3 2.113
V1005 Ori 2005-2006 0.85 4.45 11 2.472
EQ Peg 2004-2005 0.05 16.58 8 0.483
EQ Peg 2004-2005 0.15 6.87 3 0.437
EQ Peg 2004-2005 0.35 14.69 3 0.204
EQ Peg 2004-2005 0.45 17.87 9 0.504
EQ Peg 2004-2005 0.55 22.31 10 0.448
EQ Peg 2004-2005 0.65 14.01 3 0.214
EQ Peg 2004-2005 0.75 32.90 17 0.517
EQ Peg 2004-2005 0.85 30.60 7 0.229
EQ Peg 2004-2005 0.95 19.59 13 0.664
The flare data obtained from EV Lac, EQ Peg, V1005 Ori and AD Leo were used for this analysis since the photometric periods are known for these stars. Total monitoring time of AD Leo is 79.11 $h$, while it is 109.63 $h$ for EV Lac. Total monitoring time of EQ Peg is 100.26$h$. In the case of V1005 Ori, it is 44.75 $h$. V1005 Ori and EQ Peg were observed in two seasons, while others were observed in three seasons. When the phase intervals of flare monitoring are examined, it is seen that flare monitoring was acquired almost every phase intervals. On the other hand, no flare was detected in some monitoring intervals. This is not unexpected case. This is because of the nature of UV Ceti type stars and flare processes. Because the flare activity is not a periodic or cyclic variation and it is hard to predict when a flare occur. This is why we did not detect any flare in some observations.
The mean flare occurence rates versus rotational phase are demonstrated for all AD Leo flares observed in three seasons. In the figures, the lines show the histograms of mean flare occurence rates computed in intervals of 0.10 phase length.
To carry out the investigation, the seasons, in which large number of flares is detected, were chosen for each star and these seasons were only used in analyses. Thus, the distributions of detected flares can be covered almost each phase interval for each star. Nevertheless, it was seen that there was no flare some phases interval. Using the Least-Squares Method, all the histograms of $N$ were analysed with the SPSS V17.0 [Green et al. , 1999] software to determine the phase in which Maximum Flare Occurence Rates (hereafter MFOR) are seen. As it is seen in the figures, the histograms are not sometimes shown from $0.0$ to $1.0$ in phase. This is because the histograms and models are shown in the best view.
The mean flare occurence rates versus rotational phase are demonstrated for EV Lac flares observed in three seasons. In the figures, the lines show the histograms of mean flare occurence rates computed in intervals of 0.10 phase length. The histograms are for the season 2004 (a) 2005 (b) and the season 2006 (c).
AD Leo was observed for three seasons. 119 U band flares were detected in 79.61 $h$ monitoring time. The analyses do not reveal any variability out-of-flare. Because of this, we combine all 119 flares detected in three seasons and we derived one histogram for all of them. The histogram of the mean flare occurence rates versus rotational phase for AD Leo flares are shown in Figure 8. In the figures, the mean flare occurence rates versus rotational phase are demonstrated by histogram. As it is seen from the analyses of the histogram in Figure 8, the phase of MFOR is about $0^{P}.45$ for detected flares.
The mean flare occurence rates versus rotational phase are demonstrated for V1005 Ori flares observed in the season 2005/2006. In the figures, the lines show the histograms of mean flare occurence rates computed in intervals of 0.10 phase length.
EV Lac was also observed for three seasons. Although monitoring time is 109.63 $h$, but 93 U band flares were detected. On the other hand, the distributions of the flares are covered almost all phases for each observing season. Because of this, we derived histograms of the distributions of the flares versus phases for each season. The histograms are shown for EV Lac flares in Figure 9. EV Lac was observed in the seasons 2004, 2005 and 2006. According to the light curves of the rotational modulation for EV Lac, the two data sets were obtained for the season 2004. The flares were not separated into two sets for analyses. This is because the minimum phases of rotational modulations for Set 1 and 2 are the same. The histograms also exhibit the same distributions, when the flares observed in 2004 are separated into two sets. Thus, one histogram is shown in Figure 9 for the season 2004. As it is seen from the analyses of the histogram in Figure 9, the phases of MFOR are $0^{P}.75$ for 2004, $0^{P}.45$ for 2005 and $0^{P}.35$ for 2006.
The mean flare occurence rates versus rotational phase are demonstrated for EQ Peg flares observed in the seasons 2004 and 2005. In the figure, the lines show the histograms of mean flare occurence rates computed in intervals of 0.10 phase length.
Flare patrol of V1005 Ori was continued in the seasons of 2004/2005 and 2005/2006. 44 U band flares were detected in total 44.75 $h$ monitoring time. When we checked the phase distributions of detected flares for each season, it is seen that the phase distribution of flares detected in the season 2004/2005 is not covered all phases very well. Because of this, we only used the data of the season 2005/2006 in the analyses. The same histogram is shown in Figure 10 for the season 2005/2006 of V1005 Ori. As it is seen from the analyses of the histogram in Figure 10, the flare occurence rate of detected flares reaches maximum value in about $0^{P}.88$.
In the case of EQ Peg, observations were acquired for two seasons. Because of the similarity of the light curves in seasons 2004 and 2005, all flares of EQ Peg were analysed together and derived histogram is shown in Figure 11. The analyses showed that the phase of MFOR is about $0^{P}.95$
§.§ Distribution of The Fast and Slow Flares According to Each Other
In order to test the hypothesis discussed by Gurzadian , 1986, according to the rule described by Dal & Evren , 2010, the observed flares were separated into two classes as slow and fast flares. For each program star, the data sets of both slow and fast flares were composed for each observation season. Then, the same analyses mentioned above were derived with these sets to compare distribution of both flare types according to each other.
All the parameters are listed in Table 5 for the fast flares and in Table 6 for the slow flares. All the columns of these tables are the same with Table 4. It is important to note that given values of "total monitoring time in intervals of 0.10 phase length" in these tables could be slightly different from each other and from the values given in Table 4. This is because this parameter was separately computed for each flare types and for all flares (together with fast and slow flares) for each program star.
The report of the mean flare occurence rate for the fast flares for all program stars.
Observing Observing Average Observing Flare Flare
Star Season Phase Duration (h) Number Occur. Rate ($h^{-1}$)
AD Leo 2004-2006 0.05 9.09 3 0.330
AD Leo 2004-2006 0.35 6.73 5 0.743
AD Leo 2004-2006 0.45 3.02 2 0.662
AD Leo 2004-2006 0.55 7.03 5 0.711
AD Leo 2004-2006 0.65 2.07 2 0.964
AD Leo 2004-2006 0.75 5.41 4 0.740
AD Leo 2004-2006 0.95 6.40 3 0.469
EV Lac 2004 0.05 2.81 1 0.356
EV Lac 2004 0.15 2.21 1 0.452
EV Lac 2004 0.35 1.79 1 0.559
EV Lac 2004 0.45 1.71 1 0.584
EV Lac 2004 0.55 10.60 3 0.283
EV Lac 2004 0.55 5.35 1 0.187
EV Lac 2006 0.15 4.06 2 0.493
EV Lac 2006 0.25 5.01 2 0.399
EV Lac 2006 0.45 2.33 1 0.429
EV Lac 2006 0.55 2.17 1 0.461
EV Lac 2006 0.65 2.06 1 0.486
EV Lac 2006 0.95 1.59 1 0.629
V1005 Ori 2005-2006 0.05 5.78 2 0.346
V1005 Ori 2005-2006 0.25 2.48 2 0.806
V1005 Ori 2005-2006 0.35 2.96 2 0.675
V1005 Ori 2005-2006 0.65 1.42 2 1.408
V1005 Ori 2005-2006 0.85 4.45 6 1.348
EQ Peg 2004-2005 0.05 11.52 5 0.434
EQ Peg 2004-2005 0.15 4.20 2 0.476
EQ Peg 2004-2005 0.35 11.13 2 0.180
EQ Peg 2004-2005 0.45 14.31 3 0.210
EQ Peg 2004-2005 0.55 22.31 4 0.179
EQ Peg 2004-2005 0.65 14.01 3 0.214
EQ Peg 2004-2005 0.75 32.90 10 0.304
EQ Peg 2004-2005 0.85 22.36 4 0.179
EQ Peg 2004-2005 0.95 19.59 9 0.459
The report of the mean flare occurence rate for the slow flares for all program stars.
Observing Observing Average Observing Flare Flare
Star Season Phase Duration (h) Number Occur. Rate ($h^{-1}$)
AD Leo 2004-2005 0.05 10.36 8 0.772
AD Leo 2004-2005 0.15 2.74 5 1.825
AD Leo 2004-2005 0.35 5.03 10 1.990
AD Leo 2004-2005 0.45 6.31 17 2.694
AD Leo 2004-2005 0.55 9.94 17 1.710
AD Leo 2005-2006 0.65 6.68 9 1.347
AD Leo 2005-2006 0.75 9.92 15 1.512
AD Leo 2005-2006 0.85 5.11 3 0.587
AD Leo 2005-2006 0.95 2.23 1 0.448
EV Lac 2004 0.05 3.88 2 0.515
EV Lac 2004 0.15 2.21 1 0.452
EV Lac 2004 0.25 3.03 2 0.659
EV Lac 2004 0.35 2.69 2 0.743
EV Lac 2004 0.45 0.91 1 1.099
EV Lac 2004 0.55 1.08 1 0.926
EV Lac 2004 0.75 5.17 3 0.580
EV Lac 2004 0.85 3.53 2 0.567
EV Lac 2004 0.95 8.13 3 0.369
EV Lac 2006 0.05 3.51 3 0.855
EV Lac 2006 0.25 5.01 6 1.198
EV Lac 2006 0.35 4.94 7 1.417
EV Lac 2006 0.45 1.70 2 1.176
EV Lac 2006 0.55 2.17 2 0.922
EV Lac 2006 0.65 2.10 2 0.952
EV Lac 2006 0.85 1.33 1 0.752
EV Lac 2006 0.95 9.55 4 0.419
V1005 Ori 2005-2006 0.05 2.45 3 1.222
V1005 Ori 2005-2006 0.25 2.48 2 0.806
V1005 Ori 2005-2006 0.35 7.80 4 0.513
V1005 Ori 2005-2006 0.65 1.26 1 0.792
V1005 Ori 2005-2006 0.85 4.45 5 1.124
V1005 Ori 2005-2006 0.95 2.06 2 0.970
EQ Peg 2004-2005 0.05 12.40 3 0.242
EQ Peg 2004-2005 0.15 2.67 1 0.375
EQ Peg 2004-2005 0.35 3.56 1 0.281
EQ Peg 2004-2005 0.45 17.86 6 0.336
EQ Peg 2004-2005 0.55 15.31 6 0.392
EQ Peg 2004-2005 0.75 25.09 7 0.279
EQ Peg 2004-2005 0.85 10.68 3 0.281
EQ Peg 2004-2005 0.95 16.60 4 0.241
The histograms of the mean flare occurrence rates versus rotational phase are for both slow and fast flares of AD Leo. a) Fast flares observed in three seasons. b) Slow flares observed in three seasons.
All the flares of AD Leo detected in three seasons were again combined for this analysis. The same histograms were derived for both the fast and slow flares of AD Leo. They are shown in Figure 12. As it is seen from the analyses of the histogram in the figure, the phase of MFOR is $0^{P}.58$, while it is $0^{P}.41$ for the slow flares. There is a difference of $0^{P}.17$ between two types. According to Gurzadian , 1986, it is expected that there should be a difference of $0^{P}.50$ between them.
The histograms of the mean flare occurrence rates versus rotational phase are for both slow and fast flares of EV Lac. a) Fast flares of the season 2004. b) Slow flares of the season 2004. c) Fast flares of the season 2006. d) Slow flares of the season 2006.
In the case of EV Lac, it was seen that the phase distribution of the fast flares is not enough to compare it with slow flares for the season 2005. This is must be because the frequency of the fast flares is not as high as that of the slow flares, as mentioned by Dal & Evren , 2010. We only compared them for the season 2004 and 2006. All histograms of EV Lac are shown in Figure 13. The phase of MFOR for the fast flares is $0^{P}.45$, while it is $0^{P}.49$ for the slow flares in the season 2004. The difference between the phases of MFOR is $0^{P}.04$ for two types in this season. The phase of MFOR is $0^{P}.87$ for the fast flares, while it is $0^{P}.36$ for the slow flares in the season 2006. The difference between the phases of MFOR is $0^{P}.51$ for both types in the season 2006, as expected.
The histograms of the mean flare occurrence rates versus rotational phase are for both slow and fast flares of V1005 Ori. a) Fast flares of the season 2005/2006. b) Slow flares of the season 2005/2006.
In the case of V1005 Ori, comparison could be done for the season 2005/2006. The histograms of V1005 Ori are shown in Figure 14. As it is seen from the analyses of the histogram in the figure, the phase of MFOR for the fast flares is $0^{P}.64$, while it is $0^{P}.95$ for the slow flares. There is a difference of $0^{P}.31$ between two types.
The histograms of the mean flare occurrence rates versus rotational phase are for both slow and fast flares of EQ Peg. a) Fast flares of the seasons 2004 and 2005. b) Slow flares of the seasons 2004 and 2005.
The same comparison was done for the seasons 2004 and 2005 for EQ Peq. The histograms of EQ Peg are shown in Figure 15. As it is seen from the analyses of the histogram in the figure, the phase of MFOR for the fast flares is about $0^{P}.15$, while it is $0^{P}.55$ for the slow flares. There is a difference of $0^{P}.40$ between two types. This value is an acceptable value and close to the expected value according to the hypothesis discussed by Gurzadian , 1986.
§ RESULTS AND DISCUSSION
§.§ Stellar Rotational Modulation and Stellar Spot Activity
Most of the UV Ceti type stars are full convective red dwarfs with sudden-high energy emitting. As it can be seen in the literature, BY Dra Syndrome at out-of-flares is seen in a few stars among 463 flare stars catalogued by Gershberg et al. , 1999. EV Lac and V1005 Ori can be given as two examples because the studies in the literature and this study indicate that both stars show the variation due to rotational modulation at out-of-flares.
In the case of EV Lac, the time series analyses show that the period of rotational modulation found for each data set is range from $4^{d}.330$ to $4^{d}.378$. The periods found are similar to those found by Pettersen et al. , 1983 and Mahmoud & Oláh , 1981. Although the periods found for each season are a little bit different, this difference is relatively small. When the amplitudes of the light curves are examined for EV Lac, the amplitude of this variation was dramatically decreasing from the year 2004 to 2005, while the amplitude was clearly larger than ever in this study. However, the mean average of brightness in the light curves was slowly decreasing from the year 2004 to 2006. The minima phases of the light curves for the three seasons were computed and, it was found as $0^{P}.62$ for the season 2004, $0^{P}.54$ for 2005 and $0^{P}.60$ for 2006.
In the case of V1005 Ori, the periods of the rotational modulation for each season are range from $4^{d}.419$ to $4^{d}.429$. In the literature, Bopp et al. , 1978 found four possible periods varied from $1^{d}.883$ to $2^{d}.199$. On the other hand, Byrne et al. , 1984 found a period of $4^{d}.565$. As it is seen, the periods found in this study are close to the period found by Byrne et al. , 1984. When the amplitudes of the light curves were examined, the amplitude observed in the season of 2004/2005 was so smaller than the ones observed in the previous and later seasons that there was no minimum in the light curve. Although the mean average of brightness in the light curves was not changing, the minimum phases of the light curves were varying. The minimum phase of the curve for the season 2004/2005 was about $0^{P}.34$ and about $0^{P}.66$ for the season 2006/2007. It is hard to say that the minimum phase of the light curves for the season 2005/2006 was about $0^{P}.78$.
The case of AD Leo is different from the other two stars. The time series analyses do not show any regular variation over the $3\sigma$ level in one season. On the other hand, the mean brightness levels were increased a value of $0^{m}.01$ from the first season to the second and a value of $0^{m}.02$ from the second to the last season. This can be because of the stellar polar spots. If the literature is considered, the stellar spots can be carried to polar regions in the case of rapid rotation in the young stars [Schüssler & Solanki , 1992, Schüssler et al. , 1996]. According to Montes et al. , 2001, AD Leo is at the age of 200 Myr. The range of equatorial rotational velocity ($vsini$) given in the literature is between $5$ - $5.8$ and $9.0$ $kms^{-1}$ for AD Leo [Marcy & Chen , 1992, Pettersen , 1991]. Besides, considering these values of $vsini$, the real rotational velocities must be larger than these values. If both the age and equatorial rotational velocity value parameters in these papers are considered, according to Schüssler & Solanki , 1992 and Schüssler et al. , 1996, some spots might be located on the polars for AD Leo. In fact, Pettersen et al. , 1992 indicate that BY Dra had spotted area near polar region, which was stable for 14 years and EV Lac has a similar area for 10 years. If the studies made by Spiesman & Hawley , 1986 and Panov , 1993 are considered, AD Leo might sometimes show rotational modulation due to the spotted area occurring near the equatorial regions. On the other hand, there is another probability. If the colour index of V-R is considered, it is seen that the star gets bluer from a season to next one, when the star get brighter. Besides, no amplitude is seen in the light curves. These can be some indicators that all the surface of the star is covered by cool spots and the efficiency of the spots gets weaker from one season to next one.
The colour curves of both EV Lac and V1005 Ori sometimes exhibit a clear colour excess around the minimum phases of the light curves for some observing seasons. This can be an indicator of some bright areas such as faculae on the surface of these young stars. The effects of the bright areas such as faculae can be seen in the variations of B-V and sometimes V-R colour, while these effects are not seen in the variations in the light curves of BVR bands due to cool spots. The cool spots are more efficient in the light curves of B, especially V and R bands. The same effect is seen in the variations caused by the flare activity. Although there is some small effects or no effect of the flare activity in V and R bands, but there is some clear variations in U band light and U-B colour curves. In this study, we observed the stars in U band to investigate the variations out-of-flares. In a sense, U band observations is used to control whether there is any flare activity in the observing durations. If there is some variations in U band light and U-B colour, we did not used the observation to investigate the variation out-of-flares.
Norton et al. , 2007 showed that EQ Peg has a variability with the period of $1^{d}.0664$. In this study, the time series analyses supported this period. According to our analyses, EQ Peg exhibits short-term variability with the period of $1^{d}.0608$. However, it is seen that there is not any variability in the colour indexes. Analysing the light curve of EQ Peg, it was found as $0^{P}.32$ for the minimum phase of the rotational modulation.
§.§ The Relation Between Stellar Spot and Flare Activities
There are many studies about whether the flares of UV Ceti type stars showing BY Dra Syndrome are occurring at the same longitudes of stellar spots, or not. Having the same longitudes of flare and spots is an expected case for these stars, because solar flares are mostly occurring in the active regions, where spots are located on the Sun [Benz & Güdel , 2010]. In the respect of Stellar-Solar Connection, a result of the $Ca$ $II$ $H\&K$ Project of Mount Wilson Observatory [Wilson , 1978, Baliunas et al. , 1995], if the areas of flares and spots are related on the Sun, the same case might be expected for the stars. In fact, Montes et al. , 1996 have found some evidence to demonstrate this relations. Besides, Leto et al. , 1997 have found a variations of both the rotational modulation and the phase distribution of flare occurence rates in the same way for the observations in the year 1970. On the other hand, no clear relation between stellar flares and spots has been found by Bopp , 1974, Pettersen et al. , 1983. However, Pettersen et al. , 1983 did not draw firm conclusions because of being a non-uniqueness problem.
In this study, the flare occurence rates, the ratio of flare number to monitoring time, were computed in intervals of 0.10 phase length as the same method used by Leto et al. , 1997 with just one difference. The flare maximum times were used to compute the phases due to main energy emitting in this part of the flare light curves. We observed AD Leo for 79.61 $h$ and detected 119 flares in three seasons. EV Lac was observed 109.63 $h$ and 93 flares were detected in three seasons. V1005 Ori was observed for 44.75 $h$ and 44 flares were detected in two seasons. EQ Peg was observed for 100.26 $h$ and 73 U band flare were detected. Since no rotational modulation was found to compare for AD Leo, all the flares detected in three season were combined in order to just find whether there is any phase, in which the flare occurence rate gets a peak. On the other hand, we examined flare phase distributions for each season for both EV Lac and V1005 Ori. In the case of these stars, if the distribution of flares did not cover almost all phases in an observing season of a star, the season is neglected for the comparison of flare and spot activity. Consequently, for both EV Lac and V1005 Ori, we chose the seasons, in which the best flare distributions were obtained. Thus, we only used the seasons, in which there is enough data to get reliable conclusions about flare occurrence distributions. In addition, to determine the phases of MFOR, all the distributions were modelled with the polynomial function. Resolving these models, maximum flare occurrence rates and their phase were found for all program stars.
In the case of EV Lac, no relation is seen between the minimum phase of the rotational modulation and the phase, in which flare activity reaches the MFOR. The minimum phase of the rotational modulation observed in the season 2004 is $0^{P}.62$, while the phase of MFOR is $0^{P}.75$. The minimum phase of rotational modulation is $0^{P}.54$, while the flare occurrence rate reaches maximum level in about the phase of $0^{P}.45$ for the season 2005. In the last season of EV Lac, rotational modulation minimum is seen in $0^{P}.60$, as MFOR is in $0^{P}.35$. In the case of V1005 Ori, there is enough data in only one season to compare. As it is seen, the minimum phase of rotational modulation is $0^{P}.78$, while phase of MFOR is about $0^{P}.87$ for the season 2005/2006. In this study, the time series analyses indicated that AD Leo does not have any rotational modulation. Therefore, any minimum time could not have been determined from the observations of three seasons for AD Leo. Because of this, we could not compare the rotational modulation with flare activity in the case of AD Leo. On the other hand, using combined data of three seasons, we found that the MFOR is seen in $0^{P}.45$. This phases was computed with using the ephemeris given in Equation (1) taken from Panov , 1993. The time series analyses do not show any short-term variation in the light curves of AD Leo. Because of this, we waited that there is no any phase, in which the flare activity gets higher levels. On the other hand, as it is seen from the histogram and its Normal Gaussian model for AD Leo, there is a phase for MFOR. Considering the phase of MFOR, the active region(s) in some particular part of the surface can be more active than the others on the surface of the star. Considering the light and colour curves of AD Leo, almost all surface of the star may be covered by stellar spots, while it is seen that some region(s) in the surface of the star can be more active than the remainder of the surface. In the case of EQ Peg, the minimum phase of the rotational modulation is $0^{P}.32$, while the phase of MFOR is $0^{P}.95$.
The results acquired from EV Lac and V1005 Ori demonstrated that flare activity can reach high levels at almost the same longitudes, in which stellar spots occur. On the other hand, there is a considerable difference between the phases of stellar spot and MFOR for the observing season 2007 of EV Lac. In conclusion, it is seen that there is a longitudinal relation between stellar spot and flare activities in general manner. Nevertheless, there are some differences and this makes difficult to do a definite conclusion. Moreover, in the case of EQ Peg, the MFOR gets the minimum towards the minimum phase of the rotational modulation. All these cases can be because of a dynamo which is working in the red dwarf stars. In spite of the Sun, red dwarf stars are mostly known to have a different dynamo because of full convective outer atmosphere. However, in the last years, some studies showed that flares on the Sun do not have to be located upon the spotted areas on the Sun [Borovik et al. , 2007]. In addition, it should be kept in mind that most of the studies have been done with using the data obtained from white-light flare observations, but a white-light flare does not have to occur in a flare process. Recent studies have shown that non white-light flares may be so common in UV Ceti-type stars as they are in the Sun [Crespo-Chacón et al. , 2004, Crespo-Chacón et al. , 2006]. In this point, it can be mentioned that the analyses of data obtained from only white-light flare observations are not sufficiently qualified. For instance, García-Alvarez et al. , 2003 found some flares occurring in the same active area with other activity patterns with using simultaneous observations.
§.§ Phase Distribution of The Fast And Slow Flares
Using the inverse Compton event, Gurzadian , 1986 developed a hypothesis called Fast Electron Hypothesis, in which red dwarfs generate only fast flares on their surface. On the other hand, according to the flare region on the surface of the star in respect to direction of observer, the shapes of the flare light variations can be seen like a slow flare [Gurzadian , 1986]. If the scenario in this hypothesis is working, it is expected that the fast and slow flares should collected into two phases in the light curves of UV Ceti type stars showing BY Dra Syndrome. It is also expected that these two phases are separated from each other with intervals of $0^{P}.50$ in phase.
In this study, according to the rule described by [Dal & Evren , 2010], the flares are classified as fast and slow flares. Then the phase distributions of fast flares were compared with the phases of slow flares in order to find out whether there is any separation as expected in this respect. When the phases of both fast and slow flares are examined one by one, it is clear that both of them can occur in any phase. To reach a definite result, the phase distributions of both fast and slow flares are statistically investigated.
As it is stated in the previous section, if the distribution of flares did not cover almost all phases in an observing season of a star, the season is neglected for that star. Consequently, we chose the seasons, in which there is enough data to get reliable conclusions about flare occurrence distributions for both fast and slow flares. In the case of AD Leo and EQ Peg, we combined all the fast flares of three seasons as we made for the slow flares. For both fast and slow flares, using Equation (5), the number of flares occurring per an hour in intervals of 0.10 phase length was computed. The obtained occurrence rates for both fast and slow flares are shown by histograms in Figures 12, 13, 14 and 15. Once again, all the distributions were modelled with the polynomial function. Resolving these models, maximum flare occurrence rates and their phase of both slow and fast flares were found for all program stars.
In the case of AD Leo, the analyses show that both fast and slow flares have a difference of $0^{P}.17$ between the phases, in which flare occurrence rates in intervals of 0.10 phase length reach maximum amplitudes. The same difference is $0^{P}.05$ for EV Lac in the season of 2004. Although these differences are acceptable as low values according to Fast Electron Hypothesis, the difference seen in the season of 2006 is $0^{P}.50$ for EV Lac. This value is the expected value in respect of Fast Electron Hypothesis. In the case of V1005 Ori, slow and fast flares could be compared only for the season of 2005/2006. The result is that both fast and slow flares have a difference of $0^{P}.30$ between the phases of maximum flare occurrence rates. In the case of EQ Peg, the phase difference between MFORs of slow and fast flares is about $0^{P}.40$. The value obtained from EQ Peg is also the expected value in respect of Fast Electron Hypothesis. It should be noted that in the case of EQ Peg, it is seen just one clear peak for the distribution of MFOR for the fast flares, while there are several peaks for the slow flares.
As it is seen from the analyses, both the fast and the slow flares sometimes the same longitudinal distributions and sometimes different. This makes difficult to say that there is a regular longitudinal division between these two types of flares as expected according to Gurzadian , 1986. This means that, when a slow flare is observed, it does not have to be a fast flare occurred on the opposite side of the star in respect to observer direction.
§ ACKNOWLEDGMENTS
The authors acknowledge generous allotments of observing time at the Ege University Observatory. We thank both Dr. Hayal Boyacıoǧlu, who gave us important suggestions about statistical analyses, and Professor M. Can Akan, who gave us valuable suggestions that improved the language of the paper. We also thank the referee for useful comments that have contributed to the improvement of the paper. We finally thank the Ege University Research Found Council for supporting this work through grant no. 2005/FEN/051.
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|
arxiv-papers
| 2012-06-24T19:20:21 |
2024-09-04T02:49:32.222063
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, S. Evren",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.5792"
}
|
1206.5793
|
# Saturation Levels For White-Light Flares of The Flare Stars:
Variation of Minimum Flare Durations For The Saturation
H. A. Dal and S. Evren Department Of Astronomy and Space Sciences, University
of Ege,
Bornova, 35100 İzmir, Turkey ali.dal@ege.edu.tr
###### Abstract
In this study, considering the results obtained from the models and from the
statistical analyses of the obtained parameters, flare activity levels and
flare characteristics of five UV Ceti stars will be discussed. We also present
the parameters of unpublished flares detected in two years of observations of
V1005 Ori. The parameters of the U-band flares detected in several seasons of
observations of AD Leo (catalog ), EV Lac (catalog ), EQ Peg (catalog ), V1054
Oph (catalog ) and V1005 Ori (catalog ) are compared among themselves. The
flare frequencies calculated for all program stars and maximum energy levels
of the flares are compared and it is discussed which is the most correct
parameter as an indicator of the flare activity levels. Using One Phase
Exponential Association function, the distributions of the flare equivalent
durations versus the flare total durations were modelled for each programme
star. We used the Independent Samples t-Test in the statistical analyses of
the parameters obtained from the models. The results revealed some properties
of the flare processes running on the surfaces of UV Ceti type stars: 1) Flare
energies can not be higher than a specific value and it is no matter how long
flare total duration is. This must be a saturation level for white-light
flares occurring in flare processes observed in U band. Thus, in the first
time it is shown that white-light flares have a saturation in the energy
range. 2) The span values, which is the difference between the equivalent
durations of flares with the shortest and longest total durations, are almost
equal for each star. 3) Another important result to model the white-light
flares is that the half-life values, minimum flare durations for the
saturation, increases towards the later spectral types. 4) Both maximum total
durations and maximum rise times computed from the observed flares decrease
towards the later spectral types among the UV Ceti stars. According to the
maximum energy levels obtained from the models, both EV Lac (catalog ) and EQ
Peg (catalog ) are more active than other three program stars, while AD Leo
(catalog ) is the most active flare star according to the flare frequencies.
methods: data analysis — methods: statistical — stars: flare — stars:
individual(V1005 Ori, AD Leo, V1054 Oph, EV Lac, EQ Peg)
††slugcomment: Not to appear in Nonlearned J., 45.
## 1 Introduction
UV Ceti type stars are mostly pre-main sequence stars coming to the ZAMS or
the stars at the ZAMS. Most of the red dwarfs in open clusters and
associations are flare stars (Mirzoyan, 1990; Pigatto, 1990). The number of
flare stars in the clusters decreases with the increasing age of clusters.
This is an acceptable case according to Skumanich Law (Skumanich, 1972; Marcy
& Chen, 1992; Pettersen, 1991; Stauffer, 1991). Higher rotation rate cause
high flare activity level and this causes higher mass loss by flare bursts.
The studies in literature indicate that the mass loss rate of the Sun is about
$2x10^{-14}$ $M_{\odot}$ in per year (Gershberg, 2005). However, this value
for UV Ceti type stars can reach to the value of $10^{-10}$ $M_{\odot}$ in per
year because of the flare activity. The high level of mass loss for these
stars can explain that they lose 98 $\%$ of their angular momentum in the
course of main sequence (Mirzoyan, 1990).
Magnetic flare activity causing high mass loss has not been properly
explained. The average of the energies of classical flares of the Sun is
$10^{26}$ \- $10^{27}$ ergs. It is about $10^{30}$ \- $10^{31}$ ergs for two
ribbon flares, which are known as the hardest flares on the Sun (Gershberg,
2005; Benz & Güdel, 2010). When flares of the chromospherically active stars,
known as RS CVn stars, are examined, it is seen that flare energies of these
stars are about $10^{31}$ ergs (Haisch et al., 1987). On the other hand,
observations lasting about 50 years show that flare energies of UV Ceti stars
change from $10^{28}$ ergs to $10^{34}$ ergs (Gershberg, 2005). Moreover, when
flare stars in the young clusters such as Pleiades and Orion associations are
considered, flare energies can reach the value of $10^{36}$ ergs (Gershberg &
Shakhovskaya, 1983).
As it can be seen from literature, there are distinctive differences between
flare energies of different type stars. Nevertheless, although there are clear
differences between the Sun and UV Ceti type stars in point of mass loss and
flare energies, flare activity of dMe stars is modelled by basing on the
processes of Solar Flare Event. In this case, it is accepted that source of
the energy in these events is the magnetic reconnection process (Gershberg,
2005; Hudson & Khan, 1997). To understand properly all process of flare event
for dMe stars, first of all, it could be useful to find out all similarities
and differences of flare light curves by examining star-to-star. For instance,
looking over flare energy spectra can be helpful as the first step. This
distribution can show how flare energies range located from one star to other.
In this respect, both the distributions and the levels of the flare energy
spectra of UV Ceti type stars have been looked over in many studies such as
Gershberg (1972); Lacy et al. (1976); Walker (1981); Gershberg & Shakhovskaya
(1983); Pettersen et al. (1984) and, Mavridis & Avgoloupis (1986) might be
given as the examples. For instance, flare energy spectra have been studied by
Gershberg (1972) for AD Leo (catalog ), EV Lac (catalog ), UV Cet and YZ CMi.
In another study made by Gershberg & Shakhovskaya (1983), flare energy spectra
for lots of stars in galactic field have been compared with flare energy
spectra of some stars from Pleiades cluster and Orion association. As it is
seen from Figure 2b given by Gershberg & Shakhovskaya (1983), the flare energy
spectra for the stars of Orion association are located at very high levels
from others. Then Pleiades stars are located below the stars of Orion
association. Finally, the stars from galactic field are located below them.
This distinctive separation among flare energies of these stars from different
origin shows that something is different in the flare process on these stars.
It seems that the differences between the levels of the flare energy spectra
are due to different ages. On the other hand, there is some separation among
the stars located in galactic field, too.
All these differences among the different stars and star groups might be due
to the saturation level of white-light flares detected from UV Ceti type
stars. It is seen that some parameters of magnetic activity can reach the
saturation (Gershberg, 2005; Skumanich & McGregor, 1986; Vilhu and Rucinski,
1983; Vilhu et al., 1986; Doyle, 1996a, b). White-light flares are detected in
some large active regions such as compact and two-ribbon flares occurring on
the surface of the Sun (Rodonó, 1990; Benz & Güdel, 2010). We expect that the
energies or the flare equivalent durations can reach the saturation. The
analyses of some large flare data sets can demonstrate this expectation. If a
saturation level can be found, it will be a guidance to model the white-light
flares. However, the data sets using in the analyses are important. The data
sets are combined from the parameters derived with the same method from the
flares detected with the same optic system. Otherwise, some artificial
variations and differences can occur between the sets. To avoid this problem,
we used a large data sets, which were combined from the parameters derived
with the same method and the same optic system.
In the second place, flare frequencies can be another useful parameter to
understand the flare process. In some studies such as Pettersen et al. (1983);
Ishida et al. (1991); Leto et al. (1997), flare frequencies have been
examined. Two flare frequencies have been generally calculated in these
studies. As it is seen from literature, flare energies and frequencies are
varying from star to star. To understand all flare process, the reason(s) of
these variations should be find out. In this point, it must be answered to
whether these differences are because of some physical parameters such as mass
and age of stars, or are directly due to different flare processes?
In this study, we compared flare parameters among five UV Ceti type stars
given in Table 1. The physical parameters (such as mass, radius and the
distance) taken from Gershberg et al. (1999) are given in Table 1. According
to spatial velocities taken from Montes et al. (2001), it is seen that AD Leo
(catalog ) is a red dwarf member of Castor Moving Group, whose age is about
200 million years. In the same way, according to the spatial velocities, EV
Lac (catalog ) is seen as a member of 300 million years old Ursa Major Group,
which is known as the Sirius Supercluster. V1005 Ori (catalog ), whose first
flare light curve was obtained by Shakhovskaya (1974), is a member of 35
million years old IC 2391 Supercluster (Montes et al., 2001). According to
another study made by Veeder (1974), these three stars seem to belong to the
young disk population of the galaxy, too. EQ Peg (catalog ), whose distance is
6.58 pc, is classified as a metal-rich star and it belongs to the young disk
population of the galaxy. On the other hand, V1054 Oph (catalog ) is a triple-
lined spectroscopic system as known Gliese 644 (=Wolf 630+Wolf629ABab). The
distance of the system is 6.5 pc. V1054 Oph (catalog ) is classified as a
metal-rich star and a member of the old disk population of the galaxy (Veeder,
1974; Fleming et al., 1995). The flare stars, whose flare parameters are
compared in this study, are quite young stars except V1054 Oph (catalog ). The
masses were derived for each components of Wolf 629ABab by Mazeh et al.
(2001). They showed that the masses are 0.41 $M_{\odot}$ for Wolf 629A, 0.336
$M_{\odot}$ for Wolf 629Ba and 0.304 $M_{\odot}$ for Wolf 629Bb. In addition,
Mazeh et al. (2001) demonstrated that the age of the system is about 5 Gyr.
## 2 Observations and Analyses
### 2.1 Observations
The observations were acquired with the High-Speed Three Channel Photometer
attached to the 48 cm Cassegrain type telescope at Ege University Observatory.
Using a tracking star set in the second channel of photometer, the
observations of the variable star were continued in standard Johnson U band
with the exposure time between 2 and 10 seconds. The basic parameters of all
the program stars (such as standard V magnitudes and B-V colours) are given in
Table 2. The parameters given in Table 2 were taken from Dal & Evren (2010)
for AD Leo (catalog ), EV Lac (catalog ), EQ Peg (catalog ) and V1054 Oph
(catalog ). Although the program and comparison stars are so close on the
plane of sky, differential extinction corrections were applied. The extinction
coefficients were obtained from the observations of the comparison stars on
each night. Moreover, the comparison stars were observed with the standard
stars in their vicinity and the arrested differential magnitudes, in the sense
variable minus comparison, were transformed to the standard system using
procedures outlined by Hardie (1962). The standard stars are listed in the
catalogues of Landolt (1983, 1992). Heliocentric corrections were applied to
the times of the observations. The standard deviation of each observation
acquired in standard Johnson U band is about $0^{m}.15$ on each night.
Observational reports of all the program stars are given in Table 3. The
differential magnitudes in the sense comparison minus check stars were
carefully checked for each night. The comparison and check stars were found to
be constant in brightness during the period of observation. Equivalent
durations and energies of all the flares are computed from photoelectric
observations using Equation (1) and (2) taken from Gershberg (1972):
$P=\int[(I_{flare}-I_{0})/I_{0}]dt$ (1)
where, $I_{0}$ is the flux of star in the observing band during the star in
the quiet state; $I$ is the intensity at the moment of flare.
$E=P\times L$ (2)
where $E$ is the energy. $P$ is the flare equivalent duration in the observing
band. $L$ is the intensity in the observing band during the star in the quiet
state. For each observed flare, HJD of flare maximum moment, flare rise and
decay time, flare amplitude, flare equivalent duration and their energy have
been calculated. In the calculations, the quiescent level, in which there are
no flare and other variability, has been accepted as the basic level of light
curve in nightly light curves. All the parameters have been computed by
considering this level. Some flares have several peaks. If a flare has several
peaks, maximum point of the flare has been accepted as the peak, which is
close to beginning and the highest one. Instead of the flare energy, flare
equivalent duration has been used in the comparison. This is because of the
luminosity term in Equation (2). The luminosities of stars with different
spectral types have large differences. Although the equivalent durations of
two flares detected from two stars in different spectral types are the same,
calculated energies of these flares are different due to different
luminosities of these spectral types. Therefore, we could not use these flare
energies in the same analysis. On the other hand, flare equivalent duration
depends just on power of the flare. Another reason of using equivalent
duration is that the given distances of the same star in different studies are
quite different. These differences cause the calculated luminosities become
different.
In Figure 1, a fast flare sample detected in observations of V1005 Ori
(catalog ) on 6 January 2005 is seen. According to the rule described
described by (Dal & Evren, 2010), this flare is classified as a fast flare.
Consecutive two flares detected on 10 January 2005 are shown in Figure 2.
These flares are classified as a slow flare in respect to the same rule. The
consecutive flare samples of V1005 Ori (catalog ) are shown in Figure 3; these
flares were detected on 29 December 2005.
Using the method described by Dal & Evren (2010), all parameters are
calculated for each flare detected in the observations of V1005 Ori (catalog
). The calculated parameters for 41 U-band flares are given in Table 4. The
columns of table are the date of observation, HJD of flare maximum, flare rise
time, flare decay time, equivalent duration, flare amplitude and flare type,
respectively.
### 2.2 Analyses
V1005 Ori (catalog ) flare data were combined with the data set including 321
U band flares detected from other stars (AD Leo (catalog ), EV Lac (catalog ),
EQ Peg (catalog ) and V1054 Oph (catalog )). The parameters of 321 U band
flares detected from these stars were presented by Dal & Evren (2010). Using
this large data set including total 362 U band flares, program stars are
compared with each other in the analyses to find out whether there is any
differences between their flare activity behaviours. In this point, as it is
mentioned above, the data using in the analyses are important. The data must
be combined from the parameters determined by the same method. In addition,
the flares must be observed with the same optic systems. There are large data
sets including U band flares in the literature such as Moffett (1974); Ishida
et al. (1991). On the other hand, the methods used to determine the parameters
of the detected flares are not the same in these studies and there are some
differences between some optic system. This is why we used the data obtained
in this study and the data presented by Dal & Evren (2010). All the parameters
in these data sets were determined with the same method and all the flares
were detected with the same optic system.
Instead of flare energies, flare equivalent durations were used for all
statistical analyses. This is because of the luminosity term in Equation (2).
The luminosities of stars from different spectral types have great
differences. Although the equivalent durations of two flares obtained from two
stars in different spectral types are the same, calculated energies of these
flares are different due to different luminosities of these spectral types.
Therefore, we could not use these flare energies in the same analyses. On the
other hand, flare equivalent duration depends just on flare power. Another
reason of using equivalent duration is that the given distances of the same
star in different studies are quite different. Therefore, the calculated
luminosities become different because of these different distances.
When the distribution of flare equivalent durations in logarithmic scales
versus flare total durations is examined, it is seen that flare equivalent
duration is below a limit value and it is no matter how long flare total
duration is. In addition, when we compare stars among themselves, it is clear
that limit value of energy is different for each star.
To model the distributions the best function was searched. Using the SPSS
V17.0 (Green et al., 1999) and GrahpPad Prism V5.02 (Dawson & Trapp, 2004)
softwares, the regression calculations showed that the best fit is One Phase
Exponential Association (hereafter OPEA) for the distributions of flare
equivalent durations. The OPEA function given by Equation (3) (Motulsky, 2007;
Spanier & Oldham, 1987) is a special exponential function, which has a plateau
part, as it is seen in the distributions of flare equivalent durations. Using
the Least-Squares Method, for each star the distributions were modelled by the
OPEA function. All the distributions and their models with 95 $\%$ confidence
intervals are shown in Figure 4.
$y~{}=~{}y_{0}~{}+~{}(Plateau~{}-~{}y_{0})~{}\times~{}(1~{}-~{}e^{-k~{}\times~{}x})$
(3)
where $y$ is the flare equivalent duration in logarithmic scales, while $x$ is
flare total duration. $y_{0}$ is flare equivalent duration in logarithmic
scales for the least total duration. In other words, $y_{0}$ is the least
equivalent duration occurring in a flare for a star. The value of $y_{0}$
depends on the brightness of the target and the sensitivity of the optic
systems. The value of Plateau is the upper limit for the equivalent duration,
which can be occurred in a flare for a star. This parameter can be identified
as a saturation level for flare activity observed in U band. According to
Equation (2), the value of Plateau depends only on the energy of flares
occurring on the star. According to the definition of the OPEA function, the
parameter $k$ in Equation (3) is a constant depending on the $x$ values.
The parameters derived from the OPEA models are given for all the stars in
Table 5. Star name, B-V index, Plateau, $y_{0}$ and $k$ values are listed in
the table, respectively. The span value and half-life value are given in the
last two columns. B-V indexes are also found in this study. The span value is
the difference between the values of Plateau and $y_{0}$. The half-life value
is half of the first $x$ values, where the model starts to give the Plateau
values for a star. In other words, it is half of the flare total duration,
where flares with the highest energy start to seen. The half-life value is
minimum value of the flare total duration, which is needed for the saturation
level.
Considering the flare in the plateau phases of the OPEA models, all programme
stars were compared in respect of maximum flare equivalent durations.
Independent Samples t-Test (hereafter t-Test), which is a statistical analysis
method (Wall & Jenkins, 2003; Dawson & Trapp, 2004; Motulsky, 2007), was used
for comparisons. Using this test, the mean equivalent durations were computed
for each star and the mean values were compared among themselves. The results
are given in Table 6. Thus, the values of Plateau were tested with another
analysis.
Although the mean values computed by t-Test are expected to be close to the
Plateau values of the OPEA models for all stars, it is clear that there can be
some difference between these two values. This is because the OPEA models
depend on all distribution from the beginning of $x$ values to the end, while
the mean values computed by t-Test depend only on equivalent durations of
flares in the Plateau phases.
The variations of Plateau values, which are listed in Table 5, and the mean
values of equivalent durations, which are given in Table 6, are plotted versus
the B-V of each star and shown in Figure 5. As it is expected, both parameters
exhibit the same variations with a little difference in their levels. In the
figure, the mean equivalent durations and the Plateau values decrease with
increasing B-V index for three stars, namely V1005 Ori (catalog ), AD Leo
(catalog ) and V1054 Oph (catalog ). On the other hand, these two parameters
are dramatically higher for two stars, EV Lac (catalog ) and EQ Peg (catalog
), which are the reddest stars among five stars. In the figure, decreasing of
first three values is shown by linear fits for both parameters. The variation
of Plateau values indicates that the saturation level of flare activity can
change. Moreover, the saturation levels of the reddest stars among five stars
is absolutely higher than the levels of other stars.
The variations of other parameters derived from the OPEA models are shown
versus the B-V index in Figure 6. The variation of the parameter $y_{0}$ is
seen in panel (a) of this figure. Although the parameter of $y_{0}$ depends on
both the brightness of target and the sensitivity of the observing system in a
general manner due to the standard deviations of observations, here $y_{0}$
parameter exhibits a dramatic increase for two stars, which are located
towards the reddest edge. It means that the flare energy for the least total
duration of flares is higher for the stars, which are located towards the
reddest edge of M type. In panel (b), the variation of the span values, which
is the difference between $y_{0}$ and Plateau values obtained from model
curves, is seen versus B-V index. According to the models, the span value
shows no important variation versus B-V index. In panel (c), the variation of
half-life values is seen versus B-V index. The regression calculations
indicated that the polynomial function is the best fit, which is shown by
dotted line in the figure, for this variation of the value. As it is seen from
the panel, the half-life values increase towards the reddest M stars in
respect of analyses of these five stars.
In addition, when 362 flares observed from the five stars are examined, it is
seen that the longest flare total durations are varying from star to star. As
it is seen from Figure 4, the total duration of the longest flare is 2940
seconds for EV Lac (catalog ) and 3180 seconds for EQ Peg (catalog ), while it
is 5236 seconds for V1005 Ori (catalog ) and it reaches to 4164 seconds for AD
Leo (catalog ). The longest flare total duration is about 3270 seconds for
V1054 Oph (catalog ). As it is seen in Figure 7, the observed maximum flare
total durations decrease towards the later spectral types. The maximum flare
rise time was also computed for each star. For V1005 Ori (catalog ), AD Leo
(catalog ), V1054 Oph (catalog ), EV Lac (catalog ) and EQ Peg (catalog ), the
maximum rise times are 2036, 1212, 1460, 840 and 1230 seconds, respectively.
The variation of all these times is also shown in Figure 7.
Moreover, two different flare frequencies of stars were computed for each
season to examine flare activity levels. These flare frequencies are
identified by Equation (4) and (5) taken from Ishida et al. (1991).
$N_{1}~{}=~{}\Sigma n_{f}~{}/~{}\Sigma T_{t}$ (4)
$N_{2}~{}=~{}\Sigma P_{u}~{}/~{}\Sigma T_{t}$ (5)
where $\Sigma T_{t}$ is the total observing durations, $\Sigma n_{f}$ is the
total number of flares obtained in a season ($\Sigma n_{f}$) and $\Sigma
P_{u}$ is the total equivalent duration obtained from all flares detected in
that observing season ($\Sigma P_{u}$). $N_{1}$ and $N_{2}$ are the flare
frequencies. All computed frequencies are listed in Table 7. According to the
results, the higher flare frequencies are seen in AD Leo (catalog ).
## 3 Results and Discussion
The flare energy, which is expressed by Equation (2) given by Gershberg
(1972), has been generally used to examine the level of flare activity in lots
of studies in the literature. The studies of some authors such as Mazeh et al.
(2001), Lacy et al. (1976), Walker (1981), Gershberg & Shakhovskaya (1983),
Pettersen et al. (1984) and Mavridis & Avgoloupis (1986) can be given as
examples. The luminosity parameter ($L$) takes place in the expression of the
energy ($E$), which has been based by all these and other studies. The
luminosity ($L$) is different for each star. Although there are little
differences among the masses of M dwarfs, the luminosities of two M dwarfs,
whose masses are so close to each other, can be dramatically different from
each others due to their places in the Hertzsprung-Russell diagram. This means
that the computed energies of flares are very different from each other, even
if the light variations of the flares occurring on these two stars are the
same. Because of this, the equivalent durations ($P$) were used in the
analyses instead of energy ($E$) in this study. If there is a difference in
the equivalent durations of the flares, it is also seen in the energies in the
same way.
In the analyses, the distributions of flare equivalent durations versus flare
total duration were modelled by the OPEA function expressed by Equation (3)
for all stars in the programme. When the models are compared, it is seen that
there are some differences among the stars. As it is seen in Figure 4f and
Figure 5a, the Plateau parameter, which gives the maximum equivalent duration
level for flares on a star, is changing from one star to the other. It is seen
in Figure 4f that the distributions of the equivalent durations in logarithmic
scale versus flare total duration for EV Lac (catalog ) and EQ Peg (catalog )
are different from other three stars. The maximum equivalent durations seen in
these two stars are as high as 0.5 times in logarithmic scales. This
difference in logarithmic scales is equal to 683 times difference in energies.
This means that, for example, the energy of an EV Lac (catalog ) flare is 683
times higher than the energy of an AD Leo (catalog ) flares in average
generally. In addition, the energy of a flare occurring on AD Leo (catalog )
is never higher than the energy of an EV Lac (catalog ) flare, no matter how
long the total duration of AD Leo (catalog ) flare is. This result is
confirmed by another analysis, which is a statistical analysis method, t-Test.
We used the flares, whose equivalent durations in logarithmic scales are
located in the plateau phases in the OPEA models, in t-Test. In this point,
the aim is to compare the equivalent durations of flares, whose energies are
independent from lengths of their total duration. The results of t-Test
analyses are shown in Figure 5b. As it is seen from this figure, the mean
averages of equivalent durations computed by t-Test are close to the Plateau
values derived from the OPEA models. The mean averages of maximum equivalent
durations for flares of EV Lac (catalog ) and EQ Peg (catalog ) are
distinctively higher than the averages computed from other three stars. On the
other hand, the mean averages of equivalent durations are relatively different
for each star.
As it is mentioned above, some parameters in the chromospheric magnetic
activity can reach the saturation (Gershberg, 2005; Skumanich & McGregor,
1986; Vilhu and Rucinski, 1983; Vilhu et al., 1986; Doyle, 1996a, b). In the
case of the white-light flare, we expect that the energies or the flare
equivalent durations can reach the saturation, because these white-light
flares are detected in some large active regions such as compact and two-
ribbon flares occurring on the surface of the Sun (Rodonó, 1990; Benz & Güdel,
2010). Consequently, according to this approach, the Plateau value must be a
saturation level (or an indicator at least) for the white-light flares. In the
analyses, we used the data obtained with the same method and the same optic
system. In addition, we used the flare equivalent durations instead of the
flare energies. Therefore, the derived Plateau values depend just on the power
of the white-light flares. Considering the Plateau values, it is seen that the
power of the flare has a limit for a star. The flare equivalent durations can
not be higher than a value and it is no matter how long flare total durations.
Instead of the flare duration, some other parameters, such as magnetic field
flux and/or particle density in the volumes of the flare processes, must be
more efficient to determine the power of the flares. Considering thermal and
non-thermal flare events, both magnetic field flux and particle density in the
volumes of the flare processes can be efficient.
Gurzadian (1977, 1988) developed a hypothesis, called Fast Electron
Hypothesis. In this hypothesis, the source of the white-light flares on the
surfaces of UV Ceti stars is non-thermal process, such as the spontaneous
appearance of fast electrons on the surface of the flare stars. Considering
this hypothesis, the particle density in the volumes of the flare processes
must be more efficient to determine the power of the white-light flares,
instead of magnetic field. Gurzadian (1988) demonstrated that the inverse
Compton effect, non-thermal interactions of infrared photons with fast
electrons, causes some radiative losses. It is possible that the inverse
Compton effect can be more efficient after a specific flare durations for a UV
Ceti star, and this effect can limited the observed flare equivalent duration
(and energy) of a detected flare. However, considering all the flare process,
it should be noted that the particle density in the volumes depends on
magnetic field flux in some respects. The source of theme is some particles
accelerated by magnetic field (Benz & Güdel, 2010; Gershberg, 2005). In
addition, the magnetic field flux in the volumes is more efficient than the
particle density for high energy patterns of the flare process, such as soft
X-ray or radio intensities (Gershberg, 2005).
On the contrary, Doyle (1996a, b) suggested that the saturation in the active
stars does not have to be related to the filling factor of magnetic structures
on the stellar surfaces or the the dynamo mechanism under the surface, it can
be related to some radiative losses in the chromosphere, where the temperature
and density are increasing in the case of the fast rotation. Like this
phenomenon, a case can occur in the chromosphere due to the flare process
instead of the fast rotation, and this case causes that the Plateau phase
occurs in the distributions of flare equivalent durations versus flare total
duration. On the other hand, the Plateau phase can not be due to some
radiative losses in the chromosphere with increasing of the temperature and
density. This is because, Grinin (1983) demonstrated the effects of the
radiative losses in the chromosphere on the white-light photometry of the
flares. According to Grinin (1983), the negative H opacity in the chromosphere
causes the radiative losses, and these radiative losses seen as a pre-flare
dips in the light curves of the white-light flares. In addition, when the
results are considered, it is seen that the Plateau values are varying from a
star to next one. This indicates that some parameters, which cause the Plateau
in the distributions of flare equivalent durations, or their efficacies are
changing star-to-star. Moreover, in the case of EV Lac (catalog ) and EQ Peg
(catalog ), there is a distinctive difference. The efficacies of these
parameters (or the parameters themselves) must be dramatically changed for
these stars.
In the future, the reason(s) of the Plateau phases should be examined by
synchronous observations in the radio, optic and X-ray regions of the
spectrum. In these studies, we recommend that some tests should be made. When
the energy of a white-light flare detected with optical photometry reaches the
saturation level, it should be tested whether the energy reaches the
saturation in the radio or X-ray observation, or not. If the energy reaches
the saturation in the radio or X-ray observation, this indicates that the
reason of the saturation is generally magnetic field. This is because, the
energy source in the radio or X-ray is generally magnetic reconnection. If the
energy does not reach the saturation in the synchronous optical photometry, it
means that the particle density in the volumes is more efficient to determine
the power of the flares in the optical part of the spectrum.
However, it is worthy to note, that known very week correlation of optical and
radio flares at the UV Cet type variables makes such experiment hardly
realized. Moreover, the attribution of flares in saturation regions to white-
light flares seems to be weekly proved. On the Sun such flares are selected
with spectroscopic observations when there is a strong continuum. On stars
multicolour observations allow defining a phase when a black body radiation
that is a continuum dominates (Zhilyaev et al., 2007).
When variation of the parameter $y_{0}$ is examined, as it is seen from Figure
6a, it is changing from star to star. Like the parameter Plateau, $y_{0}$
parameters of EV Lac (catalog ) and EQ Peg (catalog ) are rather higher than
other stars. $y_{0}$ parameters of other three stars are almost equal to each
other. Actually, the parameter of $y_{0}$ depends on both the brightness of
the target and the sensitivity of the observing system. On the other hand,
considering that the brightness of all the targets are almost equal to each
other and all of them were observed with the same system in almost the same
time, the variation of $y_{0}$ parameters from star to star are close to the
real behaviour.
The difference between Plateau and $y_{0}$ parameters, which is derived from
the distribution modelled by Equation (3), is listed in Table 5 as the
parameter of span value. Figure 6b shows the behaviour of this parameter
versus B-V index. As it is seen, there is no regular variation in this
parameter. It is important because it means that the difference between
Plateau and $y_{0}$ parameters is constant along all B-V indexes of M type in
respect of five stars. The similarity of span values for all programme stars
shows that the difference between $y_{0}$ parameters of stars is exactly seen
between Plateau values of stars. This indicates that even if the conditions,
in which the flares occur, and so energies of flares are changing, the
difference between minimum and maximum energies is stable in flare mechanism.
At the same time, the variation of half-life values, which is an indicator of
the minimum flare total duration needed for maximum energy occurring, is seen
in Figure 6c. According to this variation, the values of half-life value
increases towards higher B-V indexes. It means that the minimum total
durations, which are needed for the flares emitting maximum energy in the
flare mechanisms, increase towards the later spectral types among the M type
stars. Consequently, this variation indicates that longer flare total
durations are needed to reach the saturation level towards the later spectral
types among M dwarfs. Like the variation of the Plateau value versus B-V
colour index, the variation of half-life values must be considered to model
the white-light flares detected from UV Ceti type stars.
When we consider maximum flare total durations seen among the flares for each
star, the maximum flare total durations seen among the flares are 2940 seconds
for EV Lac (catalog ) and 3180 seconds for EQ Peg (catalog ). It is 5236
seconds for V1005 Ori (catalog ) and 4164 seconds for AD Leo (catalog ), while
it is about 3270 seconds for V1054 Oph (catalog ). Maximum flare total
durations seen in flares of UV Ceti stars decrease towards the later spectral
types among the M stars as seen in Figure 7. Like flare total durations, the
maximum rise times are 2036, 1212, 1460, 840 and 1230 seconds for V1005 Ori
(catalog ), AD Leo (catalog ), V1054 Oph (catalog ), EV Lac (catalog ) and EQ
Peg (catalog ), respectively. The variation of all these times is also shown
in Figure 7. As it is seen from the figure, both maximum rise time and total
duration decrease towards the later spectral types.
As a result, four important properties can be summarised for the flare
processes occurring on the UV Ceti type stars: 1) Flare energies increase with
flare total duration until a specific total duration values, and then the
energies are constant no matter how long the flare total duration is. 2) The
differences between minimum and maximum energies of flares are constant and
the same for all stars. 3) The minimum total durations, which are needed for
the flares emitting maximum energy, increase towards the later spectral types
among the UV Ceti stars. 4) Maximum flare rise time and total durations
decrease towards the later spectral types among the UV Ceti stars.
On the other hand, two flare frequencies expressed by Equation (4) and (5) are
used to identify the flare activity levels in lots of studies in the
literature. The researchers have used these frequencies to find out whether
flare activity of UV Ceti stars exhibits any cyclic variation. Assuming that
all flares occurring on the star are observed; $N_{1}$ is an indicator of
flare number obtained in unite-time, as it is expressed by Equation (4).
However, according to Equation (5), $N_{2}$ is an indicator of the mean
equivalent duration average obtained in unite-time (Ishida et al., 1991). In
brief, $N_{1}$ refers how many flares occur on a star, while $N_{2}$ refers
how energetic these flare are.
In this study, both flare frequencies of $N_{1}$ and $N_{2}$ were computed
season-to-season for each programme star. All of them are listed in Table 7.
$N_{1}$ frequency of AD Leo (catalog ) is close to value of 1.0 in the season
of 2005. It means that one flare occurred on AD Leo (catalog ) per hour at
least. $N_{2}$ frequency of the star was computed as 0.086. Considering the
values obtained for each season and each stars, this value of $N_{2}$
indicates that the flares occurring on AD Leo (catalog ) were powerful in the
season of 2005. $N_{1}$ and $N_{2}$ frequencies are in agreement to each
others in the season of 2005. Conversely, $N_{1}$ is 1.331, while $N_{2}$ is
0.012 in the season of 2006. $N_{1}$ is higher than the value of 1.0, and it
can be accepted as the highest value of $N_{1}$, but $N_{2}$ is not high.
Although lots of flares were able to occur on AD Leo (catalog ) in the season
of 2006, their energies were not as high as expected values, as it was seen in
the season of 2005. According to the value of $N_{2}$, flare activity level of
AD Leo (catalog ) was not high. This is an argumentative case. Like the case
of AD Leo (catalog ), some similar cases are seen in other programme stars. In
the literature, both $N_{1}$ and $N_{2}$ frequencies are accepted as an
indicator of flare activity level in some studies. For example, Mavridis &
Avgoloupis (1986) computed both $N_{1}$ and $N_{2}$. Examining distribution of
these parameters versus time, they demonstrated that EV Lac (catalog ) had a
flare activity cycle of 5 years.
According to expression given by Equation (5), $N_{2}$ depends on equivalent
duration, in other words, energy. So, it is expected that $N_{2}$ can behave
like the Plateau or the mean average of the equivalent durations. $N_{2}$
frequencies of EV Lac (catalog ) and EQ Peg (catalog ) are expected to be
higher than the same frequencies of the other stars. However, the frequencies
of these two stars are almost the same. There is no clear difference between
them.
According to all these results, the parameters of the Plateau and the mean
average of the equivalent durations seem to be useful to determine the flare
activity levels. We assume that it is the most active star, whose Plateau and
mean parameters are the highest.
On the other hand, if the differences, which are seen between values of the
Plateau or the mean average of the equivalent durations, between the programme
stars had been caused by the age of the stars, it was expected that V1005 Ori
(catalog ) was the most active star among the others, according to Skumanich
(1972). This is because V1005 Ori (catalog ) is a member of IC 2391 (catalog )
Supercluster, which is 30-35 million years old (Montes et al., 2001).
Considering that all other stars are almost in the same age apart from V1054
Oph (catalog ), whose age is 5 Gyr, the difference can be caused by rapid
rotation or binarity (Veeder, 1974; Fleming et al., 1995; Montes et al.,
2001). The equatorial rotational velocity of EV Lac (catalog ) is 4 $kms^{-1}$
and it is between 5 - 5.8 $kms^{-1}$ for AD Leo (catalog ) (Marcy & Chen,
1992; Pettersen, 1991). Conversely, the equatorial rotational velocity of
V1005 Ori (catalog ) is 29.6 $kms^{-1}$ (Eker et al., 2008). Assuming that
Skumanich Law can be acceptable for flare activity of UV Ceti stars, as it is
in the case of chromospheric activity, the values of the Plateau or the mean
average of the equivalent durations must be higher for the stars, which are
rapidly rotating. However, according to our results, this is not a common
rule. Beside this, EV Lac (catalog ), AD Leo (catalog ) and V1005 Ori (catalog
) are single stars. There is a visual companion of EQ Peg (catalog ).
Moreover, V1054 Oph (catalog ) is a system composed from six stars (Pettersen,
1991). Therefore, if the binarity or multiplicity makes the flare activity
levels increase; the most active flare star should be V1054 Oph (catalog ).
We analysed the data obtained in three observing seasons from the observations
of five stars. Finally we have already reached to some clear results. These
results are derived from both the OPEA models and the computed the flare
frequencies. Obtained properties about flare processes occurring on UV Ceti
type stars are important to understand the general flare process for stellar
flare activity. In this respect, extending the B-V range of the programme
stars, it is needed to obtain much more data, which are obtained from lots of
different stars and their flare patrols spanning long years, in order to reach
more reliable results.
## Acknowledgments
The authors acknowledge generous allotments of observing time at Ege
University Observatory. We wish to thank Dr. Hayal Boyacıoǧlu, who gave us
important suggestions about statistical analyses. Prof. Dr. M. Can Akan gave
us valuable suggestions, which improved the language of the manuscript, we
wish to thank him. We also thank the referee for useful comments that have
contributed to the improvement of the paper. We finally thank the Ege
University Research Found Council for supporting this work through grant Nr.
2005/FEN/051.
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Figure 1: A fast flare sample detected in U band observation of V1005 Ori on 6
January 2005. In the figure, filled circles show observations, while
horizontal dashed line represents the level of quiescent state of the star in
U band for the observing night. Figure 2: A fast flare sample detected in U
band observation of V1005 Ori on 10 January 2005. In the figure, all the
symbols are the same as Figure 1. Figure 3: A fast flare sample detected in U
band observation of V1005 Ori on 29 December 2005. In the figure, all the
symbols are the same as Figure 1. Figure 4: The distributions of flare
equivalent durations in logarithmic scales versus flare total durations for
each programme star (a, b, c, d, e). Filled circles represent equivalent
durations computed from observed flares. The lines represent the models
identified with Equation (3) computed by using The Least-Squares Method. The
dotted lines represent the 95 $\%$ confidence intervals for the models for
each star. In panel (f), all models derived for each star are compared. Figure
5: The variations of both the Plateau parameters (a) and the mean equivalent
durations (b) are demonstrated versus B-V index of stars. The dotted lines
represent the linear fits for the first three points, which are used to
indicate decreasing of both values of the Plateau and the mean equivalent
durations. Figure 6: The variations of both the parameter y0 (a) and the span
value (b) are shown versus B-V index of stars. In panel (c), the variation of
the half-life values is seen. The dotted line in panel (a) represents the
linear fit for the first three points, as the same in Figure 5. In panel (b)
it represents the linear fit for all points. It is the polynomial fit of all
points in panel (c). All fits are only used for representation of variation
ways. Figure 7: The variations of maximum flare rise time (a) and maximum
flare total durations (b) are seen versus B-V index. The dotted lines are the
linear fit of them.
Table 1: Some physical parameters of program stars taken from Gershberg et al. (1999). Star | Distance (pc) | Mass ($M_{\odot}$) | Radius ($R_{\odot}$)
---|---|---|---
AD Leo | 4.9 | 0.28 | 0.54
EV Lac | 5 | 0.18 | 0.39
EQ Peg | 6.2 | 0.28 | 0.58
V1054 Oph | 5.7 | 0.42 | 0.76
V1005 Ori | 26.7 | - | 0.7
Table 2: Basic parameters for the targets studied and their comparison (C1) and check (C2) stars. Columns list: Star name; standard V mag and B-V colours for quiet phase of them. Star | V (mag) | B-V (mag)
---|---|---
V1005 Ori | 10.090 | 1.307
C1= BD+01 870 | 8.800 | 1.162
C2= HD 31452 | 9.990 | 0.920
AD Leo | 9.388 | 1.498
C1 = HD 89772 | 8.967 | 1.246
C2 = HD 89471 | 7.778 | 1.342
EV Lac | 10.313 | 1.554
C1 = HD 215576 | 9.227 | 1.197
C2 = HD 215488 | 10.037 | 0.881
EQ Peg | 10.170 | 1.574
C1 = SAO 108666 | 9.598 | 0.745
C2 = SAO 91312 | 9.050 | 1.040
V1054 Oph | 8.996 | 1.552
C1 = HD 152678 | 7.976 | 1.549
C2 = SAO 141448 | 9.978 | 0.805
Table 3: Observational reports of the program star for each observing season. Star | Season | HJD Interval | Number | Observing | Flare
---|---|---|---|---|---
| | (+24 00000) | Of Night | Duration (h) | Number
V1005 Ori | 2004-2005 | 53353-53453 | 9 | 28.13 | 10
| 2005-2006 | 53673-53769 | 9 | 26.45 | 31
Table 4: All the calculated parameters of flares detected in the observations
of V1005 Ori are listed. From the first column to the last, the date of
observation, HJD of flare maximum moment, flare rise time, decay time,
equivalent duration, flare energy (ergs), flare amplitude and flare type are
given, respectively. In the last column, flare types are listed.
Observing | HJD Of Maximum | Flare Rise | Flare Decay | Equivalent | Energy | Amplitude | Flare
---|---|---|---|---|---|---|---
Date | (+ 24 00000) | Time (s) | Time (s) | Duration (s) | (ergs) | (mag) | Type
13.12.2004 | 53353.42334 | 420 | 375 | 98 | 1.6799E+33 | 0.304 | Slow
13.12.2004 | 53353.43480 | 30 | 75 | 5 | 7.9712E+31 | 0.271 | Slow
13.12.2004 | 53353.44452 | 30 | 510 | 103 | 1.7617E+33 | 0.121 | Fast
06.01.2005 | 53377.31547 | 15 | 15 | 4 | 6.6193E+31 | 0.158 | Fast
06.01.2005 | 53377.36773 | 345 | 2055 | 980 | 1.6731E+34 | 0.919 | Fast
10.01.2005 | 53381.37637 | 60 | 765 | 276 | 4.7061E+33 | 0.834 | Fast
10.01.2005 | 53381.41977 | 450 | 1995 | 427 | 7.2863E+33 | 0.221 | Fast
12.01.2005 | 53383.26550 | 15 | 60 | 4 | 6.0373E+31 | 0.215 | Fast
12.02.2005 | 53414.34290 | 15 | 90 | 9 | 1.5447E+32 | 0.212 | Fast
12.02.2005 | 53414.35470 | 15 | 15 | 4 | 6.7371E+31 | 0.215 | Slow
29.10.2005 | 53673.50034 | 1450 | 3291 | 920 | 1.5697E+34 | 0.109 | Fast
29.10.2005 | 53673.56398 | 1661 | 2903 | 1005 | 1.7161E+34 | 0.100 | Slow
11.11.2005 | 53686.55961 | 2062 | 3174 | 827 | 1.4109E+34 | 0.189 | Slow
11.11.2005 | 53686.61910 | 1384 | 1065 | 577 | 9.8525E+33 | 0.383 | Slow
12.12.2005 | 53717.38316 | 750 | 1768 | 1824 | 3.1135E+34 | 0.309 | Fast
12.12.2005 | 53717.41844 | 636 | 1982 | 1621 | 2.7672E+34 | 0.694 | Fast
12.12.2005 | 53717.45388 | 750 | 1050 | 634 | 1.0814E+34 | 0.436 | Slow
12.12.2005 | 53717.47628 | 255 | 135 | 256 | 4.3697E+33 | 1.036 | Slow
19.12.2005 | 53724.34506 | 405 | 630 | 331 | 5.6530E+33 | 0.339 | Slow
19.12.2005 | 53724.40173 | 840 | 615 | 308 | 5.2619E+33 | 0.303 | Slow
19.12.2005 | 53724.41949 | 165 | 495 | 136 | 2.3156E+33 | 0.273 | Fast
19.12.2005 | 53724.42539 | 15 | 105 | 19 | 3.1694E+32 | 0.380 | Fast
19.12.2005 | 53724.42730 | 60 | 105 | 22 | 3.8074E+32 | 0.356 | Slow
19.12.2005 | 53724.43182 | 15 | 15 | 6 | 9.7737E+31 | 0.386 | Fast
19.12.2005 | 53724.43407 | 180 | 975 | 383 | 6.5315E+33 | 0.553 | Fast
19.12.2005 | 53724.44623 | 75 | 120 | 30 | 5.1338E+32 | 0.323 | Slow
19.12.2005 | 53724.45074 | 45 | 45 | 24 | 4.0302E+32 | 0.428 | Slow
19.12.2005 | 53724.45317 | 120 | 1017 | 810 | 1.3818E+34 | 1.889 | Fast
19.12.2005 | 53724.46633 | 30 | 405 | 91 | 1.5554E+33 | 0.412 | Fast
24.12.2005 | 53729.38460 | 1835 | 2660 | 643 | 1.0976E+34 | 0.073 | Slow
24.12.2005 | 53729.44269 | 1293 | 1971 | 479 | 8.1814E+33 | 0.150 | Slow
24.12.2005 | 53729.49801 | 30 | 135 | 27 | 4.5391E+32 | 0.319 | Fast
08.01.2006 | 53744.34648 | 136 | 289 | 166 | 2.8409E+33 | 0.662 | Slow
08.01.2006 | 53744.40960 | 1482 | 3472 | 2673 | 4.5624E+34 | 0.090 | Fast
27.01.2006 | 53763.30671 | 390 | 1384 | 296 | 5.0442E+33 | 0.267 | Fast
27.01.2006 | 53763.32655 | 330 | 1150 | 235 | 4.0098E+33 | 0.185 | Fast
27.01.2006 | 53763.35273 | 240 | 315 | 124 | 2.1100E+33 | 0.225 | Slow
02.02.2006 | 53769.28911 | 638 | 1471 | 245 | 4.1840E+33 | 0.040 | Fast
02.02.2006 | 53769.32073 | 1261 | 1802 | 270 | 4.6116E+33 | 0.118 | Slow
02.02.2006 | 53769.34732 | 495 | 920 | 114 | 1.9516E+33 | 0.118 | Slow
02.02.2006 | 53769.36786 | 855 | 864 | 126 | 2.1508E+33 | 0.010 | Slow
Table 5: The parameters derived from the OPEA models, which were derived by
using the Least-Squares Method.
Star | B-V | Plateau | $y_{0}$ | $k$ | Span Value | Half - Life
---|---|---|---|---|---|---
| (mag) | ($logP_{u}$) | ($logP_{u}$) | (Tot. Duration) | ($logP_{u}$) | (Tot. Duration)
V1005 Ori | 1.307 | 2.637 $\pm$ 0.074 | 0.428 $\pm$ 0.198 | 0.003063 $\pm$ 0.000623 | 2.209 $\pm$ 0.193 | 226.30
AD Leo | 1.498 | 2.527 $\pm$ 0.086 | 0.370 $\pm$ 0.054 | 0.002738 $\pm$ 0.000341 | 2.158 $\pm$ 0.089 | 253.10
V1054 Oph | 1.552 | 2.462 $\pm$ 0.105 | 0.385 $\pm$ 0.101 | 0.002305 $\pm$ 0.000435 | 2.077 $\pm$ 0.119 | 300.70
EV Lac | 1.554 | 3.014 $\pm$ 0.084 | 0.698 $\pm$ 0.057 | 0.002404 $\pm$ 0.000250 | 2.316 $\pm$ 0.087 | 288.40
EQ Peg | 1.574 | 2.935 $\pm$ 0.091 | 0.859 $\pm$ 0.081 | 0.002074 $\pm$ 0.000272 | 2.076 $\pm$ 0.094 | 334.30
Table 6: Using t-Test analysis, computed mean values of equivalent durations for flares in the plateau phases are listed. In the first column, star names are given, while the mean values of equivalent durations are listed with their errors in the second one. In the last column, the standard deviations are given. Star | Mean | Std. Deviation
---|---|---
V1005 Ori | 2.753 $\pm$ 0.051 | 0.196
AD Leo | 2.506 $\pm$ 0.074 | 0.286
V1054 Oph | 2.269 $\pm$ 0.078 | 0.257
EV Lac | 3.165 $\pm$ 0.101 | 0.285
EQ Peg | 3.021 $\pm$ 0.095 | 0.232
Table 7: The flare frequencies of stars for each observing season are listed. In the table, star name and observing seasons are given in first two columns. In the next column, the total observing durations ($\Sigma T_{t}$) are listed for each season. Then, the total numbers of flares obtained in a season ($\Sigma n_{f}$) are given. In $5^{th}$ column, the total equivalent durations obtained from all flares detected in that observing season ($\Sigma P_{u}$) are listed. In the last two columns, the flare frequencies ($N_{1}$ and $N_{2}$) are given. Star | Season | $\Sigma T_{t}$ (hour) | $\Sigma n_{f}$ | $\Sigma P_{u}$ (hour) | $N_{1}$ ($h^{-1}$) | $N_{2}$
---|---|---|---|---|---|---
V1005 Ori | 2004-2005 | 30.1403 | 10 | 0.529 | 0.332 | 0.018
| 2005-2006 | 31.4011 | 31 | 2.100 | 0.987 | 0.067
AD Leo | 2005 | 39.6908 | 39 | 3.396 | 0.983 | 0.086
| 2006 | 40.5623 | 54 | 0.491 | 1.331 | 0.012
| 2007 | 24.1147 | 17 | 0.168 | 0.705 | 0.007
V1054 Oph | 2004 | 42.6375 | 14 | 1.007 | 0.328 | 0.024
| 2005 | 33.1622 | 26 | 0.296 | 0.784 | 0.009
EV Lac | 2004 | 54.1250 | 31 | 2.644 | 0.573 | 0.049
| 2005 | 30.3108 | 32 | 0.940 | 1.056 | 0.031
| 2006 | 48.7625 | 35 | 2.443 | 0.718 | 0.050
EQ Peg | 2004 | 66.2792 | 38 | 1.364 | 0.573 | 0.021
| 2005 | 37.4664 | 35 | 3.314 | 0.934 | 0.088
|
arxiv-papers
| 2012-06-24T19:25:03 |
2024-09-04T02:49:32.237498
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, S. Evren",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.5793"
}
|
1206.5794
|
# The Patterns of High Level Magnetic Activity
Occurring on the Surface of V1285 Aql:
The OPEA Model of Flares and DFT Models of Stellar Spots
H. A. Dal and S. Evren Department Of Astronomy and Space Sciences, University
of Ege,
Bornova, 35100 İzmir, Turkey ali.dal@ege.edu.tr
###### Abstract
Statistically analysing Johnson UBVR observations of V1285 Aql (catalog )
during the three observing seasons, both activity level and behaviour of the
star are discussed in respect to obtained results. We also discuss the
variation out-of-flare due to rotational modulation. 83 flares were detected
in U band observations of the season 2006. First of all, depending on
statistical analyses using the Independent Samples t-Test, the flares were
divided into two classes as the fast and the slow flares. According to the
results of the test, there is a difference of about 73 s between the flare-
equivalent durations of slow and fast flares. The difference should be the
difference mentioned in the theoretical models. Secondly using the One Phase
Exponential Association function, the distribution of the flare-equivalent
durations versus the flare total durations was modelled. Analysing the model,
some parameters such as $Plateau$, $Half-Life$ values, the mean average of the
flare-equivalent durations, maximum flare rise and total duration times are
derived. The $Plateau$ value, which is an indicator of the saturation level of
white-light flares, was derived as 2.421$\pm$0.058 s in this model, while
$Half-Life$ is computed as 201 s. Analyses showed that observed maximum value
of flare total duration is 4641 s, while observed maximum flare rise time is
1817 s. According to these results, although computed energies of the flares
occurring on the surface of V1285 Aql (catalog ) are generally lower than
those of other stars, the length of its flaring loop can be higher that those
of more active stars. Moreover, the variation out-of-flare activity was
analysed with using three methods of time series analysis, a sinusoidal-like
variation with period of $3^{d}.1265$ was found for rotational modulation out-
of-flare for the first time in literature. Considering the variations of V-R
colour, these variations must be because of some dark spot(s) on the surface
of that star. In addition, using the ephemeris obtained from time series
analyses, the distribution of the flares was examined. The phase of maximum
mean flare occurrence rates and the phase of rotational modulation were
compared to investigate whether there is any longitudinal relation between
stellar flares and spots. The analyses show that there is a tendency of
longitudinal relation between stellar flares and spot(s). Finally, it was
tested whether slow flares are the fast flares occurring on the opposite side
of the stars according to the direction of the observers as mentioned in the
hypothesis developed by Gurzadian (1986). The flare occurrence rates reveal
that both slow and fast flares can occur in any rotational phases.
methods: data analysis — methods: statistical — stars: spots — stars: flare —
stars: individual(V1285 Aql)
††slugcomment: Not to appear in Nonlearned J., 45.
## 1 Introduction
Flares and flare processes observed on the surfaces of UV Ceti type stars have
not been perfectly understood yet, although they are heavily studied subjects
of astrophysics (Benz & Güdel, 2010). In this study, we obtained large data
set in UBVR bands from the observations of V1285 Aql (catalog ). Depending on
these large photometric data, which is very useful for a statistical analysis
of the flare properties, we have obtained some remarkable results. Observed
star, V1285 Aql (catalog ), is classified as a UV Ceti type star from spectral
type dM3e in SIMBAD data base. According to Veeder (1974), the star seems to
belong to the young disk population of the galaxy and is classified as a young
flare star. The flare activity of V1285 Aql (catalog ) was discovered for the
first time by Shakhovskaya & Maslennikov (1970). Apart from flare activity,
Andrews (1988) showed that V1285 Aql (catalog ) exhibits some sinusoidal-like
variations out-of-flare with period of 30 s. However, later it was found that
the star exhibits the same variations with period of 1.2 and 1.4 minutes
(Andrews, 1989). In fact, it is a debate issue whether V1285 Aql (catalog )
exhibits any rotational modulation, or not. Moreover, the period of the
equatorial rotation is found to be $2^{d}.9$, this is an other debate issue
for V1285 Aql (catalog ) (Doyle, 1987; Alekseev & Gershberg, 1997; Messina et
al., 2001).
V1285 Aql (catalog ) was observed in U band for flare patrol in 2006, and 83
white-light flares were detected in U band. Considering the studies of Haro &
Parsamian (1969); Osawa et al. (1968); Moffett (1974); Gurzadian (1988) and
following the method described by Dal & Evren (2010), first of all, we
analysed large U band flare data in order to classify flares. This is because
the classification of the flare light variations is important due to modelling
the event (Gurzadian, 1988; Gershberg, 2005). In the literature, white-light
flare events observed on the surfaces of UV Ceti type stars were usually
classified into two types as slow and fast flares (Haro & Parsamian, 1969;
Osawa et al., 1968). On the other hand, both Oskanian (1969) and Moffett
(1974) classified flares in more than two types. According to Kunkel, the
observed flare light variations should be a combination of slow and fast
flares (Gershberg, 2005). Finally Dal & Evren (2010) developed a rule, which
is depending on the ratios of flare decay times to flare rise times. According
to Dal & Evren (2010), if the decay time of a flare is 3.5 times longer than
its rise time at least, the flare is a fast flare. If not, the flare is a slow
flare. They demonstrated that the value of 3.5 is a boundary limit between the
two types of flare.
In the second step, we analysed flare data set to find general properties of
flare events occurring on the surface of V1285 Aql (catalog ). The method
described by Dal & Evren (2011a) was followed for these analyses. The energy
limit and some timescales of the flare events occurring on a star are as
important as the types of these event. In the literature, Gurzadian (1988)
stated about two processes as thermal and non-thermal processes, and mentioned
that there must be a large energy difference between these two types of
flares. Moreover, Gershberg (1972); Lacy et al. (1976); Wall (1981); Gershberg
& Shakhovskaya (1983); Pettersen et al. (1984) and, Mavridis & Avgoloupis
(1986) studied on the distributions of flare energy spectra of UV Ceti type
stars. There are significant differences between energy levels of stars from
different ages. Depending on the processes of Solar Flare Event, however,
flare activity seen on the surfaces of dMe stars is generally modelled. This
is why the magnetic reconnection process is accepted as the source of the
energy in these events (Gershberg, 2005; Hudson & Khan, 1997). According to
both some models and observations, it is seen that some parameters of magnetic
activity can reach the saturation (Gershberg, 2005; Skumanich & McGregor,
1986; Vilhu & Rucinski, 1983; Vilhu et al., 1986; Doyle, 1996a, b). Recently,
Dal & Evren (2011a) have been examined the distributions of flare equivalent
durations versus flare total durations. In the analyses, the distributions of
flare-equivalent durations were modelled by the One Phase Exponential
Association function (hereafter the OPEA). In the models, it is seen that
flare-equivalent durations can not be higher than a specific value and it is
no matter how long the flare total duration is. According to Dal & Evren
(2011a), this level, the $Plateau$ parameter, is an indicator for the
saturation level of the flare process occurring on the surface of the program
stars in some respects. In fact, white-light flares are detected in some large
active regions such as compact and two-ribbon flares occurring on the surface
of the Sun (Rodonó, 1990; Benz & Güdel, 2010). It is possibly expected that
the energies or the flare-equivalent durations of white-light flares can reach
the saturation.
Moreover, it is well known that some samples of UV Ceti stars exhibit
sinusoidal-like variations out-of-flare activity. The stars such as, EV Lac,
V1005 Ori are well known samples (Dal & Evren, 2011b). In this respect, apart
from the flare patrol, V1285 Aql (catalog ) was observed in BVR bands from
2006 to 2008 in order to examine whether there is any sinusoidal-like
variation due to rotational modulation. The variation out-of-flare activity
was analysed with using three different methods of the time series analyses.
In fact, a sinusoidal-like variation due to rotational modulation was found
out-of-flare activity.
## 2 Observations and Analyses
### 2.1 Observations
The observations were acquired with a High-Speed Three Channel Photometer
attached to the 48 cm Cassegrain type telescope at Ege University Observatory.
Observations were grouped in two schedules. Using a tracking star in second
channel of the photometer, flare observations were only continued in standard
Johnson U band with exposure times between 7 and 10 seconds. The second
observation schedule was used for determining whether there is any variation
out-of-flare. Pausing flare patrol of program stars, we observed the star once
or twice a night, when they were close to the celestial meridian. The
observations in this schedule were made with the exposure time of 10 seconds
in each band of standard Johnson BVR system, respectively. These observations
were continued from the season 2006 to 2008. The same comparison stars were
used for both types of observations.
Some properties of V1285 Aql (catalog ) and its comparisons are listed in
Table 1. Standard V magnitudes and B-V colour indexes obtained in this study
are given in Table 1. Although V1285 Aql (catalog ) and its comparison stars
are very close to one another on the celestial plane, differential extinction
corrections were applied. The extinction coefficients were obtained from
observations of the comparison stars on each night. Moreover, the comparison
stars were observed with the standard stars in their vicinity and the reduced
differential magnitudes, in the sense of variable minus comparison stars, were
transformed to the standard system using the procedures described by Hardie
(1962). The standard stars were chosen from the catalogues of Landolt (1992).
Heliocentric corrections were applied to the times of the observations. The
standard deviations of observation points acquired in the standard Johnson
UBVR bands are about $0^{m}.015$, $0^{m}.009$, $0^{m}.007$ and $0^{m}.007$ on
each night, respectively. To compute the standard deviations of observations,
we used the standard deviations of the reduced differential magnitudes in the
sense comparisons (C1) minus check (C2) stars for each night. There is no
variation in the standard brightness of the comparison stars. The flare patrol
of V1285 Aql (catalog ) was continued for 32 nights between May 5, 2006 and
August 25, 2006. 83 flares were detected in this U band patrol.
Gershberg (1972) developed a method for calculating flare energies. Flare-
equivalent durations and energies were calculated using Equations (1) and (2)
of this method,
$P=\int[(I_{flare}-I_{0})/I_{0}]dt$ (1)
where $I_{0}$ is the intensity of the star in the quiescent level and
$I_{flare}$ is the intensity during flare, and
$E=P\times L$ (2)
where $E$ is the flare energy, $P$ is the flare-equivalent duration, and $L$
is the luminosity of the stars in the quiescent level in the Johnson U band.
Some parameters such as HJD of flare maximum time, flare rise and decay times
(s), amplitude of flare (mag), and flare-equivalent duration (s) were
calculated for each flare. The procedure followed in the computations was
described in detail by Dal & Evren (2010). In brief, it is more important to
note that the brightness of a star without a flare was taken as a quiescent
level of the brightness of this star on each night. Considering this level,
all flare parameters were calculated for each night. If a flare has a few
peaks, both the maximum time and amplitude of the flare were calculated from
the first highest peak. All calculated parameters are listed in Table 2 for 83
flares. The observing date, HJD of flare maximum time, flare rise and decay
times (s), flare total duration (s), flare amplitude in U band (mag), U-B
colour index (mag), flare energy (erg) and flare type are listed in the
columns of the table, respectively. Although it was explained in detail by Dal
& Evren (2010), more important second note is that the flare-equivalent
durations were used in the analyses due to the luminosity term ($L$) in
Equation (2), in stead of flare energies.
When the observed flares are examined, it is seen that almost each flare has a
distinctive light variation shape (Figures 1 - 4). In these figures, the
horizontal dashed lines represent the level of quiescent brightness. According
to the rule described by Dal & Evren (2010), the flare shown in Figure 1 is a
sample of fast flare. This type flare is classified as a flare event in the
classification of Moffett (1974). According to Dal & Evren (2010), the flares
shown in Figure 2 and 4 are two samples of the fast flares, while these flares
could be classified as classical flares by Moffett (1974), and the flare shown
in Figure 3 is a complex flare sample.
### 2.2 Fast and Slow Flares
Following the method used by Dal & Evren (2010), all detected flares were
analysed to examine whether the limit ratio (3.50) is acceptable for the
flares detected from V1285 Aql (catalog ). Thus, using new-large data set, we
have tested whether the value of 3.50 is a general limit, or not. First of
all, in the analyses, we compared the equivalent durations of flares, whose
rise times are equal. For instance, there are 16 flares, whose rise times are
15 s. The average of their equivalent durations is 15.026 s. Apart from these
16 flares, there are 5 other flares, whose rise times are similarly 15 s.
However, the average of their equivalent durations is 5.454 s for these 5
flares. The main difference of these two example groups is seen in the shapes
of the light curves. Finally, we found 48 flares with higher energy and 35
flares with lower energy among 83 flares detected from V1285 Aql (catalog ).
In the analyses, the Independent Samples t-Test (hereafter t-Test) (Wall &
Jenkins, 2003; Dawson & Trapp, 2004) was used in the SPSS V17.0 (Green et al.,
1999) and GrahpPad Prism V5.02 (Motulsky, 2007) softwares in order to test
whether these two groups are really independent from each other. The flare
rise times were taken as a dependent variable in the t-Test, while the flare-
equivalent durations were taken as an independent variable. The value of
($\alpha$) is taken as 0.005, which gave us to test whether the results are
statistically acceptable, or not (Dawson & Trapp, 2004).
The mean averages of equivalent durations were compared for two groups. The
mean average of the equivalent durations for 35 slow flares was calculated and
found to be 1.479$\pm$0.054 s, and it was found to be 2.015$\pm$0.067 s for 48
fast flares in the logarithmic scale. This shows that there is a difference of
about 0.536 between average equivalent durations in the logarithmic scale. The
probability value (hereafter $p-value$) was computed to test the results of
the t-Test, and it was found to be $p<0.0001$. Considering $\alpha$ value,
this means that the result is statistically acceptable. All the results
obtained from the t-Test analyses are given in Table 3.
In the second step, the distributions of the equivalent durations ($logP_{u}$)
versus flare rise times ($logT_{r}$) were derived for both flare types. The
best models for the distributions were searched. Using the Least-Squares
Method, regression calculations showed that the best models of distributions
are linear functions given by Equations (3) and (4). The derived linear fits
are shown in Figure 5.
$log(P_{u})~{}=~{}1.150~{}\times~{}log(T_{r})~{}-~{}0.285$ (3)
$log(P_{u})~{}=~{}0.932~{}\times~{}log(T_{r})~{}+~{}0.385$ (4)
In the next step, it was tested whether these linear functions belong to two
independent distributions, or not. At this point, the slopes of linear
functions were principally examined. As can be seen in Table 3, the slope of
the linear function is 0.932$\pm$0.056 for slow flares, while it is
1.150$\pm$0.095 for fast flares. This shows that the increase in equivalent
durations versus flare rise times for both fast and slow flares is almost
parallel. When the $p-value$ was calculated to test whether two fits can be
statistically accepted as parallel, it was found to be $p=0.670$. This value
indicates that there is no significant difference between the slopes of fits,
and they are statistically parallel.
Finally, the y-intercept values of two linear fits were calculated and
compared. While the $y-intercept$ value is -0.385 for the slow flares, it is
-0.285 for the fast flares in the logarithmic scale. There is a difference of
about 0.100 between them. When the $p-value$ was calculated for the
$y-intercept$ values to say whether there is a statistically significant
difference, it was found that $p<0.0001$. This result indicates that the
difference between two $y-intercept$ values is clearly important.
To test whether there is a difference between maximum energy levels and
timescales of the two flare types, the distributions of the equivalent
durations in the logarithmic scale versus flare rise times were derived. The
derived distributions are shown in Figure 6. Using the Least-Squares Method,
regression calculations showed that the averaged value of upper limit is
2.928$\pm$0.251 for the fast flares, while it is 2.217$\pm$0.075 for the slow
flares. Moreover, the lengths of flare rise times for both types of flares can
be compared in Figure 6. While the lengths of rise times for slow flares can
reach to 1817 s, they are not longer than 510 s for fast flares.
### 2.3 The One-Phase Exponential Association Models of the Distribution of
the Flares
In order to test whether there are any upper limits for the distributions of
the equivalent durations ($logP_{u}$), the distributions of the equivalent
durations in the logarithmic scale versus flare total durations were examined.
Using regression calculations, the best model fit was identified by SPSS V17.0
software for this distribution. Analyses showed that the OPEA function
(Motulsky, 2007; Spanier & Oldham, 1987) given by Equation (5) is the best
model fit. According to Dal & Evren (2011a), this is an expected case, and
this demonstrated that the flares occurring on the surface of V1285 Aql
(catalog ) have an upper limit for producing energy. Using the Least-Squares
Method, the OPEA model of the distributions of the equivalent durations in the
logarithmic scale versus flare total durations was derived by GrahpPad Prism
V5.02.
$y~{}=~{}y_{0}~{}+~{}(Plateau~{}-~{}y_{0})~{}\times~{}(1~{}-~{}e^{-k~{}\times~{}x})$
(5)
Although the details of the OPEA function have been given by Dal & Evren
(2011a), in brief, there are some important parameters derived from this
function, which are some indicators for the condition of the occurring flare
processes. One of them is $y_{0}$, which is the lower limit of equivalent
durations for observed flares in the logarithmic scale. In contrast to
$y_{0}$, the parameter of $Plateau$ is the upper limit. The value of $y_{0}$
depends on the quality of observations as well as flare power, while the value
of $Plateau$ depends only on power of flares. This parameter is identified as
a saturation level for flare activity observed in U band by Dal & Evren
(2011a). The derived OPEA model is shown in Figure 7, while the parameters of
the model are listed in Table 4. The $Span$ value listed in the table is the
difference between the values of $Plateau$ and $y_{0}$. The $Half-Life$ value
is half of the first $x$ values, where the model reaches the $Plateau$ values
for a star. In other words, it is half of the flare total duration, where
flares with the highest energy start to be seen. Moreover, statistical
analyses showed that maximum flare rise time obtained from these 83 flares is
1817 s, while the maximum flare total duration is 4641 s.
On the other hand, using the t-Test, the $Plateau$ value derived from the
model was tested whether the $Plateau$ value is statistically acceptable, or
not. The flares in the $Plateau$ phases were only used to test. The mean
average of the equivalent durations was computed and found to be
2.500$\pm$0.076. In fact, the $Plateau$ value had been found to be
2.421$\pm$0.058. Considering standard deviations of the parameters, these two
parameters are almost equal to each other.
### 2.4 The Rotational Modulation Out of Flares
In order to determining whether there is any sinusoidal-like variation due to
rotational modulation out-of-flare, we observed the star once or twice a night
for three observing seasons. To purify the data from any flares or flare like
variations, we used U band light and U-B colour as an indicators. This is
because the U-B colour indexes are much more sensitive to the flare activity
on the surface of the star. If a flare is too small to be detected in respect
to the threshold, no sing is seen in V light, B-V and V-R colours. On the
other hand, some distinctive sign could be seen in U band light and U-B
colour. According to the results of these controls, some data and observations
were disregarded for the analyses of sinusoidal-like variation.
After these eliminations, all the data sets in BVR bands were analysed with
the method of Discrete Fourier Transform (DFT) (Scargle, 1982). The results
obtained from DFT were tested by two other methods. One of them is CLEANest,
which is another Fourier method (Foster, 1995). The second method is the Phase
Dispersion Minimization (PDM), which is a statistical method (Stellingwerf,
1978). These methods confirmed the results obtained by DFT. Found photometric
periods are 3.1269$\pm$0.0005 in B band, 3.1265$\pm$0.0005 in V band and
3.1268$\pm$0.0005 in R band. The photometric periods found from each sets are
almost equal to each other.
Using the ephemeris given in Equation (6), which was found from V band by the
DFT method (Scargle, 1982), the phases were computed for each season of three
years. The obtained light curves in V band and colour curves of B-V and V-R
are shown in Figure 8.
$JD(Hel.)=~{}24~{}53905.49106+3^{d}.1265~{}\times~{}E.$ (6)
From the DFT models of V band, the minimum phase of sinusoidal-like variation
is $0^{P}.47$ in the season 2006, while it is $0^{P}.19$ for 2008 and
$0^{P}.51$ for 2008. Although no variation in B-V colour curves is seen above
the values of $3\sigma$ for each season, there are some variations in the V-R.
If the colour curves of V-R are considered, it is seen that the star gets
redden toward the minimum phase of the light curves in 2006 and 2008, while
there is not any reddening or bluer in the V-R curve in 2007. Although it
seems that there is a variation in the V-R curves of 2007, its amplitude is
close to the values of $3\sigma$, and is almost lower than it. Considering the
mean B-V value of $1^{m}.469$ and the variations in both light and colour
curves, all these sinusoidal-like variation must be due to the rotational
modulation of some region(s) covered by dark stellar spots.
### 2.5 The Phase Distributions of the Flares
The phases of flare maxima were computed for all flare types (together with
fast and slow flares) with the same method used for the phases of light
curves. The flare maximum times were used to compute the phases due to main
energy emitting in this part of the flare light curves. In addition, the
photometric periods of the stars are too long according to the average of
flare total durations. In the second step, we computed the flare occurrence
rates, the ratio of total flare number ($\Sigma n_{f}$) to total monitoring
time ($\Sigma T_{t}$), in intervals of 0.10 phase length. Using Equation (7),
we followed the method used by Leto et al. (1997) and Dal & Evren (2011b).
$N~{}=~{}\Sigma n_{f}~{}/~{}\Sigma T_{t}$ (7)
The computed mean flare occurrence rates were plotted versus the rotational
phase as a histogram. The plotted histogram is shown in Figure 9. Using the
Least-Squares Method, the histogram of $N$ was analysed with the SPSS V17.0
(Green et al., 1999) software to determine the phase in which Maximum Flare
Occurrence Rates (hereafter MFOR) are seen. The analysis shows that the phase
of MFOR is between the phases of $0^{P}.40$ \- $0^{P}.50$ in the season 2006.
This phase range is the same range of the minimum phase of sinusoidal-like
variation due to rotational modulation found in the season 2006.
Gurzadian (1986) developed a hypothesis called Fast Electron Hypothesis, in
which red dwarfs generate only fast flares on their surface. On the other
hand, according to the flare region on the surface of the star in respect to
direction of observer, the shapes of the flare light variations can be seen
like a slow flare. If the scenario in this hypothesis is working, it is
expected that the fast and slow flares should be collected into two phases in
the light curves of UV Ceti type stars, which exhibit BY Dra Syndrome. It is
also expected that these two phases are separated from each other with
intervals of $0^{P}.50$ in phase.
Then the phase distributions of fast flares were compared with the phases of
slow flares in order to find out whether there is any separation as expected
in this respect. When the phases of both fast and slow flares were examined
one by one, it was clearly seen that both of them can occur in any phase. To
reach a definite result, the phase distributions of both fast and slow flares
were statistically investigated. The same method described above was used for
these groups, too. The obtained histograms of both the fast and slow flares
are shown in Figure 10. The histogram of the fast flares is shown in upper
panel, while it is shown in bottom panel for the slow flares. These histograms
were analysed with using the Least-Squares Method, the phase of MFOR was found
to be $0^{P}.45$ for the fast flares, while it was found to be $0^{P}.30$ for
the slow flares. The difference between the distributions of the two groups is
about $0^{P}.15$.
## 3 Results and Discussion
### 3.1 Flare Activity and Flare Types
In this study, we detected 83 flares in U band observations of V1285 Aql
(catalog ). The samples of the fast and slow flares, whose rise times are
equal, were determined as the suitable sets to analysis with using the t-Test.
The results of the statistical t-Test analyses show that there are some
distinctive differences between two data sets. The slope of the linear fit is
0.932 for slow flares, which are low energy flares, and it is 1.150 for fast
flares, which are high energy flares. The values are almost close to each
other. It demonstrates that the flare-equivalent durations versus the flare
rise times increase in similar ways for both groups.
When the averages of equivalent durations for two types of flares were
computed in the logarithmic scale, it was found that the average of equivalent
durations is 1.479 for slow flares, and it is 2.205 for fast flares. The
difference of 0.536 between these values in the logarithmic scale is equal to
the 73.384 s difference between the equivalent durations. As can be seen from
Equation (2), this difference between average equivalent durations affects the
energies in the same way. Therefore, there is a difference of 73.384 between
the energies of these two types of flares. This difference must be the
difference mentioned by Gurzadian (1988). On the other hand, according to Dal
& Evren (2010), this difference between two flare types is about 157 s.
Although the value found in this study is about half of the value found by Dal
& Evren (2010), but there are several reasons for this difference. The value
given by Dal & Evren (2010) is an average derived from five flare stars.
Moreover, there are EV Lac and EQ Peg among these five stars in the study of
Dal & Evren (2010). These two stars could cause some difference on the values,
because they are seen to be dramatically different among other stars, as seen
in Figures 5 and 6 given by Dal & Evren (2011a). There must be some
differences in the flare processes occurring on the surfaces of these stars.
Comparing the $y-intercept$ values of the linear fits, it is seen that there
is a 0.100 times difference in the logarithmic scale, while there is a 0.536
times difference between general averages. Also considering Figure 5, it is
seen that equivalent durations of fast flares can increase more than the
equivalent durations of slow flare toward long rise times. Some other effects
should be involved in the fast flare process for long rise times. These
effects can make fast flares seem more powerful than they actually are.
There is another difference between these two types due to both the lengths of
their rise times and their amplitudes. The lengths of rise times can reach to
1817 s for slow flares, but are not longer than 510 s for fast flares. In
addition, when the flare amplitudes are examined for both types of flares, an
adverse difference is seen contrary to rise times. While the amplitudes of
slow flares reach to $0^{m}.480$ at most, the amplitudes of fast flares can
exceed $1^{m}.430$.
Finally, computing the ratios of flare decay times to flare rise times for two
types of flares, it is seen that the ratios do not exceed a specific value for
all the slow flares. On the other hand, the ratios are always above this
specific value for the fast flares. It means that the type of an observed
flare can be determined by considering this value of the ratio. The limit
value of this ratio is about 2.0 for the flares detected from V1285 Aql
(catalog ). However, Dal & Evren (2010) demonstrated that the limit values of
the ratio of flare decay time to flare rise time is 3.5. Like the difference
seen between the values of the equivalent durations given by Dal & Evren
(2010) and the values of the equivalent durations given in this study for the
fast and slow flares, the limit values of the ratios of flare decay times to
flare rise times are also different. This must be again because of the same
reasons.
Providing that the limit value of the ratios of flare decay times to flare
rise times between flare types, the fast flare rate is 58.5$\%$ of the 83
flares observed in this study, while the slow flare rate is 41.5$\%$. It means
that one of every two flares is the fast flare, other one is slow flare. This
result diverges from what Gurzadian (1988) stated. According to Gurzadian
(1988), slow flares with low energies and low amplitudes make up 95$\%$ of all
flares, and the remainders are fast flares.
It must be noted that comparing the correlation coefficients of the linear
fits, it is seen that the correlation coefficient of the slow flares is quite
higher than the fast flares. The same results was revealed by Dal & Evren
(2010). The difference is due to the equivalent durations of fast flares
taking values in a wide range. Considering the non-thermal processes dominated
in the fast flare events, Dal & Evren (2010) tried to explain it with the
magnetic reconnection processes.
### 3.2 The Saturation Level in the Detected White-Light Flare
The distributions of flare-equivalent durations versus flare total duration
were modelled by the OPEA function expressed by Equation (5) for 83 flares
detected in U band observations of V1285 Aql (catalog ). The regression
calculations demonstrated that the best model function is the OPEA function
for the distributions of flare-equivalent durations versus flare total
duration. The derived model shows that flare-equivalent durations increase
with the flare total duration until a specific total duration value, and then
the flare-equivalent durations are constant no matter how long the flare total
duration is.
According to the OPEA model of the flares detected from V1285 Aql (catalog ),
observed flare-equivalent durations start to reach maximum value in the value
of 402.2 s of the flare total durations (note that the $Half-Life$ value is
201.1 s). The maximum value of the flare-equivalent durations is 2.421 s in
logarithmic scales for the flare detected from V1285 Aql (catalog ). It means
that the flare processes occurring on the surface of V1285 Aql (catalog ) do
not generate a flare more powerful than the specific value. This specific
value of the flare-equivalent durations can be defined as a saturation level
for the white-light flares occurring on the surface of V1285 Aql (catalog ).
The white-light flares occur in the regions, where the compact and two-ribbon
flare events are seen (Rodonó, 1990; Benz & Güdel, 2010). In the analyses, we
used data obtained by the same method and the same optical system. In
addition, we used the flare-equivalent durations instead of the flare
energies. In fact, the derived $Plateau$ values depend only on the power of
the white-light flares. Considering the $Plateau$ values, the flare-equivalent
durations cannot be higher than a particular value no matter how long the
flare total duration is. Instead of the flare duration, some other parameters,
such as magnetic field flux and/or particle density in the volumes of the
flare processes, must be more efficient in determining the power of the
flares. Considering thermal and non-thermal flare events, both these
parameters can be more efficient. However, Both Doyle (1996a) and Doyle
(1996b) suggested that the saturation in the active stars does not have to be
related to the filling factor of magnetic structures on the stellar surfaces
or the dynamo mechanism under the surface. It can be related to some radiative
losses in the chromosphere, where the temperature and density are increasing
in the case of fast rotation. This phenomenon can occur in the chromosphere
due to the flare process instead of fast rotation, and this causes the
$Plateau$ phase to occur in the distributions of flare-equivalent duration
versus flare total duration.
On the other hand, the $Plateau$ phase cannot be due to some radiative losses
in the chromosphere with increasing temperature and density. This is because
Grinin (1983) demonstrated the effects of radiative losses in the chromosphere
on the white-light photometry of the flares. According to Grinin (1983), the
negative H opacity in the chromosphere causes the radiative losses, and these
are seen as pre-flare dip in the light curves of the white-light flares.
Unfortunately, considering the results of Dal & Evren (2011a), it is seen that
the $Plateau$ values vary from one star to the next. This indicates that some
parameters or their efficacies, which make the $Plateau$ increase, are
changing from star to star. It is seen that V1285 Aql (catalog ) is among its
analogues. According to Standard Magnetic Reconnection Model developed by
Petschek (1964), there are several important parameters giving shape to flare
events, such as Alfvén velocity ($\nu_{A}$), $B$, the emissivity of the plasma
($R$) and the most important one, the electron density of the plasma ($n_{e}$)
(van den Oord & Barstow, 1988; van den Oord et al., 1988). All these
parameters are related with both heating and cooling processes in a flare
event. van den Oord et al. (1988); van den Oord & Barstow (1988) have defined
the radiative loss timescale ($\tau_{d}$) as $E_{th}/R$. Here $E_{th}$ is the
total thermal energy, while $R$ is emissivity of the plasma. $E_{th}$ depends
on the magnetic energy, which is defined as $B^{2}/8\pi$, and $R$ depends on
the electron density ($n_{e}$) of the plasma. $\tau_{d}$ is firmly correlated
with $B$ and $n_{e}$, while $\tau_{r}$ is proportional to a larger loop length
($\ell$) and smaller $B$ values. Consequently, it is seen that both the shape
and power of a flare event depend on mainly two parameters, $n_{e}$ and $B$.
In addition, obtained maximum flare duration for V1285 Aql (catalog ) flares
is 4641 s. This duration is in agreement with those found by Dal & Evren
(2011a). Observed maximum duration is 2940 s for EV Lac (catalog ), and 3180 s
for EQ Peg (catalog ). The flares of both stars are dramatically lower than
that of V1285 Aql (catalog ). Maximum flare duration of V1285 Aql (catalog )
is almost 1.5 times of them. This case reveals some clues about the flaring
loop geometry on these stars Reeves & Warren (2002); Imanishi et al. (2003);
Favata et al. (2005); Pandey & Singh (2008).
### 3.3 Rotational Modulation and the Flare Distributions versus Rotational
Phase
V1285 Aql (catalog ) was observed in BVR bands apart from U band. Using the
DFT method (Scargle, 1982), the data sets of BVR bands were separately and
together analysed for each observing season. It was found that the photometric
period of the sinusoidal-like variation due to rotational modulation out-of-
flare is $3^{d}.1269$ in B band, $3^{d}.1265$ in V band, $3^{d}.1268$ in R
band. All the results obtained from the DFT method were tested by PDM and
CLEANest methods. Examining the light variations, it is seen that the
amplitudes are lower in some degree, but all the amplitudes are higher than
the level of $3\sigma$. Besides, there are some variations in the colour
curves, too. Considering the B-V index of V1285 Aql (catalog ), the
sinusoidal-like variation due to rotational modulation out-of-flare must be
due to the heterogeneous temperature distribution on the surface of V1285 Aql
(catalog ). There must be some dark stellar spots on the surface. The
variation of the V-R colour also supports this case. However, we do not see
any variation above the level of $3\sigma$ in the B-V colour.
In this study, it was found that B-V colour index is $1^{m}.469$. It was given
as $1^{m}.53$ by Pettersen (1991), while it was given as $1^{m}.75$ by Messina
et al. (2001). Different B-V indexes have been given in three studies for
V1285 Aql (catalog ). This could be due to both the methods and parameters
used to find B-V index. If the methods and/or parameters are a little bit
different, the obtained B-V can be slightly different. On the other hand, the
main reason of these differences must be the magnetic activity seen on the
surface of the star. The level of the flare activity observed on the star is
very high. Although the same activity level is not seen in the light curves
for the stellar rotational modulation, but all the surface of the star could
be covered by dark stellar spots. If this is the case, it means that the level
of the magnetic activity is rather higher than it is observed in the light
curves out-of-flares. This caused to vary the observed B-V colour indexes from
one study to next. Apart from B-V colour index of the star, found photometric
period is also a bit different from the period found by Doyle (1987). The
photometric period found in this study is about $3^{d}.127$, while the period
found by Doyle (1987) is $2^{d}.19$. This difference must be due to the
variations of the locations of the spotted area(s). It is possible that the
photometric period might be changing because of this.
Using the models shown with dashed lines in Figure 8, which were derived with
analysing of each V band data set, the minimum, maximum, mean levels and the
amplitudes of the light curves were computed. Examining these parameters
listed in Table 5, the levels of the brightness slowly varies season to
season. The variations are slow. This must be because of the small developing
of the structures on the surface of the star.
There are many studies about whether the flares of UV Ceti type stars, which
exhibit BY Dra Syndrome, are occurring at the same longitudes of stellar
spots, or not (Dal & Evren, 2011b). Having the same longitudes of flare and
spots is an expected case for these stars, because solar flares are mostly
occurring in the active regions, where spots are located on the Sun (Benz &
Güdel, 2010). In the respect of Stellar-Solar Connection, a result of the $Ca$
$II$ $H\&K$ Project of Mount Wilson Observatory (Wilson, 1978; Baliunas et
al., 1995), if the areas of flares and spots are related on the Sun, the same
case might be expected for the stars. In fact, Montes et al. (1996) found some
evidence to demonstrate this relation. Besides, Leto et al. (1997) found a
variation of both the rotational modulation and the phase distribution of
flare occurrence rates in the same way for the observations in the year 1970.
On the other hand, no clear relation between stellar flares and spots was
found by Bopp (1974); Pettersen et al. (1983) and Dal & Evren (2011b).
However, Pettersen et al. (1983) did not draw firm conclusions because of
being a non-uniqueness problem.
In this study, using the method described in Section 2.5, we derived the
distribution of flares versus rotational phase for all the flares detected in
the observing season 2006. The derived distribution is shown in Figure 9.
According to the distribution, the phase of MFOR is seen between $0^{P}.40$
and $0^{P}.50$. Almost 6 flares were detected per hour in this phase interval,
while 2 flares were detected per hour at most in all other phase intervals.
This is a more important result, because this phase interval is where the
minimum phase of the sinusoidal-like variation due to rotational modulation
out-of-flare is seen in the V band light curves of the observing season 2006.
The phase of the sinusoidal-like variation is $0^{P}.47$.
Using the inverse Compton event, Gurzadian (1986) developed the Fast Electron
Hypothesis. According to this hypothesis, UV Ceti type stars should generate
only fast flares on their surface. However, the shapes of the flare light
variations can be seen like a slow flare in respect to direction of observer
(Gurzadian, 1986). According to the Fast Electron Hypothesis, it should be
expected that both the fast and slow flares get groups, which are separated
$0^{P}.50$ from each other. In this study, using the method described by Dal &
Evren (2011b), 48 fast and 35 slow flares were defined and their phases were
computed. As seen from the analyses, the slow flares occur more frequently
around $0^{P}.30$, while the fast flares are occurring more frequently around
$0^{P}.45$. Unfortunately, the difference between the phases of two flare
types is $0^{P}.15$ instead of $0^{P}.50$. Apart from this unexpected value,
both the fast and the slow flares occur almost all phase intervals.
In conclusion, some parameters can be computed from flares observed in
photoelectric photometry and if the behaviours between these parameters can be
analysed by suitable methods, flare types can be determined. In this study, we
analysed the distributions of equivalent durations versus flare rise times by
using a t-Test. Finally, it is seen that using the ratios of flare decay times
to flare rise times, flares can be classified. It is seen that there are
considerable differences between these two types of flares. Moreover, analyses
demonstrated the detected flares have some critical energy level. There is no
flare, whose energy is much more than this level. In addition, analyses
demonstrated that V1285 Aql (catalog ) exhibits stellar spot activity apart
from flares. Comparing the phase distribution of the flares with the phase of
the sinusoidal-like variation demonstrated that the flares have a tendency to
occur in the same longitudes with stellar spots. On the other hand, according
to the statistical analyses, the slow flares and fast flares are not separated
with some definite rules from each other. In this respect, extending the B-V
range of observed stars, required to obtain more data, which should be
obtained from many different stars and flare patrols spanning many years, in
order to obtain more reliable results.
The authors acknowledge generous allotments of observing time at the Ege
University Observatory. We thank the staffs working at EUO. We thank the
referee for useful comments that have contributed to the improvement of the
paper. We also thank Professor Dr. M. Can Akan who gave us valuable
suggestions that improved the language of the manuscript. Facilities: HSTCP,
A48, Ege University Observatory (EUO).
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Figure 1: A flare light curve sample for fast flares obtained from U-band
observations of V1285 Aql on 2006, May 21. Figure 2: A flare light curve
sample for fast flares obtained from U-band observations of V1285 Aql on 2006,
June 17. Figure 3: A flare light curve sample for fast flares obtained from
U-band observations of V1285 Aql on 2006, July 21. Figure 4: A flare light
curve sample for fast flares obtained from U-band observations of V1285 Aql on
2006, July 21. Figure 5: Distributions for the mean averages of the equivalent
durations ($logP_{u}$) vs. flare rise times ($logT_{r}$) in the logarithmic
scale. In the figure, open circles represent slow flares, while filled circles
show the fast flares. Lines represent fits given in Equations (3) and (4).
Figure 6: Distributions of the equivalent durations ($logP_{u}$) in the
logarithmic scale vs. flare rise times ($T_{r}$) for all 83 flares detected in
observations of program stars. In the figure open circles represent slow
flares, while filled circles show the fast flares. Figure 7: Distributions of
flare-equivalent duration on a logarithmic scale vs. flare total duration.
Filled circles represent equivalent durations computed from flares detected
from V1285 Aql. The line represents the model identified with Equation (5)
computed using the least-squares method. The dotted lines represent 95$\%$
confidence intervals for the model. Figure 8: V band light and B-V and V-R
colour curves obtained in this study are seen for three data sets composed
from the observations of V1285 Aql. a) Observing season 2006, b) Observing
season 2007, c) Observing season 2008. In the panels of the light curves,
dashed lines represent the fits derived from the Discrete Fourier Transform.
Figure 9: The mean flare occurrence rates versus rotational phase are
demonstrated for all V1285 Aql flares observed in the season 2006. In the
figure, the line shows the histogram of mean flare occurrence rates computed
in intervals of 0.10 phase length. All 83 flares (combining the fast and slow
flares together) were counted in the calculation. Figure 10: The mean flare
occurrence rates versus rotational phase are demonstrated for the V1285 Aql
flares observed in the season 2006. In the figures, the lines show the
histograms of mean flare occurrence rates computed in intervals of 0.10 phase
length. The histogram of the fast flares is shown in upper panel, while it is
shown for the slow flares in bottom panel.
Table 1: Basic parameters for the star studied and its comparison (C1) and check (C2) stars. Stars | V (mag) | B-V (mag)
---|---|---
V1285 Aql | 10.142 | 1.469
C1 = BD +08 3899 | 9.651 | 1.137
C2 = BD +08 3900 | 9.994 | 1.410
Table 2: The parameters derived from analyses of the detected flares. Observing | HJD of Flare | Rise | Decay | Total | Equivalent | Flare | Flare | Flare | Flare
---|---|---|---|---|---|---|---|---|---
| Maximum | Time | Time | Duration | Duration | Amplitude | U-B | Energy |
Date | (+24 00000) | (s) | (s) | (s) | (s) | (mag) | (mag) | (erg) | Type
19.05.2006 | 53875.53888 | 32 | 32 | 64 | 15.06684 | 0.357 | 0.095 | 7.10348E+30 | Slow
21.05.2006 | 53877.47941 | 30 | 20 | 50 | 9.54445 | 0.477 | 0.073 | 4.49987E+30 | Slow
21.05.2006 | 53877.50898 | 350 | 1157 | 1507 | 418.16843 | 0.462 | -0.043 | 1.97152E+32 | Fast
26.05.2006 | 53882.46588 | 10 | 50 | 60 | 17.16708 | 0.353 | 0.122 | 8.09367E+30 | Fast
26.05.2006 | 53882.46901 | 30 | 80 | 110 | 19.68734 | 0.259 | 0.203 | 9.28188E+30 | Fast
26.05.2006 | 53882.47190 | 10 | 10 | 20 | 4.82332 | 0.417 | 0.057 | 2.27402E+30 | Slow
26.05.2006 | 53882.48450 | 10 | 20 | 30 | 5.07340 | 0.289 | 0.170 | 2.39193E+30 | Fast
26.05.2006 | 53882.48902 | 20 | 40 | 60 | 7.63124 | 0.359 | 0.147 | 3.59786E+30 | Fast
30.05.2006 | 53886.36985 | 20 | 70 | 90 | 20.72296 | 0.341 | 0.065 | 9.77014E+30 | Fast
30.05.2006 | 53886.37100 | 30 | 70 | 100 | 19.67298 | 0.230 | 0.268 | 9.27511E+30 | Fast
30.05.2006 | 53886.37228 | 10 | 10 | 20 | 4.85286 | 0.438 | 0.080 | 2.28795E+30 | Slow
30.05.2006 | 53886.37575 | 10 | 30 | 40 | 15.19447 | 0.370 | 0.033 | 7.16365E+30 | Fast
30.05.2006 | 53886.37887 | 20 | 40 | 60 | 19.41469 | 0.317 | 0.116 | 9.15334E+30 | Fast
30.05.2006 | 53886.38026 | 80 | 90 | 170 | 48.04427 | 0.461 | -0.041 | 2.26512E+31 | Slow
30.05.2006 | 53886.40733 | 90 | 50 | 140 | 37.68882 | 0.453 | 0.059 | 1.77689E+31 | Slow
30.05.2006 | 53886.41347 | 50 | 130 | 180 | 25.92122 | 0.274 | 0.210 | 1.22209E+31 | Fast
17.06.2006 | 53904.37813 | 180 | 1291 | 1471 | 306.95989 | 0.282 | -0.072 | 1.44721E+32 | Fast
17.06.2006 | 53904.39776 | 30 | 45 | 75 | 12.57656 | 0.328 | 0.197 | 5.92940E+30 | Slow
17.06.2006 | 53904.39984 | 60 | 30 | 90 | 15.51715 | 0.350 | 0.167 | 7.31579E+30 | Slow
17.06.2006 | 53904.40366 | 15 | 30 | 45 | 7.49366 | 0.281 | 0.151 | 3.53300E+30 | Fast
17.06.2006 | 53904.40922 | 195 | 135 | 330 | 44.64127 | 0.241 | 0.252 | 2.10468E+31 | Slow
17.06.2006 | 53904.41338 | 30 | 45 | 75 | 10.39874 | 0.251 | 0.215 | 4.90264E+30 | Slow
17.06.2006 | 53904.46263 | 15 | 105 | 120 | 12.07430 | 0.221 | 0.336 | 5.69260E+30 | Fast
17.06.2006 | 53904.46437 | 30 | 75 | 105 | 11.53398 | 0.305 | 0.230 | 5.43786E+30 | Fast
17.06.2006 | 53904.46680 | 15 | 15 | 30 | 2.92073 | 0.234 | 0.228 | 1.37702E+30 | Slow
17.06.2006 | 53904.46923 | 30 | 105 | 135 | 14.06378 | 0.227 | 0.283 | 6.63057E+30 | Fast
17.06.2006 | 53904.47131 | 30 | 90 | 120 | 10.98540 | 0.204 | 0.277 | 5.17923E+30 | Fast
17.06.2006 | 53904.47589 | 276 | 2632 | 2908 | 2107.73499 | 1.437 | -0.745 | 9.93722E+32 | Fast
18.06.2006 | 53905.33006 | 165 | 135 | 300 | 70.78081 | 0.411 | 0.093 | 3.33706E+31 | Slow
18.06.2006 | 53905.33474 | 270 | 885 | 1155 | 272.17565 | 0.397 | 0.135 | 1.28321E+32 | Fast
18.06.2006 | 53905.35978 | 15 | 30 | 45 | 7.19677 | 0.349 | 0.160 | 3.39302E+30 | Fast
18.06.2006 | 53905.36047 | 30 | 90 | 120 | 22.84462 | 0.254 | 0.244 | 1.07704E+31 | Fast
18.06.2006 | 53905.36204 | 15 | 90 | 105 | 13.29829 | 0.291 | 0.216 | 6.26967E+30 | Fast
18.06.2006 | 53905.36796 | 30 | 30 | 60 | 9.75576 | 0.318 | 0.168 | 4.59949E+30 | Slow
18.06.2006 | 53905.37533 | 15 | 30 | 45 | 15.85617 | 0.354 | 0.153 | 7.47562E+30 | Fast
18.06.2006 | 53905.38783 | 795 | 495 | 1290 | 190.68324 | 0.190 | 0.229 | 8.99004E+31 | Slow
18.06.2006 | 53905.39862 | 15 | 60 | 75 | 17.70708 | 0.387 | 0.076 | 8.34826E+30 | Fast
18.06.2006 | 53905.40060 | 30 | 30 | 60 | 12.71769 | 0.258 | 0.181 | 5.99594E+30 | Slow
21.06.2006 | 53908.33621 | 30 | 90 | 120 | 14.55190 | 0.299 | 0.153 | 6.86070E+30 | Fast
21.06.2006 | 53908.33743 | 15 | 105 | 120 | 16.35125 | 0.241 | 0.158 | 7.70903E+30 | Fast
21.06.2006 | 53908.33899 | 15 | 45 | 60 | 6.98155 | 0.331 | 0.196 | 3.29155E+30 | Fast
21.06.2006 | 53908.33986 | 30 | 30 | 60 | 9.38434 | 0.299 | 0.154 | 4.42438E+30 | Slow
21.06.2006 | 53908.34055 | 15 | 30 | 45 | 7.28724 | 0.288 | 0.189 | 3.43567E+30 | Fast
21.06.2006 | 53908.34107 | 15 | 15 | 30 | 5.41634 | 0.314 | 0.134 | 2.55361E+30 | Slow
21.06.2006 | 53908.34142 | 15 | 15 | 30 | 4.51176 | 0.395 | 0.092 | 2.12713E+30 | Slow
21.06.2006 | 53908.34194 | 15 | 30 | 45 | 4.16976 | 0.268 | 0.201 | 1.96589E+30 | Fast
21.06.2006 | 53908.34368 | 30 | 15 | 45 | 6.69016 | 0.390 | 0.088 | 3.15417E+30 | Slow
21.06.2006 | 53908.34437 | 15 | 90 | 105 | 14.64679 | 0.289 | 0.166 | 6.90544E+30 | Fast
21.06.2006 | 53908.34559 | 15 | 15 | 30 | 7.04872 | 0.349 | 0.095 | 3.32322E+30 | Slow
21.06.2006 | 53908.34732 | 60 | 30 | 90 | 13.51126 | 0.296 | 0.202 | 6.37008E+30 | Slow
21.06.2006 | 53908.34819 | 45 | 15 | 60 | 6.55336 | 0.372 | 0.173 | 3.08968E+30 | Slow
21.06.2006 | 53908.37044 | 75 | 405 | 480 | 142.99419 | 0.728 | -0.210 | 6.74167E+31 | Fast
21.06.2006 | 53908.44178 | 1817 | 2824 | 4641 | 229.22026 | 0.171 | -0.022 | 2.30310E+32 | Slow
21.06.2006 | 53908.48645 | 150 | 405 | 555 | 110.87499 | 0.434 | 0.066 | 5.22736E+31 | Fast
21.06.2006 | 53908.49322 | 75 | 195 | 270 | 39.24736 | 0.368 | 0.154 | 1.85037E+31 | Fast
27.06.2006 | 53914.35370 | 360 | 450 | 810 | 109.39455 | 0.293 | 0.319 | 5.15756E+31 | Slow
27.06.2006 | 53914.37720 | 90 | 120 | 210 | 46.60488 | 0.322 | 0.242 | 2.19725E+31 | Slow
27.06.2006 | 53914.41381 | 75 | 465 | 540 | 80.08406 | 0.363 | 0.239 | 3.77568E+31 | Fast
27.06.2006 | 53914.45440 | 150 | 390 | 540 | 95.19813 | 0.332 | 0.212 | 4.48825E+31 | Fast
28.06.2006 | 53915.45427 | 15 | 135 | 150 | 23.80623 | 0.464 | 0.140 | 1.12238E+31 | Fast
28.06.2006 | 53915.46989 | 705 | 695 | 1400 | 163.30309 | 0.390 | 0.157 | 7.69916E+31 | Slow
21.07.2006 | 53938.34925 | 510 | 1726 | 2236 | 765.66892 | 0.821 | -0.234 | 3.60986E+32 | Fast
21.07.2006 | 53938.44548 | 15 | 90 | 105 | 17.96942 | 0.402 | 0.122 | 8.47195E+30 | Fast
21.07.2006 | 53938.45556 | 30 | 270 | 300 | 107.79387 | 0.831 | -0.323 | 5.08210E+31 | Fast
23.07.2006 | 53940.34400 | 15 | 30 | 45 | 6.47911 | 0.304 | 0.249 | 3.05467E+30 | Fast
23.07.2006 | 53940.35286 | 30 | 45 | 75 | 7.19161 | 0.381 | 0.230 | 3.39059E+30 | Slow
23.07.2006 | 53940.35563 | 150 | 584 | 734 | 78.86360 | 0.355 | 0.141 | 3.71814E+31 | Fast
23.07.2006 | 53940.36430 | 30 | 90 | 120 | 13.56160 | 0.296 | 0.140 | 6.39382E+30 | Fast
23.07.2006 | 53940.36708 | 30 | 15 | 45 | 6.34242 | 0.251 | 0.143 | 2.99023E+30 | Slow
23.07.2006 | 53940.37350 | 30 | 30 | 60 | 7.77530 | 0.220 | 0.182 | 3.66578E+30 | Slow
23.07.2006 | 53940.37559 | 45 | 30 | 75 | 13.49058 | 0.263 | 0.232 | 6.36033E+30 | Slow
27.07.2006 | 53944.34396 | 15 | 195 | 210 | 54.53049 | 0.368 | 0.213 | 2.57092E+31 | Fast
27.07.2006 | 53944.34743 | 60 | 60 | 120 | 24.11020 | 0.388 | 0.244 | 1.13671E+31 | Slow
27.07.2006 | 53944.34951 | 30 | 300 | 330 | 73.31641 | 0.366 | 0.324 | 3.45661E+31 | Fast
27.07.2006 | 53944.35368 | 15 | 30 | 45 | 14.56958 | 0.410 | 0.185 | 6.86904E+30 | Fast
27.07.2006 | 53944.36449 | 60 | 270 | 330 | 80.76671 | 0.517 | 0.053 | 3.80786E+31 | Fast
27.07.2006 | 53944.37213 | 30 | 90 | 120 | 47.89915 | 0.574 | 0.098 | 2.25828E+31 | Fast
29.07.2006 | 53946.36976 | 45 | 60 | 105 | 16.50538 | 0.288 | 0.231 | 7.78170E+30 | Slow
29.07.2006 | 53946.42235 | 45 | 15 | 60 | 7.88789 | 0.291 | 0.179 | 3.71886E+30 | Slow
29.07.2006 | 53946.42373 | 45 | 120 | 165 | 22.43408 | 0.277 | 0.233 | 1.05769E+31 | Fast
29.07.2006 | 53946.42651 | 30 | 15 | 45 | 8.10063 | 0.344 | 0.191 | 3.81916E+30 | Slow
02.08.2006 | 53950.34266 | 30 | 90 | 120 | 19.39572 | 0.352 | 0.133 | 9.14439E+30 | Fast
02.08.2006 | 53950.35061 | 15 | 15 | 30 | 7.37094 | 0.328 | 0.147 | 3.47514E+30 | Slow
Table 3: For both fast and slow flares whose rise times are the same. The results obtained from both the regression calculations and the t-Test analyses performed to the mean averages of the equivalent durations ($logP_{u}$) versus flare rise times ($logT_{r}$) in the logarithmic scale are listed. Flare Groups : | Slow Flare | Fast Flare
---|---|---
Best Representation Values | |
Slope : | 0.932$\pm$0.056 | 1.150$\pm$0.095
$y-intercept$ when $x=0.0$ : | -0.385$\pm$0.096 | -0.285$\pm$0.151
$x-intercept$ when $y=0.0$ : | 0.414 | 0.248
Mean Average of All Y Values | |
Mean Average : | 1.479 | 2.015
Mean Average Error : | 0.054 | 0.067
Goodness of Fit | |
$r^{2}$ : | 0.896 | 0.761
Is slope significantly non-zero? | |
$p-value$ : | $<$ 0.0001 | $<$ 0.0001
Deviation from zero? : | $Significant$ | $Significant$
Table 4: Using the Least-Squares Method, the parameters were obtained from the OPEA function. Parameter | Value | Error
---|---|---
$y_{0}$ | 0.656 | 0.048
$Plateau$ | 2.421 | 0.058
$k$ | 0.003447 | 0.000365
$Tau$ | 290.1 | -
$Half-Life$ | 201.1 | -
$Span$ | 1.765 | 0.064
Table 5: The maximum, minimum, mean brightness levels and amplitudes of V band light curves obtained with using the light elements given in Equation (6), which was found by the DFT method Scargle (1982). Observing | $V_{min}$ | $V_{max}$ | $V_{mean}$ | Amplitude
---|---|---|---|---
Season | (mag) | (mag) | (mag) | (mag)
2006 | 10.164 | 10.137 | 10.152 | 0.027
2007 | 10.152 | 10.128 | 10.139 | 0.024
2008 | 10.158 | 10.115 | 10.135 | 0.043
|
arxiv-papers
| 2012-06-24T19:27:53 |
2024-09-04T02:49:32.248155
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal, S. Evren",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.5794"
}
|
1206.6122
|
* # The Statistical Analyses and The OPEA Model of The White-Light Flares Occurring on Krüger 60B (DO Cep)
Hasan Ali DAL∗
∗ Department of Astronomy and Space Sciences, University of Ege, Bornova,
35100 İzmir, Turkey
∗ Email: ali.dal@ege.edu.tr
Abstract: In this study, new observations and some results of the statistical
analyses are presented. The largest flare data set of DO Cep in the literature
have been obtained with 89 flares detected in 67.61 hours $U$-band flare
patrol. First of all, the observations demonstrated that the star is one of
the most active flare stars in respect to the computed flare frequency.
Secondly, using the independent samples t-test, the detected flares were
classified into two subtypes, and then they were modelled. Analysing the
models demonstrated that the fast and slow flares occurring on the star can be
separated with a critical value of the ratio of their decay time to rise time.
The critical value was computed as 3.40. According to this value, the fast
flare rate is 20.22$\%$, while the slow flare rate is 79.78$\%$. Besides,
there is a 39.282 times difference between the energies of these two types of
flares. However, the flare-equivalent durations versus the flare rise times
increase in similar ways for both groups. In addition, all the flares were
modelled with the one-phase exponential association function. Analysing this
model, the $plateau$ value was found to be 2.810. Moreover, the $half-life$
value was computed as 433.1 s from the model. The maximum flare rise time was
found to be 1164 s, while the maximum flare total duration was found to be
3472 s. The results of the flare timescales indicate that the geometry of
flaring loop on the surface of the star might be similar to those seen on
analogues of DO Cep. Consequently, considering the both $half-life$ value and
flare timescales, the flares detected on the surface of the DO Cep get maximum
energy in longer times, while the geometries of flaring loops or areas get
smaller.
Keywords:methods: data analysis, methods: statistical, stars: spots, stars:
flare, stars: individual(DO Cep).
## 1 Introduction
The white-light flares observed on the surfaces of UV Ceti-type stars and
their process could not be absolutely understood, although these subjects have
been heavily studied (Benz and Güdel, 2010). In this study, we obtained the
largest data set from the observations of DO Cep in the literature. The data
are very useful for a statistical analysis of the white-light flare property.
Observed star, DO Cep (= Krüger 60B = KR 60B), is classified as a UV Ceti-type
star in the SIMBAD database from spectral-type dM4V (Henry et al., 1969). The
star is a component of HD 239960, which is a visual binary (Lacy, 1977;
Söderhjelm, 1999). The other component of the binary is GJ 860 A, which
classified as dM3V by Henry et al., (1969); Tamazian et al., (1969). Unlike
DO Cep, GJ 860 A (= KR 60A) is not active as seen from the literature. In the
studies of the system, the semi-major axis of the orbit ($a$) was found to be
2.420 $arcsec$, while the orbital inclination ($i$) was calculated as 172
$deg$ by Söderhjelm, (1999). The orbital period was found to be 44.64 years,
while the orbit eccentricity ($e$) was computed as 0.41 in the same study. The
distance of the system is given as 4.0 $pc$ by Pettersen, (1991), while it is
given as 4.04 $pc$ by Schmitt and Liefke, (2004). Some basic properties taken
from (Lacy, 1977) are listed in Table 1 for each component.
According to Veeder, (1974), DO Cep is an old disk star. In fact, taking
$M_{bol}=9.72$ mag and $log(T_{eff})=3.525$, its age was computed as about
$5.0\times 10^{8}$ years by Vandenberg et al., (1983). The equatorial
rotational velocity ($v\sin i$) of DO Cep was found to be 4.7 $kms^{-1}$ by
Glebocki and Gnacinski, (2005); Jenkins et al., (2009). There are several
flare patrols for DO Cep in the literature (Haro and Chavira, 1955; Herr and
Brcich, 1969; Nicastro, 1975; Contadakis et al., 1982). For the first time,
Haro and Chavira, (1955) suspected that DO Cep is a flare star. Then, Herr
and Brcich, (1969) observed the star along 27.8 hours, and they detected 10
flares. Secondly, Nicastro, (1975) detected 22 flares in 57.4 hours flare
patrol, while Contadakis et al., (1982) detected no flare in 59.13 hours
flare patrol. As seen from the literature, DO Cep has a high level flare
activity.
In this study, DO Cep was observed in $U$-band for flare patrol in 2006 and
2007, and 89 white-light flares were detected. In order to classify the flares
detected from DO Cep, the method described by Dal and Evren, (2010) was used.
In the literature, there are several studies about classifying the white-light
flares (Haro and Parsamian, 1969; Osawa et al., 1968; Moffett, 1974;
Gurzadian, 1988). The classification of the flare light-variations is
important due to modelling these events (Gurzadian, 1988; Gershberg, 2005).
The white-light flare events were generally classified into two subtypes as
slow and fast flares (Haro and Parsamian, 1969; Osawa et al., 1968).
However, some studies, such as Oskanian, (1969) and Moffett, (1974),
revealed that the white-light flares can be classified in more than two
subtypes. Kunkel, (1967) revealed that the observed flare light-curves should
be a combination of slow and fast flares. Recently, Dal and Evren, (2010)
developed a rule to classifying white-light flares. The rule depended on the
ratios of flare decay times to flare rise times demonstrates that the flare,
whose decay time is 3.5 times longer than its rise time, is a fast flare. If
the decay time of a flare is shorter than 3.5 times of its rise time, the
flare can be classified as a slow flare. In fact, there are two possible
energy sources for the white-light flares (Gurzadian, 1988). According to the
author, the thermal processes are dominant in the slow flare events, while the
nonthermal processes are dominant in the fast flare events. A rapid increasing
is generally seen in light curves, if the energy source is caused by the
nonthermal processes (Benz and Güdel, 2010; Gershberg, 2005; Gurzadian,
1988).
Apart from the shapes of the flare light-variations, the upper and lower
limits of both flare-power and flare timescales are also important to
understand the flare processes occurring on a star. In order to compare the
flare-powers of different stars, several studies have been done. In these
studies, the flare energy spectra were derived for each star (Gershberg,
1972; Lacy et al., 1976; Wall, 1981; Gershberg and Shakhovskaya, 1983;
Pettersen et al., 1984; Mavridis and Avgoloupis, 1986). According to the
results of these studies, the energy levels of stars vary from a star to next
one. The variations seem to be caused by different ages of stars. On the other
hand, the analyses based on the flare energies could not give the real
results. Because the flare energy depends on the luminosity of a star as well
as the power of the flare. Thus, if the stars are from different spectral-
types, the energies of these flare will be different, even if their real
powers are the same. In this respect, the flare-equivalent duration was based
on the analyses in this study in order to determined the behaviour of the
white-light flares of DO Cep. These method recently developed by Dal and
Evren, 2011a is based on the modelling the distributions of the flare-
equivalent durations versus flare total durations. The authors demonstrated
that the best function is the one-phase exponential association function
(hereafter the OPEA) to model the distribution. As it is seen in the OPEA
model, the flare-equivalent durations can not be higher than a specific value,
and the flare’s total duration does not matter. Dal and Evren, 2011a defined
this level as an indicator for the saturation level for the white-light flare
processes. In fact, white-light flares are detected in some large active
regions, where compact and two-ribbon flares are occurring on the surface of
the Sun (Rodonó, 1990; Benz and Güdel, 2010). It is possibly expected that
the energies or the flare-equivalent durations of white-light flares can also
reach the saturation. Generally, flare activity seen on the surfaces of dMe
stars is modelled concerning the processes of the solar flare event. This is
why the magnetic reconnection process is accepted as the source of the energy
in these events (Gershberg, 2005; Hudson and Khan, 1997). According to both
some models and observations, it is seen that some parameters of magnetic
activity can reach the saturation (Gershberg, 2005; Skumanich and McGregor,
1986; Vilhu and Rucinski, 1983; Vilhu et al., 1986; Doyle, 1996a ; Doyle,
1996b ).
## 2 Observations and Analyses
### 2.1 Observations
The observations of the flare patrol were acquired with a High-Speed Three
Channel Photometer attached to the 48 cm Cassegrain type telescope at Ege
University Observatory. Using a tracking star in second channel of the
photometer, flare observations were continued in standard Johnson $U$-band
with exposure times between 7 and 10 seconds in the time resolution of 0.01
seconds. Considering the technical properties of the HSTCP given by Meištas,
(2002) and following the procedures outlined by Kirkup & Frenkel, (2006), the
mean average of the standard deviations of observation times was computed as
0.08 seconds for the U band observations. Some properties of DO Cep and its
comparisons are listed in Table 2. Standard $V$ magnitudes and $B-V$ color
indexes obtained in this study are given in Table 2. Although DO Cep and its
comparison stars are very close to one another on the celestial plane,
differential extinction corrections were applied. The extinction coefficients
were obtained from observations of the comparison stars on each night.
Moreover, the variable and its comparison stars were observed in the standard
Johnson $UBVR$ bands with the standard stars in their vicinity and the reduced
differential magnitudes, in the sense of variable minus comparison stars, were
transformed to the standard system using the procedures described by Hardie,
(1962). The standard stars were chosen from the catalogues of Landolt,
(1992). Heliocentric corrections were applied to the times of the
observations. The standard deviations of observation points acquired in the
standard Johnson $UBVR$ bands are about 0.015 mag, 0.009 mag, 0.007 mag and
0.007 mag on each night, respectively. To compute the standard deviations of
observations, we used the standard deviations of the reduced differential
magnitudes in the sense comparisons (C1) minus check (C2) stars for each
night. There is no variation in the standard brightness of the comparison
stars. The flare patrol of DO Cep was continued for 9 nights between September
9 and November 26 in 2006, and 11 nights between July 31 and October 17 in
2007. The total duration of the $U$-band flare patrol is 22.76 h in 2006,
while it is 44.85 h in 2007. 4446 observing pints were obtained in 2006, while
9841 observing points were obtained in 2007. According to the $3\sigma$ of the
$U$ band standard deviation in each night, it was decided whether an event
observed in that night is a flare, or not. Therefore, 88 flares were detected
in 2007, while only one flare could be detected in 2006.
Gershberg, (1972) developed a method for calculating flare energies. Flare-
equivalent durations (s) and energies (erg) were calculated using equations
(1) and (2) of this method,
$P=\int[(I_{flare}-I_{0})/I_{0}]dt$ (1)
where $I_{0}$ is the intensity of the star in the quiescent level and
$I_{flare}$ is the intensity during flare, and
$E=P\times L$ (2)
where $E$ is the flare energy (erg), $P$ is the flare-equivalent duration (s),
and $L$ is the luminosity of the stars in the quiescent level in the Johnson
$U$-band. It must be noted that the flare-equivalent duration has a time unit
(s).
The parameters of the flare light curves were calculated for each flare. All
the parameters were computed following the procedure described in detail by
Dal and Evren, (2010). There are some important points in the procedure. We
firstly separated each flare light curve into three parts. One of them is the
part indicating the quiescent level of the brightness before the first flare
on each nigh. The brightness level without any variations (such as a flare or
any oscillation) was taken as a quiescent level of the brightness of this
star. To determine this level, we used the standard deviation of each
observation point, considering the mean average of all the observation points
until this last point. If the standard deviations of that one and following
points get over the $3\sigma$ level, this point was taken as the beginning of
a flare. Thus, the quiescent levels of each star were determined from all the
observation points before the first flare on each night. We fitted this level
with a linear function, and then, using this linear function, we computed the
flare-equivalent duration, flare amplitude and all the flare time-scales (rise
and decay times). The part of the light curve above the quiescent level was
also separated into two sub-parts. First of them is the impulsive phase, in
which the flare increases. Second one is the decay phase. The impulsive and
decay phases were separated according to the maximum brightness observed in
this part. It must be noted that some flares have a few peaks. In this case,
the point of the first-highest peak was assumed as the flare maximum. To
determine the flare time-scales, we fitted the impulsive and decay phases with
the polynomial functions. The best polynomial functions were chosen according
to the correlation coefficients ($r^{2}$) of fits. To determine the beginning
and end of each flare, we computed the intersection points of the polynomial
fits with the linear fit of the quiescent level and their standard deviations.
In this study, the intersection points were taken as the beginning and end of
each flare. The flare rise time was taken the duration between the beginning
and the flare maximum point. In the same way, the flare decay time was taken
the duration between the flare maximum point and the flare end. The height of
the observed-maximum points from the quiescent level was taken as the
amplitudes of this flare. The same procedure was used for each flare, and
Grahp-Pad Prism V5.02 (Motulsky, 2007) software was performed in all
calculations.
All calculated parameters are listed in Table 3 for 89 flares. The observing
date, HJD of flare maximum time, flare rise and decay times (s), flare total
duration (s), flare-equivalent durations (s), flare amplitude in $U$-band
(mag), $U-B$ color index (mag), flare energy (erg) and flare type are listed
in the columns of the table, respectively. As it is explained in Section 1,
other important point is that the flare-equivalent durations were used in the
analyses due to the luminosity term ($L$) in equation (2), in stead of flare
energies.
$N~{}=~{}\Sigma n_{f}~{}/~{}\Sigma T_{t}$ (3)
In addition to these parameters, following the method used by Leto et al.,
(1997) and Dal and Evren, 2011b , the flare frequency ($N$) was computed for
each observing season. In equation (3), $n_{f}$ is the total number of the
flare detected in a season, and $T_{t}$ is total time of the flare patrol in
that season. Using equation (3), the value of $N$ was found to be 0.044 for
2006. It was found to be 3.866 for 2007.
If the detected flares are examined, it will be seen that the light curve of
each flare has a distinctive light-variation shape. Five light curve parts
from the observations are seen in Figures 1 - 5 for the examples. The
horizontal dashed lines seen in these figures represent the level of quiescent
brightness. Three flares detected on 2007, August 1 are seen in Figure 1.
According to the rule described by Dal and Evren, (2010), the Flare A and B
shown in Figure 1 are two slow flare samples, while the Flare C is a sample of
the fast flare. The flares seen in Figure 2 were detected on 2007, August 3.
In this figure, the Flare A and B are the samples of the fast flares, while
the Flare C is a slow flare. Figure 3 shows two flares, and both of them are
the fast flares. The Flare A in this figure is the most powerful flare
detected in this study. Its amplitude is 1.90 mag in the U band. Its rise time
is 270 s, and decay time is 3202 s. The flare seen in figure 4 is a combined
flare. There should be actually two flares (Part A and Part B), but their
light variations are combined in the light curves. Moffett, (1974) classified
the flares like this as a complex flares. It should be noted that the flares
like this one were not ignored in the analyses described in Section 2.2.
Another interesting samples are seen in Figure 5. In the figure, Flare A and B
are the slow flare, while Flare C is the fast flare. Apart from these three
flares, there are three spikes, which are combined with Flare A and B.
### 2.2 Fast and Slow Flares
The flares detected from DO Cep were analysed using the method developed by
Dal and Evren, (2010). It was tested whether the limit ratio (3.50) is also
acceptable for the white-light flares detected from DO Cep. Thus, using new-
large data set, it was tested whether the value of 3.50 is a general limit, or
not.
In first step, the equivalent durations of flares, whose rise times are equal,
were compared. For example, there are 13 flares, whose rise time are 15 s. The
light variations of 9 flares among these 13 flares are similar to Flare A and
B seen in Figure 1. The light variations of other 4 fares among these 13
flares are similar to Flare C shown in Figure 1. The average of their
equivalent durations is 6.206 s for 9 flares, which are similar Flare A and B.
However, the average of their equivalent durations is 29.551 s for other 4
flares. The main difference of these two example groups is seen in the shapes
of the light curves. Finally, we found 18 flares with higher energy and 71
flares with lower energy among 83 flares detected from DO Cep.
Using the independent samples t-test (hereafter t-test) (Wall and Jenkins,
2003; Dawson and Trapp, 2004) in the SPSS V17.0 (Green et al., 1999) and
Grahp-Pad Prism V5.02 (Motulsky, 2007) software, data sets were analysed in
order to test whether these two groups are statistically independent from each
other. In the analyses, the flare rise times were taken as a dependent
variable, and the flare-equivalent durations were taken as an independent
variable. The value of ($\alpha$) is taken as 0.005, which allowed us to test
whether the results are statistically acceptable, or not (Dawson and Trapp,
2004).
The mean average of the equivalent durations for 71 slow flares was found to
be 1.544$\pm$0.067 s, and it was computed as 1.871$\pm$0.130 s for 18 fast
flares in the logarithmic scale. This shows that there is a difference of
about 0.327 s between average equivalent durations in the logarithmic scale.
The probability value (hereafter $p-value$) was computed to test the results
of the t-test, and it was found to be $p<0.0001$. Considering $\alpha$ value,
this means that the result is statistically acceptable. All the results
obtained from the t-test analyses are given in Table 4.
In the second step, the distributions of the equivalent durations ($logP_{u}$)
versus flare rise times ($logT_{r}$) were modelled for both flare types. Using
the least-squares method, the best models for the distributions were examined
in the SPSS V17.0 and Grahp-Pad Prism V5.02 software. The regression
calculations demonstrated that the best fits of distributions are linear
functions. The derived linear fits given by equations (4) and (5) are shown in
Figure 6.
$log(P_{u})~{}=~{}1.232~{}\times~{}log(T_{r})~{}-~{}0.137$ (4)
$log(P_{u})~{}=~{}1.046~{}\times~{}log(T_{r})~{}-~{}0.450$ (5)
In the next step, the linear functions were compared. The slope of the linear
function was found to be 1.046$\pm$0.048 for slow flares, while it was
computed as 1.232$\pm$0.181 for fast flares. Then, the $p-value$ was
calculated and found to be $p=0.650$. The $p-value$ indicates that there is no
significant difference between the slopes of fits, and it can be assumed that
they are statistically parallel.
Finally, the $y-intercept$ values of both linear fits were compared. The
$y-intercept$ value was found to be -0.450 for the slow flares, and it was
found to be -0.137 for the fast flares in the logarithmic scale. There is a
difference of about 0.313 between them. Then, the $p-value$ was computed for
the $y-intercept$ values to say whether there is a statistically significant
difference, it was found to be $p<0.0001$. The result demonstrated that the
difference between two $y-intercept$ values is obviously important.
The distributions of the equivalent durations in the logarithmic scale versus
flare rise times were modelled with the OPEA function to find the maximum
energy levels and timescales of the two flare types. The derived distributions
are shown in Figure 7. Using the least-squares method, the regression
calculations showed that the averaged value of upper limit is 2.559$\pm$0.096
for the slow flares. On the other hand, the regression calculations indicated
that the distribution can not be modelled with the OPEA function due to the
linear increasing. The linear fit derived for the fast flares is also seen in
Figure 7. Beside the averaged value of upper limit, it was found that the
lengths of rise times for slow flares can reach to 1164 s, while they are not
longer than 270 s for fast flares.
### 2.3 The One-Phase Exponential Association Models of the Distribution of
the Flares
The distribution of the flare-equivalent durations in the logarithmic scale
versus the flare total durations indicates that the flare mechanism occurring
on the surface of DO Cep has a upper limit for the flare energy. To examine
this case, the distribution was modelled and statistically analysed. First of
all, the distribution of the equivalent durations ($logP_{u}$) in the
logarithmic scale versus the flare total durations was obtained. Then, using
regression calculations, the best function was determined to fit the
distribution by SPSS V17.0 software. The regression analyses demonstrated that
the OPEA function (Motulsky, 2007; Spanier & Oldham, 1987) given by equation
(6) is the best model fit. According to Dal and Evren, 2011a , this is
actually an expected case. The case demonstrates that the flares occurring on
the surface of DO Cep have an upper limit for producing energy. In the final
step, the OPEA model of the distributions was derived by Grahp-Pad Prism V5.02
using the least-squares method.
$y~{}=~{}y_{0}~{}+~{}(plateau~{}-~{}y_{0})~{}\times~{}(1~{}-~{}e^{-k~{}\times~{}x})$
(6)
The details of the OPEA function have been given by Dal and Evren, 2011a . In
brief, some important parameters can be derived from the OPEA function, and
these parameters reveal the condition of the flare mechanism occurring on the
surface of the star. One of them is $y_{0}$, which is the lower limit of
equivalent durations for observed flares in the logarithmic scale. In contrast
to $y_{0}$, the parameter of $plateau$ is the upper limit. It should be noted
that the $y_{0}$ parameter depends on the quality of observations as well as
flare power. However, $plateau$ parameter depends only on power of flares. Dal
and Evren, 2011a identified the $plateau$ parameter as a saturation level for
the white-light flare activity observed in $U$-band. The derived OPEA model is
shown in Figure 8, while the parameters of the model are listed in Table 5.
The $span$ value listed in the table is the difference between the values of
$plateau$ and $y_{0}$. One of the most important parameters is the $half-life$
value. This parameter is half of the first $x$ values, where the model reaches
the $plateau$ values. In other words, it is half of the flare total duration,
where flares with the highest energy start to be seen. In order to test the
the $plateau$ values derived from the OPEA model, the upper limit of the
equivalent durations was computed using the t-test. Thus, the $plateau$ value
was tested whether it is statistically acceptable, or not. The flares in the
$plateau$ phases of the model were only used to test. The mean average of the
equivalent durations was computed and found to be 2.808$\pm$1.149.
As seen from the data distributions, the maximum flare rise time obtained from
these 89 flares is 1164 s, while the maximum flare total duration is 3472 s.
## 3 Results and Discussion
### 3.1 Flare Activity and Flare Types
In this study, 89 white-light flares were detected in $U$-band observations of
DO Cep. 88 flare were detected in 44.85 h flare patrol of 2007, while only one
flare was detected in 22.76 h flare patrol of 2006. Therefore, 0.044 flares
were detected per hour in 2006, while 3.866 flares were detected per hour in
2007. There is a large difference between the flare frequencies ($N_{2006}$
and $N_{2007}$) of consecutive observing seasons. A large differences between
the flare frequencies obtained in different years are also seen in the
literature (Herr and Brcich, 1969; Nicastro, 1975; Contadakis et al.,
1982). According to the results of Herr and Brcich, (1969), $N$ value is
0.360 in 1968. The flare frequency ($N$) is 0.509 flares per hour in 1970, and
it is 0.208 flares per hour in the observing season of 1972 - 1973 (Nicastro,
1975). However, Contadakis et al., (1982) detected no flare in 1975.
Consequently, the largest frequency was obtained with 3.866 flares per hour in
this study. It seems that DO Cep is active as well as EV Lac, EQ Peg and AD
Leo (Moffett, 1974; Dal and Evren, 2010). The flare frequency variation of
UV Ceti type stars has been examined in several studies (Ishida et al., 1991;
Leto et al., 1997). Ishida et al., (1991) found no variation for a few
stars, while Leto et al., (1997) demonstrated that the flare frequency of EV
Lac is dramatically increasing. It must be noted that DO Cep should be taken
to observing programs, because its flare frequency is remarkably varying.
The flares detected in this study are examined one by one. The flares, whose
rise times are equal, were determined. It was seen that the flares are
accumulating into two groups. It was seen that even if the rise times of two
flares are equal, their equivalent durations can be different from each other.
Apart from their equivalent durations, the main difference between two type
flares is light-variation shapes. As seen from Figures 1 - 5, the some flares
slowly increase and slowly decrease, while some of them rapidly increase, but
slowly decrease.
In logarithmic scale, the flare distributions were obtained for both groups.
First of all, two group flares were analysed with t-test. Then, they were
modelled with the linear function, and the models of two groups were analysed
to compare. Using t-test, the averages of equivalent durations for two types
of flares were computed. The average of equivalent durations was found to be
1.871 s for the fast flares, and it is 1.544 s for the slow flares. The
difference of 0.327 s between these values in the logarithmic scale is equal
to the 39.282 s difference between the equivalent durations. As can be seen
from equation (2), this difference between average equivalent durations
affects the energies in the same way. Therefore, there is 39.282 times
difference between the energies of these two types of flares. This difference
must be the difference mentioned by Gurzadian, (1988). On the other hand,
according to Dal and Evren, (2010), this difference between two flare types
is about 157 times. In the case of DO Cep, the energies of the slow and fast
flares occurring on the surface of the star seem to be closer to each other.
Apart from the average of equivalent durations, the parameters of the linear
fits were also compared. The slope of the linear fit is 1.046 for the slow
flares, which are low-energy flares, and it is 1.232 for the fast flares,
which are high-energy flares. According to the $p-value$, the slopes are
almost close to each other. It demonstrates that the flare-equivalent
durations versus the flare rise times increase in similar ways for both
groups. However, the fast flare (Flare A) seen in Figure 3 is seen out of the
general trend of the fast flares. This flare is the most powerful flare
detected in the study. It seems to be an extreme example. In the case of the
extreme examples, some effects must be involved in the fast flare process
towards the long rise times. These effects can make fast flares seem more
powerful than they actually are. However, comparing the $y-intercept$ values
of the linear fits, it is seen that there is a 0.313 times difference in the
logarithmic scale. There is a 0.327 times difference between general averages.
Both values are close to each other. It means that the energy emitting
processes behave similarly except the extreme flare. Apart from the equivalent
durations, the differences between these two flare types are seen in the
lengths of their rise times and their amplitudes. The maximum rise time seen
among the slow flares is 1164 s, but it is 270 s for fast flares. In addition,
the amplitudes of slow flares can reach to 0.791 mag at most, the amplitudes
of fast flares can reach to 1.900 mag.
Dal and Evren, (2010) computed the ratios of flare decay times to flare rise
times for two types of flares. They have demonstrated that there is a limit
values between two flare types. This limit value of the ratio of flare decay
time to flare rise time is 3.50. In this study, the limit value of this ratio
is found to be 3.40 for the flares detected from DO Cep. Providing this limit
value between flare types, it was found that the fast flare rate is 20.22$\%$
of the 89 flares observed in this study, while the slow flare rate is
79.78$\%$. It means that one of every five flares is the fast flare, other
four flares are the slow flare. This result is close to what Gurzadian,
(1988) stated. According to Gurzadian, (1988), slow flares with low energies
and low amplitudes make up 95$\%$ of all flares, and the remainder are fast
flares.
### 3.2 The Saturation Level in the Detected White-Light Flare
The distributions of flare-equivalent durations versus flare total duration
were modelled by the OPEA function expressed by equation (6) for 89 white-
light flares detected in observations of DO Cep. To model the distribution,
the best model curve was searched. Considering $p-value$ and the correlation
coefficient ($r^{2}$) parameters, the OPEA function was found as the best
model function. The main characteristic feature of the OPEA is that this
function has a $plateau$ phase. According to the observations, the flare-
equivalent durations increase with the flare total duration until a specific
total duration value. After the specific total duration, the flare-equivalent
durations are constant, and the total duration does not matter. There is just
one flare among all of them. This flare is Flare A seen in Figure 3. It must
be an extreme sample.
Some parameters such as $plateau$ value, $half-life$, etc., were derived from
the OPEA model. The $plateau$ value was found to be 2.810. The value is in
agreement with the mean average of the equivalent durations. It had been found
to be 2.808. Considering the standard deviations of two values, they can be
assumed to be equal. Besides, the found $plateau$ value is also in agreement
with the $plateau$ values found from other stars by Dal and Evren, 2011a .
According to the $B-V$ colour index of DO Cep, it is seen that the star is
among its analogues. This result supports that the upper limit of the energy
producing by white-light flare mechanism really increase towards the later
spectral types.
It is well known that the white-light flares occur in the regions, where the
compact and two-ribbon flare events are seen (Rodonó, 1990; Benz and Güdel,
2010). In the analyses, the flare-equivalent durations were used instead of
the flare energies. In fact, the derived $plateau$ values depend only on the
power of the white-light flares. According to observations, the $plateau$
phase exists in the model. The flare-equivalent durations can not be higher
than a particular value, and the flare’s total duration does not matter. Apart
from the timescales, the power of the flares must depend on some other
parameters, such as magnetic field flux and/or particle density in the volumes
of the flare processes. However, Doyle, 1996a and Doyle, 1996b suggested that
the saturation in the active stars does not have to be related to the filling
factor of magnetic structures on the stellar surfaces or the dynamo mechanism
under the surface. It can be related to some radiative losses in the
chromosphere, where the temperature and density are increasing in the case of
fast rotation. This phenomenon can occur in the chromosphere due to the flare
process instead of fast rotation, and this causes the $plateau$ phase to occur
in the distributions of flare-equivalent duration versus flare total duration.
On the other hand, the $plateau$ phase cannot be due to some radiative losses
in the chromosphere with increasing temperature and density. This is because
Grinin, (1983) demonstrated the effects of radiative losses in the
chromosphere on the white-light photometry of the flares. According to Grinin,
(1983), the negative H opacity in the chromosphere causes the radiative
losses, and these are seen as pre-flare dip in the light curves of the white-
light flares. Unfortunately, considering the results of Dal and Evren, 2011a ,
it is seen that the $plateau$ values vary from one star to the next. This
indicates that some parameters or their efficacies, which make the $plateau$
increase, are changing from star to star. According to Standard Magnetic
Reconnection Model developed by Petschek, (1964), there are several important
parameters giving shape to flare events, such as Alfvén velocity ($\nu_{A}$),
$B$, the emissivity of the plasma ($R$) and the most important one, the
electron density of the plasma ($n_{e}$) (Van Den Oord and Barstow, 1988; Van
Den Oord et al., 1988). All these parameters are related with both heating
and cooling processes in a flare event. Van Den Oord et al., (1988); Van Den
Oord and Barstow, (1988) have defined the radiative loss timescale
($\tau_{d}$) as $E_{th}/R$. Here $E_{th}$ is the total thermal energy, while
$R$ is emissivity of the plasma. $E_{th}$ depends on the magnetic energy,
which is defined as $B^{2}/8\pi$, and $R$ depends on the electron density
($n_{e}$) of the plasma. $\tau_{d}$ is firmly correlated with $B$ and $n_{e}$,
while $\tau_{r}$ is proportional to a larger loop length ($\ell$) and smaller
$B$ values. Consequently, it is seen that both the shape and power of a flare
event depend on mainly two parameters, $n_{e}$ and $B$.
As seen from the OPEA model, the flare-equivalent durations start to reach
maximum value in a specific total duration, and the $half-life$ value was
found to be 433.1 s from the model. In addition, the maximum flare rise time
was found to be 1164 s, while the maximum flare total duration was found to be
3472 s. These results demonstrated that the flare timescales of the flares
detected from DO Cep are in fact shorter than they are in the earlier spectral
types. However, the flares get the maximum energy limits in longer times. It
is well known from the X-Ray observations of the flares that the timescales of
the X-Ray flares give some clues about the flaring loop geometry on the stars
(Reeves and Warren, 2002; Imanishi et al., 2003; Favata et al., 2005;
Pandey and Singh, 2008). The white-light flares can exhibit the same
behaviour with its counterpart observed in X-Ray (Gershberg, 2005; Benz and
Güdel, 2010). If this case is valid, the timescales derived from the white-
light flares can also give some clues about the the flaring loop geometry or
the flaring area geometry (at least for the photosphere). The obtained
timescales from the observations of DO Cep demonstrated that the flaring loop
or area are smaller than those seen on the stars from the earlier spectral
types. Because, the obtained maximum flare duration for DO Cep flares is 3472
s. The observed maximum duration is 5236 s for V1005 Ori, and 4164 s for AD
Leo (Dal and Evren, 2011a ). The flare timescales of both stars are
dramatically longer than that of DO Cep. However, considering the $half-life$
value, the flares detected from DO Cep get maximum energy in longer times,
while the geometries of flaring loops or areas get smaller.
## Acknowledgments
The author acknowledges generous allotments of observing time at the Ege
University Observatory. I also thank the referee for useful comments that have
contributed to the improvement of the paper.
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Figure 1: The flare light curves detected in U-band observations of DO Cep on
2007, August 1. In figure, Flare A and B are two samples for the slow flares,
while Flare C is a sample for the fast flares. Figure 2: The flare light
curves detected in U-band observations of DO Cep on 2007, August 3. In figure,
Flare A and B are two samples for the fast flares, while Flare C is a sample
for the slow flares. Figure 3: The flare light curves detected in U-band
observations of DO Cep on 2007, September 13. In figure, both Flare A and B
are two samples for the fast flares. Figure 4: A flare light curve detected in
U-band observations of DO Cep on 2007, September 23. In figure, this flare
light curve is a sample for the combined flares. Part A is a flare part, while
Part B is a part of another flare. Both flares would be probably a slow flare.
Figure 5: The flare light curves detected in U-band observations of DO Cep on
2007, October 10. In figure, all three flares are the samples for the slow
flares. Besides, three impulsive spikes are seen during the first two slow
flares. Figure 6: Distributions for the mean averages of the equivalent
durations ($logP_{u}$) vs. flare rise times ($logT_{r}$) in the logarithmic
scale. In the figure, open circles represent slow flares, while filled circles
show the fast flares. Lines represent fits given in equations (4) and (5).
Figure 7: Distributions of the equivalent durations ($logP_{u}$) in the
logarithmic scale vs. flare rise times ($T_{r}$) for all 89 flares detected in
observations of program stars. In the figure open circles represent slow
flares, while filled circles show the fast flares. Figure 8: Distributions of
flare-equivalent duration on a logarithmic scale vs. flare total duration.
Filled circles represent equivalent durations computed from flares detected
from DO Cep. The line represents the model identified with equation (6)
computed using the least-squares method. The dotted lines represent 95$\%$
confidence intervals for the model.
Table 1: Basic parameters for the components of visual binary Krüger 60. All the parameters were taken from Lacy, (1977). Parameter | KR 60A | KR 60B
---|---|---
| (GJ 860 A) | (DO Cep)
V (mag) | 9.850 | 11.220
V-R (mag) | 1.760 | 1.890
$\pi\times 10^{3}$ (arcsec) | 251$\pm$5 | 251$\pm$5
$log(R_{star}/R_{\odot})$ | -0.45$\pm$0.05 | -0.65$\pm$0.05
$log(M_{star}/M_{\odot})$ | -0.57$\pm$0.03 | 0.80$\pm$0.03
Table 2: Basic parameters for the star studied and its comparison (C1) and check (C2) stars. Stars | V (mag) | B-V (mag)
---|---|---
DO Cep | 9.615 | 1.604
C1 = HD 239952 | 9.528 | 1.378
C2 = SAO 34476 | 7.943 | 0.530
Table 3: The parameters derived from analyses of the detected flares. Observing | HJD of Flare | Rise | Decay | Total | Equivalent | Flare | Flare | Flare | Flare
---|---|---|---|---|---|---|---|---|---
| Maximum | Time | Time | Duration | Duration | Amplitude | U-B | Energy |
Date | (+24 00000) | (s) | (s) | (s) | (s) | (mag) | (mag) | (erg) | Type
09.10.2006 | 54049.31335 | 45 | 75 | 120 | 27.31375 | 0.386 | 0.933 | 1.28775E+31 | Slow
31.07.2007 | 54313.46502 | 45 | 30 | 75 | 23.82013 | 0.725 | 0.831 | 1.12303E+31 | Slow
31.07.2007 | 54313.50060 | 45 | 15 | 60 | 47.32471 | 0.609 | 0.919 | 2.23119E+31 | Slow
31.07.2007 | 54313.50355 | 75 | 15 | 90 | 40.59164 | 0.530 | 0.909 | 1.91375E+31 | Slow
31.07.2007 | 54313.51657 | 960 | 1032 | 1992 | 774.43561 | 0.538 | 0.715 | 3.65119E+32 | Slow
01.08.2007 | 54314.43809 | 555 | 1140 | 1695 | 449.94280 | 0.601 | 0.867 | 2.12132E+32 | Slow
01.08.2007 | 54314.46118 | 465 | 405 | 870 | 265.55134 | 0.539 | 0.832 | 1.25198E+32 | Slow
01.08.2007 | 54314.46604 | 15 | 120 | 135 | 62.38419 | 0.487 | 0.851 | 2.94119E+31 | Fast
01.08.2007 | 54314.46760 | 15 | 45 | 60 | 30.62136 | 0.791 | 0.596 | 1.44369E+31 | Slow
03.08.2007 | 54316.44609 | 300 | 945 | 1245 | 179.91426 | 0.261 | 1.075 | 8.48232E+31 | Slow
03.08.2007 | 54316.46772 | 15 | 30 | 45 | 4.42329 | 0.230 | 1.135 | 2.08543E+30 | Slow
03.08.2007 | 54316.48127 | 15 | 15 | 30 | 7.04602 | 0.331 | 1.008 | 3.32195E+30 | Slow
03.08.2007 | 54316.48213 | 45 | 60 | 105 | 18.89710 | 0.280 | 1.094 | 8.90931E+30 | Slow
03.08.2007 | 54316.48682 | 60 | 60 | 120 | 16.21439 | 0.284 | 1.089 | 7.64451E+30 | Slow
03.08.2007 | 54316.50990 | 45 | 60 | 105 | 16.65787 | 0.295 | 1.008 | 7.85360E+30 | Slow
03.08.2007 | 54316.51355 | 45 | 60 | 105 | 21.59144 | 0.055 | 1.007 | 1.01796E+31 | Slow
03.08.2007 | 54316.52072 | 90 | 735 | 825 | 236.56860 | 0.651 | 0.711 | 1.11534E+32 | Fast
03.08.2007 | 54316.53010 | 15 | 60 | 75 | 14.24462 | 0.317 | 0.948 | 6.71583E+30 | Fast
04.09.2007 | 54348.36803 | 15 | 15 | 30 | 3.67867 | 0.286 | 1.162 | 1.73436E+30 | Slow
04.09.2007 | 54348.37497 | 15 | 30 | 45 | 5.97366 | 0.057 | 1.252 | 2.81637E+30 | Slow
04.09.2007 | 54348.41749 | 225 | 270 | 495 | 59.74276 | 0.243 | 1.090 | 2.81666E+31 | Slow
04.09.2007 | 54348.42113 | 45 | 180 | 225 | 27.28352 | 0.154 | 1.189 | 1.28632E+31 | Fast
04.09.2007 | 54348.42339 | 15 | 45 | 60 | 8.12777 | 0.246 | 1.108 | 3.83195E+30 | Slow
04.09.2007 | 54348.42894 | 435 | 1260 | 1695 | 245.28684 | 0.252 | 1.101 | 1.15644E+32 | Slow
04.09.2007 | 54348.49088 | 30 | 555 | 585 | 68.49528 | 0.353 | 1.029 | 3.22931E+31 | Fast
04.09.2007 | 54348.53259 | 105 | 105 | 210 | 22.31739 | 0.251 | 1.150 | 1.05219E+31 | Slow
04.09.2007 | 54348.54624 | 15 | 150 | 165 | 25.48638 | 0.327 | 1.043 | 1.20159E+31 | Fast
04.09.2007 | 54348.57241 | 45 | 45 | 90 | 20.44655 | 0.259 | 1.060 | 9.63983E+30 | Slow
13.09.2007 | 54357.45391 | 270 | 3202 | 3472 | 4260.46668 | 1.900 | -0.324 | 2.00866E+33 | Fast
13.09.2007 | 54357.50611 | 90 | 525 | 615 | 164.36076 | 0.413 | 0.748 | 7.74903E+31 | Fast
13.09.2007 | 54357.51479 | 225 | 255 | 480 | 64.11248 | 0.221 | 1.058 | 3.02268E+31 | Slow
13.09.2007 | 54357.51792 | 15 | 30 | 45 | 4.89558 | 0.217 | 1.061 | 2.30809E+30 | Slow
13.09.2007 | 54357.52603 | 90 | 225 | 315 | 31.39548 | 0.195 | 1.105 | 1.48019E+31 | Slow
13.09.2007 | 54357.53159 | 45 | 45 | 90 | 31.21739 | 0.578 | 0.700 | 1.47179E+31 | Slow
13.09.2007 | 54357.53367 | 135 | 495 | 630 | 139.53195 | 0.320 | 0.942 | 6.57844E+31 | Fast
13.09.2007 | 54357.54027 | 15 | 90 | 105 | 16.08904 | 0.274 | 1.031 | 7.58541E+30 | Fast
13.09.2007 | 54357.54391 | 225 | 608 | 833 | 115.29866 | 0.259 | 1.021 | 5.43592E+31 | Slow
13.09.2007 | 54357.55564 | 375 | 330 | 705 | 86.03669 | 0.243 | 1.047 | 4.05633E+31 | Slow
13.09.2007 | 54357.56033 | 75 | 90 | 165 | 21.26797 | 0.218 | 1.091 | 1.00271E+31 | Slow
13.09.2007 | 54357.56310 | 150 | 90 | 240 | 28.46478 | 0.222 | 1.077 | 1.34201E+31 | Slow
13.09.2007 | 54357.56467 | 15 | 15 | 30 | 2.73049 | 0.230 | 1.060 | 1.28733E+30 | Slow
13.09.2007 | 54357.56744 | 195 | 345 | 540 | 51.13014 | 0.223 | 1.066 | 2.41060E+31 | Slow
13.09.2007 | 54357.58622 | 135 | 422 | 557 | 76.76394 | 0.267 | 0.958 | 3.61915E+31 | Slow
13.09.2007 | 54357.59180 | 60 | 45 | 105 | 14.50886 | 0.225 | 1.030 | 6.84041E+30 | Slow
13.09.2007 | 54357.59492 | 75 | 60 | 135 | 12.12054 | 0.238 | 1.031 | 5.71440E+30 | Slow
18.09.2007 | 54362.30638 | 12 | 12 | 24 | 4.85403 | 0.307 | 1.075 | 2.28850E+30 | Slow
18.09.2007 | 54362.32015 | 138 | 252 | 390 | 42.15909 | 0.273 | 1.190 | 1.98765E+31 | Slow
18.09.2007 | 54362.53984 | 96 | 60 | 156 | 25.40758 | 0.244 | 1.167 | 1.19788E+31 | Slow
18.09.2007 | 54362.55264 | 36 | 108 | 144 | 16.09805 | 0.289 | 1.133 | 7.58966E+30 | Slow
18.09.2007 | 54362.55430 | 36 | 36 | 72 | 6.24738 | 0.266 | 1.169 | 2.94542E+30 | Slow
18.09.2007 | 54362.55597 | 108 | 36 | 144 | 17.35319 | 0.306 | 1.134 | 8.18142E+30 | Slow
18.09.2007 | 54362.56389 | 648 | 538 | 1186 | 141.62994 | 0.180 | 1.241 | 6.67735E+31 | Slow
18.09.2007 | 54362.57109 | 24 | 36 | 60 | 7.93551 | 0.382 | 1.055 | 3.74131E+30 | Slow
18.09.2007 | 54362.57386 | 48 | 36 | 84 | 14.60747 | 0.303 | 1.138 | 6.88690E+30 | Slow
18.09.2007 | 54362.60384 | 96 | 185 | 281 | 41.67595 | 0.356 | 1.051 | 1.96487E+31 | Slow
23.09.2007 | 54367.31400 | 30 | 150 | 180 | 58.40560 | 0.624 | 0.947 | 2.75362E+31 | Fast
23.09.2007 | 54367.31834 | 15 | 15 | 30 | 10.36101 | 0.611 | 0.936 | 4.88485E+30 | Slow
Table 3: The parameters derived from analyses of the detected flares. Observing | HJD of Flare | Rise | Decay | Total | Equivalent | Flare | Flare | Flare | Flare
---|---|---|---|---|---|---|---|---|---
| Maximum | Time | Time | Duration | Duration | Amplitude | U-B | Energy |
Date | (+24 00000) | (s) | (s) | (s) | (s) | (mag) | (mag) | (erg) | Type
23.09.2007 | 54367.31956 | 30 | 15 | 45 | 12.05673 | 0.587 | 1.019 | 5.68432E+30 | Slow
23.09.2007 | 54367.32268 | 45 | 45 | 90 | 31.64597 | 0.590 | 0.909 | 1.49200E+31 | Slow
23.09.2007 | 54367.36543 | 90 | 180 | 270 | 72.42159 | 0.454 | 0.991 | 3.41442E+31 | Slow
23.09.2007 | 54367.36890 | 90 | 90 | 180 | 57.38554 | 0.499 | 0.974 | 2.70552E+31 | Slow
23.09.2007 | 54367.37872 | 255 | 135 | 390 | 98.81383 | 0.545 | 0.919 | 4.65872E+31 | Slow
23.09.2007 | 54367.38237 | 75 | 270 | 345 | 91.03458 | 0.533 | 0.979 | 4.29196E+31 | Fast
23.09.2007 | 54367.38741 | 165 | 165 | 330 | 112.60634 | 0.554 | 1.005 | 5.30899E+31 | Slow
23.09.2007 | 54367.39001 | 60 | 90 | 150 | 43.89776 | 0.526 | 1.072 | 2.06962E+31 | Slow
23.09.2007 | 54367.39348 | 195 | 345 | 540 | 157.07515 | 0.516 | 0.988 | 7.40554E+31 | Slow
23.09.2007 | 54367.39869 | 105 | 315 | 420 | 161.72549 | 0.698 | 0.880 | 7.62478E+31 | Slow
23.09.2007 | 54367.42463 | 120 | 240 | 360 | 78.94735 | 0.372 | 1.099 | 3.72209E+31 | Slow
23.09.2007 | 54367.43088 | 75 | 150 | 225 | 56.78978 | 0.661 | 0.845 | 2.67744E+31 | Slow
23.09.2007 | 54367.43366 | 45 | 30 | 75 | 33.35158 | 0.630 | 0.874 | 1.57241E+31 | Slow
23.09.2007 | 54367.43938 | 30 | 60 | 90 | 21.49201 | 0.522 | 1.042 | 1.01327E+31 | Slow
23.09.2007 | 54367.44095 | 30 | 45 | 75 | 13.82339 | 0.513 | 1.129 | 6.51724E+30 | Slow
30.09.2007 | 54374.46141 | 75 | 495 | 570 | 117.72863 | 0.416 | 1.063 | 5.55049E+31 | Fast
30.09.2007 | 54374.46888 | 60 | 210 | 270 | 55.56658 | 0.428 | 1.047 | 2.61977E+31 | Fast
30.09.2007 | 54374.47217 | 75 | 255 | 330 | 76.63445 | 0.425 | 1.040 | 3.61304E+31 | Fast
10.10.2007 | 54384.31223 | 577 | 744 | 1321 | 482.01474 | 0.784 | 0.751 | 2.27253E+32 | Slow
10.10.2007 | 54384.32514 | 372 | 744 | 1116 | 288.43829 | 0.275 | 1.278 | 1.35988E+32 | Slow
10.10.2007 | 54384.33889 | 36 | 48 | 84 | 14.26053 | 0.462 | 1.119 | 6.72333E+30 | Slow
10.10.2007 | 54384.34167 | 36 | 24 | 60 | 21.88830 | 0.448 | 1.103 | 1.03196E+31 | Slow
10.10.2007 | 54384.34709 | 132 | 156 | 288 | 56.78634 | 0.347 | 1.181 | 2.67727E+31 | Slow
10.10.2007 | 54384.35084 | 168 | 204 | 372 | 59.84651 | 0.328 | 1.215 | 2.82155E+31 | Slow
10.10.2007 | 54384.37766 | 1057 | 1164 | 2221 | 617.18031 | 0.444 | 1.103 | 2.90979E+32 | Slow
10.10.2007 | 54384.41952 | 12 | 60 | 72 | 23.38733 | 0.498 | 1.027 | 1.10263E+31 | Fast
10.10.2007 | 54384.46799 | 24 | 12 | 36 | 7.28310 | 0.521 | 0.994 | 3.43372E+30 | Slow
10.10.2007 | 54384.48174 | 60 | 108 | 168 | 32.81632 | 0.603 | 0.881 | 1.54717E+31 | Slow
10.10.2007 | 54384.49910 | 24 | 84 | 108 | 31.69380 | 0.609 | 0.845 | 1.49425E+31 | Fast
10.10.2007 | 54384.50133 | 108 | 36 | 144 | 51.14498 | 0.564 | 0.941 | 2.41130E+31 | Slow
17.10.2007 | 54391.25731 | 48 | 540 | 588 | 72.54632 | 0.317 | 1.052 | 3.42030E+31 | Fast
17.10.2007 | 54391.35890 | 1164 | 2075 | 3239 | 418.76582 | 0.190 | 1.156 | 1.97433E+32 | Slow
Table 4: For both fast and slow flares whose rise times are the same. The results obtained from both the regression calculations and the t-test analyses performed to the mean averages of the equivalent durations ($logP_{u}$) versus flare rise times ($logT_{r}$) in the logarithmic scale are listed. Flare Groups : | Slow Flare | Fast Flare
---|---|---
Best Representation Values | |
Slope : | 1.046$\pm$0.048 | 1.232$\pm$0.181
$y-intercept$ when $x=0.0$ : | -0.450$\pm$0.095 | -0.137$\pm$0.302
$x-intercept$ when $y=0.0$ : | 0.430 | 0.111
Mean Average of All Y Values | |
Mean Average : | 1.544 | 1.871
Mean Average Error : | 0.067 | 0.130
Goodness of Fit | |
$r^{2}$ : | 0.871 | 0.744
Is slope significantly non-zero? | |
$p-value$ : | $<$ 0.0001 | $<$ 0.0001
Deviation from zero? : | $Significant$ | $Significant$
Table 5: Using the least-squares method, the parameters were obtained from the OPEA function. Parameter | Value | Error
---|---|---
$y_{0}$ | 0.972 | 0.057
$plateau$ | 2.810 | 0.057
$k$ | 0.001601 | 0.000258
$tau$ | 624.8 | -
$half-life$ | 433.1 | -
$span$ | 1.837 | 0.115
|
arxiv-papers
| 2012-06-24T19:30:35 |
2024-09-04T02:49:32.269370
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. A. Dal",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1206.6122"
}
|
1206.6170
|
arxiv-papers
| 2012-06-27T04:51:33 |
2024-09-04T02:49:32.280376
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hero Modares (corresponding author), Rosli Salleh, Amirhosein\n Moravejosharieh, Hossein Malakootikhah",
"submitter": "Hassan Keshavarz",
"url": "https://arxiv.org/abs/1206.6170"
}
|
|
1206.6250
|
# Effect of a non-volatile cosolvent on crack pattern induced by desiccation
of a colloidal gel111Soft Matter, DOI:10.1039/C2SM25663K
F. Boulogne, L. Pauchard, and F. Giorgiutti-Dauphiné222CNRS, UMR 7608, Lab
FAST, Bat 502, Campus Univ - F-91405, Orsay, France, EU. Fax: +33 1 69 15 80
60; Tel: +33 1 69 15 80 49; E-mail: pauchard@fast.u-psud.fr
###### Abstract
Consolidation of colloidal gels results in enormous stresses that are usually
released in the formation of undesirable cracks. The capacity of a gel network
to crack during drying depends on the existence and significance of a pressure
gradient in the pore liquid; in addition it depends on the way the gel relaxes
the resulting drying stresses. In this paper the effect of a binary mixture of
solvents saturating the gel network on the crack patterns formation is
investigated. Indeed, incorporation of a small quantity of non-volatile
cosolvent, i.e. glycerol, inhibits drying-induced cracks; moreover addition of
a concentration greater than $10\%$ to a colloidal dispersion leads to a crack
free coating in room conditions. Mass variation with time reveals that both
evaporation rate and cracking time are not affected by glycerol, in the range
of added glycerol contents studied. In addition measurements of mechanical
properties show that the elastic modulus is reduced with glycerol content. The
decrease of the number of cracks with the glycerol content is related to the
flattening of the pressure gradient in the pore liquid. The mechanism is shown
to be due to the combination of two processes: flow driven by the pressure
gradient and diffusion mechanisms in accordance with Scherer work (1989).
## 1 Introduction
Most coatings are made by depositing a volatile liquid that contains dispersed
colloidal particles on a surface. The liquid is then evaporated until a dry
film is obtained. Common examples in industrial fields are paints, protective
coatings (anti-corrosion), ceramic membranes [3]. Coatings are required to be
homogeneous and smooth. Particularly they have to be free of heterogeneities
possibly leading to cracking due to high stresses developing during the
consolidation. In a liquid film, evaporation of the volatile solvent causes
the solid particles to be confined into a smaller volume, until they come into
direct contact with each other, and form a solid porous network, so called gel
[4]. Then, further evaporation causes the liquid/air interface to curve into a
set of menisci that join the particles. As a consequence the capillary forces
exert extremely strong compressive stress on the particle network. When the
drying stress development in the gel exceeds the strength of the material,
undesirable effects occur such as the formation of various modes of
cracking[1, 21, 31, 32]. As a consequence the stress in the gel is relaxed.
Since numerous studies on cracks induced by drying of colloidal suspensions
have been investigated for the last two decades, the stress development and
the organization of colloidal particles in a drying film remain unclear. One
of the main questions that still stands: how can the stress be relaxed before
cracks can form? In addition, the development of stresses, hence of cracking
notably depends on the type of the particles (size[12], chemical
composition[21, 26]) and mechanical properties of the gel formed ([2, 18]),
the substrate, the drying conditions (temperature[15], relative humidity[11],
air velocity[17]) and the film thickness[2, 18, 7]. In particular the capacity
of a gel network to crack during drying depends on the existence and
significance of a pressure gradient in the pore liquid and the way the gel
relaxes the induced drying stresses.
In the present work, we focus on the effect of cosolvent added to the
colloidal dispersion on crack patterns induced by drying. Such additive
reveals crack-free drying when added to ceramic foams[14]. Glycerol has been
chosen as a cosolvent for interesting properties of high miscibility in water,
low volatility, and because it exhibits a highly different intrinsic diffusion
coefficient from that of water. Also glycerol is an element composing many
essential fine arts products[16]: due to its transparency properties and
because it is used as a softener, avoiding creases and cracks. Crack patterns
have been quantified as a function of the added glycerol content. Drying
kinetics have been investigated by mass variation with time measurements. Also
mechanical properties of colloidal gels are measured by indentation testing as
a function of the added cosolvent content. Experimental results can be
explained by the combination of two processes occuring during the drying
process: flow driven by the pressure gradient accordingly with Darcy law and
diffusion mechanism accordingly with Fick law.
## 2 Experimental
### 2.1 Materials and samples preparation
Figure 1: (a) Experimental set-up (side view): during solvent loss the layer
consolidates. (b) Sketch of the solid network saturated by solvent: curvature
of the solvent/air menisci occurs at the evaporation surface during solvent
loss and exerts an enormous stress on this network.
We used colloidal silica dispersion, Ludox HS-40 purchased from Sigma-Aldrich.
The radius of the particles is $9\pm 2$ nm, the density $2.36\times 10^{3}$
kg/m3. The mass fraction of the initial dispersion was estimated to
$\phi_{m}=40.5$% by a dry extract. Thus, we could deduce the particle volume
fraction for this solution to be $\phi_{v}\simeq 22$%. Particles are
polydisperse enough not to crystallize. In the absence of evaporation, the
suspension is stable due to the competition between the attractive Van der
Waals forces and the electrostatic repulsion[20] between particles (pH =
$9.8$).
Glycerol (purity: $99.0$%, from Merck) was added to the aqueous dispersion as
a cosolvent. This solvent exhibits high miscibility properties in water and
low vapor pressure ($2.2\times 10^{-4}$ mbar at $25^{\circ}$C)[9] compared to
water ($31.7$ mbar). Thus, we assume glycerol as a non-volatile compound. The
surface tension of pure glycerol is $62.5$mN/m at $20^{\circ}$C [22].
Glycerol/water weight ratio was prepared using the following process: The
resulting solution was filtered to $0.5$ $\mu$m to eliminate eventual
aggregates; the mass fraction is estimated by a dry extract at $140^{\circ}$C.
In order to keep the particle mass fraction constant for each sample, a given
quantity of water is evaporated, then replaced by an equal mass of glycerol;
the evaporation of water is achieved by stirring the aqueous dispersion at
$35^{\circ}$C in an Erlenmeyer. Finally, we checked the final pH of the
solution.
We note $\kappa$ the ratio between the mass of glycerol added and the mass of
the resulting dispersion. Six solutions have been investigated: $\kappa=1.1$,
$2.1$, $3.1$, $5.1$, $7.2$ and $8.9$ $\%$ wt. This leads to glycerol fraction
in solvent denoted by $\chi_{i}$ (the subscript stands for initial). In the
range of glycerol content added, the solvent viscosity increases until $1.3$
mPa.s.
### 2.2 Drying geometry
Experiments are performed on films of fixed thickness. Also a small amount of
the solution (initial weight $m_{i}=0.52\pm 0.02$ g) is deposited inside a
circular container (diameter $=15$ mm) whose bottom is a glass microscope
slide carefully cleaned with ethanol and the lateral wall is a rigid rubber
ring (figure 1 (a)). The contact line of the solution is quenched at the upper
edge of the circular wall and remains there during the whole experiment. In
this geometry the layer dries at room temperature ($T=23^{\circ}$C) under
controlled relative humidity ($R_{H}\sim 50\pm 2\%$). In this way, the
desiccation takes place in the absence of convection in the vapor so that the
evaporation is limited by diffusion of the solvent into the air. The drying
kinetics is obtained using a scale (Sartorius) with an accuracy of $0.01$ mg.
When the gelled phase is formed, the layer exhibits an approximately constant
thickness in a central region (figure 1 (a)). In this region, covering about
$70\%$ of the total surface area, the evaporation is assumed to be uniform. A
typical value of the thickness far from the border is $h\sim 0.7$ mm. The
colloidal gel is transparent allowing us to observe easily the dynamics of
crack patterns in the layer. Also the crack patterns formation is recorded
using an interval timer with an AVT’s Marlin camera positioned on the top of
the sample.
### 2.3 Measurements of the mechanical properties of the gel phase
To characterize the mechanical properties of the gelled layers relevant
elastic modulus, $E_{p}$, has been performed by indentation testing. This
quantity has been determined using a CSM Instruments Micro Indentation Tester
(MHT) with a four-sided pyramidal Vickers indenter. Indentation testing is a
local method of mechanical investigation. The indenter, initially in contact
with the surface of the gel, is driven in the material until a maximal $70$ mN
load with a loading speed $70$ mN/min. The maximal force is held during $10$
s. Then, the load is decreased until zero with the same speed. Figure 5a shows
a typical load/unload curve. The standard way of estimating the elastic
modulus from the indentation load-displacement curve uses the initial slope of
the unloading curve. Based on Sneddon work [30], the following formula has
been derived for the elastic modulus, $E_{p}$, to the unloading slope [24,
23]:
$E_{p}=\frac{\sqrt{\pi}}{2}\frac{S}{\sqrt{A}}$
In this equation, $S$ is the slope of the unloading curve at the start of
unloading, and $A$ is the projected area of contact between the indenter and
the material at that point (inset in figure 5a). Note that the contact between
the indenter and the gel is always chosen in a location where the layer
adheres on the substrate. In the case of successive measurements of the
elastic modulus with time, indentation testing was performed in the same
adhering region to obtain comparable values as effectively as possible.
Particularly, the difficulties of these measurements lie in gel films without
glycerol addition exhibiting the smallest size of adhering regions. Indeed it
has been observed that glycerol reduces the capacity of the gel to delaminate.
In addition indentation testing applied to drying gelled layers has to be
carefully analyzed since the system is not homogeneous in the thickness and
evolves with time. Also measurements of elastic moduli in gels by this way do
not give absolute values but provide on the one hand a process of comparing
mechanical properties of different materials, on the other hand a
characterization of the time evolution of the mechanical properties of the
system. Therefore measurements for the elastic modulus have been reproduced
several times and performed on 2 or 3 samples; bars in figure 5b take into
account the minimum and maximum measurements obtained.
## 3 Results
### 3.1 Drying
Once the layer is deposited in the circular trough, evaporation of the water
takes place. The drying process of a colloidal dispersion is usually separated
into two stages, that can be roughly distinguished by measuring mass, $m$, and
drying rate, $\frac{dm}{dt}$, variations with time (figure 2)[29]. In the
first regime (constant rate period) the evaporation rate is constant and
evaporation is mainly controlled by the external conditions of relative
humidity and temperature in the surroundings. During this stage, water removal
concentrates the colloidal particles into a closed packed array: a porous
network saturated with solvent. Due to geometry of the planar film,
evaporation at the gel-air interface and adhesion at the gel-substrate
interface result in the development of high stresses in the gel (figure 1(b)).
Capillary pressure that is caused by the liquid menisci formed between the
particles at the top of the packed region is responsible for the shrinkage of
the porous network. The shrinkage of the gel at the gel-air interface is
frustrated by the adhesion to the substrate. As a consequence tensile stresses
progressively build up in the layer. Then a non-linear drying rate follows:
the falling rate period, when liquid/air interfaces recede into the porous
medium and when the drying process is limited by flow of liquid to the top
layer through the porosity. The transient between the constant rate period and
the falling rate period is highlighted in the lower inset in Figure 2: it
shows the drying rate derivative $\frac{d^{2}m}{dt^{2}}$. In our experimental
conditions we observe that the transient always occurs at $\sim 32000$s in the
studied $\kappa$ range. Consequently the drying rate decreases as clearly
shown in figure 2. Similar behaviors occur when glycerol content is added to
the aqueous dispersion (figure 2).
Figure 2: Drying curves. Drying rate, $\frac{dm}{dt}$, for silica films for
different additional glycerol concentrations, $\kappa$. The drying rate is
calculated from the mass variation with time (upper inset). The variation
$\frac{\partial^{2}m}{\partial t^{2}}$ with time evidences the transient time
close to $32000$s.
### 3.2 Cracking
As the tensile stress reaches a threshold value, cracks appear in the film and
invade the plane of the layer. First cracks appear at $t\in[24000,27000]$s
after deposition of the layer on the substrate, without any correlation with
the glycerol concentration nor the initial mass disparity. Moreover the cracks
invade the layer during only a few minutes. Thus the crack network develops
during the initial stage of the drying process, in constant rate period 2, and
does not evolve afterwards. Plot in inset in figure 3 shows the ratio,
$m_{f}/m_{i}$, of the final (at $t=60000$s) and the initial mass of material
on the glass plate (raw data). Futhermore, we know that silica particles
contributes to $40$% of the mass. Thus, the excess of mass is due to the
solvent. As previously mentioned only water could evaporate and glycerol
remains in the porous medium. Also we can calculate the mass fraction of
glycerol $\chi_{f}$ composing the solvent when the mass variations with time
do not evolve anymore ($t\sim 60000$s):
$\chi_{f}=1-\frac{m_{f}/m_{i}-0.4-\kappa}{m_{f}/m_{i}-0.4}$ (1)
We found that the higher the initial fraction $\kappa$ is, the higher
$\chi_{f}$ reaches (see inset of figure 3), leading to a saturation of
glycerol in the gel for $\kappa>8.9\pm 0.3\%$ in accordance with the fit.
Crack patterns are strongly modified by the presence of a glycerol within the
solvent. Indeed, in the case of a silica gel without additional glycerol, the
successive formation of cracks results in the typical pattern shown in figure
4a. This pattern is usually continued by further crack generations:
particularly delamination process takes place when the gel detaches from the
substrate [25].
Figure 3: Mass ratio of samples for different glycerol concentration $\kappa$.
The inset shows the final concentration of glycerol in the solvent wetting the
porous medium as a function of $\kappa$. The solid line is a linear adjustment
(slope$=11.3\pm 0.3$).
Moreover final crack patterns are shown in figure 4a,b,c,d for increasing
glycerol concentrations $\kappa$ up to $10.2\%$ in our experimental
conditions. For each pattern, cracks still interconnect leading to complete
network: no broken network is observable suggesting that if crack nucleation
is less favourable in presence of additional glycerol, crack propagation is
still not inhibited. The number of cracks generated diminishes when the
quantity of additional glycerol increases; this results in increasing the
crack spacing as shown in figure 4e. Figure 4f shows a magnification of a
typical crack tip propagating in a gel with glycerol content. The sharpness of
this crack tip, also observed for every $\kappa\lesssim 9.5\%$, suggests that
the gel still exhibits brittleness properties. In addition buckle-driven
delamination process, that usually takes place after the formation of a crack
network, is inhibited by increasing additional glycerol concentration in the
gel.
Figure 4: Crack patterns after 16 hours in gelled layers when adding a second
solvent to the system. (a) Without any additional glycerol. (b)
$\kappa=5.1\%$. (c) $\kappa=8.9\%$. (d) $\kappa=10.2\%$: the layer is crack-
free at the mesoscopic scale. (e) Crack spacing plotted as a function of the
concentration in glycerol (drying conditions $R_{H}=50\%$ and $T=23^{\circ}$C;
the initial weight deposited in each trough is the same). (f) Propagating
crack tip in a gel with additional glycerol $\kappa=8.9\%$.
Above a threshold concentration of additional glycerol, the layer remains
crack free (figure 4d): no crack forms because the gel could not reach the
compaction (or the threshold stress) for which cracks propagate. It results in
a flat gel film saturated in glycerol.
### 3.3 Gel consolidation
Figure 5: Macroscopic elastic response of the gel obtained by indentation
testing. (a) Loading/unloading curve: force applied on the tip versus indenter
displacement ($\mu m$). Insets: variation of the applied force versus time;
image of indentation print at the surface of the gel after a loading/unloading
cycle observed using optical microscopy (maximum load: $70mN$, bar $=25\mu
m$): the dashed square limit the projected area of contact, $A$. (b) Variation
of the elastic modulus, $E_{p}$, for seven glycerol-Ludox solutions. (c)
Variation of the elastic modulus, $E_{p}$, as a function of time for Ludox
HS-40 without glycerol addition ($\kappa=0\%$) and for two different
concentrations of glycerol ($\kappa=3.1\%,8.9\%$). Measurements start after
crack propagation, $t>t_{m}$.
The macroscopic elastic response of the gel is characterized by indentation
testing for various binary mixtures of solvents (figure 5). Based on the
procedure depicted in figure 5a, elastic moduli have been measured for gels of
various glycerol contents. Each measurement has been obtained $\sim 8$ hours
after the cracks formation since the gel evolves slowly after this duration
(figure 2). Measurements in figure 5b reveal that an increase of the quantity
of glycerol in the gel results in a decrease of its elastic modulus. Since
glycerol remains inside the porous medium, it could act as a wetting agent
lowering the material stiffness.
The elastic modulus of silica gels has been measured as a function of time
during the drying process; measurements started after crack propagation,
$t>t_{m}$. These measurements are shown for the silica gel without any
glycerol addition and for two glycerol contents in figure 5c. In any cases,
gel films clearly stiffens as they consolidate with time; for the silica gel
without any glycerol addition, this process is more rapid than in presence of
glycerol content. Two factors could contribute to increase the modulus as a
function of time: stiffening of the solid phase of the gel and reduction in
the porosity [27]. Also the elastic modulus increases faster for gels having
lower concentrations of glycerol.
## 4 Discussion
The capacity of a gel to crack during drying depends on the existence and
significance of a pressure gradient in the pore liquid. Indeed the spatial
variation in pressure causes a variation in contraction of the gel network
[28]. This modifies the way the drying stresses are relaxed and consequently
the crack are formed. In the following the spatial variation in pressure in
the liquid pore is characterized in gels saturated by binary mixtures of
solvents during the first drying regime since cracks form during this stage.
Particularly we have chosen to compare the pressure distribution for different
glycerol contents at time $t_{m}=25500$s. At this moment, close to the typical
cracking time of the gelled layers, gels can be considered as non-shrinking,
that is the gel network does not change during time[28]; pores contain the
binary mixture of solvents and the liquid-air interfaces are already curved.
In drying gels, liquid usually moves through the network in response to a
pressure gradient of liquid, $\frac{\partial P}{\partial z}$, accordingly to
the Darcy law[4]. Also, in the case of a gel saturated with a single volatile
solvent, the flux of evaporating liquid at the gel-air interface comes
entirely from flow of liquid. It expresses as:
$J_{Darcy}=\frac{k}{\eta}\frac{\partial P}{\partial z}$ (2)
where $k$ is the network permeability of the gel, and $\eta$ is the dynamic
viscosity of the solvent in the pore; note that the pressure of the liquid
pore is regarded as positive.
Let us now consider the incorporation of a soluble and non-volatile cosolvent,
i.e. glycerol, to the aqueous dispersion. The drying process leads to a gel
phase saturated by a binary mixture of water and glycerol. This results in a
combination of the pressure flow component (2) and a diffusion mechanism[8,
28, 4]. In that way, the diffusive flux for each component, $i$, expresses in
accordance with the Fick law:
$J_{Fick}^{i}=-D_{i}\frac{\partial C_{i}}{\partial z}$ (3)
where $D_{i}$ is the intrinsic diffusion coefficient refers in terms of the
rate of transfer of component $i$ across a fixed section. Note that $P$ and
$C_{i}$ are time dependent.
The total fluid flux in the porous network combines the flow driven by the
pressure gradient (2) and the diffusive flux (3). Also, the relative change in
volume of liquid over time, $\dot{V_{L}}/V_{L}$, is due to the change in both
local flow, $\partial_{z}J_{Darcy}$, and local diffusion flux,
$\partial_{z}J_{Fick}^{i}$. As a result the following conservation equation
can be written[28]:
$(1-\rho)\frac{\dot{V_{L}}}{V_{L}}=-\partial_{z}J_{Darcy}-K\partial_{z}(J_{Fick}^{1}+J_{Fick}^{2})$
(4)
where $\rho$ is the relative density of the gel, $K$ denotes the porosity of
the gel, $J_{Fick}^{i}$ is the diffusive flux of component $i$, stating $i=1$
for water and $i=2$ for glycerol.
Since the change in the volume of liquid has to equal the change in pore
volume, the following relation can be written in accordance with
reference[28]:
$(1-\rho)\frac{\dot{V_{L}}}{V_{L}}=-\frac{1}{E_{p}}\left(\beta\dot{P}+(1-\beta)<\dot{P}>\right)$
(5)
where $\beta$ is a function of the Poisson ratio of the gel and $\dot{P}$
denotes the time derivative of the pressure. Also the flux from the gel
surface equals the rate of change in volume of the gel; this expresses as
$<\dot{P}>=E_{p}\frac{V_{E}}{h}$ for a layer of thickness $h$ accordingly to
reference[28]. Here the evaporation rate, $V_{E}$, determines the rate of
volume loss by water evaporation.
By equating the right sides of expressions 4 and 5 and taking into account
relations 2 and 3, it comes:
$\frac{\beta}{E_{p}}\frac{\partial P}{\partial
t}+(1-\beta)\frac{V_{E}}{h}=\frac{k}{\eta}\frac{\partial^{2}P}{\partial
z^{2}}+K\left(D_{2}-D_{1}\right)\frac{\partial^{2}C_{1}}{\partial z^{2}}$ (6)
Equation 6 reveals that the diffusion mechanism can become significant only
when the intrinsic diffusion coefficients of cosolvents are sufficiently
different from each other. Note that equation 6 is only valid in the case of a
solvent made of two components: in the case of a pure solvent, the diffusive
term (second term of the right side) disappears and the flow is only governed
by the Darcy law. Applying equation 6 to our system allows us to plot the
pressure in a pore liquid of the gel for different glycerol contents. In that
purpose the concentration of component 1 (water), $C_{1}$, has to be
incorporated in equation 6; $C_{1}$ satisfies the diffusion equation using the
boundary condition of constant flux at the evaporation surface, that expresses
as (reference[8] p. 61):
$-(\bar{C_{1}}D_{1}+(1-\bar{C_{1}})D_{2})\frac{\partial C_{1}}{\partial
z}\mid_{z=h}=V_{E}$, where $\bar{C_{1}}$ is the mean concentration of water in
the layer defined as:
$\bar{C_{1}}(t_{m})=\frac{1}{h}\int_{0}^{h}C_{1}(z,t_{m})dz$ at time $t_{m}$.
As a result an estimation of each quantity of equation 6 from the experimental
results is required. $\bar{C_{1}}(t_{m})$ is precisely estimated from the
inset in figure 3. In the first drying regime, $V_{E}$ can be measured from
the mass variation with time in figure 2 ( $V_{E}\simeq 4\times 10^{-8}$ m/s).
The values of the intrinsic coefficients are obtained from reference [10]:
$\frac{D_{2}}{D_{1}}\sim 50$. The porosity $K$ is roughly estimated to $0.40$
for the random close-packed phase while $\beta=1/2$ (reference [28]). We
assume that the permeability of the gel keeps a constant value during the
crack formation; this quantity is estimated using the Carman-Kozeny relation
($k=4.9\times 10^{-19}m^{2}$)[13]. Finally the elastic moduli of gels composed
with various glycerol contents have been estimated from experimental results
shown in figure 5(b) and (c) extrapolating values a short time after cracking.
Consequently, the concentration distribution $C_{1}$ and the pressure, $P$,
are plotted along liquid pores at time $t_{m}$ in figure 6 for different
glycerol contents. The flattening of the pressure gradient when the
concentration of glycerol increases is clearly observable and is mainly
attributed to the diffusion mechanism. One should note that the reduction in
the capillary forces near the surface should be of minor importance as the
surface tension from one mixture to another varies only from $\sim 68$mN/m to
$\sim 72$mN/m [22]. These results highlight the effect of the addition of a
non-volatile solvent to the pressure in the liquid pore since diffusion can
extract liquid from the interior and flatten the pressure gradient in the pore
liquid. The pressure gradient causes a shrinkage that is transmitted in the
hard particles network. In the present paper, the adhesion with the substrate
is assumed to be a constant and to play a determinant role as responsible for
the development of high stresses in the gel[5, 6]. Nevertheless the adhesion
is seen as a necessary but not sufficient condition for cracking. Thus we
eliminate cracking by modifying the pressure gradient and keeping constant the
adhesion. The increase of glycerol content, through diffusion process, will
imply that the resulting drying stresses will be distributed more uniformly in
the gel. Consequently, the stress concentration in the gel network is limited
and the crack formation can be inhibited.
Figure 6: Concentration distribution in water, $C_{1}$, (left) and normalized
pressure distribution, $P/P_{m}$, (right) along the pore liquid, for several
initial glycerol contents, $\kappa$. These quantities are plotted at time
$t_{m}$ when the highest pressure value reaches $P_{m}$ at the evaporation
surface. Inset: sketch of the pressure gradient in the liquid pore.
The effect of addition of glycerol content results in the flattening of the
pressure gradient in the pore liquid due to a diffusion process as depicted by
equation 6, for a constant value of the elastic modulus. Moreover, the effect
of addition of glycerol content results in the tendency of a decrease of the
elastic modulus of the gel. As a consequence a decrease of the elastic modulus
in equation 6 leads to a flattening of the pressure gradient, and, in that
sense, emphasizes the effect of diffusion.
## 5 Conclusion
The incorporation of a small percentage of non-volatile cosolvent modifies the
way the drying stresses relaxe in a colloidal gel. Experimental data show that
the drying rate and cracking time is not significantly modified by
incorporation of the cosolvent. However the presence of non-volatile decreases
the stiffness of the gel at a given consolidation time and inhibits crack
formation: incorporation of a concentration of glycerol greater than $10\%$ in
the colloidal dispersion results in a crack free drying gel. The mechanism is
shown to be due to the combination of both flow driven by the pressure
gradient and diffusion of solvent into a another; in this scenario the
diffusion induced by evaporation causes a demixing effect of solvents. This
results in a flattening of the pressure gradient in the pore liquid and
consequently the distribution of the drying stresses is more uniform, limiting
stress concentration in the gel network possibly cause of cracking. Moreover,
the effect of a cosolvent added to a silica dispersion on the crack pattern
has been highlighted in the case of glucose. As in the case of glycerol, the
presence of glucose in the gel film reveals an increase of the crack spacing.
However, contrary to the case of glycerol, cracks are not completely inhibited
above a threshold glucose content: the crack spacing tends to a maximum value
(typically $0.45$cm in the same experimental conditions as for glycerol
cosolvent).
Finally, the effect of the flattening of the pressure gradient is surely not
the only process leading to the diminution of the number of cracks in the gel
film. Particularly chemical effects between glycerol and silanol groups
covering the surfaces of silica particles could change the strength of the gel
film. This process is suggested by aggregation rate measurements of silica
particles with polyols[19] and would be of interest for further works applied
to gel film cracks induced by desiccation with chemical additives.
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## 6 Acknowledgment
The authors thank A. Aubertin, L. Auffray, C. Borget and R. Pidoux for
technical supports. In addition we thank the referees for their valuable
comments which improved the consistancy of the manuscript.
|
arxiv-papers
| 2012-06-27T12:56:14 |
2024-09-04T02:49:32.289674
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fran\\c{c}ois Boulogne, Ludovic Pauchard, and Fr\\'ed\\'erique\n Giorgiutti-Dauphin\\'e",
"submitter": "Fran\\c{c}ois Boulogne",
"url": "https://arxiv.org/abs/1206.6250"
}
|
1206.6519
|
# A Permutation Approach to Testing Interactions in Many Dimensions
Noah Simon Department of Statistics, Stanford University, nsimon@stanford.edu
Rob Tibshirani Department of Statistics, Stanford University, Department of
Health Research and Policy, Stanford University
###### Abstract
To date, testing interactions in high dimensions has been a challenging task.
Existing methods often have issues with sensitivity to modeling assumptions
and heavily asymptotic nominal p-values. To help alleviate these issues, we
propose a permutation-based method for testing marginal interactions with a
binary response. Our method searches for pairwise correlations which differ
between classes. In this manuscript, we compare our method on real and
simulated data to the standard approach of running many pairwise logistic
models. On simulated data our method finds more significant interactions at a
lower false discovery rate (especially in the presence of main effects). On
real genomic data, although there is no gold standard, our method finds
apparent signal and tells a believable story, while logistic regression does
not. We also give asymptotic consistency results under not too restrictive
assumptions.
Keywords: correlation, high dimensional, logistic regression, false discovery
rate
## 1 Introduction
In many areas of modern science, massive amounts of data are generated. In the
biomedical sciences, examples arise in genomics, proteomics, and flow
cytometry. New high-throughput experiments allow researchers to look at the
dynamics of very rich systems. With these vast increases in data accumulation,
scientists have found classical statistical techniques in need of improvement,
and classical notions of error control (type 1 error) overwhelmed.
Consider the following two class situation: our data consists of $n$
observations, each observation with a known class label of 1 or 2, with $p$
covariates measured per observation. Let $y$ denote the $n$-vector
corresponding to class (with $n_{1}$ observations in class $1$ and $n_{2}$ in
class $2$), and $X$, the $n\times p$ matrix of covariates. We often assume
each row of $X$ is independently normally distributed with some class specific
mean $\mu_{y(i)}\in\mathbb{R}^{p}$ and covariance $\Sigma_{y(i)}$ (for
instance in quadratic discriminant analysis). Here, we are interested in
differences between classes. A common example is gene expression data on
healthy and diseased patients: the covariates are the genes ($p\sim 20,000$),
the observations are patients ($n\sim 100$) belonging to either the healthy or
diseased class. Here, one might look at differences between classes to develop
a genetic prognostic test of the disease, or to better understand its
underlying biology. Recent high dimensional procedures have focused on
detecting differences between $\mu_{1}$ and $\mu_{2}$ by considering them one
covariate at a time.
In this paper we consider the more difficult problem of testing marginal
interactions. In a fashion similar to the approaches used in large scale
testing of main effects (see e.g Dudoit et al. (2003), Tusher et al. (2001)
and Efron (2010)), we do this on a pair by pair basis.
The standard approach for this problem has been to run many bivariate logistic
regressions and then conduct a post-hoc analysis on the nominal p-values.
Buzkova et al. (2011) has a nice summary of the subtle issues that arise in
testing for just a single interaction in a regression framework. In
particular, a permutation approach cannot be simply applied because it tests
the null hypothesis of both no interaction and no main effects at the same
time. In the high-dimensional setting with FDR estimates, these issues are
compounded.
The logistic regression based methods are all derived from what we call a
forward model, that is, a model for the conditional distribution of $Y|X$. In
contrast, a backward model (discussed below) is a model for the conditional
distribution of $X|Y$. We propose a method, based on a backwards model, to
approach this same problem. By using this backwards framework we avoid many of
the pitfalls of standard approaches: we have a less model-based method, we
attack a potentially more scientifically interesting quantity, and we can use
a permutation null for FDR estimates. Our approach is unfortunately only for
binary response — the backwards model is more difficult to work with for
continuous $y$.
In this paper we develop our method, and show its efficacy as compared to
straightforward logistic regression on real and simulated data. We explain how
to deal with nuisance variables, and give insight into our permutation-based
estimates of FDR. We also give some asymptotic consistency results.
## 2 Existing Methods
We begin by going more in-depth on the standard approach and its issues. In
general one might like to specify a generative logistic model for the data (a
forward model) of the form
$\operatorname{logit}\left[\operatorname{P}(y_{i}=1|X_{i,\cdot})\right]=\beta_{0}+\sum_{j=1}^{p}\beta_{j}X_{i,j}+\sum_{k<j}\gamma_{j,k}X_{i,j}X_{i,k}$
(1)
where $X_{i,\cdot}$ is the $i$-th row of $X$, and test if the $\gamma_{j,k}$
are nonzero in this model. Here $i$ indexes the observations and $j,k$ index
the predictors. However, because it is a joint rather than a marginal model,
this does not easily allow us to test individual pairs of covariates
separately from the others. Furthermore in the scenario with $n<p(p+1)/2$, the
MLE for this model is not well defined (one can always get perfect separation)
and non-MLE estimates are very difficult to use for testing.
Alternatively, for each pair $(X_{i,j},X_{i,k})$ one might assume a generative
logistic model of the form
$\operatorname{logit}\left[\operatorname{P}(y_{i}=1|X_{i,j},X_{i,k})\right]=\beta_{0}+\beta_{j}X_{i,j}+\beta_{k}X_{i,k}+\gamma_{j,k}X_{i,j}X_{i,k}$
(2)
and estimate or test $\gamma_{j,k}$ using the MLE $\hat{\gamma}_{j,k}$.
A standard approach to this problem in the past has been to fit pairwise
logistic models (2) independently for every pair $(j,k)$, and then use
standard tools (ie. asymptotic normality of the MLE) to calculate approximate
$P$-values. Once the $p(p-1)/2$ $p$-values are calculated, the approach of
Benjamini and Hochberg (1995) or some other standard procedure can be used to
estimate/control FDR.
This approach has a number of problems. First of all, while the approach is
very model-based, one cannot even ensure that all of the bivariate logistic
models are consistent with one another (i.e. that there is a multivariate
model with the given marginals). In particular, model misspecification will
often cause over-dispersion resulting in anti-conservative FDR estimates.
Also, if the true model contained quadratic terms (which we do not have in our
model) then for correlated pairs of features this approach will compensate by
trying to add false interactions. Even if we did believe the model, the
p-values are only approximate, and this approximation grows worse as we move
into the tails.
One might hope to avoid some of these issues by using permutation p-values,
however, as shown in Buzkova et al. (2011) permutation methods are incongruous
with this approach — they test the joint null hypothesis of no main effect or
interaction, which is not the hypothesis of interest. This difficulty is also
discussed in Pesarin (2001). In an attempt to resolve this, Kooperberg and
LeBlanc (2008) regress out the main effects before permuting the residuals.
This is a nice adjustment, but is still heavily model-based.
To deal with these issues, we take a step back and use a different generative
model. Our generative model has an equivalent logistic model and this
correspondence allows us to sidestep many of the issues with the standard
logistic approach.
### 2.1 Forward vs Backward Model
We propose to begin with a “backward” generative model — as mentioned in
Section 1, we assume that observations are Gaussian in each class
$\left(x_{i}|y_{i}\right)\sim N(\mu_{y(i)},\Sigma_{y(i)})$ with a class
specific mean and covariance matrix. We argue that the most natural test of
interaction is a test of equality of correlations between groups.
Toward this end, let us apply Bayes theorem to our backwards generative model,
to obtain
$\displaystyle\operatorname{P}(y=1|x)$
$\displaystyle=\frac{\pi_{1}\operatorname{exp}\left(l_{1}\right)}{\pi_{2}\operatorname{exp}\left(l_{2}\right)+\pi_{1}\operatorname{exp}\left(l_{1}\right)}$
$\displaystyle=\frac{\operatorname{exp}\left[\operatorname{log}(\pi_{1}/\pi_{2})+l_{1}-l_{2}\right]}{1+\operatorname{exp}\left[\operatorname{log}(\pi_{1}/\pi_{2})+l_{1}-l_{2}\right]}$
where
$l_{m}=-p\operatorname{log}\left(2\pi\right)/2-\operatorname{logdet}\left(\Sigma_{m}\right)/2-(x-\mu_{m})^{\top}\Sigma_{m}^{-1}(x-\mu_{m})/2$
and $\pi_{m}$ is the overall prevalence of class $m$. We can simplify this to
$\displaystyle\operatorname{logit}\left(P\right)$
$\displaystyle=\operatorname{logdet}\left(\Sigma_{2}\right)/2-\operatorname{logdet}\left(\Sigma_{1}\right)/2+\operatorname{log}(\pi_{1}/\pi_{2})+\mu_{2}^{\top}\Sigma_{2}^{-1}\mu_{2}/2$
$\displaystyle-\mu_{1}^{\top}\Sigma_{1}^{-1}\mu_{1}/2+\left(\Sigma_{1}^{-1}\mu_{1}-\Sigma_{2}^{-1}\mu_{2}\right)^{\top}x+x^{\top}\left(\Sigma_{2}^{-1}-\Sigma_{1}^{-1}\right)x/2.$
This is just a logistic model with interactions and quadratic terms, and in
the form of (1) (with additional quadratic terms) we have
$\displaystyle\beta_{0}$
$\displaystyle=\operatorname{logdet}\left(\Sigma_{2}\right)/2-\operatorname{logdet}\left(\Sigma_{1}\right)/2+\operatorname{log}(\pi_{1}/\pi_{2})$
$\displaystyle+\mu_{2}^{\top}\Sigma_{2}^{-1}\mu_{2}/2-\mu_{1}^{\top}\Sigma_{1}^{-1}\mu_{1}/2$
$\displaystyle\beta_{j}$
$\displaystyle=\left(\Sigma_{1}^{-1}\mu_{1}-\Sigma_{2}^{-1}\mu_{2}\right)_{j}$
$\displaystyle\gamma_{j,k}$
$\displaystyle=\left(\Sigma_{2}^{-1}-\Sigma_{1}^{-1}\right)_{j,k}.$
From here we can see that traditional logistic regression interactions in the
full model correspond to nonzero off-diagonal elements of
$\Sigma_{2}^{-1}-\Sigma_{1}^{-1}$. Testing for non-zero elements here is not
particularly satisfying for a number of reasons. Because coordinate estimates
are so intertwined, there is no simple way to marginally test for non-zero
elements in $\Sigma_{2}^{-1}-\Sigma_{1}^{-1}$ — in particular there is no
straightforward permutation test. Also, for $n<p$ the MLEs for the precision
matrices are not well defined.
As in the logistic model (2) we may condition on only a pair of covariates $j$
and $k$ in our backwards model. Using Bayes theorem as above, our equivalent
bivariate forward model is
$\displaystyle\operatorname{P}(y=1|\,\tilde{x}=\left(x_{j},x_{k}\right)^{\top})$
$\displaystyle=\operatorname{log}(\pi_{1}/\pi_{2})+\tilde{\mu}_{2}^{\top}\tilde{\Sigma}_{2}^{-1}\tilde{\mu}_{2}/2-\tilde{\mu}_{1}^{\top}\tilde{\Sigma}_{1}^{-1}\tilde{\mu}_{1}/2$
$\displaystyle+\left(\tilde{\Sigma}_{1}^{-1}\tilde{\mu}_{1}-\tilde{\Sigma}_{2}^{-1}\tilde{\mu}_{2}\right)^{\top}\tilde{x}+\tilde{x}^{\top}\left(\tilde{\Sigma}_{2}^{-1}-\tilde{\Sigma}_{1}^{-1}\right)\tilde{x}/2$
where $\tilde{\mu}_{m}$ and $\tilde{\Sigma}_{m}$ are the mean vector and
covariance matrix in class $m$ for only $X_{j}$ and $X_{k}$. Hence the
backwards model has an equivalent logistic model similar to (2) but with
quadratic terms included as well. One should note that the main effect and
interaction coefficients in this marginal model _do not_ match those from the
full model (i.e. the marginal interactions and conditional interactions are
different).
Our usual marginal logistic interaction between covariates $j$ and $k$
corresponds to a nonzero off-diagonal entry in
$\tilde{\Sigma}_{2}^{-1}-\tilde{\Sigma}_{1}^{-1}$. Simple algebra gives
$\tilde{\Sigma}^{-1}_{m(1,2)}=-\left(\frac{R_{m(j,k)}}{\sigma_{m(j)}\sigma_{m(k)}\left(1-R_{m(j,k)}^{2}\right)}\right)$
where $R_{m(j,k)}$ is the correlation between features $j$ and $k$ in class
$m$, and $\sigma_{m(j)}$ is the standard deviation of variable $j$ in class
$m$.
Thus, if we were to test for “logistic interactions” in our pairwise backwards
model, we would be testing:
$\frac{R_{1(j,k)}}{\sigma_{1(j)}\sigma_{1(k)}\left(1-R_{1(j,k)}^{2}\right)}=\frac{R_{2(j,k)}}{\sigma_{2(j)}\sigma_{2(k)}\left(1-R_{2(j,k)}^{2}\right)}$
Now, if $\sigma_{1(j)}=\sigma_{2(j)}$, and $\sigma_{1(k)}=\sigma_{2(k)}$, then
this is equivalent to testing if $R_{1(j,k)}=R_{2(j,k)}$. If not, then a
number of unsatisfying things may happen. For example if the variance of a
single variable changes between classes, then, even if its correlation with
other variables remains the same, it still has an “interaction” with all
variables with which it is correlated. This change of variance is a
characteristic of a single variable, and it seems scientifically misleading to
call this as an “interaction” between a pair of features.
Toward this end, we consider a restricted set of null hypotheses — rather than
testing for an interaction between each pair of features $(j,k)$, we test the
null $R_{1(j,k)}=R_{2(j,k)}$. Not all logistic interactions will have
$R_{1(j,k)}\neq R_{2(j,k)}$, but we believe this is the property which makes
an interaction physically/scientifically interesting.
To summarize, there are a number of issues in the forward model which are
alleviated through the use of the backwards model:
* •
The marginal forward models are not necessarily consistent (one cannot always
find a “full forward model” with the given marginals).
* •
Omitted quadratic terms may be mistaken for interactions between correlated
covariates.
* •
Interesting interactions are only those for which $R_{1(j,k)}\neq R_{2(j,k)}$.
* •
$P$-values are approximate and based on parametric assumptions.
## 3 Proposal
We begin with the generative model described in Section 2.1— we assume
observations are Gaussian in each class $\left(x_{i}|y_{i}\right)\sim
N(\mu_{y(i)},\Sigma_{y(i)})$ with a class specific mean and covariance matrix.
As argued above, we test for interactions by testing
$\mathbf{H}_{j,k}:\,R_{1(j,k)}=R_{2(j,k)}$
for each $j<k$, where again, $R_{m(j,k)}$ denotes the $(j,k)$-th entry of the
correlation matrix for class $m$.
If we were only testing one pair of covariates $(j,k)$, a straightforward
approach would be to compare the sample correlation coefficients
$\hat{R}_{1(j,k)}$ to $\hat{R}_{2(j,k)}$. In general, because the variance of
$\hat{R}_{m(j,k)}$ is dependent on $R_{m(j,k)}$, it is better to make
inference on a Fisher transformed version of $\hat{R}_{m(j,k)}$:
$U_{m(j,k)}=\operatorname{arctanh}\left(\hat{R}_{m(j,k)}\right)\dot{\sim}N\left(\operatorname{arctanh}\left(R_{m(j,k)}\right),\frac{1}{n_{m}-3}\right)$
This is a variance stabilizing transformation. Now, to compare the two
correlations we consider the statistic
$T_{(j,k)}=U_{1(j,k)}-U_{2(j,k)}\dot{\sim}N\left(\operatorname{arctanh}\left(R_{1(j,k)}\right)-\operatorname{arctanh}\left(R_{2(j,k)}\right),\frac{1}{n_{1}-3}+\frac{1}{n_{2}-3}\right)$
(3)
Under the null hypothesis: $R_{1(j,k)}=R_{2(j,k)}$, this statistic is
distributed $N\left(0,\frac{1}{n_{1}-3}+\frac{1}{n_{2}-3}\right)$. To test if
the correlations are equal we need only compare our statistic $T_{(j,k)}$ to
its null distribution and find a $p$-value. While this approach works well for
single tests, because we are in the high dimensional setting we use a
different approach which doesn’t rely on the statistic’s asymptotic normal
distribution.
We are interested in testing differences between two large correlation
matrices in higher dimensional spaces. We again calculate the differences of
our transformed sample correlations — we now calculate $p(p-1)/2$ statistics;
one for each pair $(j,k)$ with $j<k$. However to assess significance we no
longer just compare each statistic to the theoretical null distribution and
find a p-value. Instead we directly estimate false discovery rates (FDR): we
choose some threshold for our statistics, $t$, and reject (/call significant)
all $(j,k)$ with $|T_{(j,k)}|>t$. Clearly, not all marginal interactions
called significant in this way will be truly non-null and it is important to
estimate the FDR of the procedure for this cutoff, that is
$\operatorname{FDR}=E\left[\frac{\textrm{\\# false rejections}}{\textrm{\\#
total rejections}}\right],$
where ‘#’ is short-hand for “number of”. It is standard to approximate this
quantity by
$\frac{\hat{E}[\textrm{\\# false rejections}]}{\textrm{\\# total
rejections}}.$ (4)
The denominator is just the number of $|T_{(j,k)}|>t$ (which we know). If we
knew which hypotheses were null and their distributions then one could find
the numerator by
$E[\textrm{\\# false rejections}]=\sum_{(j,k)\textrm{
null}}\operatorname{P}(|T_{(j,k)}|>t)$ (5)
Clearly we don’t know which hypotheses are null. To estimate (6) we propose
the following permutation approach.
We first center and scale our variables within class: for each observation we
subtract off the class mean for each feature and divide by that feature’s
within-class standard deviation — let $\tilde{X}$ denote this standardized
matrix. This standardization doesn’t change our original statistics, $T_{j,k}$
(the correlation calculated from $X$ and $\tilde{X}$ are identical), but is
important for our null distribution. Now, let $\pi$ be some random permutation
of $\\{1,\ldots,n\\}$. Thus, $\pi(y)$ is a random permutation of the class
memberships of the standardized variables (we keep the standardization from
before the permutation). With these new class labels we calculate a new set of
$p(p-1)/2$ statistics, $\\{T^{*a}_{(j,k)}\\}_{j<k}$. We can permute our data
$A$ times, and gather a large collection of these null statistics,
($Ap(p-1)/2$) of them. To estimate $E[\textrm{\\# false rejections}]$, we take
the average number of these statistics that lie above our cutoff
$\hat{E}[\textrm{\\# false
rejections}]=\frac{1}{A}\sum_{a=1}^{A}\\#\\{|T^{*a}_{(j,k)}|>t\\}$
Often, one is interested in the FDR of the $l$ most significant interactions.
In this case the cutoff, $t$, is chosen to be the absolute value of the $l$-th
most significant statistic, denoted $T(l)$. We refer to this procedure as
Testing Marginal Interactions through correlation (TMIcor) and summarize it
below.
TMIcor: Algorithm for Testing Marginal Interactions
1. 1.
Mean center and scale $X$ within each group.
2. 2.
Calculate the feature correlation matrices $\hat{R}_{1}$ and $\hat{R}_{2}$
within each class.
3. 3.
Fisher transform the entries (for $j<k$):
$U_{m(j,k)}=\operatorname{arctanh}\left(\hat{R}_{m(j,k)}\right)$
and take their coordinate-wise differences: $T_{(j,k)}=U_{1(j,k)}-U_{2(j,k)}$
4. 4.
for $a=1,\ldots\,,A$ execute the following
1. (a)
Randomly permute class labels of the standardized variables.
2. (b)
Using the new class labels, reapply steps 2-4 to calculate new statistics
$\\{T^{*a}_{(j,k)}\\}_{j<k}$
5. 5.
Estimate FDR for any $l$ most significant interactions by
$\widehat{{\rm
FDR}}=\frac{\left(\frac{1}{A}\right)\sum_{a=1}^{A}\\#\\{|T^{*a}_{(j,k)}|>T(l)\\}}{l}$
Using this approach, one gets a ranking of pairs of features and an FDR
estimate for every position in the ranking. Furthermore, rather than testing
for interactions between all pairs of variables, one may instead test for
interactions between variables in one set (such as genes) and variables in
another (such as environmental variables). To do this, one would only need
restrict the statistics considered in steps $3$, $4b$ and $5$.
Standardizing in step $(1)$ before permuting may seem strange, but in this
case is necessary. If we do not standardize first, we are testing the joint
null that the means, variances and correlations are the same between classes.
This is precisely what we moved to the backward model to avoid — by
standardizing we avoid permuting the “main effects”. We discuss this
permutation-based estimate of FDR in more depth in appendix A.
## 4 Comparisons
In this section we apply TMIcor and the standard logistic approach to real and
simulated data. On simulated data we see that in some scenarios (in particular
with main effects) the usual approach has serious power issues as compared to
TMIcor. Similarly on our real dataset we see that the usual approach does a
poor job of finding interesting interactions, while TMIcor does well.
### 4.1 Simulated Data
We attempt to simulate a simplified version of biological data. In general,
groups of proteins or genes act in concert based on biological processes. We
model this with a block diagonal correlation matrix — each block of
proteins/genes is equi-correlated. This can be interpreted as a latent factor
model — all the proteins in a single block are highly correlated with the same
latent variable (maybe some unmeasured cytokine), and conditional on this
latent variable, the proteins are all uncorrelated. In our simulations we use
$10$ blocks, each with $10$ proteins ($100$ total proteins). We simulate the
proteins for our healthy controls as jointly Gaussian with $0$ mean and
covariance matrix
$\Sigma_{1}=\begin{pmatrix}R_{1}&0&\cdots&0\\\ 0&R_{2}&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&0&R_{10}\end{pmatrix}$
where each $R_{i}$ is a $10\times 10$ matrix with $1$s along the diagonal, and
a fixed $\rho_{i}>0$ for all off-diagonal entries. Now, for our diseased
patients we again use mean $0$ proteins, but change our covariance matrix to
$\Sigma_{2}=\begin{pmatrix}\tilde{R}_{1}&0&\cdots&0\\\ 0&R_{2}&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&0&R_{10}\end{pmatrix}$
where $\tilde{R}_{1}$ has $1$s on the diagonal and $\tilde{\rho}_{1}$ for all
off-diagonal entries (with $0\leq\tilde{\rho}_{1}\neq\rho_{1}$). This
correlation structure would be indicative of a mutation in the cytokine for
the first group causing a change in the association between that signaling
protein and the rest of the group.
Within each class (diseased and healthy) we simulated $250$ patients and
applied TMIcor and the usual logistic approach. We averaged the true and
estimated false discovery rates of these methods over $10$ trials. As we can
see from Figure 1 TMIcor outperforms the logistic approach. This difference is
particularly pronounced in the second plot of Figure 1. In this plot, because
the correlations are large but different in both groups ($\rho_{1}=0.3$,
$\tilde{\rho}_{1}=0.6$), there are some moderate quadratic effects in the true
model — this induces a bias in the logistic approach and its FDR suffers. In
contrast, these quadratic effects are not problematic in the backward
framework.
Figure 1: Plots of estimated and true FDR for TMIcor and logistic regression
averaged over $10$ trials. Error bars contain the mean value $\pm$ 1 se of the
mean. For controls, $\rho_{i}=0.3$ for all $i$. On the left
$\tilde{\rho}_{1}=0$, while on the right $\tilde{\rho}_{1}=0.6$. There is no
main effect in either panel.
We also consider a second set of simulations. This set used $\rho_{i}=0.3$ for
all $i$ and $\tilde{\rho}_{1}=0$. However, instead of mean $0$ in both
classes, we set the mean for all proteins in block 1 for diseased patients to
be some $\tilde{\mu}_{1}$ ($>0$). The results are plotted in Figure 2. This
mean shift had no effect on TMIcor (the procedure is meanshift invariant), but
as the mean difference grows, it becomes increasingly difficult for the
logistic regression to find any interactions. This issue is especially
important as, biologically, one might expect that genes with main effects to
be more likely to have true marginal interactions (and these interactions may
also be more scientifically interesting).
Figure 2: Plots of estimated and true FDR for TMIcor and logistic regression
averaged over $10$ trials. Error bars contain the mean value $\pm$ 1 se of the
mean. For both plots $\tilde{\rho}_{1}=0$ and $\rho_{i}=0.3$ for all $i$. Both
panels have main effects — on the left $\tilde{\mu}_{1}-\mu_{1}=0.5$, while on
the right $\tilde{\mu}_{1}-\mu_{1}=1$.
While these simulations are not exhaustive, they give an indication of a
number of scenarios in which TMIcor significantly outperforms logistic
regression. More exhaustive simulations were run and the results mirrored
those in this section.
### 4.2 Real Data
We also applied both TMIcor and logistic regression to the colitis gene
expression data of Burczynski et al. (2006). In this dataset, there are $127$
total patients, $85$ with colitis ($59$ Crohn’s patients + $26$ ulcerative
colitis patients) and $42$ healthy controls. We restricted our analysis to the
$101$ patients without ulcerative colitis. Each patient had expression data
for $22283$ genes run on an Affymetrix U133A microarray. Because chromosomes
$5$ and $10$ have been indicated in Crohn’s disease, we enriched our dataset
by using only the genes on these chromosomes, along with the $NOD2$ and
$ATG16L1$ genes (chromosomes as specified by the $C1$ geneset from Subramanian
et al. (2005)). In total $663$ genes were used. Some of these genes were
measured by multiple probesets — the final expression values used for those
genes were the average of all probesets.
From these $663$ genes we have $219,453$ of interactions to consider. Figure 3
shows the estimated FDR curves for the two methods. TMIcor finds many more
significant interactions — at an FDR cutoff of $0.1$, TMIcor finds $2570$
significant interactions, while the logistic approach finds $15$. The
significant $15$ from the logistic approach may not even be entirely
believeable — the smallest p-value of the $15$ is roughly $1/219453$, which is
what we would expect it to be if all null hypotheses were true. Because the
smallest p-value is large, we see that the FDR for logistic regression begins
surprisingly high. The FDR subsequently drops because there are a number of
p-values near the smallest, however, the significance of these hypotheses is
still suspect.
Figure 3: Corhn’s data; FDR estimates for TMIcor and logistic approaches for
the $5000$ most significant marginal interactions
Unfortunately interpreting $2570$ marginal interactions is difficult (even if
all are true). Toward this end we consider the graphical representation of our
analysis in Figure 4. Each gene is a node in our graph, and edges between
genes signify marginal interactions. In this plot we considered only the
$1250$ of the $2570$ significant marginal interactions indicative of a
decrease in correlation from healthy control to Crohn’s (ie. $T_{j,k}>0$).
There is one large connected component, a few connected pairs and a large
number of isolated genes. The connected component appears to be split into $2$
clusters. To get a better handle on this, we considered a more stringent
cutoff for significant interactions — at an FDR cutoff of $0.03$, we are left
with $832$ significant interactions of which only $402$ have $T_{j,k}>0$. We
plot this graph in Figure 5: we see that our large connected component has
divided into $2$. From here we further zoomed in on each component (now
displaying only the $50$ most significant interactions per component), and can
actually see which genes are are most important (in figure 6).
Figure 4: Graph of $1250$ marginal interactions (with decreasing correlation)
significant at FDR cutoff of $0.1$. Genes with no significant interactions not
shown
Figure 5: Graph of $402$ marginal interactions (with decreasing correlation)
significant at FDR cutoff of $0.03$. Genes with no significant interactions
not shown
Figure 6: Graphs of the top $50$ marginal interactions in each cluster (and
corresponding genes)
It appears, from this analysis, that there are two genetic pathways which are
modified in Crohn’s disease. Many of the genes in each cluster are already
known to be indicated in Crohn’s, but to our knowledge these interactions have
not been considered.
## 5 Dealing with Nuisance Variables
Often, aside from the variables of interest, one may believe that other
nuisance variables play a role in complex interactions. For example, it seems
reasonable that many genes are conditionally independent given age, but are
each highly correlated with age. Ignoring age, these genes would appear to be
highly correlated, but this correlation is uninteresting to us. TMIcor can be
adapted to deal with these nuisance variables provided there are few compared
to the number of observations, they are continuous, and they are observed.
We resolve this issue by using partial correlations. Assume $x_{j}$ and
$x_{k}$ are our variables of interest, and $z$ is a vector of potential
confounders. Rather than comparing
$\operatorname{cor}\left(x_{j},x_{k}\right)$ in groups $1$ and $2$, we compare
the partial correlations,
$\operatorname{cor}\left(\left[x_{j}|z\right],\left[x_{k}|z\right]\right)$.
This is done by first regressing our potential confounders, $Z$, out of all
the other features, then running the remainder of the analysis as usual.
To adapt the original algorithm in Section 3 to deal with nuisance variables
we need only replace step $(1)$ by:
1. 1.
Replace our feature matrices $X_{1}$ and $X_{2}$ by
$\tilde{X}_{m}=\left[I-Z_{m}\left(Z_{m}^{\top}Z_{m}\right)Z_{m}^{\top}\right]X_{m}$
Now, mean center and scale $\tilde{X}$ within each group.
We give more details motivating this approach and discussing potential
computational advantages in appendix B.
## 6 Asymptotics
In this section we give two asymptotic results. We show that if
$n\rightarrow\infty$, and $\frac{\log p_{n}}{n}\rightarrow 0$, then under
certain regularity conditions our procedure for testing marginal interactions
(in the absence of nuisance variables) is asymptotically consistent — with
probability approaching $1$ it calls significant all true marginal
interactions and makes no false rejections. Furthermore, using the permutation
null, it also consistently estimates that the true FDR is converging to $0$.
Because we only need $\frac{\log p_{n}}{n}\rightarrow 0$, $p_{n}$ may increase
very rapidly in $n$.
We first give a result showing that for sub-Gaussian variables our null
statistics converge to $0$ and our alternative statistics are asymptotically
bounded away from $0$. The proof of this theorem is based on several technical
lemmas which we relegate to appendix C.
###### Theorem 6.1
Let $\tilde{x}_{1(j)}$ and $\tilde{x}_{2(j)}$, $j=1,\ldots$ be random
variables. Assume there is some $C>0$ such that for all $t\geq 0$
$\operatorname{P}\left(\left|x_{m(j)}-\operatorname{E}[x_{m(j)}]\right|>t\right)\leq\operatorname{exp}\left(1-t^{2}/C^{2}\right)$
for each $m=1,2$ . Let $\mu_{i(j)}$ denote the mean of $\tilde{x}_{m(j)}$ and
$\sigma_{m(j)}^{2}$ its variance. For each $i\leq\infty$, let $x_{m(i,\cdot)}$
be independent realizations with the same distribution as
$\tilde{x}_{m(\cdot)}$.
Let $p_{n}$ be a sequence of integers such that $\frac{\log
p_{n}}{n}\rightarrow 0$. Let $R_{m}$ be the correlation “matrix” (an infinite
but countably indexed matrix) of the covariates from group $m$. Let $I$ denote
the set of ordered pairs $(j,k)$ for which $R_{1(j,k)}\neq R_{2(j,k)}$, and
$C_{n}$ denote the set of ordered pairs $(j,k)$ with $j,k\leq p_{n}$.
Assume for every $m$ and $j$, $\sigma_{m(j)}^{2}\geq\sigma_{min}^{2}$ (for
some $\sigma_{min}^{2}>0$). Furthermore, assume that for all $(j,k)$ in each
$I$, $\left|R_{1(j,k)}-R_{2(j,k)}\right|>\Delta_{\min}$ for some
$\Delta_{\min}>0$ and that for $m=1,2$,
$\operatorname{sup}_{j<k}\left|R_{m(j,k)}\right|<\rho_{\max}$ for some fixed
$\rho_{\max}<1$.
Now, given any $\epsilon_{p}>0$, and $0<t<\Delta_{\min}$, if we choose $n$
sufficiently large, then with probability at least $1-\epsilon_{p}$
$\left|T_{(j,k)}\right|\leq t$
for all $(j,k)$ in $C_{n}-I$, and
$\left|T_{(j,k)}\right|\geq t$
for all $(j,k)$ in $C_{n}\cap I$.
The notation here is a little bit tricky, but the result is very
straightforward: under some simple conditions, we find all marginal
interactions and make no false identifications.
While there were a number of assumptions in the above theorem, most of these
are fairly trivial and will almost always be found in practice: the variance
must be bounded away from $0$ and the correlations bounded away from $\pm 1$.
The assumption that the correlation differences are bounded below by a fixed
$\Delta_{\min}$ for true marginal interactions is a bit more cumbersome, but
may easily be relaxed to $\Delta_{\min}\rightarrow 0$ at a slow enough rate
that $\Delta_{min}/\left[\log p/n\right]^{1/2}\rightarrow\infty$.
The astute reader might note that our assumption bounding the variance away
from $0$ seems strange — the distribution of the sample correlation is
independent of the variance. This is necessary only because we assumed the
covariates have a subgaussian tail with a shared constant $C$. One could have
relaxed the bounded variance assumption to the assumption that
$\left\\{x_{j}/\sigma_{j}\right\\}_{j=1,\ldots}$ have a sub-Gaussian tail with
a shared constant $C$.
### 6.1 Permutation Consistency
Now that we have shown our procedure has FDR converging to $0$, we would like
to show that it asymptotically estimates FDR consistently as well. In
particular we show that as $n\rightarrow\infty$, if $\frac{\log
p}{n}\rightarrow 0$, then with probability approaching $1$, for a random
permutation, our permuted statistics converge to $0$ uniformly in probability
($\max_{j,k}\left|T_{(j,k)}^{*}\right|\leq t$ for any fixed $t>0$ with
probability converging to $1$). Thus our estimated FDR converges to $0$ under
the same conditions as our true FDR.
We begin with some notation. Let us consider an arbitrary permutation of class
labels, $\Pi$. Let $\hat{\pi}$ denote the proportion of observations from
class $1$ that remain in class $1$ after permuting.
We discuss a somewhat simplified procedure in our proof, as otherwise the
algebra becomes significantly more painful (without any added value in
clarity), but it is straightforward to carry the proof through to the full
procedure. In our original procedure, after permuting class labels we recenter
and rescale our variables within each class. Because we already centered and
scaled variables before permuting, this step will have very little effect on
our procedure (though it does have the nice effect of never giving
$|\rho^{*}|>1$). In this proof we consider a procedure identical in every way
except without recentering and rescaling within each permutation.
Before we give the theorem, we would like to define a few new terms for
clarity. For a given permutation $\Pi$, let
$\Pi_{i}(m)\in\left\\{0,1\right\\}$ be the permuted class of the $i$-th
observation originally in class $m$. Furthermore, let $\Pi\left(m,l\right)$ be
the set of observations in class $m$ that are permuted to class $l$, and let
$\Pi\left(\cdot,l\right)$ be the set of observations in both classes permuted
to class $l$, ie.
$\displaystyle\Pi\left(m,l\right)$
$\displaystyle=\left\\{i:\,\Pi_{i}(m)=l\right\\}$
$\displaystyle\Pi\left(\cdot,l\right)$
$\displaystyle=\left\\{(i,m):\,\Pi_{i}(m)=l\right\\}$
Now, we give a result which shows that for any fixed $t>0$ if our variables
are sub-Gaussian with some other minor conditions, then for
$n\rightarrow\infty$ and $\log p/n\rightarrow 0$ with probability approaching
$1$, none of our permuted statistics will be larger than $t$, or in other
words, as our true converged to $0$, so will our estimated FDR $0$. As before,
the proof of this theorem is based on several technical lemmas which we again
leave to appendix C.
###### Theorem 6.2
Let $\tilde{x}_{1(j)}$ and $\tilde{x}_{2(j)}$, $j=1,\ldots$ be random
variables with
$\operatorname{P}\left(|x_{m(j)}-\operatorname{E}\left[x_{m(j)}\right]|\geq
t\right)\leq 1-e^{t^{2}/C}$
for all $t>0$, and each $m=1,2$, with some fixed $C>0$. Let $\mu_{m(j)}$
denote the mean of $\tilde{x}_{m(j)}$ and $\sigma_{m(j)}^{2}$ its variance.
For each $i\leq\infty$, let $x_{m(i,\cdot)}$ be independent realizations with
the same distribution as $\tilde{x}_{m(\cdot)}$.
Let $p_{n}$ be a sequence of integers such that $\frac{\log
p_{n}}{n}\rightarrow 0$. Let $R_{m}$ be the correlation “matrix” (an infinite
but countably indexed matrix) of the covariates from class $m$.
Assume for every $m,\,j$, $\sigma_{m(j)}^{2}\geq\sigma_{min}^{2}$ (for some
$\sigma_{min}^{2}>0$). Furthermore, assume that for $m=1,2$,
$\operatorname{sup}_{j<k}\left|R_{m(j,k)}\right|<\rho_{\max}$ for some fixed
$\rho_{\max}<1$.
Now, given any $\epsilon_{p}>0$, and $0<t$, if we choose $n$ sufficiently
large and let $\Pi$ be a random permutation, then with probability at least
$1-\epsilon_{p}$
$\left|T_{(j,k)}^{*}\right|\leq t$
for all $(j,k)$ with $j,k\leq p_{n}$ where
$T_{(j,k)}^{*}=\operatorname{arctanh}\left(\hat{R}_{\operatorname{perm:1(j,k)}}\right)-\operatorname{arctanh}\left(\hat{R}_{\operatorname{perm:2(j,k)}}\right)$
and
$\hat{R}_{\textrm{perm}:m(j,k)}=\frac{1}{n}\sum_{(i,l)\in\Pi(\cdot,m)}\left(\frac{x_{l(i,j)}-\hat{\mu}_{l(j)}}{\hat{\sigma}_{l(j)}}\right)\left(\frac{x_{m(i,k)}-\hat{\mu}_{l(k)}}{\hat{\sigma}_{l(k)}}\right)$
The notation is again somewhat ugly, but the result is very straightforward:
under some simple conditions, our permuted statistics are very small. In
particular from the proof one can see that
$\operatorname{sup}\left\\{T_{(j,k)}^{*}\right\\}=O_{p}\left(\sqrt{\log
p_{n}/n}\right)$.
Note there is an implicit indexing of $n$ in $\hat{R}_{\textrm{perm}:m(j,k)}$
(it seemed unneccessary to add more indices). As in theorem 6.1, some of our
conditions may be relaxed. Instead of bounding $\sigma_{j}^{2}$ below, we need
only bound $C\sigma_{j}$ below. Also, rather than choose a fixed cutoff,
$t>0$, we may use any sequence $\left\\{t_{n}\right\\}$ with $t_{n}/\left(\log
p_{n}/n\right)^{1/2}\rightarrow\infty$. Also, as noted before, the result we
have just shown ignores the restandardizing within each permutation, however
it is straightforward (though algebraicly arduous, and not insightful) to
extend this result to that case as well.
As a last note, in theorem 6.2, we gave our consistency result for only a
single permutation. This result can easily be extended to any fixed number of
permutations using a union bound. This was left out of the original
statement/proof as the notation is already clunky and the extension is
straightforward.
Through theorems 6.1 and 6.2 we have shown that, under fairly relaxed
conditions, our procedure is asymptotically consistent at discovering marginal
interactions and that the permutation null reflects this.
## 7 Discussion
In this paper we have discussed marginal interactions for logistic regression
in the framework of forward and backward models. We have developed a
permutation based method, TMIcor, which leverages the backward model. We have
shown its efficacy on real and simulated data and given asymptotic results
showing its consistency and convergence rate. We also plan to release a
publically available R implementation.
## 8 Appendix A
In this section we give more details on our permutation-based estimate of FDR,
and discuss a potential alternative. Recall that we are using the permutations
to approximate
$\sum_{(j,k)\textrm{ null}}\operatorname{P}(|T_{(j,k)}|>t).$ (6)
For the moment, assume that all covariates in both classes have mean $0$ and
variance $1$, and that we did not do any sample standarization. Then, under
the null hypothesis that $R_{1(j,k)}=R_{2(j,k)}$, $T_{(j,k)}$ calculated under
the original class assignments and $T^{*}_{(j,k)}$ calculated under any
permuted class assignments have the same distribution, so
$\sum_{(j,k)\textrm{ null}}\operatorname{P}(|T_{(j,k)}|>t)=\sum_{(j,k)\textrm{
null}}\operatorname{P}(|T^{*}_{(j,k)}|>t)$
which is reasonably (and unbiasedly) approximated by
$\sum_{(j,k)\textrm{ null}}\frac{1}{A}\sum_{a=1}^{A}I(|T^{*a}_{(j,k)}|>t).$
Because we do not know which genes are null, our actual estimate of (6) is
$\displaystyle\sum_{(j,k)}\frac{1}{A}\sum_{a=1}^{A}I(|T^{*a}_{(j,k)}|>t)$
$\displaystyle=\sum_{(j,k)\textrm{
null}}\frac{1}{A}\sum_{a=1}^{A}I(|T^{*a}_{(j,k)}|>t)$ (7)
$\displaystyle+\sum_{(j,k)\textrm{
alternative}}\frac{1}{A}\sum_{a=1}^{A}I(|T^{*a}_{(j,k)}|>t)$ (8)
This gives a slight conservative bias (especially small if most marginal
interactions are null). One should also note that unlike the null statistics,
for the alternative $(j,k)$, $T^{*}_{(j,k)}$ are not distributed
$N\left(0,\frac{2}{n-3}\right)$; they are still mean $0$, but the variance is
increased. However, this conservative bias is very slight — in general there
are few alternative hypotheses, and the variance increase is not large.
Because in practice we do not have mean $0$, variance $1$ for all covariates
in both classes, we must standardize before running our procedure. Otherwise,
instead of testing for a changing correlation, we are actually testing for a
different mean, variance, or correlation between classes. The effect of
standardizing with the sample mean and variance rather than the true values is
asymptotically washed out, and while the variance of our tests is increased
for small samples, this increase is only minimal.
An alternative to permutations, as discussed in Efron (2010), is to directly
estimate the numerator using the approximate theoretical distribution of the
null statistics. Each null statistic is asymptotically
$N\left(0,\frac{1}{n_{1}-3}+\frac{1}{n_{2}-3}\right)$, so for $(j,k)$ null
$\operatorname{P}(|T_{(j,k)}|>t)=2\Phi\left(-\frac{t(n_{1}-3)(n_{2}-3)}{n_{1}+n_{2}-6}\right).$
Now we can conservatively approximate the quantity in Eq (6) by
$\displaystyle\sum_{(j,k)\textrm{ null}}P\left(|T_{(j,k)}|>t\right)$
$\displaystyle\leq p(p-1)/2\cdot P\left(|T_{\textrm{null}}|>t\right)$
$\displaystyle=p(p-1)\cdot\Phi\left(-\frac{t(n_{1}-3)(n_{2}-3)}{n_{1}+n_{2}-6}\right)$
While this approach is reasonable and simple, it is less robust than using
permutations, and in practice, even for truly Gaussian data, it is only
slightly more efficient.
## 9 Appendix B
Before proceeding, we remind the reader that $x$ are our variables of interest
and $z$ are potential confounding variables. Furthermore we are interested in
comparing
$\operatorname{cor}\left(\left[x_{j}|z\right],\left[x_{k}|z\right]\right)$
between groups. From basic properties of the Gaussian distribution we know
that
$x|z\sim
N\left[\mu_{x}+\Sigma_{(x,z)}\Sigma_{z}^{-1}\left(z-\mu_{z}\right),\Sigma_{(x|z)}\right]$
where $\Sigma_{(x|z)}$ is the variance/covariance matrix of $x$ given $z$,
$\Sigma_{(x,z)}$ is the covariance matrix between $x$ and $z$, $\Sigma_{z}$ is
the variance matrix of $z$, and $\mu_{x}$ and $\mu_{z}$ are the means of $x$
and $z$. Now, if $\mu_{x},\,\mu_{z},\,\Sigma_{(x,z)},$ and $\Sigma_{z}$ were
known, then the MLE for $\Sigma_{(x|z)}$ would be
$\hat{\Sigma}_{(x|z)}=\frac{1}{n}\left[X-1\mu_{x}^{\top}-\left(Z-1\mu_{z}^{\top}\right)\Sigma_{z}^{-1}\Sigma_{(z,x)}\right]^{\top}\left[X-1\mu_{X}^{\top}-\left(Z-1\mu_{Z}^{\top}\right)\Sigma_{Z}^{-1}\Sigma_{(z,x)}\right].$
Unfortunately, these nuisance parameters are unknown. However we can also
estimate them by maximum likelihood. This gives us the estimate
$\displaystyle\hat{\Sigma}_{(X|Z)}$
$\displaystyle=\frac{1}{n}\left[\tilde{X}-\tilde{Z}\left(\tilde{Z}^{\top}\tilde{Z}\right)^{-1}\tilde{Z}^{\top}\tilde{X}\right]^{\top}\left[\tilde{X}-\tilde{Z}\left(\tilde{Z}^{\top}\tilde{Z}\right)^{-1}\tilde{Z}^{\top}\tilde{X}\right]$
$\displaystyle=\frac{1}{n}\left[\operatorname{P}_{\tilde{Z}\perp}\left(\tilde{X}\right)\right]^{\top}\left[\operatorname{P}_{\tilde{Z}\perp}\left(\tilde{X}\right)\right]$
where $\tilde{Z}$ is the standardized version of $Z$, and $\tilde{X}$ is the
standardized version of $X$, and $\operatorname{P}_{\tilde{Z}\perp}$ is the
projection onto the orthogonal complement of the column space of $\tilde{Z}$.
So, our estimate of partial correlation is just an estimate of correlation
with $Z$ regressed out of both covariates. We use this to contruct our
permutation null. In the orginal algorithm, we mean centered and scaled before
permuting; here we do the equivalent — we project our variables of interest
onto the orthogonal complement of our nuisance variables, and then
center/scale them. Now we are ready to permute. We permute these “residuals”,
and calculate permuted correlations as before.
Before proceeding, we note that for $n$ sufficiently large $n$ ($n>>p$) one
might use a similar approach to consider partial correlations rather than
marginal correlations in our original algorithm (conditioning out all
covariates except any particular $2$). However, in general $n<<p$ and thus
$\operatorname{P}_{\perp}\equiv 0$ rendering this approach ineffective — this
approach only works for nuisance variables because we assume that there are
very few relative to the number of observations.
As stated in the text, to adapt the original algorithm to deal with nuisance
variables we need only replace step $(1)$ by:
1. 1.
Replace our feature matrices $X_{1}$ and $X_{2}$ by
$\tilde{X}_{m}=\left[I-Z_{m}\left(Z_{m}^{\top}Z_{m}\right)Z_{m}^{\top}\right]X_{m}$
Now, mean center and scale $\tilde{X}$ within each group.
One may note that we only calculate $\tilde{X}$ once per class, at the
beginning of our procedure, not in each permutation. We do this for a similar
reason that we standardize our variables before permuting — because we are not
testing the hypothesis that the relationship between $X$ and $Z$ is the same
in both groups. If we relcalulate after each permutation then we are
implicitly assuming that this relationship is the same in both groups under
the null.
Even with nuisance variables this approach is very computationally fast.
Projecting our original variables onto $Z\perp$ can be done in
$O\left(npp_{\textrm{nuis}}\right)$ operations where $p_{\textrm{nuis}}$ is
the number of nuisance variables. Thus the total runtime of this algorithm is
$O\left(npp_{\textrm{nuis}}+Anp(p-1)/2\right)$ where $A$ is the number of
permutations — this is dominated by the second term, which is independent of
the number of nuisance parameters. In contrast, if we were to use the standard
approach (fitting pairwise logistic regressions with nuisance variables), its
runtime would be
$O\left[\left(iter\right)(3+p_{\textrm{nuis}})^{2}np(p-1)/2\right]$ where
$iter$ is the number of iterations of the algorithm for finding the MLE. In
general $A\sim 100$ and $iter\sim 5$. Now, since $(3+p_{\textrm{nuis}})^{2}$
grows very quickly in $p_{\textrm{nuis}}$, for even a small number of nuisance
parameters the logistic approach becomes much slower.
## 10 Appendix C
This appendix contains the technical details from the theorems in section $7$
of the main manuscript. We begin with a number of technical lemmas:
First, as one might imagine, if we can consistently estimate our correlation
matrices, applying a Fisher transformation should not change much. We
formalize this with the next lemma.
###### Lemma 10.1
Let $R_{1}$, $R_{2}$ be correlation matrices, and $\hat{R}_{1}$, $\hat{R}_{2}$
be estimates of $R_{1}$ and $R_{2}$.
Let $I$ be the set of ordered pairs $(j,k)$ where $R_{1(j,k)}\neq R_{2(j,k)}$.
Assume for all $(j,k)$ in $I$,
$\left|R_{1(j,k)}-R_{2(j,k)}\right|>\Delta_{\min}$ for some $\Delta_{\min}>0$
and that for $m=1,2$ we have
$\operatorname{sup}_{j<k}\left\|R_{m(j,k)}\right\|_{\infty}<\rho_{\max}$ for
some fixed $\rho_{\max}<1$.
Further assume that for $m=1,2$,
$\left\|R_{m}-\hat{R}_{m}\right\|_{\infty}\leq\delta$ (for some
$\delta<1-\rho_{\max}$). Then for all $(j,k)$ in $I^{c}$ with $j\neq k$ we
have
$\left|\operatorname{arctanh}\left(\hat{R}_{1(j,k)}\right)-\operatorname{arctanh}\left(\hat{R}_{2(j,k)}\right)\right|\leq\frac{2\delta}{1-\left(\rho_{\max}+\delta\right)^{2}}$
(9)
and for all $(j,k)$ in $I$ with $j\neq k$ we have
$\left|\operatorname{arctanh}\left(\hat{R}_{1(j,k)}\right)-\operatorname{arctanh}\left(\hat{R}_{2(j,k)}\right)\right|\geq\Delta_{\min}-2\delta$
(10)
One immediate consequence of this lemma is that as $\delta\rightarrow 0$, for
$(j,k)$ in $I^{C}$ our statistics $T_{(j,k)}$ converge to $0$ (at rate
O($\delta)$), and for $(j,k)$ in $I$, $T_{(j,k)}$ are bounded away from $0$
(at a rate of at least O($\delta)$).
###### Proof 10.2 (Proof of Lemma 10.1)
We begin by showing that for all $(j,k)$ in $I^{c}$ with $j\neq k$ we have
$\left|\operatorname{arctanh}\left(\hat{R}_{1(j,k)}\right)-\operatorname{arctanh}\left(\hat{R}_{2(j,k)}\right)\right|\leq\frac{2\delta}{1-\left(\rho_{\max}+\delta\right)^{2}}$
The mean value theorem gives us that
$\left|\operatorname{arctanh}\left(\hat{R}_{1(j,k)}\right)-\operatorname{arctanh}\left(\hat{R}_{2(j,k)}\right)\right|\leq\operatorname{sup}_{r}\left|\frac{1}{1-r^{2}}\right|\left|\hat{R}_{1(j,k)}-\hat{R}_{2(j,k)}\right|$
where the supremum is taken over $r$ in
$\left[\hat{R}_{1(j,k)},\,\hat{R}_{2(j,k)}\right]$. Note that for $m=1,2$, we
have $|\hat{R}_{m(j,k)}|<\rho_{\max}+\delta$, and
$\left|\hat{R}_{1(j,k)}-\hat{R}_{2(j,k)}\right|\leq 2\delta$, for $(j,k)$ not
in $I$. Thus,
$\operatorname{sup}_{r}\left|\frac{1}{1-r^{2}}\right|\left|\hat{R}_{1(j,k)}-\hat{R}_{2(j,k)}\right|\leq\frac{2\delta}{1-\left(\rho_{\max}+\delta\right)^{2}}.$
Now for $(j,k)$ in $I$, we again use the mean value theorem:
$\left|\operatorname{arctanh}\left(\hat{R}_{1(j,k)}\right)-\operatorname{arctanh}\left(\hat{R}_{2(j,k)}\right)\right|\geq\operatorname{inf}_{r}\left|\frac{1}{1-r^{2}}\right|\left|\hat{R}_{1(j,k)}-\hat{R}_{2(j,k)}\right|$
and our result follows because
$\left|\hat{R}_{1(j,k)}-\hat{R}_{2(j,k)}\right|\geq\Delta_{\min}-2\delta$.
Now we consider convergence of these sample correlation matrices. We show that
their convergence depends only on the convergence of the sample means
($\hat{\mu}_{j}$), variances ($\hat{\sigma}_{j}^{2}$), and pairwise inner
products. We formalize this in the following lemma.
###### Lemma 10.3
Let $\tilde{x}_{j}$, $j=1,\ldots$ be random variables. Let $\mu_{j}$ denote
the mean of $\tilde{x}_{j}$ and $\sigma_{j}^{2}$ its variance. Let $R_{j,k}$
be the correlation between $\tilde{x}_{j}$ and $\tilde{x}_{k}$. For each $i$,
let $x_{i,\cdot}$ be independent realizations with the same distribution as
$\tilde{x}_{\cdot}$ (eg. $x_{i,j}$ has the marginal distribution of
$\tilde{x}_{j}$).
For any given $\epsilon>0$, there exists $\delta>0$ such that if
$\operatorname{sup}\left\\{\left|\hat{\sigma}_{j}-\sigma_{j}\right|,\,\left|\hat{\mu}_{j}-\mu_{j}\right|,\,\left|\frac{(1/n)\sum_{i\leq
n}x_{i,j}x_{i,k}}{\sigma_{j}\sigma_{k}}-\frac{\mu_{j}\mu_{k}}{\sigma_{j}\sigma_{k}}-R_{j,k}\right|\right\\}_{j,k}\leq\delta$
(11)
then
$\operatorname{sup}_{j<k\leq p}\left|\hat{R}_{j,k}-R_{j,k}\right|\leq\epsilon$
(12)
Furthermore, one can choose $\delta=O(\epsilon)$
###### Proof 10.4 (Proof of Lemma 10.3)
We begin by noting that the distribution of $\hat{R}_{j,k}$ is independent of
$\mu_{j}$, $\mu_{k}$, $\sigma_{j}$ and $\sigma_{k}$. For ease of notation we
assume $\mu_{j}=\mu_{k}=0$ and $\sigma_{j}=\sigma_{k}=1$.
To see that (11) is sufficient for (12) we write $\hat{R}_{j,k}-R_{j,k}$ as
$\displaystyle\left|\hat{R}_{j,k}-R_{j,k}\right|$
$\displaystyle=\left|\frac{\left(1/n\right)\sum_{i=1}^{n}x_{i,j}x_{i,k}}{\hat{\sigma}_{j}\hat{\sigma}_{k}}-\frac{\hat{\mu}_{j}\hat{\mu}_{k}}{\hat{\sigma}_{j}\hat{\sigma}_{k}}-R_{j,k}\right|$
$\displaystyle\leq\left|\frac{1}{n}\sum_{i=1}^{n}x_{i,j}x_{i,k}\right|\left|\left(\frac{1}{\hat{\sigma}_{j}\hat{\sigma}_{k}}-1\right)\right|$
$\displaystyle+\left|\frac{1}{n}\sum_{i=1}^{n}x_{i,j}x_{i,k}-R_{j,k}\right|+\left|\frac{\hat{\mu}_{j}\hat{\mu}_{k}}{\hat{\sigma}_{j}\hat{\sigma}_{k}}\right|$
We first note that
$\left|\frac{1}{n}\sum_{i=1}^{n}x_{i,j}x_{i,k}-R_{j,k}\right|<\delta$. Thus we
need only consider
$\left|\frac{\hat{\mu}_{j}\hat{\mu}_{k}}{\hat{\sigma}_{j}\hat{\sigma}_{k}}\right|$
and $\left|\left(\frac{1}{\hat{\sigma}_{j}\hat{\sigma}_{k}}-1\right)\right|$.
Expanding these terms using the fact that $1/(1-\delta)=1+O(\delta)$, it is
straightforward to see that the whole expression converges to $0$ at rate
$O(\delta)$. This completes our proof.
Now that we have reduced convergence to that of the sample mean, variance, and
inner products, we show particular circumstances under which our estimation is
consistent, and give rates of convergence.
###### Lemma 10.5
Let $\tilde{x}_{j}$, $j=1,\ldots$ be random variables. Assume there is some
$C>0$ such that for all $t\geq 0$
$\operatorname{P}\left(\left|x_{j}-\operatorname{E}[x_{j}]\right|>t\right)\leq\operatorname{exp}\left(1-t^{2}/C^{2}\right)$
(These are known as sub-Gaussian random variables). Let $\mu_{j}$ denote the
mean of $\tilde{x}_{j}$ and $\sigma_{j}^{2}$ its variance. Let $R_{j,k}$ be
the correlation between $\tilde{x}_{j}$ and $\tilde{x}_{k}$. For each $i$, let
$x_{i,\cdot}$ be independent realizations with the same distribution as
$\tilde{x}$.
Let $\delta,\,\epsilon_{p}>0$ be given. Then for $n$ sufficiently large and
$\frac{\log p}{n}$ sufficiently small we have that
$\operatorname{sup}\left\\{\left|\hat{\sigma}_{j}-\sigma_{j}\right|,\,\left|\hat{\mu}_{j}-\mu_{j}\right|,\,\left|\frac{(1/n)\sum_{i\leq
n}x_{i,j}x_{i,k}}{\sigma_{j}\sigma_{k}}-\frac{\mu_{j}\mu_{k}}{\sigma_{j}\sigma_{k}}-R_{j,k}\right|\right\\}_{j,k\leq
p}\leq\delta$ (13)
with probability greater than $1-\epsilon_{p}$. In particular one can choose
$\delta=O\left(\log p/n\right)^{1/2}$.
The class of subgaussian random variables is rather broad, containing gaussian
random variables and all bounded random variables. Applying this lemma, we are
able to show consistency for the wide class of variables with sufficiently
light tails.
In the proof of this lemma we get a convergence rate of $\delta=O\left(\log
p/n\right)^{1/2}$. This rate agrees with the literature for other similar
problems in covariance estimation (Bickel and Levina (2008) among others).
###### Proof 10.6 (Proof of Lemma 10.5)
We will begin by bounding $\left|\hat{\mu}_{j}-\mu_{j}\right|$. If we consider
Lemma $5.10$ of Vershynin (2010) we see that
$\operatorname{P}\left(\left|\hat{\mu}_{j}-\mu_{j}\right|>t\right)\leq
e\cdot\operatorname{exp}\left[-\left(\tilde{C}t^{2}\right)n\right]$
where $\tilde{C}$ is some function of $C$ (one can prove this Hoeffding type
inequality by an exponential Markov argument). Applying the union bound to
this we see that
$\operatorname{P}\left(\operatorname{sup}_{j\leq
p}\left|\hat{\mu}_{j}-\mu_{i}\right|>t\right)\leq
3p\operatorname{exp}\left[-\left(\tilde{C}t^{2}\right)n\right]$
If we set $t=\left(\sqrt{1/C}\right)\sqrt{\frac{q+\log p}{n}}$ then we have
$\operatorname{P}\left(\operatorname{sup}_{j\leq
p}\left|\hat{\mu}_{j}-\mu_{j}\right|>t\right)\leq e^{1-q},$
bounding $\left|\hat{\mu}_{j}-\mu_{j}\right|$.
Next we bound $\left|\hat{\sigma}_{j}-\sigma_{i}\right|$. We first note that
$\left|\hat{\sigma}_{j}-\sigma_{j}\right|=\frac{\left|\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}\right|}{\hat{\sigma}_{j}+\sigma_{j}}\leq\frac{\left|\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}\right|}{\sigma_{j}}$
because $\hat{\sigma_{j}},\sigma_{j}>0$. so we need only consider convergence
of $\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}$. Next note that
$\frac{1}{n}\sum_{i}\left(x_{i,j}-\bar{x}_{j}\right)^{2}-\frac{1}{n}\sum_{i}\left(x_{i,j}-\mu_{j}\right)^{2}=-\left(\bar{x}_{j}-\mu_{j}\right)^{2}$
So now if we can bound
$\left|\frac{1}{n}\sum_{i}\left(x_{i,j}-\mu_{j}\right)^{2}-\sigma_{j}^{2}\right|$
and $\left(\bar{x}_{j}-\mu_{j}\right)^{2}$, then we can bound
$|\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}|$.
To bound
$\left|\frac{1}{n}\sum_{i}\left(x_{i,j}-\mu_{j}\right)^{2}-\sigma_{j}^{2}\right|$,
we first note that if $x_{i,j}$ is sub-Gaussian then $(x_{i,j}-\mu_{j})^{2}$
is subexponential; ie
$\operatorname{P}\left(\left(x_{i,j}-\mu_{j}\right)^{2}-\sigma_{i}>t\right)\leq\operatorname{exp}\left(-C_{1}t\right)$
for some fixed $C_{1}$. Now we apply Corollary $5.17$ of Vershynin (2010), and
get that for any $t$ sufficiently small (independent of $n$)
$\operatorname{P}\left(\frac{1}{n}\sum_{i}\left(x_{i,j}-\mu_{j}\right)^{2}>t\right)\leq
2\operatorname{exp}\left(-\tilde{C}_{1}t^{2}\right)$
for some fixed $\tilde{C}_{1}$. Bounding
$\left(\bar{x}_{j}-\mu_{j}\right)^{2}$ is also quite straightforward (we just
use the bound for $\left|\bar{x}_{j}-\mu_{j}\right|$)
$P\left(\left(\bar{x}_{j}-\mu_{j}\right)^{2}\geq t\right)\leq
e\operatorname{exp}\left[-\left(\tilde{C}t\right)n\right]$
We note that for $t<1$, $t^{2}<t$. Let
$\bar{C}=\min\\{\tilde{C}_{1},\tilde{C}\\}$. Now, combining these inequalities
with the triangle inequality we have
$\displaystyle P\left(\left|\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}\right|\geq
t\right)$ $\displaystyle\leq
e\operatorname{exp}\left[-\left(\tilde{C}t\right)n\right]+2\operatorname{exp}\left(-\tilde{C}_{1}t^{2}\right)$
$\displaystyle\leq 5\operatorname{exp}\left[-\bar{C}t^{2}n\right]$
for $t$ sufficiently small. Now finally,
$P\left(\left|\hat{\sigma}_{j}-\sigma_{j}\right|\geq t\right)\leq
P\left(\left|\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}\right|\geq
t\sigma_{\min}\right)\leq
5\operatorname{exp}\left[-\bar{C}\sigma_{\min}^{2}t^{2}n\right].$
Using the union bound again, we get
$P\left(\operatorname{sup}_{j}\left|\hat{\sigma}_{j}^{2}-\sigma_{j}^{2}\right|\geq
t\right)\leq 5p\operatorname{exp}\left[-\bar{C}t^{2}n\right].$
so
$P\left(\operatorname{sup}_{j}\left|\hat{\sigma}_{j}-\sigma_{j}\right|\geq
t\right)\leq
5p\operatorname{exp}\left[-\bar{C}\sigma_{\min}^{2}t^{2}n\right].$
Finally, we need to bound $\left|\frac{(1/n)\sum_{i\leq
n}x_{i,j}x_{i,k}}{\sigma_{j}\sigma_{k}}-\frac{\mu_{j}\mu_{k}}{\sigma_{j}\sigma_{k}}-\rho_{j,k}\right|$.
This is slightly trickier but still not terrible. We first note that
$(1/n)\sum_{i\leq n}x_{i,j}x_{i,k}-\mu_{j}\mu_{k}=(1/n)\sum_{i\leq
n}\left(x_{i,j}-\mu_{j}\right)\left(x_{i,k}-\mu_{k}\right)$
We also see that
$\displaystyle 2\sum_{i\leq
n}\left(x_{i,j}-\mu_{j}\right)\left(x_{i,k}-\mu_{k}\right)$
$\displaystyle=\sum_{i\leq
n}\left[\left(x_{i,j}-\mu_{j}\right)+\left(x_{i,k}-\mu_{k}\right)\right]^{2}$
$\displaystyle-\sum_{i\leq n}\left(x_{i,j}-\mu_{j}\right)^{2}-\sum_{i\leq
n}\left(x_{i,k}-\mu_{k}\right)^{2}$
Now to bound the above quantity we consider the moment generating function of
$x_{i,j}-\mu_{j}+x_{i,k}-\mu_{k}$. This not necessarily the sum of independent
random variables, still by Cauchy Schwartz we have
$\displaystyle\operatorname{E}\left[\operatorname{exp}\left[t\left(x_{i,j}-\mu_{j}+x_{i,k}-\mu_{k}\right)\right]\right]$
$\displaystyle\leq\operatorname{max}\left\\{\operatorname{E}\left[\operatorname{exp}\left[2t\left(x_{i,j}-\mu_{j}\right)\right]\right],\operatorname{E}\left[\operatorname{exp}\left[2t\left(x_{i,k}-\mu_{k}\right)\right]\right]\right\\}$
It is a well known fact that sub-gaussan random variables can be charaterized
by their MGF (shown in Vershynin (2010)), and this is still the moment
generating function of a subgaussian random variable. Thus,
$\left(x_{i,j}-\mu_{j}+x_{i,k}-\mu_{k}\right)^{2}$ is sub-exponential, and
again by Corollary $5.17$ of Vershynin (2010) we have that
$\displaystyle\operatorname{P}\left(\left|\frac{1}{n}\sum_{i}\left(x_{i,j}-\mu_{j}+x_{i,k}-\mu_{k}\right)^{2}-\sigma_{j}^{2}-\sigma_{k}^{2}-2\sigma_{j}\sigma_{k}\rho_{j,k}\right|>t\right)$
$\displaystyle\leq 2\operatorname{exp}\left[-C_{2}t^{2}n\right].$
for $t>0$ sufficiently small and some fixed $C_{2}>0$. Now, stringing all of
these together with the triangle inequality we have that
$\displaystyle\operatorname{P}\left(\left|\frac{2}{n}\sum_{i\leq
n}\left(x_{i,j}-\mu_{j}\right)\left(x_{i,k}-\mu_{k}\right)-2\rho\sigma_{j}\sigma_{k}\right|>3t\right)$
$\displaystyle\leq\operatorname{P}\left(\left|\frac{1}{n}\sum_{i\leq
n}\left(x_{i,j}-\mu_{j}+x_{i,k}-\mu_{k}\right)^{2}-\sigma_{j}^{2}+\sigma_{k}^{2}-2\sigma_{j}\sigma_{k}\rho_{j,k}\right|>t\right)$
$\displaystyle+\operatorname{P}\left(\left|\frac{1}{n}\sum_{i\leq
n}\left(x_{i,j}-\mu_{j}\right)^{2}-\sigma_{j}^{2}\right|>t\right)+\operatorname{P}\left(\left|\frac{1}{n}\sum_{i\leq
n}\left(x_{i,k}-\mu_{k}\right)^{2}-\sigma_{k}^{2}\right|>t\right)$
$\displaystyle\leq
2\operatorname{exp}\left[-C_{2}t^{2}n\right]+2*5\operatorname{exp}\left[-\bar{C}t^{2}n\right]$
$\displaystyle\leq 12\operatorname{exp}\left[-\bar{C}_{1}t^{2}n\right]$
for all $t>0$ sufficiently small with some fixed $\bar{C}_{1}>0$. Taking this
a step further, and applying the union bound, we see that
$P\left(\operatorname{sup}_{j,k}\left|\frac{(1/n)\sum_{i\leq
n}x_{i,j}x_{i,k}}{\sigma_{j}\sigma_{k}}-\frac{\mu_{j}\mu_{k}}{\sigma_{j}\sigma_{k}}-\rho_{j,k}\right|>t\right)\leq
12p^{2}\operatorname{exp}\left[-\bar{C}_{2}t^{2}n\right]$
for some fixed $\bar{C}_{2}$.
Now that we have bounded each term, we see that (13) happens with probability
at most
$\displaystyle
12p^{2}\operatorname{exp}\left[-\bar{C}_{2}\delta^{2}n\right]+2*5p\operatorname{exp}\left[-\bar{C}\sigma_{\min}^{2}\delta^{2}n\right]+2*3p\operatorname{exp}\left[-\tilde{C}\delta^{2}n\right]$
$\displaystyle\leq
28p^{2}\operatorname{exp}\left[-\mathbf{C}\delta^{2}n\right]$
for $\delta$ sufficiently small where
$\mathbf{C}=\min\left\\{\bar{C}\sigma_{\min}^{2},\bar{C_{2}},\tilde{C}\right\\}$.
Thus, if $\delta=\left(\frac{q+2\log p}{\mathbf{C}n}\right)^{1/2}$ then we
have (13) with probability at least $1-28e^{-q}$. If $n$ is sufficiently
large, and $\frac{\log p}{n}$ sufficiently small, then for any $q$, $\delta$
can be made arbitrarily small.
Now, we combine these lemmas to show that under certain conditions, for a
given cutoff $t$, as $n\rightarrow\infty$ if $\log p/n\rightarrow 0$ then,
with probability approaching $1$, all true marginal interactions have
$|T_{i,j}|>t$, and all null statistics will have $|T_{i,j}|<t$ (ie. we
asymptotically find all true interactions and make no false rejections).
Before we begin, it deserves mention that we use slightly different notation
than in the discussion of our algorithm in Section $3$. Rather than having
$X_{i,\cdot}$ denote the $i$-th observation overall, and letting $y(i)$ denote
its group (where $i$ ranged from $1$ to the total number of observations in
both groups), we split up our observations by group, letting $x_{m(i,\cdot)}$
denote the $i$-th observation from group $m$ (now $i$ ranges from $1$ to the
total number of observations in group $m$). This change simplifies notation in
the statement of the theorem and its proof. We also assume equal group sizes
($n_{1}=n_{2}=n$), this again simplifies notation but can be relaxed to
$n_{1}/(n_{1}+n_{2})\rightarrow\alpha\in(0,1)$.
###### Proof 10.7 (Proof of Theorem 6.1)
This result is a straightforward corollary of our $3$ lemmas:
First choose an arbitrary $\epsilon_{p}>0$, and $0<t<\Delta_{\min}$. If we
consider Lemma 10.3, we see that the conclusion of our theorem holds if we can
find a bound on the sup-norm distance between each correlation matrix and its
MLE (a bound I will call $\delta_{1}$) which satisfies
$\max\left\\{\frac{2\delta_{1}}{1-\left(\rho_{\max}+\delta_{1}\right)^{2}},\,\Delta_{\min}-2\delta_{1}\right\\}\leq
t.$
Because $\rho_{\max}<1$, $\delta_{1}>0$ sufficiently small will satisfy this.
Now applying Lemma 10.5: if we choose $\delta_{2}$ sufficiently small (but
still of $O(\delta_{1})$), then if
$\operatorname{sup}\left\\{\left|\hat{\sigma}_{j}-\sigma_{j}\right|,\,\left|\hat{\mu}_{j}-\mu_{j}\right|,\,\left|\frac{(1/n)\sum_{i\leq
n}x_{i,j}x_{i,k}}{\sigma_{j}\sigma_{k}}-\frac{\mu_{j}\mu_{k}}{\sigma_{j}\sigma_{k}}-\rho_{j,k}\right|\right\\}_{j,k}\leq\delta_{2}$
(14)
we have that the sup norm distance between each correlation matrix and its MLE
is bounded by $\delta_{1}$: for $m=1,2$
$\left\|\hat{R}_{m}-R_{m}\right\|_{\infty}\leq\delta_{1}$
Finally, by Lemma!10.1, we see that if $n$ is sufficiently large and $\log
p/n$ is sufficiently small then (14) holds with probability at least
$1-\epsilon_{p}$. This finishes our proof.
### 10.1 Proofs of Permutation Results
To begin, we prove a Lemma which does most of the leg-work for our eventual
theorem. It says that for a reasonably balanced permutation, for $n$
sufficiently large and $\log p/n$ sufficiently small, both of our permuted
sample correlation matrices will be very close to the average of the $2$
population correlation matrices.
###### Lemma 10.8
Let $\tilde{x}_{1(j)}$ and $\tilde{x}_{2(j)}$, $j=1,\ldots$ be random
variables with
$\operatorname{P}\left(|x_{m(j)}-\operatorname{E}\left[x_{m(j)}\right]|\geq
t\right)\leq 1-e^{t^{2}/C}$
for all $t>0$, and each $m=1,2$, with some fixed $C>0$. Let $\mu_{m(j)}$
denote the mean of $\tilde{x}_{m(j)}$ and $\sigma_{m(j)}^{2}$ its variance.
For each $i<\infty$, let $x_{m(i,\cdot)}$ be independent realizations with the
same distribution as $\tilde{x}_{m(\cdot)}$.
Let $p_{n}$ be a sequence of integers such that $\frac{\log
p_{n}}{n}\rightarrow 0$. Let $R_{m}$ be the correlation “matrix” (an infinite
but countably indexed matrix) of the covariates from class $m$. Define
$R_{\operatorname{perm}}$ to be the average of the two,
$R_{\textrm{perm}}=\frac{1}{2}R_{1}+\frac{1}{2}R_{2}$
Let $\hat{\mu}_{m(j)}$ and $\hat{\sigma}_{m(j)}^{2}$ be the pre-permuted
estimates of the mean and variance (in each class):
$\hat{\mu}_{m(j)}=\frac{1}{n}\sum_{i\leq n}x_{m(i,j)}$
and
$\hat{\sigma}_{m(j)}^{2}=\frac{1}{n}\sum_{i\leq
n}\left(x_{m(i,j)}-\hat{\mu}_{m(j)}\right)^{2}.$
Further, define
$\hat{R}_{\textrm{perm}:m(j,k)}=\frac{1}{n}\sum_{(i,l)\in\Pi(\cdot,m)}\left(\frac{x_{m(i,j)}-\hat{\mu}_{m(j))}}{\hat{\sigma}_{m(j)}}\right)\left(\frac{x_{m(i,k)}-\hat{\mu}_{m(k)}}{\hat{\sigma}_{m(k)}}\right)$
our permuted correlation between covariates $j$ and $k$ in class $m$.
Assume for every $j$, $\sigma_{j}^{2}\geq\sigma_{min}^{2}>0$. Now for any
$\epsilon>0$, $\delta>0$, one can find $n$ sufficiently large such that for
any permutation, $\Pi$ with
$\left|\hat{\pi}-\frac{1}{2}\right|\leq\frac{\delta}{12}$
(where $\hat{\pi}$ is the proportion of class $1$ that remains fixed under
$\Pi$). We have
$\left\|R_{\textrm{perm}}-\hat{R}_{\textrm{perm}:m}\right\|_{\infty}\leq\delta$
(15)
for both $m=1,2$ with probability at least $1-\epsilon$.
###### Proof 10.9 (Proof of Lemma 10.8)
We first consider only $m=1$. If we can show that
$\left\|R_{\textrm{perm}}-\hat{R}_{\textrm{perm}:m}\right\|_{\infty}\leq\delta$
(16)
with high probability for $m=1$, then by symmetry we have it for $m=2$, and by
a simple union bound we have it for both simultaneously.
Now, we begin by decomposing our sample permuted correlation matrix
$\displaystyle\hat{R}_{\textrm{perm}:1}$
$\displaystyle=\frac{1}{n}\sum_{(i,m)\in\Pi(\cdot,1)}\left(\frac{x_{m(i,j)}-\hat{\mu}_{m(j)}}{\hat{\sigma}_{m(j)}}\right)\left(\frac{x_{m(i,k)}-\hat{\mu}_{m(k)}}{\hat{\sigma}_{m(k)}}\right)$
$\displaystyle=\hat{\pi}\hat{R}_{\textrm{perm}:1}^{(1)}+\left(1-\hat{\pi}\right)\hat{R}_{\textrm{perm}:1}^{(2)}$
where $\hat{R}_{\textrm{perm}:1}^{(l)}$ is a matrix defined by
$\hat{R}_{\textrm{perm}:1(j,k)}^{(l)}=\frac{1}{\tilde{n}_{l}}\sum_{i\in\Pi(l,1)}\left(\frac{x_{l(i,j)}-\hat{\mu}_{1(j)}}{\hat{\sigma}_{1(j)}}\right)\left(\frac{x_{l(i,k)}-\hat{\mu}_{1(k)}}{\hat{\sigma}_{1(k)}}\right)$
(17)
where $\tilde{n}_{l}$ is the number of elements from group $l$ permuted to
group $1$ (ie. the cardinality of $\Pi(l,l)$ or more explicitly
$\tilde{n}_{1}=\hat{\pi}n$ and $\tilde{n}_{2}=(1-\hat{\pi}n$). The quantity
(17) is just the contribution from observations originally in class $l$ to the
permuted correlation matrix for class $1$. Thus by the triangle inequality
$\displaystyle\left\|R_{\textrm{perm}}-\hat{R}_{\textrm{perm}:1}\right\|_{\infty}$
$\displaystyle\leq\left\|\frac{1}{2}R_{1}-\hat{\pi}\hat{R}_{\textrm{perm}:1}^{(1)}\right\|_{\infty}+\left\|\frac{1}{2}R_{2}-\left(1-\hat{\pi}\right)\hat{R}_{\textrm{perm}:1}^{(1)}\right\|_{\infty}$
(18)
$\displaystyle\leq\frac{1}{2}\left\|R_{1}-\hat{R}_{\textrm{perm}:1}^{(1)}\right\|_{\infty}+\frac{1}{2}\left\|R_{2}-\hat{R}_{\textrm{perm}:1}^{(2)}\right\|_{\infty}$
$\displaystyle+\left|\hat{\pi}-\frac{1}{2}\right|\left(\left\|\hat{R}_{\textrm{perm}:1}^{(1)}\right\|_{\infty}+\left\|\hat{R}_{\textrm{perm}:1}^{(2)}\right\|_{\infty}\right)$
If we consider $\hat{R}_{\textrm{perm}:1}^{(1)}$, we see that it is
essentially a sample correlation matrix (using only the $\hat{\pi}n$
observations that were fixed in class $1$ by $\Pi$ for the inner product). We
can make a similar observation for $\hat{R}_{\textrm{perm}:1}^{(2)}$. Now, for
$n$ sufficiently large, because $|\frac{1}{2}-\hat{\pi}|$ is small, we can
make $\hat{\pi}n$ and $\left(1-\hat{\pi}\right)n$ as large as we would like.
Thus, by a combination of Lemma 10.1 and Lemma 10.5, we have that
$\left\|R_{l}-\hat{R}_{\textrm{perm}:1}^{(l)}\right\|_{\infty}<\delta/3$
with probability greater than $1-\epsilon/3$. Furthermore, using the same
Lemmas we get
$\left\|\hat{R}_{\textrm{perm}:1}^{(1)}\right\|_{\infty}+\left\|\hat{R}_{\textrm{perm}:1}^{(2)}\right\|_{\infty}\leq
4$
with probability at least $1-\epsilon/3$ (this bound can easily be made
tighter, and if we were to standardize within permutation this bound is
trivial). Plugging this in with the assumed bound on
$\left|\hat{\pi}-\frac{1}{2}\right|$ completes the proof.
Now, we use this Lemma (along with some of our previous Lemmas) to show that
for any fixed $t>0$ if our variables are subgaussian with some other minor
conditions, then for $n\rightarrow\infty$ and $\log p/n\rightarrow 0$ with
probability approaching $1$, none of our permuted statistics will be larger
than $t$, or in other words our estimated FDR will converge to $0$.
###### Proof 10.10 (Proof of Theorem 6.2)
First we choose an arbitrary $2\epsilon_{p}>0$ and $t>0$. If we consider Lemma
$7.1$, we see that if we find some $\delta>0$ satisfying
$\left\|R_{\textrm{perm}}-\hat{R}_{\textrm{perm}:m}\right\|_{\infty}$ (19)
for $m=1,2$ with probability at least $1-\epsilon_{p}$ and
$\frac{2\delta}{1-(\rho_{\max}+\delta)^{2}}\leq t$ (20)
then we have satisfied our claim. Because, $\rho_{\max}<1$, there exists some
$\delta>0$ satisfying (20). Now, we first note that, for $n$ sufficiently
large, standard concentration inequalities give us that
$\left|\hat{\pi}-\frac{1}{2}\right|\leq\delta/12$
with probability greater than $1-\epsilon_{p}$. If we apply Lemma $7.5$ with
this bound on $\hat{\pi}$ and combine the probabilities with the union bound,
we get that for $n$ sufficiently large (19) is violated with at most
probability $2\epsilon_{p}$. This completes our proof.
## 11 Acknowledgments
We would like to thank Jonathan Taylor and Trevor Hastie for their helpful
comments and insight.
## References
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* Buzkova et al. [2011] P. Buzkova, T. Lumley, and K. Rice. Permutation and parametric bootstrap tests for gene - gene and gene–environment interactions. _Annals of Human Genetics_ , 2011.
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|
arxiv-papers
| 2012-06-27T20:38:20 |
2024-09-04T02:49:32.309323
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Noah Simon and Robert Tibshirani",
"submitter": "Noah Simon",
"url": "https://arxiv.org/abs/1206.6519"
}
|
1206.6541
|
# A Monotone Function Given by a Low-Depth Decision Tree that is not an
Approximate Junta
Daniel M. Kane
## 1 Introduction
In [3], O’Donnell and Servedio show that any monotone function given by a
depth-$d$ decision tree can be learned to constant accuracy from random
samples in $\textrm{poly}(n,2^{d})$ time. The impact of this result is
somewhat lessened by an apparent lack of interesting monotone functions given
by low-depth decision trees. In particular, it was independently suggested by
Elad Verbin and by Rocco Servedio and Li-Yang Tan, that all such functions
might be approximated by functions on few variables (see [2], page 10).
###### Conjecture 1.
For every $\epsilon>0$ and every monotone function
$f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ given by a depth-$d$ decision tree,
there is a $k$-junta, $g$, for $k=\textrm{poly}_{\epsilon}(d)$ so that $f$ and
$g$ agree on all but an $\epsilon$-fraction of inputs.
In this note, we disprove the above conjecture, and in particular provide an
example of a monotone low-degree function that is not well approximated by any
small junta. In particular we prove:
###### Theorem 2.
There exists a constant $\epsilon>0$ so that for every positive integer $d$,
there exists a $k=\exp(\Omega(\sqrt{d}))$ and a monotone function
$f:\\{0,1\\}^{n}\rightarrow\\{0,1\\}$ given by a depth-$d$ decision tree, so
that for every $k$-junta $g$, $f$ and $g$ disagree on at least an
$\epsilon$-fraction of inputs.
In fact it is known that the bound on $k$ in Theorem 2 is tight up to the
constant in the exponent. In particular, it is shown in [3] that any monotone
function given by a depth-$d$ decision tree has total influence
$I(f)=O(\sqrt{d})$. We combine this with the main result of [1], which says
that any boolean function $f$ can be $\epsilon$-approximated by a $k$-junta
for $k=\exp(O(I(f)/\epsilon)).$ Combining these results we find that:
###### Observation.
If $f$ is a monotone function given by a depth-$d$ decision tree, and if
$\epsilon>0$, then there is a $k$-junta $g$ that agrees with $f$ on all but an
$\epsilon$ fraction of inputs for $k=\exp(O(\sqrt{d}/\epsilon))$.
The function we construct to show Theorem 2 will combine ideas from two
previous constructions, the monotone addressing function and Talagrand’s
function.
The monotone addressing function is defined by
$f(x_{1},\ldots,x_{d-1},y_{0},\ldots,y_{2^{d-1}-1})=\begin{cases}1&\textrm{ if
}\sum x_{i}>\left\lfloor(d-1)/2\right\rfloor\\\ y_{x_{0}\ldots
x_{d-1}}&\textrm{ if }\sum x_{i}=\left\lfloor(d-1)/2\right\rfloor\\\
0&\textrm{ if }\sum x_{i}<\left\lfloor(d-1)/2\right\rfloor\end{cases}.$
This is an example of a monotone function given by a depth-$d$ decision tree
that depends on exponentially many variables, and thus provides us with a good
starting point. The monotone addressing function fails to provide a counter-
example to Conjecture 1 though since it agrees with the majority function
except on a set of measure $O(1/\sqrt{d})$.
Given the bound on the total sensitivity of a low-depth monotone function, we
know that any $f$ satisfying the conditions of Theorem 2 must not only have
near the maximum possible total influence for a low-depth monotone function,
but also must not be approximable by a function with much lower total
influence. Because of this restriction, our construction will look somewhat
similar to a construction of Talagrand in [4]. In particular, Talagrand
constructs a monotone function $f$ on $\\{0,1\\}^{d}$ so that on a constant
fraction of inputs, $f$ has sensitivity (i.e. the number of coordinates such
that changing the input at that coordinate would change the output of $f$)
$\Omega(\sqrt{d})$. Since, as is easily seen, the average sensitivity over all
inputs is equal to the total influence, this is as large as possible. On the
other hand, this condition tells us that $f$ retains large average sensitivity
even after ignoring any $\epsilon$-fraction of inputs for sufficiently small
constant $\epsilon$. Talagrand’s function fails to provide a counter-example
to Conjecture 1 on its own, because it is already a $d$-junta.
## 2 The Construction
In order to define the function $f$ with the properties specified by Theorem
2, we first introduce some background notation. We let $d,t$ and $m$ be
integers with $t=\Theta(\sqrt{d})$ and $m=\Theta(2^{t})$. We furthermore
assume that $2^{-t}m$ is sufficiently small given the value of $t/\sqrt{d}$.
We let $\mathcal{S}=(S_{1},\ldots,S_{m})$ be a random sequence of sets, where
the $S_{i}$ are chosen independently and uniformly from the set of subsets of
$\\{1,2,\ldots,d-1\\}$ of size exactly $t$. Given this $\mathcal{S}$, we
define the function $T_{\mathcal{S}}$ on $\\{0,1\\}^{d-1}$ as follows:
$T_{\mathcal{S}}(x_{1},\ldots,x_{d-1})=\\{1\leq i\leq m:x_{j}=1\textrm{ for
all }j\in S_{i}\\}.$
We will hereafter abbreviate $T$ by suppressing the explicit dependence on
$\mathcal{S}$, and abbreviate $(x_{1},\ldots,x_{d-1})$ by $x$.
We finally define $f$ as
$f_{\mathcal{S}}(x_{1},\ldots,x_{d-1},y_{1},\ldots,y_{m})=\begin{cases}1&\textrm{if
}|T(x)|\geq 2\\\ 0&\textrm{if }|T(x)|=0\\\ y_{i}&\textrm{if
}T(x)=\\{i\\}\end{cases}.$
Again, we will often suppress the dependence of $f$ on $\mathcal{S}$. It is
clear that $f$ is monotone. Furthermore, $f$ is given by a depth-$d$ decision
tree, since after fixing the values of the $x_{i}$, the value of $f$ depends
on at most one more coordinate. In the next Section, we show that $f$ cannot
be approximated by any $k$-junta for small $k$.
Note that Talagrand’s function is given (for appropriately chosen
$\mathcal{S}$) by
$G(x_{1},\ldots,x_{d-1})=\begin{cases}1&\textrm{if }|T(x)|\geq 1\\\
0&\textrm{if }|T(x)|=0\end{cases}.$
## 3 Approximation Bounds
Theorem 2 will follow from the following Proposition:
###### Proposition 3.
There exists an $\epsilon>0$ so that for $f_{\mathcal{S}}$ defined as above,
with constant probability over the choice of $\mathcal{S}$, $f$ is not
$\epsilon$-approximated by any $k$-junta for $k=o(2^{t})$.
A key step in our proof will be to show that with constant probability $f$
actually depends on one of the $y_{i}$.
###### Lemma 4.
With $T$ as above,
$\textrm{Pr}_{\mathcal{S},x}(|T_{\mathcal{S}}(x)|=1)=\Omega(1).$
###### Proof.
We will show the further claim that
$\mathbb{E}\left[|T_{\mathcal{S}}(x)|(2-|T_{\mathcal{S}}(x)|)\right]=\Omega(1).$
(1)
Since the term in the expectation is positive only if $|T|=1$, this will
complete our proof. We note that
$\displaystyle\mathbb{E}\left[|T_{\mathcal{S}}(x)|\right]$
$\displaystyle=\sum_{i=1}^{m}\textrm{Pr}(i\in T_{\mathcal{S}}(x))$
$\displaystyle=\sum_{i=1}^{m}\textrm{Pr}(x_{j}=1\textrm{ for all }j\in S_{i})$
$\displaystyle=m2^{-t}.$
On the other hand, we have that
$\displaystyle\mathbb{E}\left[|T_{\mathcal{S}}(x)|(|T_{\mathcal{S}}(x)|-1)\right]$
$\displaystyle=\sum_{i\neq j}\textrm{Pr}(i,j\in T_{\mathcal{S}}(x))$
$\displaystyle=\sum_{i\neq j}\textrm{Pr}(i\in
T_{\mathcal{S}}(x))\textrm{Pr}(j\in T_{\mathcal{S}}(x)|i\in
T_{\mathcal{S}}(x))$ $\displaystyle=\sum_{i\neq
j}2^{-t}\textrm{Pr}(x_{\ell}=1\textrm{ for all }\ell\in
S_{j}|x_{\ell}=1\textrm{ for all }\ell\in S_{i}).$
To compute this conditional probability we let
$S_{j}=\\{a_{1},\ldots,a_{t}\\}$ where the $a_{i}$ are picked randomly from
$\\{1,2,\ldots,d-1\\}$ without replacement. We compute it as the product
$\prod_{k=1}^{t}\textrm{Pr}(x_{a_{k}}=1|x_{a_{1}}=\ldots=x_{a_{k-1}}=1\textrm{
and }x_{\ell}=1\textrm{ for all }\ell\in S_{i}).$
These probabilities are approximated by first fixing the values of $S_{i}$ and
$a_{1},\ldots,a_{k-1}$. After additionally fixing the value of $a_{k}$, the
probability in question becomes $1$ if $a_{k}\in S_{i}$ and $1/2$ otherwise.
Thus the probability that $x_{a_{r}}=1$ is
$(1+\textrm{Pr}(a_{r}\in
S_{i}))/2=\left(1+\frac{|S_{i}\backslash\\{a_{1},\ldots,a_{r-1}\\}|}{d-r}\right)/2=1/2+O(t/d).$
Hence the probability that $j\in T_{\mathcal{S}}(x)$ given that $i\in
T_{\mathcal{S}}(x)$ is
$(1/2+O(t/d))^{t}=2^{-t}\exp(O(t^{2}/d)).$
Therefore, we have that
$\mathbb{E}\left[|T_{\mathcal{S}}(x)|(|T_{\mathcal{S}}(x)|-1)\right]=\sum_{i\neq
j}2^{-2t}\exp(O(t^{2}/d))\leq(2^{-t}m)^{2}\exp(O(t^{2}/d)).$
Therefore, we have that
$\displaystyle\mathbb{E}\left[|T_{\mathcal{S}}(x)|(2-|T_{\mathcal{S}}(x)|)\right]$
$\displaystyle=\mathbb{E}\left[|T_{\mathcal{S}}(x)|\right]-\mathbb{E}\left[|T_{\mathcal{S}}(x)|(|T_{\mathcal{S}}(x)|-1)\right]$
$\displaystyle\geq(2^{-t}m)-(2^{-t}m)^{2}\exp(O(t^{2}/d))$
$\displaystyle=(2^{-t}m)\left(1-(2^{-t}m)\exp(O(t^{2}/d))\right).$
As long as $2^{-t}m$ is bounded below by a constant and above by
$\exp(-O(t^{2}/d))/2$, this is $\Omega(1)$. ∎
We are now ready to prove Proposition 3. By Lemma 4, we note that with
constant probability over $\mathcal{S}$, that
$\textrm{Pr}_{x}(|T(x)|=1)=\Omega(1)$. For such $\mathcal{S}$, we claim that
$f$ has the desired property. In particular we claim the following:
###### Lemma 5.
If $f$ is as above and $g$ is a $k$-junta, then
$\textrm{Pr}(f(x,y)\neq
g(x,y))\geq\frac{\textrm{Pr}_{x}(|T(x)|=1)-k2^{-t}}{2}.$
###### Proof.
This follows from the simple observation that if, after fixing the value of
$x$, we have that $T=\\{i\\}$ where $g$ does not depend on $y_{i}$, then
$\textrm{Pr}_{y}(f(x,y)\neq g(x,y))=1/2$. This is because after further
conditioning on the values of all $y_{j}$ for $j\neq i$, $g$ becomes a
constant function (by assumption) and $f$ takes the values $0$ and $1$ each
with probability $1/2$. Therefore we have that
Pr $\displaystyle(f(x,y)\neq g(x,y))$
$\displaystyle\geq\frac{\textrm{Pr}(T(x)=\\{i\\}\textrm{ for some }i,\textrm{
and }g\textrm{ does not depend on }y_{i})}{2}$
$\displaystyle=\frac{\textrm{Pr}(|T(x)|=1)-\textrm{Pr}(T(x)=\\{i\\}\textrm{
for some }i,\textrm{ and }g\textrm{ depends on }y_{i})}{2}$
$\displaystyle=\frac{\textrm{Pr}(|T(x)|=1)-\sum_{i:g\textrm{ depends on
}y_{i}}\textrm{Pr}(T(x)=\\{i\\})}{2}$
$\displaystyle\geq\frac{\textrm{Pr}(|T(x)|=1)-\sum_{i:g\textrm{ depends on
}y_{i}}\textrm{Pr}(i\in T(x))}{2}$
$\displaystyle=\frac{\textrm{Pr}(|T(x)|=1)-\sum_{i:g\textrm{ depends on
}y_{i}}2^{-t}}{2}$
$\displaystyle\geq\frac{\textrm{Pr}_{x}(|T(x)|=1)-k2^{-t}}{2}.$
∎
Proposition 3 and Theorem 2 now follow immediately.
## Acknowledgements
I would like to thank Ryan O’Donnell for making me aware of this problem, and
for his help with finding appropriate references for this paper. This research
was done with the support of an NSF postdoctoral fellowship.
## References
* [1] E. Friedgut _Boolean Functions with Low Average Sensitivity Depend on Few coordinates_ , Combinatorica Vol. 18(1), pp. 27-36, 1998.
* [2] R. O’Donnell _Open Problems in Analysis of Boolean Functions_ http://arxiv.org/abs/1204.6447.
* [3] R. O’Donnell, R. Servedio _Learning monotone decision trees in polynomial time_ , SIAM Journal on Computing Vol. 37(3), pp. 827-844, 2007.
* [4] M. Talagrand _How much are increasing sets positively correlated?_ , Combinatorica Vol. 16(2), pp. 243-258, 1996.
|
arxiv-papers
| 2012-06-27T23:56:31 |
2024-09-04T02:49:32.321559
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniel M. Kane",
"submitter": "Daniel Kane",
"url": "https://arxiv.org/abs/1206.6541"
}
|
1206.6596
|
# Brauer algebras of type ${\rm F}_{4}$
Shoumin Liu
###### Abstract
We present an algebra related to the Coxeter group of type ${\rm F}_{4}$ which
can be viewed as the Brauer algebra of type ${\rm F}_{4}$ and is obtained as a
subalgebra of the Brauer algebra of type ${\rm E}_{6}$. We also describe some
properties of this algebra.
## 1 Introduction
When studying tensor decompositions for orthogonal groups, Brauer ([2])
introduce algebras which we now call Brauer algebras of type ${\rm A}$. Cohen,
Frenk and Wales ([4]) extended the definition to simply laced types, including
type ${\rm E}_{6}$. Tits ([18]) described how to obtain the Coxeter group of
type ${\rm F}_{4}$ as the fixed subgroup of the Coxeter group of type ${\rm
E}_{6}$ under a diagram automorphism (also seen in [3]). For this, Mühlherr
gave a more general way by admissible partitions to obtain Coxeter groups as
subgroups in Coxeter groups in [16]. Here we will apply a similar method to
the Brauer algebra ${\rm Br}({\rm E}_{6})$. This is a part of a project to
define Brauer algebras of spherical types ([6], [7]). It turns out that the
presentation by generators and relations obtainable from the Dynkin diagram of
type ${\rm F}_{4}$ in the same way as was done for type ${\rm B}_{n}$ ([7])
and ${\rm C}_{n}$ ([6]).
First we give the definition of ${\rm Br}({\rm F}_{4})$ using a presentation.
Let $\delta$ be the generator of the infinite cyclic group.
###### Definition 1.1.
The Brauer algebra of type ${\rm F}_{4}$, denoted by ${\rm Br}({\rm F}_{4})$,
is a unital associative $\mathbb{Z}[\delta^{\pm 1}]$-algebra generated by
$\\{r_{i},\,e_{i}\\}_{i=1}^{4}$, subject to the following relations.
$\displaystyle r_{i}^{2}$ $\displaystyle=$ $\displaystyle
1\qquad\qquad\,\,\,\kern 0.20004pt\mbox{for}\,\mbox{any}\ i$ (1.1)
$\displaystyle r_{i}e_{i}$ $\displaystyle=$ $\displaystyle
e_{i}r_{i}\,=\,e_{i}\,\,\,\,\,\,\kern 0.50003pt\mbox{for}\,\mbox{any}\ i$
(1.2) $\displaystyle e_{i}^{2}$ $\displaystyle=$ $\displaystyle\delta
e_{i}\qquad\quad\,\,\kern 0.20004pt\mbox{for}\ i>2$ (1.3) $\displaystyle
e_{i}^{2}$ $\displaystyle=$ $\displaystyle\delta^{2}e_{i}\qquad\quad\,\,\kern
0.20004pt\mbox{for}\ i<3$ (1.4) $\displaystyle r_{i}r_{j}$ $\displaystyle=$
$\displaystyle r_{j}r_{i},\qquad\quad\mbox{for}\ i\nsim j$ (1.5)
$\displaystyle e_{i}r_{j}$ $\displaystyle=$ $\displaystyle
r_{j}e_{i},\qquad\quad\kern-0.29999pt\mbox{for}\ i\nsim j$ (1.6)
$\displaystyle e_{i}e_{j}$ $\displaystyle=$ $\displaystyle
e_{j}e_{i},\qquad\quad\kern-0.59998pt\mbox{for}\ i\nsim j$ (1.7)
$\displaystyle r_{i}r_{j}r_{i}$ $\displaystyle=$ $\displaystyle
r_{j}r_{i}r_{j},\qquad\,\kern-0.39993pt\mbox{for}\ {i\sim j}\,$ (1.8)
$\displaystyle r_{j}r_{i}e_{j}$ $\displaystyle=$ $\displaystyle
e_{i}e_{j},\quad\qquad\kern-1.1pt\mbox{for}\ i\sim j$ (1.9) $\displaystyle
r_{i}e_{j}r_{i}$ $\displaystyle=$ $\displaystyle
r_{j}e_{i}r_{j},\quad\quad\kern 0.59998pt\mbox{for}\ i\sim j$ (1.10)
and for $\mathrel{\mathop{\kern 0.0pt\circ}\limits_{\hbox
to0.0pt{\hss$\scriptstyle 2$\hss}}^{\hbox
to0.0pt{\hss$$\hss}}}\kern-4.0pt{{\vrule
height=2.0pt,depth=-1.6pt,width=36.135pt}\hbox to0.0pt{\hss\vrule
height=4.0pt,depth=-3.6pt,width=36.135pt}\kern-25.0pt<}\kern
8.0pt\mathrel{\mathop{\kern 0.0pt\circ}\limits_{\hbox
to0.0pt{\hss$\scriptstyle 3$\hss}}^{\hbox to0.0pt{\hss$$\hss}}}\kern-1.0pt$ ,
$\displaystyle r_{2}r_{3}r_{2}r_{3}$ $\displaystyle=$ $\displaystyle
r_{3}r_{2}r_{3}r_{2}$ (1.11) $\displaystyle r_{2}r_{3}e_{2}$ $\displaystyle=$
$\displaystyle r_{3}e_{2}$ (1.12) $\displaystyle r_{2}e_{3}r_{2}e_{3}$
$\displaystyle=$ $\displaystyle e_{3}e_{2}e_{3}$ (1.13)
$\displaystyle(r_{2}r_{3}r_{2})e_{3}$ $\displaystyle=$ $\displaystyle
e_{3}(r_{2}r_{3}r_{2})$ (1.14) $\displaystyle e_{2}r_{3}e_{2}$
$\displaystyle=$ $\displaystyle\delta e_{2}$ (1.15) $\displaystyle
e_{2}e_{3}e_{2}$ $\displaystyle=$ $\displaystyle\delta e_{2}$ (1.16)
$\displaystyle e_{2}r_{3}r_{2}$ $\displaystyle=$ $\displaystyle e_{2}r_{3}$
(1.17) $\displaystyle e_{2}e_{3}r_{2}$ $\displaystyle=$ $\displaystyle
e_{2}e_{3}$ (1.18)
Here $i\sim j$ means that $i$ and $j$ are connected by a simple bond and
$i\nsim j$ means that there is no bond (simple or multiple) between $i$ and
$j$ in the Dynkin Diagram of type ${\rm F}_{4}$ depicted in the Figure 1. The
submonoid of the multiplicative monoid of ${\rm Br}({\rm F}_{4})$ generated by
$\delta$, $\\{r_{i},\,e_{i}\\}_{i=1}^{4}$ is denoted by ${\rm BrM}({\rm
F}_{4})$. This is the monoid of monomials in ${\rm Br}({\rm F}_{4})$.
The defining relations (1.11)–(1.18) can be found in ${\rm Br}({\rm C}_{2})$
in [6] and ${\rm Br}({\rm B}_{2})$ in [7] by renumbering indices. Note that
these relations are not symmetric for $2$ and $3$. Their relations are fully
determined by the Dynkin diagram in the sense that all relations depend only
on the vertices and bonds of the Dynkin diagram and the lengths of their
roots.
It is well known that the Coxeter group $W({\rm F}_{4})$ of type ${\rm
F}_{4}$, can be obtained as the subgroup from the Coxeter group $W({\rm
E}_{6})$ of type ${\rm E}_{6}$, of elements invariant under the automorphism
of $W({\rm E}_{6})$ determined by the diagram automorphism $\sigma=(1,6)(3,5)$
indicated as a permutation on the generators of $W({\rm E}_{6})$ whose Dynkin
diagram are labeled and presented in Figure 1.
Figure 1: Dynkin diagrams of ${\rm E}_{6}$ and ${\rm F}_{4}$
The action $\sigma$ can be extended to an automorphism of the Brauer algebra
of type ${\rm E}_{6}$ by acting on the Temperley-Lieb generators $E_{i}$
([17]) by the same permutation as for Weyl group generators. We denote by
${\rm SBr}({\rm E}_{6})$ the subalgebra generated by $\sigma$-invariant
elements in ${\rm BrM}({\rm E}_{6})$. The main theorem of this paper is the
following. In order to avoid confusion with the above generators, the
generators of ${\rm Br}({\rm E}_{6})$ have been capitalized.
###### Theorem 1.2.
There is an algebra isomorphism
$\phi:\,{\rm Br}({\rm F}_{4})\longrightarrow{\rm SBr}({\rm E}_{6})$
determined by $\phi(r_{1})=R_{1}R_{6}$, $\phi(r_{2})=R_{3}R_{5}$,
$\phi(r_{3})=R_{4}$, $\phi(r_{4})=R_{2}$, and $\phi(e_{1})=E_{1}E_{6}$,
$\phi(e_{2})=E_{3}E_{5}$, $\phi(e_{3})=E_{4}$, $\phi(e_{4})=E_{2}$.
Furthermore, the algebra ${\rm Br}({\rm F}_{4})$ is free over
$\mathbb{Z}[\delta^{\pm 1}]$ of rank $14985$.
The proof of the theorem is finished in Section 5. Moreover, in the last
section we will prove that the algebra ${\rm Br}({\rm F}_{4})\otimes R$ for a
field $R$ is cellularly stratified. This paper is included as chapter $4$ in
the author’s PhD thesis ([15]).
## 2 Basic properties of ${\rm Br}({\rm F}_{4})$
By the properties of ${\rm Br}({\rm B}_{3})$ in [7] or ${\rm Br}({\rm C}_{3})$
in [6], we have more relations between $\\{r_{2},\,r_{3},\,e_{2},\,e_{3}\\}$
Such as those of [6, Lemma 4.1]. Just as [6, Remark 3.5] for type ${\rm C}$,
there is an anti-involution on ${\rm Br}({\rm F}_{4})$.
###### Proposition 2.1.
There is a unique anti-involution on ${\rm Br}({\rm F}_{4})$ that fixes the
generators $r_{i}$, $e_{i}$ $(1\leq i\leq 4)$.
###### Definition 2.2.
Let $Q$ be a graph. The Brauer monoid ${\rm BrM}(Q)$ is the monoid generated
by the symbols $R_{i}$ and $E_{i}$, for each node $i$ of $Q$ and $\delta$,
$\delta^{-1}$ subject to the relation (1.1)–(1.3) and (1.5)–(1.10). The Brauer
algebra ${\rm Br}(Q)$ is the free $\mathbb{Z}[\delta^{\pm 1}]$-algebra for
Brauer monoid ${\rm BrM}(Q)$.
###### Proposition 2.3.
The map $\phi$ determined on generators in Theorem 1.2 induces an algebra
homomorphism from ${\rm Br}({\rm F}_{4})$ to ${\rm Br}({\rm E}_{6})$.
###### Proof.
It suffices to prove that the relations in Definition 1.1 still holds when the
generators are replaced by their images under $\phi$. The difficult ones are
(1.11)–(1.18); which have been proved in the meanwhile of the homomorphism
from ${\rm Br}({\rm C}_{2})$ to ${\rm Br}({\rm A}_{3})$ in [6]. ∎
To distinguish them from the generators of ${\rm Br}({\rm F}_{4})$, we denote
the generators of ${\rm Br}({\rm C}_{3})$ ($\mathrel{\mathop{\kern
0.0pt\circ}\limits_{\hbox to0.0pt{\hss$\scriptstyle 2$\hss}}^{\hbox
to0.0pt{\hss$$\hss}}}\kern-1.0pt\vrule
height=2.4pt,depth=-2.0pt,width=36.135pt\kern-1.0pt\mathrel{\mathop{\kern
0.0pt\circ}\limits_{\hbox to0.0pt{\hss$\scriptstyle 1$\hss}}^{\hbox
to0.0pt{\hss$$\hss}}}\kern-4.0pt{{\vrule
height=2.0pt,depth=-1.6pt,width=36.135pt}\hbox to0.0pt{\hss\vrule
height=4.0pt,depth=-3.6pt,width=36.135pt}\kern-25.0pt<}\kern
8.0pt\mathrel{\mathop{\kern 0.0pt\circ}\limits_{\hbox
to0.0pt{\hss$\scriptstyle 0$\hss}}^{\hbox to0.0pt{\hss$$\hss}}}\kern-1.0pt$)
as in [6] by $\\{r^{\prime}_{i},e^{\prime}_{i}\\}_{i=0}^{2}$ and the
generators of ${\rm Br}({\rm B}_{3})$ ($\mathrel{\mathop{\kern
0.0pt\circ}\limits_{\hbox to0.0pt{\hss$\scriptstyle 2$\hss}}^{\hbox
to0.0pt{\hss$$\hss}}}\kern-1.0pt\vrule
height=2.4pt,depth=-2.0pt,width=36.135pt\kern-1.0pt\mathrel{\mathop{\kern
0.0pt\circ}\limits_{\hbox to0.0pt{\hss$\scriptstyle 1$\hss}}^{\hbox
to0.0pt{\hss$$\hss}}}\kern-4.0pt{{\vrule
height=2.0pt,depth=-1.6pt,width=36.135pt}\hbox to0.0pt{\hss\vrule
height=4.0pt,depth=-3.6pt,width=36.135pt}\kern-25.0pt>}\kern
8.0pt\mathrel{\mathop{\kern 0.0pt\circ}\limits_{\hbox
to0.0pt{\hss$\scriptstyle 0$\hss}}^{\hbox to0.0pt{\hss$$\hss}}}\kern-1.0pt$)
in [7] by $\\{r^{\prime\prime}_{i},e^{\prime\prime}_{i}\\}_{i=0}^{2}$. By
checking their defining relations, we have the following proposition.
###### Proposition 2.4.
There are injective algebra homomorphisms
$\phi_{1}:\quad{\rm Br}({\rm C}_{3})\rightarrow{\rm Br}({\rm F}_{4})$
$\phi_{2}:\quad{\rm Br}({\rm B}_{3})\rightarrow{\rm Br}({\rm F}_{4}),$
defined on generators as follows.
$\displaystyle\phi_{1}(r^{\prime}_{i})$ $\displaystyle=$ $\displaystyle
r_{3-i},\,\phi_{1}(e^{\prime}_{i})=e_{3-i},\,\quad\mbox{for}\quad\,0\leq i\leq
2,$ $\displaystyle\phi_{2}(r^{\prime\prime}_{i})$ $\displaystyle=$
$\displaystyle
r_{2+i},\,\phi_{2}(e^{\prime\prime}_{i})=e_{2+i},\quad`\mbox{for}\quad\,0\leq
i\leq 2.$
###### Proof.
Just checking the defining relations of two algebras, we find that $\phi_{1}$
and $\phi_{2}$ are algebra morphisms. We see that $\phi\phi_{1}({\rm Br}({\rm
C}_{3}))$ is contained in the subalgebra of ${\rm Br}({\rm E}_{6})$ generated
by $\\{R_{1},\,E_{1}\\}\cup\\{R_{i},\,E_{i}\\}_{i=3}^{6}$, which is isomorphic
to ${\rm Br}({\rm A}_{5})$ according to [8]. By the main theorem of [6] and
the behavior of $\phi\phi_{1}$ on generators, the homomorphism $\phi\phi_{1}$
coincides with the embedding of ${\rm Br}({\rm C}_{3})$ in ${\rm Br}({\rm
A}_{5})$. Similarly, with [8] and the main theorem of [7], the homomorphism
$\phi\phi_{2}$ coincides with the embedding of ${\rm Br}({\rm B}_{3})$ in
${\rm Br}({\rm D}_{4})$; by application of these two theorems, it follows that
the homomorphisms $\phi_{1}$ and $\phi_{2}$ are injective. ∎
## 3 Admissible root sets
In this section, we give the definition and description for admissible sets of
type ${\rm F}_{4}$ and study some of their basic properties.
Let $\\{\beta_{i}\\}_{i=1}^{4}$ be simple roots of $W({\rm F}_{4})$. They can
be realized in $\mathbb{R}^{4}$ as
$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{2},\,\beta_{2}=\epsilon_{2},$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle\epsilon_{3}-\epsilon_{2},\,\beta_{4}=\epsilon_{4}-\epsilon_{3},$
with $\\{\epsilon_{i}\\}_{i=1}^{4}$ being the standard orthonormal basis of
$\mathbb{R}^{4}$. The set
$\Psi^{+}=\\{\frac{\epsilon_{1}\pm\epsilon_{2}\pm\epsilon_{3}\pm\epsilon_{4}}{2}\\}\cup\\{\epsilon_{i}\\}_{i=1}^{4}\cup\\{\epsilon_{j}\pm\epsilon_{i}\\}_{1\leq
i<j\leq 4}$
of cardinality $24$ is the set of positive roots of a root system $\Psi$ of
$W({\rm F}_{4})$ having $\beta_{1}$, $\ldots$, $\beta_{4}$ as simple roots. We
call a vector $\beta\in\Psi^{+}$ a short root if its Euclidean length is $1$,
a long root if its Euclidean length is $\sqrt{2}$.
Let $\\{\alpha_{i}\\}_{i=1}^{6}$ be simple roots of $W({\rm E}_{6})$. The
$\\{\alpha_{i}\\}_{i=1}^{6}$ span a linear space over $\mathbb{R}$ of
dimension $6$. We define a linear map
$\mathfrak{p}:\mathbb{R}^{6}\rightarrow\mathbb{R}^{4}$ by specifying its
images on the given basis :
$\mathfrak{p}(\alpha_{1})=\beta_{1}$, $\mathfrak{p}(\alpha_{6})=\beta_{1}$,
$\mathfrak{p}(\alpha_{3})=\beta_{2}$, $\mathfrak{p}(\alpha_{5})=\beta_{2}$,
$\mathfrak{p}(\alpha_{4})=\beta_{3}$, $\mathfrak{p}(\alpha_{2})=\beta_{4}$.
Now $\mathfrak{p}(\mathbb{R}^{6})$ is the $\sigma$-invariant space of
$\mathbb{R}^{6}$, where $\sigma$ is the linear transformation of
$\mathbb{R}^{6}$ determined by:
$\sigma(\alpha_{1})=\alpha_{6}$, $\sigma(\alpha_{6})=\alpha_{1}$,
$\sigma(\alpha_{3})=\alpha_{5}$, $\sigma(\alpha_{5})=\alpha_{3}$,
$\sigma(\alpha_{4})=\alpha_{4}$, $\sigma(\alpha_{2})=\alpha_{2}$.
Let $\Phi\subset\mathbb{R}^{6}$ be the root system of ${\rm E}_{6}$ with
simple roots $\\{\alpha_{i}\\}_{i=1}^{6}$, and $\Phi^{+}$ the positive roots
of $\Phi$. In [4], the admissible root sets for simply-laced type are given.
Now we will define the admissible for type ${\rm F}_{4}$.
###### Definition 3.1.
Let $X\subset\Psi^{+}$ be a set of mutually orthogonal roots. The set $X$ is
called admissible if $\mathfrak{p}^{-1}(X)\cap\Phi^{+}$ is an admissible set.
###### Proposition 3.2.
There is a one-to-one correspondence between $\sigma$-invariant admissible
root sets of type ${\rm E}_{6}$ and the admissible root sets of type ${\rm
F}_{4}$. Collections of all admissible sets can be partitioned into six
$W({\rm F}_{4})$-orbits given by the following representatives.
1. (I)
$\emptyset$,
2. (II)
$\\{\epsilon_{4}-\epsilon_{3}\\}$,
3. (III)
$\\{\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{2}\\}$,
4. (IV)
$\\{\epsilon_{3}-\epsilon_{2},\epsilon_{3}+\epsilon_{2}\\}$,
5. (V)
$\\{\frac{\epsilon_{1}-\epsilon_{2}-\epsilon_{3}-\epsilon_{4}}{2},\epsilon_{3}-\epsilon_{2},\epsilon_{4}+\epsilon_{1}\\}$,
6. (VI)
$\\{\epsilon_{3}-\epsilon_{2},\epsilon_{3}+\epsilon_{2},\epsilon_{4}+\epsilon_{1},\epsilon_{4}-\epsilon_{1}\\}$.
Furthermore, their cardinalities are respectively, $1$, $12$, $12$, $18$,
$36$, $3$.
###### Proof.
In the admissible root sets of type ${\rm E}_{6}$, there are four $W({\rm
E}_{6})$-orbits, which are the orbits of $\emptyset$, $\\{\alpha_{2}\\}$,
$\\{\alpha_{3},\,\alpha_{5}\\}$,
$\\{\alpha_{2},\,\alpha_{3},\,\alpha_{5},\,2\alpha_{4}+\alpha_{2}+\alpha_{3}+\alpha_{5}\\}$
of respective sizes. The $W({\rm E}_{6})$-orbits are easily seen to possess
$1$ ((I)), $12$ ((II)), $30$ ((III), (IV)), $39$ ((V), (VI))
$\sigma$-invariant admissible root sets which can be decomposed into $W({\rm
F}_{4})$-orbits with representatives (I), (II), (III), (IV),(V), (VI) as
listed in the proposition. ∎
As in [4], [6], and [7], we have the following lemmas.
###### Lemma 3.3.
Let $i$ and $j$ be nodes of the Dynkin diagram ${\rm F}_{4}$. If $w\in W({\rm
F}_{4})$ satisfies $w\beta_{i}=\beta_{j}$, then $we_{i}w^{-1}=e_{j}$.
Consider a positive root $\beta$ and a node $i$ of type ${\rm F}_{4}$.
If there exists $w\in W$ such that $w\beta_{i}=\beta$, then due to Lemma 3.3
we can define the element $e_{\beta}$ in ${\rm BrM}({\rm F}_{4})$ by
$e_{\beta}=we_{i}w^{-1}.$
In general,
$we_{\beta}w^{-1}=e_{w\beta},$
for $w\in W({\rm F}_{4})$ and $\beta$ a root of $W({\rm F}_{4})$, and here we
just consider the natural action of $W({\rm F}_{4})$ on $\Psi^{+}$ by negating
negative roots. Analogous to the argument in [7, Lemma 4.5], the following
lemma can be obtained by checking case by case listed in Proposition 3.2.
###### Lemma 3.4.
Let $\gamma_{1}$, $\gamma_{2}\in\Psi^{+}$ and $\gamma_{1}$ orthogonal to
$\gamma_{2}$ such that $\\{\gamma_{1},\,\gamma_{2}\\}$ can be a subset of some
admissible root set, then
$e_{\gamma_{1}}e_{\gamma_{2}}=e_{\gamma_{2}}e_{\gamma_{1}}.$
If $X\subset\Psi^{+}$ is a subset of some admissible root set, then by the
lemma we can define
$\displaystyle e_{X}=\Pi_{\beta\in X}e_{\beta}.$ (3.1)
###### Definition 3.5.
Suppose that $X\subset\Psi^{+}$ is a mutually orthogonal root set. If $X$ can
be contained in some admissible root set, Then the minimal admissible set
containing $X$ is called the $admissible$ $closure$ of $X$, denoted by $X^{\rm
cl}$.
Thanks to an argument similar to [7, Lemma 4.7], the following lemma holds.
###### Lemma 3.6.
Let $X\subset\Psi^{+}$ be a mutually orthogonal root set. If $X^{\rm cl}$
exists, then
$e_{X^{\rm cl}}=\delta^{\\#(X^{\rm cl}\setminus X)}e_{X}.$
## 4 An upper bound for the rank
We introduce notation for the following admissible sets of type ${\rm F}_{4}$
corresponding to those listed in Proposition 3.2.
$\displaystyle X_{0}$ $\displaystyle=$ $\displaystyle\emptyset$ $\displaystyle
X_{1}$ $\displaystyle=$ $\displaystyle\\{\beta_{4}\\}$ $\displaystyle X_{2}$
$\displaystyle=$ $\displaystyle\\{\beta_{1}\\}$ $\displaystyle X_{3}$
$\displaystyle=$ $\displaystyle\\{\beta_{3},\beta_{3}+2\beta_{2}\\}$
$\displaystyle X_{4}$ $\displaystyle=$
$\displaystyle\\{\beta_{1},\beta_{3}\\}^{\rm cl}$ $\displaystyle X_{5}$
$\displaystyle=$
$\displaystyle\\{\beta_{3},\beta_{3}+2\beta_{2},\beta_{3}+2\beta_{2}+2\beta_{1}\\}^{\rm
cl}$
Let
1. (I)
$N_{0}=W({\rm F}_{4})$, $A_{0}=\\{1\\}$, $C_{0}=N_{0}$,
2. (II)
$N_{1}=\left<r_{1},\,r_{2},\,r_{\epsilon_{4}+\epsilon_{3}},\,r_{4}\right>$
$C_{1}=\left<r_{1},\,r_{2},\,r_{\epsilon_{4}+\epsilon_{3}},\right>$,
$A_{1}=\left<r_{4}\right>$,
3. (III)
$N_{2}=\left<r_{1},\,r_{3},\,r_{4},r_{(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{4})/2}\right>$,
$C_{2}=\left<r_{3},\,r_{4}\right>$,
$A_{2}=\left<r_{1},r_{(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{4})/2},\,r_{(\epsilon_{1}+\epsilon_{2}-\epsilon_{3}+\epsilon_{4})/2},r_{(\epsilon_{1}-\epsilon_{2}+\epsilon_{3}+\epsilon_{4})/2}\right>$,
4. (IV)
$N_{3}=\left<r_{2},\,r_{3},\,r_{\epsilon_{4}},\,r_{\epsilon_{4}-\epsilon_{1}}\right>$,
$C_{3}=\left<r_{\epsilon_{4}-\epsilon_{1}}\right>$,
$A_{3}=\left<r_{2},\,r_{3},\,r_{\epsilon_{4}},\,r_{\epsilon_{1}}\right>$,
5. (V)
$N_{4}=\left<r_{3},\,r_{1},\,r_{(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{4})/2},\,r_{(\epsilon_{1}-\epsilon_{2}+\epsilon_{3}+\epsilon_{4})/2}\right>$,
$C_{4}=\\{1\\}$, $A_{4}=N_{4}$,
6. (VI)
$N_{5}=\left<r_{1},r_{2},r_{3},r_{\epsilon_{4}},r_{\epsilon_{1}}\right>$,
$C_{5}=\\{1\\}$, $A_{5}=N_{5}$.
The structure of these groups can be determined below.
###### Lemma 4.1.
For $N_{i}$, $A_{i}$, $C_{i}$, $i=1$, $\ldots$, $5$, the following holds,
1. (I)
$N_{0}\cong W({\rm F}_{4})$, $A_{0}\cong\\{1\\}$, $C_{0}\cong N_{0}$,
2. (II)
$N_{1}\cong W({\rm B}_{3})\times W({\rm A}_{1})$, $C_{1}\cong W({\rm B}_{3})$,
$A_{2}\cong W({\rm A}_{1})$,
3. (III)
$N_{2}\cong W({\rm B}_{3})\times W({\rm A}_{1})$, $C_{2}\cong W({\rm A}_{2})$,
$A_{2}\cong W({\rm A}_{1})^{4}$,
4. (IV)
$N_{3}\cong W({\rm B}_{2})^{2}$, $C_{3}\cong W({\rm A}_{1})$, $A_{3}\cong
W({\rm B}_{2})\times W({\rm A}_{1})^{2}$,
5. (V)
$A_{4}=N_{4}\cong W({\rm B}_{2})\times W({\rm A}_{1})^{2}$, $C_{4}=\\{1\\}$,
6. (VI)
$A_{5}=N_{5}\cong W({\rm B}_{3})\times W({\rm B}_{2})$, $C_{5}=\\{1\\}$.
###### Proof.
We do not give the full proof but restrict to the case $i=1$. It can be
checked that $\left<\epsilon_{3}+\epsilon_{4},\beta_{1}\right>=-1$ and
$\left<\epsilon_{3}+\epsilon_{4},\beta_{2}\right>=0$, and
$\\{\epsilon_{3}+\epsilon_{4},\beta_{1},\beta_{2}\\}$ are linearly
independent, hence $C_{1}\cong W({\rm B}_{3})$. Since each element of
$\\{\epsilon_{3}+\epsilon_{4},\beta_{1},\beta_{2}\\}$ is orthogonal to
$\beta_{4}$, so we get that $N_{1}\cong W({\rm B}_{3})\times W({\rm A}_{1})$.
∎
When we consider these groups in ${\rm BrM}({\rm F}_{4})$, the following lemma
can be obtained.
###### Lemma 4.2.
For $i=0$, $\ldots$, $5$, the following holds.
1. (I)
The group $N_{i}$ is the normalizer of $X_{i}$ in $W({\rm F}_{4})$.
2. (II)
The group $N_{i}$ is the semidirect product of $A_{i}$ and $C_{i}$, with
$C_{i}$ normalized by $A_{i}$.
3. (III)
For $x\in A_{i}$, we have $xe_{X_{i}}=e_{X_{i}}$.
4. (IV)
For $x\in C_{i}$, we have $xe_{X_{i}}=e_{X_{i}}x$.
###### Proof.
Clearly, $N_{i}$ normalizes $X_{i}$, so $N_{i}\leq N(X_{i})$ (the normalizer
of $X_{i}$ in $W({\rm F}_{4})$), and the equality follows from Lagrange’s
Theorem by verification in the table below. Here $\\#N_{i}$ is known from
Lemma 4.1, and the lengths of $W({\rm F}_{4})$-orbits are given in Proposition
3.2. Therefore the first claim hold. The proof of the left conclusions is as
arguments in [6, Section 6] and [7, Section 6]. ∎
Suppose $D_{i}$ is a set of left coset representatives of $N_{i}$ in $W({\rm
F}_{4})$. We have the table below. In the table the product of the three
entries in each row is equal to $1152$.
$i$ | $\\#D_{i}$ | $\\#C_{i}$ | $\\#A_{i}$
---|---|---|---
0 | 1 | 1152 | 1
1 | 12 | 48 | 2
2 | 12 | 6 | 16
3 | 18 | 2 | 32
4 | 36 | 1 | 32
5 | 3 | 1 | 384
We find that for the Brauer monoid action of some monomial $\phi(a)$ for
$a\in{\rm BrM}({\rm F}_{4})$, the admissible root sets
$\mathfrak{p}(\phi(a)\emptyset)$ and $\mathfrak{p}(\phi(a)^{\rm op}\emptyset)$
belong to different $W({\rm F}_{4})$-orbits; for example
$\mathfrak{p}(\phi(e_{2}e_{3})\emptyset)=\\{\beta_{2}\\}$, and
$\mathfrak{p}(\phi(e_{2}e_{3})^{\rm
op}\emptyset)=\\{\beta_{3},\beta_{2}+\beta_{3}\\}$. Some more groups as
follows are needed. Let
$\displaystyle N_{6}^{L}$ $\displaystyle=$
$\displaystyle\left<r_{2},\,r_{3},\,r_{\epsilon_{4}},\,r_{\epsilon_{4}-\epsilon_{1}}\right>,$
(4.1) $\displaystyle N_{6}^{R}$ $\displaystyle=$
$\displaystyle\left<r_{2},\,r_{\epsilon_{3}},\,r_{\epsilon_{4}},\,r_{\epsilon_{4}-\epsilon_{1}}\right>,$
(4.2) $\displaystyle C_{6}$ $\displaystyle=$
$\displaystyle\left<r_{\epsilon_{4}-\epsilon_{1}}\right>,$ (4.3)
$\displaystyle N_{8}$ $\displaystyle=$
$\displaystyle\left<r_{2},\,r_{4},\,r_{\epsilon_{1}},r_{\epsilon_{3}}\right>.$
(4.4)
Additionally, we choose $D_{6}^{L}$, $D_{6}^{R}$, and $D_{8}$ to be sets of
left coset representatives of $N_{6}^{L}$, $N_{6}^{R}$, and $N_{8}$ in $W({\rm
F}_{4})$, respectively.
Let $N_{7}^{L}=N_{6}^{R}$, $N_{7}^{R}=N_{6}^{L}$, $D_{7}^{L}=D_{6}^{R}$,
$D_{7}^{R}=D_{6}^{L}$, $C_{7}=C_{6}$,
$N_{9}^{L}=N_{5}$, $N_{9}^{R}=N_{4}$, $D_{9}^{L}=D_{5}$, $D_{9}^{R}=D_{4}$,
$N_{10}^{L}=N_{4}$, $N_{10}^{R}=N_{5}$, $D_{10}^{L}=D_{4}$,
$D_{10}^{R}=D_{5}$.
In view of [6] and [7], the following lemma holds.
###### Lemma 4.3.
For above groups, we have
1. (I)
$N_{6}^{L}\cong W({\rm B}_{2})\times W({\rm B}_{2})$, $N_{6}^{R}\cong W({\rm
B}_{2})\times W({\rm A}_{1})^{2}$,
$C_{6}\cong W({\rm A}_{1})$, $N_{8}\cong W({\rm B}_{2})\times W({\rm
A}_{1})^{2}$.
2. (II)
For each $a\in N_{6}^{L}$ $(b\in N_{6}^{R})$, there exists some $c\in C_{6}$
such that
$ae_{3}e_{2}=e_{3}e_{2}c$ $(e_{3}e_{2}b=ce_{3}e_{2})$.
3. (III)
For each $a\in N_{8}$ we have that
$ae_{4}r_{3}e_{2}e_{3}e_{4}=e_{4}r_{3}e_{2}e_{3}e_{4}$.
4. (IV)
For each $a\in N_{9}^{L}$ and $b\in N_{9}^{R}$, we have
$ae_{3}e_{2}e_{1}e_{3}=e_{3}e_{2}e_{1}e_{3}$ and
$e_{3}e_{2}e_{1}e_{3}b=e_{3}e_{2}e_{1}e_{3}$.
###### Theorem 4.4.
Up to some power of $\delta$, each monomial in ${\rm BrM}({\rm F}_{4})$ can be
written in one of the following forms.
1. (I)
$ue_{X_{i}}vw$, $u\in D_{i}$, $w\in D_{i}^{-1}$, $v\in C_{i}$, $0\leq i\leq
5$.
2. (II)
$ue_{3}e_{2}vw$, $u\in D_{6}^{L}$, $w\in(D_{6}^{R})^{-1}$, $v\in C_{6}$.
3. (III)
$ue_{2}e_{3}vw$, $u\in D_{7}^{L}$, $w\in(D_{7}^{R})^{-1}$, $v\in C_{7}$.
4. (IV)
$ue_{4}r_{3}e_{2}e_{3}e_{4}w$, $u\in D_{8}$, $w\in D_{8}^{-1}$.
5. (V)
$ue_{3}e_{2}e_{1}e_{3}w$, $u\in D_{9}^{L}$, $w\in(D_{9}^{R})^{-1}$.
6. (VI)
$ue_{3}e_{1}e_{2}e_{3}w$, $u\in D_{10}^{L}$, $w\in(D_{10}^{R})^{-1}$.
###### Proof.
From [7], $(e_{4}r_{3}e_{2}e_{3}e_{4})^{\rm op}=e_{4}r_{3}e_{2}e_{3}e_{4}$. In
view of Proposition 2.1, it suffices to prove that the claim that the result
of a left multiplication by each $r_{i}$ and $e_{\beta}$ for
$\beta\in\Psi^{+}$ at the left of each element of $S$ can be written as in
(I)-(VI), where
$S=\\{e_{X_{i}}\\}_{i=0}^{5}\cup\\{e_{3}e_{2},\,e_{2}e_{3},e_{4}r_{3}e_{2}e_{3}e_{4},\,e_{3}e_{2}e_{1}e_{3},\,e_{3}e_{1}e_{2}e_{3}\\}.$
According to 4.2 and Lemma 4.3, the above holds for $\\{r_{i}\\}_{i=1}^{4}$.
By Proposition 2.4, and special cases ${\rm Br}({\rm C}_{3})$ in [6] and ${\rm
Br}({\rm B}_{3})$ in [7], we have that
$\phi_{1}({\rm Br}({\rm C}_{3}))=\bigoplus_{s\in S_{1}}\mathbb{Z}[\delta^{\pm
1}]W({\rm C}_{3})sW({\rm C}_{3}),$ $\phi_{2}({\rm Br}({\rm
B}_{3}))=\bigoplus_{s\in S_{2}}\mathbb{Z}[\delta^{\pm 1}]W({\rm B}_{3})sW({\rm
B}_{3}),$
Where
$S_{1}=\\{1,e_{3},e_{X_{2}},e_{3}e_{2},e_{2}e_{3},e_{X_{3}},e_{X_{4}},e_{3}e_{2}e_{1}e_{3},e_{1}e_{3}e_{2}e_{3},e_{X_{5}}\\}$
$S_{2}=\\{1,e_{X_{1}},e_{2},e_{3}e_{2},e_{2}e_{3},e_{X_{3}},e_{4}e_{4}^{*},e_{4}e_{2},e_{4}r_{3}e_{2}e_{3}e_{4}\\}$
with $e_{4}^{*}=r_{3}r_{2}r_{3}e_{4}r_{3}r_{2}r_{3}$. It can be seen that each
element of $S_{1}$ and $S_{2}$ is in $S$ or conjugate to some element of $S$
under $W({\rm F}_{4})$. Therefore the proof is reduced to cases of ${\rm
Br}({\rm B}_{3})$ and ${\rm Br}({\rm C}_{3})$, which can be found in [6] and
[7]. ∎
Aa a consequence, we obtain some information about rank of ${\rm Br}({\rm
F}_{4})$ over $\mathbb{Z}[\delta^{\pm 1}]$.
###### Corollary 4.5.
As a $\mathbb{Z}[\delta^{\pm 1}]$-algebra, the algebra ${\rm Br}({\rm F}_{4})$
is spanned by $14985$ elements.
###### Proof.
For $i=6$, $\ldots$, $10$, the following holds.
set | cardinaltity | set | cardinality
---|---|---|---
$D_{6}^{L}$, $D_{7}^{R}$ | 18 | $D_{6}^{R}$, $D_{7}^{L}$ | 36
$C_{6}$, $C_{7}$ | 2 | $D_{8}$ | 36
$D_{9}^{L}$, $D_{10}^{R}$ | 3 | $D_{9}^{R}$, $D_{10}^{L}$ | 36
By Theorem 4.4 and numerical information from the above two tables, the
algebra ${\rm Br}({\rm F}_{4})$ over $\mathbb{Z}[\delta^{\pm 1}]$ has rank at
most
$\sum_{i=0}^{5}(\\#D_{i})^{2}\\#C_{i}+2\\#D_{6}^{L}\\#D_{6}^{R}\\#C_{6}+\\#D_{8}^{2}+2\\#D_{9}^{L}\\#D_{9}^{R}=14985.$
∎
## 5 $\phi({\rm Br}({\rm F}_{4}))$ in ${\rm Br}({\rm E}_{6})$
We keep notation as in [8, Section 2] and first introduce some basic concepts.
Let $M$ be the diagram of a connected finite simply laced Coxeter group (type
${\rm A}$, ${\rm D}$, ${\rm E}_{6}$, ${\rm E}_{7}$, ${\rm E}_{8}$). ${\rm
BrM}(M)$ is the associated Brauer monoid as Definition 2.2. An element
$a\in{\rm BrM}(M)$ is said to be of _height_ $t$ if the minimal number of
$R_{i}$ occurring in an expression of $a$ is $t$, denoted by $\rm{ht}$$(a)$.
By $B_{Y}$ we denote the admissible closure ([4]) of $\\{\alpha_{i}|i\in
Y\\}$, where $Y$ is a coclique of $M$. The set $B_{Y}$ is a minimal element in
the $W(M)$-orbit of $B_{Y}$ which is endowed with a poset structure induced by
the partial ordering $<$ defined on $W({\rm E}_{6})$-orbits in $\mathcal{A}$
(the set of all admissible sets) in [5]. If $d$ is the Hasse diagram distance
for $W(M)B_{Y}$ from $B_{Y}$ to the unique maximal element ([5, Corollary
3.6]), then for $B\in W(M)B_{Y}$ the height of $B$, already used in Definition
notation $\rm{ht}$$(B)$, is $d-l$, where $l$ is the distance in the Hasse
diagram from $B$ to the maximal element.
In [4], a Brauer monoid action is defined as follows. For any mutually
orthogonal positive root set $B$, we define $B^{\rm cl}$ to be the _admissible
closure_ of $B$, already used in Lemma 3.4 and Definition 3.5 for a more
difficult type, namely the minimal admissible root set([4]) containing $B$.
The generator $R_{i}$ acts by the natural action of Coxeter group elements on
its root sets, where negative roots are negated so as to obtain positive
roots, the element $\delta$ acts as the identity, and the action of
$\\{E_{i}\\}_{i=1}^{n+1}$ is defined below.
$E_{i}B:=\begin{cases}B&\text{if}\ \alpha_{i}\in B,\\\
(B\cup\\{\alpha_{i}\\})^{\rm cl}&\text{if}\ \alpha_{i}\perp B,\\\
R_{\beta}R_{i}B&\text{if}\ \beta\in B\setminus\alpha_{i}^{\perp}.\end{cases}$
(5.1)
By the natural involution, we can define a right monoid action of ${\rm
BrM}(M)$ on $\mathcal{A}$.
Considering our $\sigma$ and table 3 in [8], let
$Y\in\mathcal{Y}=\\{\emptyset,\\{2\\},\\{1,6\\},\\{2,3,5\\}\\}.$
Obviously, each element of $\mathcal{Y}$ is $\sigma$-invariant. From [8,
Theorem 2.7], it is known that each monomial $a$ in ${\rm BrM}({\rm E}_{6})$
can be uniquely written as $\delta^{i}a_{B}\hat{e}_{Y}ha_{B^{\prime}}^{\rm
op}$ for some $i\in\mathbb{Z}$ and $h\in W(M_{Y})$ in [8, table 3], where
$B=a\emptyset$, $B^{{}^{\prime}}=\emptyset a$, $a_{B}\in{\rm BrM}({\rm
E}_{6})$, $a_{B^{\prime}}^{\rm op}\in{\rm BrM}({\rm E}_{6})$ and
1. (i)
$a\emptyset=a_{B}\emptyset=a_{B}B_{Y}$, $\emptyset a=\emptyset
a_{B^{\prime}}^{\rm op}=B_{Y}a_{B^{\prime}}^{\rm op}$,
2. (ii)
$\rm{ht}$$(B)=$ht$(a_{B})$, $\rm{ht}$$(B^{\prime})=$ht$(a_{B^{\prime}}^{\rm
op})$.
Now we can apply this to obtain the following corollary immediately.
###### Corollary 5.1.
Let $B$, $Y$, $B^{\prime}$ be as above. Then
$a=a_{B}\hat{e}_{Y}ha_{B^{\prime}}^{\rm op}$ is $\sigma$-invariant monomial in
${\rm BrM}({\rm E}_{6})$ if and only if
$\rm(i)$ the sets $B$, $B^{{}^{\prime}}$ are $\sigma$-invariant,
$\rm(ii)$ the element $h\in W(M_{Y})$ is $\sigma$-invariant.
In type ${\rm E}_{6}$, we see that
$E_{2}E_{4}E_{5}\\{\alpha_{6}\\}=\alpha_{2}$,
$E_{1}E_{3}\\{\alpha_{4},\alpha_{6}\\}=\\{\alpha_{1},\alpha_{6}\\}.$ Then
$W(M_{\\{2\\}})=E_{2}E_{4}E_{5}W(M_{\\{6\\}})E_{5}E_{4}E_{2},$
$W(M_{\\{1,6\\}})=E_{1}E_{3}W(M_{\\{4,6\\}})E_{3}E_{1}.$
Let $\hat{E}_{i}=\delta^{-1}E_{i}$. By the 6th column list of [8, table 3],
the group $W(M_{\\{2\\}})\cong W({\rm A}_{5})$ has generators
$R_{1}\hat{E}_{2}$, $R_{3}\hat{E}_{2}$, $R_{5}\hat{E}_{2}$,
$R_{6}\hat{E}_{2}$, $E_{2}E_{4}R_{3}E_{5}E_{4}\hat{E}_{2}$ and their relation
is corresponding to subdiagram of ${\rm E}_{6}$ by deleting the node $2$ and
the edge between $2$ and $4$ with $E_{2}E_{4}R_{3}E_{5}E_{4}\hat{E}_{2}$
corresponding to node $4$, $R_{i}\hat{E}_{2}$ corresponding to node $i$ for
$i=1$,$3$, $5$, $6$; the group $W(M_{\\{1,6\\}})\cong W({\rm A}_{2})$ has
generators $R_{2}\hat{E}_{1}\hat{E}_{6}$, $R_{4}\hat{E}_{1}\hat{E}_{6}$ and
whose relation is the subdiagram of ${\rm E}_{6}$ of nodes $2$, $4$ and the
edge between them. When $\sigma$ acts on the generators of $W(M_{\\{2\\}})$
and $W(M_{\\{1,6\\}})$, we have that
$\displaystyle\sigma(E_{2}E_{4}R_{3}E_{5}E_{4}\hat{E}_{2})$ $\displaystyle=$
$\displaystyle E_{2}E_{4}R_{3}E_{5}E_{4}\hat{E}_{2},$
$\displaystyle\sigma(R_{i}\hat{E}_{2})$ $\displaystyle=$ $\displaystyle
R_{\sigma(i)}\hat{E}_{2},\,\,(i=1,3,5,6,)$
$\displaystyle\sigma(R_{j}\hat{E}_{1}\hat{E}_{6})$ $\displaystyle=$
$\displaystyle R_{j}\hat{E}_{1}\hat{E}_{6},\,\,(j=2,4,)$
which implies that the $\sigma$-action on those two groups is determined the
$\sigma$-action on the subdigrams of ${\rm E}_{6}$ described in the above.
Hence $W(M_{\\{2\\}})^{\sigma}\cong W({\rm B}_{3})$, and
$W(M_{\\{1,6\\}})^{\sigma}=W(M_{\\{1,6\\}})\cong W({\rm A}_{2})$. This
conclusion can be summarized in the table below with the second column from
our GAP [9] code.
$Y$ | $\\#((W({\rm E}_{6})B_{Y})^{\sigma})$ | $M_{Y}$ | $M_{Y}^{\sigma}$
---|---|---|---
$\emptyset$ | $1$ | ${\rm E}_{6}$ | ${\rm F}_{4}$
$2$ | $12$ | ${\rm A}_{5}$ | ${\rm B}_{3}$
$1$, $6$ | $30$ | ${\rm A}_{2}$ | ${\rm A}_{2}$
$2$, $3$, $5$ | 39 | $\emptyset$ | $\emptyset$
###### Corollary 5.2.
The algebra ${\rm Br}({\rm E}_{6})^{\sigma}$ is free over
$\mathbb{Z}[\delta^{\pm 1}]$ with rank
$1152+12^{2}\times 48+30^{2}\times 6+39^{2}=14985.$
###### Proof.
By Corollary 5.1, we have
$\displaystyle\rm{rk}({\rm Br}({\rm
E}_{6})^{\sigma})=\Sigma_{Y\in\mathcal{Y}}(\\#((W({\rm
E}_{6})B_{Y})^{\sigma}))^{2}\\#W(M_{Y})^{\sigma}.$
Hence the corollary holds. ∎
The following can be checked by computation and application of Theorem 4.4.
###### Lemma 5.3.
Let $K_{Y}=\\{a\in{\rm BrM}({\rm E}_{6})\mid\sigma(a)=a,a\emptyset\in W({\rm
E}_{6})B_{Y}\\}$ for $Y\in\mathcal{Y}$. Then $K_{Y}=\\{\phi(b)\mid b\in{\rm
BrM}({\rm F}_{4}),\,\phi(b)\emptyset\in W({\rm E}_{6})B_{Y}\\}$.
###### Proof.
If $Y=\\{2\\}$, then $\\{\phi(b)\mid b\in{\rm BrM}({\rm
F}_{4}),\,\phi(b)\emptyset\in W({\rm E}_{6})B_{Y}\\}$ are the elements in (I)
for $i=1$ in Theorem 4.4, which has cardinality $12^{2}\times 48$, and
${\hat{E}}_{2}\phi(C_{1}){\hat{E}}_{2}=W(M_{Y})^{\sigma}$ (in the proof of
Corollary 5.1), and the $\phi(W({\rm F}_{4}))$-orbit of $\\{\alpha_{2}\\}$ is
$\sigma$-invariant elements in $W({\rm E}_{6})B_{Y}$. Hence the lemma holds
for $Y=\\{2\\}$.
If $Y=\\{1,6\\}$, then $\\{\phi(b)\mid b\in{\rm BrM}({\rm
F}_{4}),\,\phi(b)\emptyset\in W({\rm E}_{6})B_{Y}\\}$ are those monomials in
(I) for $i=2$ and $i=3$, (II), (III) and (IV) in Theorem 4.4. Analogous to the
above argument for $Y=\\{2\\}$, we see that the image of those monomials under
$\phi$ correspond to different normal forms for ${\rm BrM}({\rm E}_{6})$ up to
some powers of $\delta$, and those monomials of in (I) for $i=2$ and $i=3$,
(II), (III) and (IV) in Theorem 4.4 have cardinality $12^{2}\times
6+18^{2}\times 2+18\times 36\times 2+18\times 36\times 2+36^{2}=30^{2}\times
6$. Hence the lemma holds for $Y=\\{1,6\\}$.
If $Y=\\{2,3,5\\}$, then $\\{\phi(b)\mid b\in{\rm BrM}({\rm
F}_{4}),\,\phi(b)\emptyset\in W({\rm E}_{6})B_{Y}\\}$ are those monomials in
(I) for $i=4$ and $i=5$, (V), and (VI) in Theorem 4.4. Similarly, we see those
monomials are corresponding to different normal forms for ${\rm BrM}({\rm
E}_{6})$ up to some powers of $\delta$, and they have cardinality
$36^{2}+3^{2}+36\times 3+36\times 3=39^{2}$. Hence the lemma holds for
$Y=\\{2,3,5\\}$. ∎
Now, we can give the proof of our main Theorem 1.2.
###### Proof.
Proposition 2.3 implies $\phi$ is a homomorphism. Corollary 4.5 indicates that
the $\phi({\rm Br}({\rm F}_{4}))$ has rank at most $14985$. Corollary 5.2
implies that the ${\rm Br}({\rm E}_{6})^{\sigma}$ has rank $14985$. Lemma 5.3
indicates that $\phi$ has image ${\rm SBr}({\rm E}_{6})$; therefore $\phi$ is
surjective. The homomorphism $\phi$ is an isomorphism and ${\rm Br}({\rm
F}_{4})$ is free over $\mathbb{Z}[\delta^{\pm 1}]$ of rank $14985$ because of
the freeness of ${\rm SBr}({\rm E}_{6})$. ∎
## 6 Cellularity
Recall from [11] and [12] that an associative algebra ${A}$ over a commutative
ring $R$ is cellular if there is a quadruple $(\Lambda,T,C,*)$ satisfying the
following three conditions.
* (C1)
$\Lambda$ is a finite partially ordered set. Associated to each
$\lambda\in\Lambda$, there is a finite set $T(\lambda)$. Also, $C$ is an
injective map
$\coprod_{\lambda\in\Lambda}T(\lambda)\times T(\lambda)\rightarrow{A}$
whose image is an $R$-basis of ${A}$.
* (C2)
The map $*:{A}\rightarrow{A}$ is an $R$-linear anti-involution such that
$C(x,y)^{*}=C(y,x)$ whenever $x,y\in T(\lambda)$ for some $\lambda\in\Lambda$.
* (C3)
If $\lambda\in\Lambda$ and $x,y\in T(\lambda)$, then, for any element
$a\in{A}$,
$aC(x,y)\equiv\sum_{u\in T(\lambda)}r_{a}(u,x)C(u,y)\ \ \ {\rm mod}\
{A}_{<\lambda},$
where $r_{a}(u,x)\in R$ is independent of $y$ and where ${A}_{<\lambda}$ is
the $R$-submodule of ${A}$ spanned by $\\{C(x^{\prime},y^{\prime})\mid
x^{\prime},y^{\prime}\in T(\mu)\mbox{ for }\mu<\lambda\\}$.
Such a quadruple $(\Lambda,T,C,*)$ is called a cell datum for ${A}$.
There is also an equivalent definition due to Köning and Xi.
###### Definition 6.1.
Let $A$ be $R$-algebra. Assume there is an anti-automorphism $i$ on $A$ with
$i^{2}=id$. A two sided ideal $J$ in $A$ is called cellular if and only if
$i(J)=J$ and there exists a left ideal $\Delta\subset J$ such that $\Delta$
has finite rank and there is an isomorphism of $A$-bimodules
$\alpha:J\simeq\Delta\otimes_{R}i(\Delta)$ making the following diagram
commutative:
$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{i}$$\textstyle{\Delta\otimes_{R}i(\Delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x\otimes
y\rightarrow i(y)\otimes
i(x)}$$\textstyle{J\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\Delta\otimes_{R}i(\Delta)}$
The algebra $A$ is called _cellular_ if there is a vector space decomposition
$A=J^{\prime}_{1}\oplus\cdots\oplus J^{\prime}_{n}$ with
$i(J^{\prime}_{j})=J^{\prime}_{j}$ for each $j$ and such that setting
$J_{j}=\oplus_{k=1}^{j}J^{\prime}_{j}$ gives a chain of two sided ideals of
$A$ such that for each $j$ the quotient $J^{\prime}_{j}=J_{j}/J_{j-1}$ is a
cellular ideal of $A/J_{j-1}$.
Also recall definitions of iterated inflations from [14] and cellularly
stratified algebra from [13] in Definition 6.4. Given an $R$-algebra $B$, a
finitely generated free $R$-module $V$, and a bilinear form
$\varphi:V\otimes_{R}V\longrightarrow B$ with values in $B$, we define an
associative algebra (possibly without unit) $A(B,V,\varphi)$ as follows: as an
$R$-module, $A(B,V,\varphi)$ equals $V\otimes_{R}V\otimes_{R}B$. The
multiplication is defined on basis element as follows:
$\displaystyle(a\otimes b\otimes x)(c\otimes d\otimes y):=a\otimes d\otimes
x\varphi(b,c)y.$
Assume that there is an involution $i$ on $B$. Assume, moreover, that
$i(\varphi(v,w))=\varphi(w,v)$. If we can extend this involution $i$ to
$A(B,V,\varphi)$ by defining $i(a\otimes b\otimes x)=b\otimes a\otimes i(x)$.
Then We call $A(B,V,\varphi)$ is an _inflation_ of $B$ along $V$. Let $B$ be
an inflated algebra (possible without unit) and $C$ be an algebra with unit.
We define an algebra structure in such a way that $B$ is a two-sided ideal and
$A/B=C$. We require that $B$ is an ideal, the multiplication is associative,
and that there exists a unit element of $A$ which maps onto the unit of the
quotient $C$. The necessary conditions are outlined in [14, Section 3]. Then
we call $A$ an inflation of $C$ along $B$, or iterated inflation of $C$ along
$B$. We present Proposition 3.5 and Theorem 4.1 of [14].
###### Proposition 6.2.
An inflation of a cellular algebra is cellular again. In particular, an
iterated inflation of $n$ copies of $R$ is cellular, with a cell chain of
length $n$ as in Definition 6.1.
More precisely, the second statement has the following meaning. Start with $C$
a full matrix ring over $R$ and $B$ an inflation of $R$ along a free
$R$-module, and from a new $A$ which is an inflation of the old $A$ along the
new $B$, and continue this operation. Then after $n$ steps we have produced a
cellular algebra $A$ with a cell chain of length $n$.
###### Theorem 6.3.
Any cellular algebra over $R$ is the iterated inflation of finitely many
copies of $R$. Conversely, any iterated inflation of finitely many copies of
$R$ is cellular.
Let $A$ be cellular(with identity) which can be realized as an iterated
inflation of cellular algebras $B_{l}$ along vector spaces $V_{l}$ for
$l=1,\ldots,n.$ This implies that as a vector space
$\displaystyle A=\oplus_{l=1}^{n}V_{l}\otimes V_{l}\otimes B_{l},$
and $A$ is cellular with a chain of two sided ideals ${0}=J_{0}\subset
J_{1}\cdots\subset J_{n}=A$, which can be refined to a cell chain, and each
quotient $J_{l}/J_{l-1}$ equals $V_{l}\otimes V_{l}\otimes B_{l}$ as an
algebra without unit. The involution $i$ of $A$,is defined through the
involution $i_{l}$ of the algebra $B_{l}$ where $i(a\otimes b\otimes
x)=b\otimes a\otimes j_{l}(x)$. The multiplication rule of a layer
$V_{l}\oplus V_{l}\oplus B_{l}$ is indicated by
$\displaystyle(a\otimes b\otimes x)(c\otimes d\otimes y):=a\otimes d\otimes
x\varphi(b,c)y+\text{lower terms}.$
Here lower terms refers to element in lower layers $V_{h}\otimes V_{h}\otimes
B_{h}$ for $h<l$. Let $1_{B_{l}}$ be the identity of the algebra $B_{l}$.
Let that $R$ is a field.
###### Definition 6.4.
A finite dimensional associative algebra $A$ is called cellularly stratified
with stratification data $(B_{1},V_{1},\ldots,B_{n},V_{n})$ if and only if the
following conditions are satisfied:
* (1)
The algebra is an iterated inflation of cellular algebra $B_{l}$ along vector
spaces $V_{l}$ for $l=1$,$\ldots$, $n$.
* (2)
For each $l=1$,$\ldots$, $n$, there exist $u_{l}$, $v_{l}$ such that
$e_{l}=u_{l}\otimes v_{l}\otimes 1_{B_{l}}$ is an idempotent.
* (3)
If $l>m$, then $e_{l}e_{m}=e_{m}=e_{m}e_{l}$.
As [1], the following theorem can be obtained.
###### Theorem 6.5.
Let $R$ be a field with $2$ and $3$ being invertible in $R$ and containing
$\mathbb{Z}[\delta^{\pm 1}]$ as a subring. Then the algebra ${\rm Br}(R,{\rm
F}_{4})={\rm Br}({\rm F}_{4})\otimes_{\mathbb{Z}[\delta^{\pm 1}]}R$ is a
cellularly stratified algebra over $R$.
###### Proof.
To prove this, it suffices to prove it for ${\rm Br}(R,{\rm E}_{6})={\rm
SBr}({\rm E}_{6})\otimes_{\mathbb{Z}[\delta^{\pm 1}]}R$ because of Theorem
1.2. Let $Z_{0}\subset Z_{1}\subset Z_{2}\subset Z_{3}$ be a
$\sigma$-invariant and admissible root set sequence of type ${\rm E}_{4}$,
where $Z_{0}=\emptyset$, $Z_{1}=\\{\alpha_{2}\\}$,
$Z_{2}=\\{\alpha_{2},\alpha_{2}+\alpha_{3}+\alpha_{5}+2\alpha_{4}\\}$,
$Z_{3}=\\{\alpha_{2},\alpha_{3},\alpha_{5},\alpha_{2}+\alpha_{3}+\alpha_{5}+2\alpha_{4}\\}$.
As $E_{2}E_{4}E_{5}E_{3}\\{\alpha_{1},\alpha_{6}\\}=Z_{2}$, we have $\rm
ht$$(Z_{2})=0$.
For $0\leq i\leq 3$, let $B_{Z_{i}}$ be the group algebras of
$W_{Z_{0}}=W(M_{\emptyset}^{\sigma})$, $W_{Z_{1}}=W(M_{\\{2\\}}^{\sigma})$,
$W_{Z_{2}}=E_{2}E_{4}E_{5}E_{3}W(M_{\\{1,6\\}}^{\sigma})E_{3}E_{5}E_{4}E_{2}$,
$W_{Z_{3}}=W(M_{\\{2,3,5\\}}^{\sigma})$ over $R$, respectively, whose group
rings over $R$ are cellular algebras due to [10, Theorem 1.1].
Then each monomial $a$ in ${\rm BrM}({\rm E}_{6})^{\sigma}$ can be uniquely
written as $\delta^{i}a_{Z_{i},B}\hat{e}_{Y}ha_{Z_{i},B^{\prime}}^{\rm op}$
for some $i\in\\{0,1,2,3\\}$ and $h$ is from the above four groups, where
$B=a\emptyset$, $B^{{}^{\prime}}=\emptyset a$ being $\sigma$-invariant,
$a_{Z_{i},B}\in{\rm BrM}({\rm E}_{6})^{\sigma}$, $a_{Z_{i},B^{\prime}}^{\rm
op}\in{\rm BrM}({\rm E}_{6})^{\sigma}$ and
1. (i)
$a\emptyset=a_{Z_{i},B}\emptyset=a_{B}Z_{i}$, $\emptyset a=\emptyset
a_{Z_{i},B^{\prime}}^{\rm op}=Z_{i}a_{B^{\prime}}^{\rm op}$,
2. (ii)
$\rm{ht}$$(B)=$ht$(a_{Z_{i},B})$,
$\rm{ht}$$(B^{\prime})=$ht$(a_{Z_{i},B^{\prime}}^{\rm op})$.
For each $Z_{i}$, let $V_{Z_{i}}$ be a linear space over $R$ with basis
$u_{Z_{i},B}$ where $B\in W({\rm E}_{6})Z_{i}$ and $\sigma(B)=B$. and let
$\varphi_{Z_{i}}$ be a bilinear map defined as
$\displaystyle V_{Z_{i}}\otimes_{R}V_{Z_{i}}$ $\displaystyle\longrightarrow$
$\displaystyle B_{Z_{i}}$
$\displaystyle\varphi_{Z_{i}}(u_{Z_{i},B},u_{Z_{i},B^{\prime}})$
$\displaystyle=$ $\displaystyle a_{Z_{i},B}^{\rm
op}a_{Z_{i},B^{\prime}},\text{ }\,if\,\text{ }Z_{i}=a_{Z_{i},B}^{\rm
op}a_{Z_{i},B^{\prime}}\emptyset,$
$\displaystyle\varphi_{Z_{i}}(u_{Z_{i},B},u_{Z_{i},B^{\prime}})$
$\displaystyle=$ $\displaystyle 0,\text{ }\,if\text{ }\,Z_{i}\subsetneqq
a_{Z_{i},B}^{\rm op}a_{Z_{i},B^{\prime}}\emptyset.$
We first prove that $\varphi$ is well defined. As $a_{Z_{i},B}^{\rm
op}=Z_{i}$, we find $Z_{i}\subset a_{Z_{i},B}^{\rm
op}a_{Z_{i},B^{\prime}}\emptyset$, similarly $Z_{i}\subset\emptyset
a_{Z_{i},B}^{\rm op}a_{Z_{i},B^{\prime}}$. If $Z_{i}=a_{Z_{i},B}^{\rm
op}a_{Z_{i},B^{\prime}}\emptyset$, this indicates that $Z_{i}=\emptyset
a_{Z_{i},B}^{\rm op}a_{Z_{i},B^{\prime}}$ and that $a_{Z_{i},B}^{\rm
op}a_{Z_{i},B^{\prime}}$ will be in $W_{Z_{i}}$ up to some power of $\delta$.
Therefore our $\varphi_{Z_{i}}$ is well defined. Observe that
$(a_{Z_{i},B}^{\rm op}a_{Z_{i},B^{\prime}})^{\rm op}=a_{Z_{i},B^{\prime}}^{\rm
op}(a_{Z_{i},B}^{\rm op})^{\rm op}=a_{Z_{i},B^{\prime}}^{\rm op}a_{Z_{i},B},$
so $(\varphi_{Z_{i}}(u_{Z_{i},B},u_{Z_{i},B^{\prime}}))^{\rm
op}=\varphi_{Z_{i}}(u_{Z_{i},B^{\prime}},u_{Z_{i},B})$. By linearly extension,
we find $(\varphi_{Z_{i}}(u,v))^{\rm op}=\varphi_{Z_{i}}(v,u)$, for $u$,$v\in
V_{Z_{i}}$. By the proof of Lemma 5.3, the algebra ${\rm SBr}(R,{\rm
E}_{6})\otimes_{Z[\delta^{\pm 1}]}$ is an iterated inflation of cellular
algebra $B_{Z_{i}}$ along vector space $V_{Z_{i}}$ for $Z_{0}$, $\ldots$,
$Z_{3}$, namely ${\rm SBr}(R,{\rm E}_{6})\otimes_{Z[\delta^{\pm 1}]}$
satisfies (1) of cellulary stratified algebra. We take
$e_{Z_{i}}=u_{Z_{i},Z_{i}}\otimes u_{Z_{i},Z_{i}}\otimes 1_{B_{Z_{i}}}$, where
$1_{B_{Z_{i}}}=\delta^{-\\#Z_{i}}E_{Z_{i}}$. Because
$E_{Z_{i}}E_{Z_{j}}=\delta^{\\#Z_{i}}E_{Z_{j}}$ for $Z_{i}\subset Z_{j}$,
hence the condition (2) and (3) follows since that that $Z_{i}>Z_{j}$ means
$Z_{i}\subsetneqq Z_{j}$. Finally, ${\rm SBr}(R,{\rm
E}_{6})\otimes_{Z[\delta^{\pm 1}]}$ is a cellularly stratified algebra. ∎
## References
* [1] C. Bowman, Brauer algebras of type $C$ are cellulary stratified algebras, arXiv:1102.0438v1.
* [2] R. Brauer, On algebras which are connected with the semisimple continous groups, Annals of Mathematics, 38 (1937), 857–872.
* [3] R. Carter, Simple groups of Lie type, Wiley classics library.
* [4] A.M. Cohen, B. Frenk and D.B. Wales, Brauer algebras of simply laced type, Israel Journal of Mathematics, 173 (2009) 335–365.
* [5] A.M. Cohen, Dié A.H. Gijsbers and D.B. Wales, A poset connected to Artin monoids of simply laced type, Journal of Combinatorial Theory, Series A 113 (2006) 1646–1666.
* [6] A.M. Cohen, Shoumin. Liu and Shona.H. Yu, Brauer algebras of type C, Journal of Pure and Applied Algebra, 216 (2012), 407–426.
* [7] A.M. Cohen, S. Liu, Brauer algebras of type B, arXiv:1112.4954, December 2012.
* [8] A.M. Cohen and D.B. Wales, The Birman-Murakami-Wenzl algebras of type ${\rm E}_{n}$, arXiv:1101.3544v1.
* [9] The GAP group (2002), GAP-Groups, Algorithms and Programming, Aachen, St Andrews, available at http://www-gap.dcs.st-and.ac.uk/gap
* [10] M. Geck, Hecke algebras of finite type are cellular, Inventiones Mathematicae, 169 (2007), 501–517.
* [11] J. J. Graham, Modular representations of Hecke algebras and related algebras, Ph. D. thesis, University of Sydney (1995).
* [12] J.J. Graham and G.I. Lehrer, Cellular algebras, Inventiones Mathematicae 123 (1996), 1–44.
* [13] R. Hartmann, A. Henke, S. König and R. Paget, Cohomological stratification of diagram algebras, Math. Ann. (2010) 347:765804
* [14] S. König and C. Xi, Cellular algebras: inflations and Morita equivalences, J. London Math. Soc. (2)60 (1999), 700–722 123 (1996), 1–44.
* [15] Shoumin Liu, Brauer algebras of non-simply laced type, PhD thesis, 2012.
* [16] B. Mühlherr, Coxeter groups in Coxeter groups, pp. 277–287 in Finite Geometry and Combinatorics (Deinze 1992). London Math. Soc. Lecture Note Series 191, Cambridge University Press, Cambridge, 1993.
* [17] H.N.V. Temperley and E. Lieb, Relation between percolation and colouring problems and other graph theoretical problems associated with regular planar lattices: some exact results for the percolation problems, Proc. Royal. Soc. A. 322 (1971) 251–288.
* [18] J.Tits, Groupes algébriques semi-simples et géométries associées. In Algebraic and topological Foundations of Geometry (Proc. Colloq., Utrecht, 1959). Pergamon, Oxford, 175–192.
Shoumin Liu
Eindhoven University of Technology
liushoumin2003@gmail.com
|
arxiv-papers
| 2012-06-28T08:58:36 |
2024-09-04T02:49:32.329693
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shoumin Liu",
"submitter": "Shoumin Liu",
"url": "https://arxiv.org/abs/1206.6596"
}
|
1206.6716
|
# Long-range adiabatic quantum state transfer through a linear array of
quantum dots
CHEN Bing chenbingphys@gmail.com FAN Wei XU Yan x1y5@hotmail.com College of
Science, Shandong University of Science and Technology, Qingdao 266510, China
Centre for Quantum Technologies, National University of Singapore, 3 Science
drive 2, Singapore 117543
###### Abstract
We introduce an adiabatic long-range quantum communication proposal based on a
quantum dot array. By adiabatically varying the external gate voltage applied
on the system, the quantum information encoded in the electron can be
transported from one end dot to another. We numerically solve the schrödinger
equation for a system with a given number of quantum dots. It is shown that
this scheme is a simple and efficient protocol to coherently manipulate the
population transfer under suitable gate pulses. The dependence of the energy
gap and the transfer time on system parameters is analyzed and shown
numerically. We also investigate the adiabatic passage in a more realistic
system in the presence of inevitable fabrication imperfections. This method
provides guidance for future realizations of adiabatic quantum state transfer
in experiments.
###### keywords:
adiabatic passage , tight-binding model , quantum state transfer PACS:
03.65.-w, 03.67.Hk, 73.23.Hk
††journal: Sci China Ser G-Phys Mech Astron
## 1 Introduction
In quantum information science, quantum state transfer (QST), as the name
suggests, refers to the transfer of an arbitrary quantum state
$\alpha\left|0\right\rangle+\beta\left|1\right\rangle$ from one qubit to
another. There are two major mechanisms for QST in quantum mechanics. The
first approach is usually characterized by preparing the quantum channel with
an always-on interaction where QST is equivalent to the time evolution of the
quantum state in the time-independent Hamiltonian [1, 2, 3]. However, these
approaches require precise control of distance and timing. Any deviation may
lead to significant errors. The other approach has paid much attention to
adiabatic passage for coherent QST in time-evolving quantum systems, which is
a powerful tool for manipulating a quantum system from an initial state to a
target state. This method of population transfer has the important property of
being robust against small variations of the Hamiltonian and the transport
time, which is crucial experimentally since the system parameters are often
hard to control. Recently, the adiabatic method has been applied to a variety
of physical systems to realize coherent QST. Among these, the typical scheme
for coherently spatial population transfer has been independently proposed for
neutral atoms in optical traps [4] and for electrons in quantum dot (QD)
systems [5] via a dark state of the system, which is termed coherent tunneling
via adiabatic passage (CTAP) following [5]. In such a scheme, the tunneling
interaction between adjacent quantum units is dynamically tuned by changing
either the distance or the height of the neighboring potential wells following
a counterintuitive scheme, which is a solid-state analog of the well-known
stimulated Raman adiabatic passage (STIRAP) protocol [6] of quantum optics.
Since then, the CTAP technique has been proposed in a variety of physical
systems for transporting single atoms [7, 8], spin states [9], electrons [10,
11] and Bose-Einstein condensates [12, 13, 14]. It has also been proposed as a
crucial element in the scale up to large quantum processors [15, 16].
Recently, Ref. [17] presented a scheme to adiabatically transfer an electron
from the left end to the right end of a three dot chain using the ground state
of the system. This technique is a copy of the frequency chirping method [18,
19], which is used in quantum optics to transfer the population of a three-
level atom of the Lambda configuration. The scheme [17] is presented as an
alternative to a well known transfer scheme (CTAP) [5]. However, different
from the CTAP process, the protocol in Ref. [17] considers a three QD array
with an _always-on_ interaction that can be manipulated by the external gate
voltage applied on the two external dots (sender and receiver). Through
maintaining the system in the ground state, it shows that it is a high-
fidelity process for a proper choice of system parameters and also robust
against experimental parameter variations. In this paper we will consider the
passage through the $N$-site coupled QDs array (tight-binding model), which is
schematically illustrated in Fig. 1. Gates applied on the two end dots control
the on-site energy of each dot. In particular, the nearest-neighbor hopping
amplitudes are set to be uniform. We first investigate the effect of system
parameters on the minimum energy gap between the ground state and the first
excited state. Taking a 5-dot structure as an example, we show that the
electron can be robustly transported from one end of the chain to the other by
slowly varying the gate voltages. This structure is easy to extend to an
arbitrary number of sites.
The paper is organized as follows. In Sec. II the model is setup and we
describe the adiabatic transfer of an electron between QDs. In Sec. III we
show numerical results that substantiate the analytical results. The last
section is the summary and discussion of the paper.
Figure 1: Schematic illustrations of adiabatic QST in a multi-dot array. The
system is controlled by gate voltages, $\mu_{\alpha}(t)$ $(\alpha=A,$ $B)$. By
adiabatically varying the gate voltages, one can achieve long-range QST from
the left end to the right QD of the proposal.
## 2 Model setup
We introduce a simple tight-binding chain with uniform nearest-neighbor
hopping integral $-J$ as a quantum data bus for long-range quantum transport,
see Fig. 1. The sender (Alice) and the receiver (Bob) can only control the
external gate voltages $\mu_{\alpha}(t)$ $(\alpha=A,$ $B)$, which are applied
on the two end QDs. In this proposal, the quantum information
$\cos\theta\left|\uparrow\right\rangle+\sin\theta\left|\downarrow\right\rangle$
encoded in the polarization of the electron can be transported from Alice to
Bob. In this scheme the spin state of the electron is a conserved quantity for
the Hamiltonian of the medium, so the spin state cannot be influenced during
the propagation. For simplicity, we will disregard the electron’s spin degrees
of freedom in the following discussion and just illustrate the principles of
QST.
Accounting only for the occupation of the lowest single-particle state of each
dot, the system is described by an $N$-site tight-binding chain. The
Hamiltonian can be written as
$\displaystyle\mathcal{H}(t)$ $\displaystyle=$
$\displaystyle\mathcal{H}_{M}+\mathcal{H}_{C},$ $\displaystyle\mathcal{H}_{M}$
$\displaystyle=$
$\displaystyle-J\sum_{j=1}^{N-1}\left(a_{j}^{{\dagger}}a_{j+1}+\text{h.c.}\right),$
(1) $\displaystyle\mathcal{H}_{C}$ $\displaystyle=$
$\displaystyle\mu_{A}\left(t\right)a_{1}^{{\dagger}}a_{1}+\mu_{B}\left(t\right)a_{N}^{{\dagger}}a_{N},$
where $a_{j}^{{\dagger}}(a_{j})$ denotes the spinless fermion creation
(annihilation) operator at the $j$-th quantum site. $-J$ $(J>0)$is the
coupling constant, which accounts for the hopping of the electron between dots
$j$ and $j+1$. The on-site energies $\mu_{A}(t)$ and $\mu_{B}(t)$ are
modulated in Gaussian pulses to realize the adiabatic transfer, according to
[shown in Fig. 2(a)]
$\displaystyle\mu_{A}(t)$
$\displaystyle=-\mu_{A,max}\exp\left[-\alpha^{2}t^{2}/2\right],$ (2a)
$\displaystyle\mu_{B}(t)$
$\displaystyle=-\mu_{B,max}\exp\left[-\alpha^{2}\left(t-\tau\right)^{2}/2\right],$
(2b)
where $\tau$ and $\alpha$ are the total adiabatic evolution time and standard
deviation of the control pulse. For simplicity we set the two peak values to
be equal, i.e., $\mu_{A,max}=\mu_{B,max}=\mu_{0}$ $(\mu>0)$, in the discussion
that follows and choose $\mu_{0}\gg J$.
In this proposal, we will concentrate on the single-particle problem and use
the ground state $\left|\psi_{g}(t)\right\rangle$ of the Hamiltonian
$\mathcal{H}(t)$ to induce a population transfer from state
$\left|1\right\rangle$ to $\left|N\right\rangle$. Starting from $t=0$, we have
$\mu_{A}(0)=-\mu_{0}$ and $\mu_{B}(0)\approx 0$. The Hamiltonian at $t=0$
reads
$\mathcal{H}(t=0)=-\mu_{0}a_{1}^{{\dagger}}a_{1}-J\sum_{j=1}^{N-1}\left(a_{j}^{{\dagger}}a_{j+1}+\text{h.c.}\right).$
(3)
The ground state of Eq. (3) is a bound state, which can be obtained via the
Bethe ansatz method. A straightforward calculation shows that
$\left|\psi_{g}(t=0)\right\rangle=\sqrt{1-\zeta^{2}}\sum_{j=1}^{N}\zeta^{j-1}\left|j\right\rangle,$
(4)
where $\zeta=J/\mu_{0}$. By choosing a sufficiently large value of $\mu_{0}$,
the ground state $\left|\psi_{g}(0)\right\rangle$ can be reduced to
$\left|\psi_{g}(0)\right\rangle\approx\left|1\right\rangle=a_{1}^{{\dagger}}\left|0\right\rangle$.
To illustrate with an example, the probability of $\left|1\right\rangle$ in
$\left|\psi_{g}(t=0)\right\rangle$ can achieve 99.75% when the peak voltage is
set to be $\mu_{0}/J=20$.
With the same reasoning, in the time limit $t=\tau$, the parameter
$\mu_{A}(t)$ goes to zero and $\mu_{B}(t)$ goes to $-\mu_{0}$. Due to the
reflection symmetry (relabelling sites from right to left) of the system, we
can see that
$\left|\psi_{g}(t=\tau)\right\rangle=\sqrt{1-\zeta^{2}}\sum_{j=1}^{N}\zeta^{N-j}\left|j\right\rangle.$
(5)
One can see that the ground state of Eq. (1) evolves to be
$\left|\psi_{g}(t=\tau)\right\rangle\approx\left|N\right\rangle$. Preparing
the system in state
$\left|\Psi\left(t=0\right)\right\rangle=\left|1\right\rangle\,$and
adiabatially changing $\mu_{A}(t)$ and $\mu_{B}(t)$, one can see that the
system will end up in $\left|N\right\rangle$
$\left|\Psi\left(t=0\right)\right\rangle=\left|1\right\rangle\rightarrow\left|\Psi\left(t=\tau\right)\right\rangle=\left|N\right\rangle.$
(6)
Figure 2: (a) Gate voltages (in units of $\mu_{0}$) as a function of time (in
units of $\tau$) described in Eq. (2). $\mu_{A}(t)$ is the solid line and
$\mu_{B}(t)$ is the dashed line. (b) The instantaneous eigenenergy of the
lowest two states $\psi_{1}$ and $\psi_{g}$ though the gate pulse shown in (a)
for the values of $\mu_{0}=20$, $J=1.0$, and $\alpha=5/\tau$ in an $N=5$
structure. The gap is minimum at $t=\tau/2$,
$\Delta=\varepsilon_{1}(\tau/2)-\varepsilon_{g}(\tau/2)$.
The analysis above is based on the assumption that the adiabaticity is
satisfied. The crucial requirement for adiabatic evolution is
$\left|\varepsilon_{g}(t)-\varepsilon_{1}(t)\right|\gg\left|\langle\dot{\psi}_{g}(t)|\psi_{1}(t)\rangle\right|,$
(7)
which greatly suppresses the quantum transition from the ground state
$|\psi_{g}(t)\rangle$ to the first-excited state $|\psi_{1}(t)\rangle$.
Firstly, one must make sure that no level crossings occur, i.e.,
$\varepsilon_{g}(t)-\varepsilon_{1}(t)<0$. To evaluate instantaneous
eigenvalues of the Hamiltonian is generally only possible numerically. In Fig.
2(b) we present the results showing the eigenenergy gap between the
instantaneous first-excited state and ground state undergoing evolution due to
modulation of the gate voltage according the pulses given in Eq. (2) for
$\mu_{0}=20$, $J=1.0$ and $\alpha=5/\tau$. The eigenvalues shown in this
figure exhibit pronounced avoided crossing and approach nonzero minimum
$\Delta=\varepsilon_{1}(\tau/2)-\varepsilon_{g}(\tau/2)$ at $t=\tau/2$. This
minimum energy gap plays a significant role in the transfer, because the total
evolution time $\tau$ should be large compared to $1/\Delta$. In this scheme,
the energy gap $\Delta$ depends both on the number of QDs and gate voltages.
To study the relationship between the total evolution time $\tau$ and system
parameters is one of the important contributions of this paper.
Figure 3: Effect the system parameters have on the energy difference. (a) The
gap $\Delta$ obtained using a numerical method for the systems $N=5$, $6$,
$7$, $8$, $9$, and $10$, with $\mu_{0}=16$, $20$, $24$ and $J=1.0$ are
plotted. It indicates that $\Delta\sim J^{2}/\left(\mu_{0}N^{2}\right)$. (b)
The gap $\Delta$ as a function of $\alpha$. It shows the increase of the gap
with increasing $\alpha$.
Fig. 3 shows the effect four factors have on the energy gap $\Delta$. In Fig.
3(a), for a given $\alpha=5/\tau$ and $J=1.0$, we plot the energy gap $\Delta$
as a function of $N^{-2}$ for $\mu_{0}=$16, 20, and 24. The numerical results
indicate that the gap $\Delta\sim J^{2}/\left(\mu_{0}N^{2}\right)$, which
implies that the adiabatic transfer time of QST grows quadratically with the
spatial separation of the two end states because the minimum gap plays an
opposite role for the adiabatic QST. The other thing is that $\Delta$ is also
determined by the dimensionless parameter $\alpha$. As an example, Fig. 3(b)
shows the numerically computed behavior of $\Delta$ as a function of $\alpha$
with $N=5$ and $\mu_{0}/J=20$. From Fig. 3(b) we see that the gap becomes
larger as $\alpha$ increases and then tends to be a constant $0.732$, which is
the gap of tight-binding chain ($N=5$) without any on-site energy. The reason
is that the bigger $\alpha\tau$ is, the smaller the overlap amplitude of the
two pulses, i.e., $\exp(-\alpha^{2}\tau^{2}/8)\rightarrow 0$. The energy gap
$\Delta$ of $\mathcal{H}(\tau/2)$ then approaches the maximum value
$2J[\cos\pi/(N+1)-\cos 2\pi/(N+1)]$.
## 3 Numerical Examples
In this section let us firstly review the transfer process of this protocol.
At $t=0$ we initialize the device so that the electron occupies site-$1$,
i.e., the total initial state is
$\left|\Psi\left(0\right)\right\rangle=\left|1\right\rangle$, and slowly apply
gate pulses, which results in robust transport of the electron from one end of
the chain to the other. The consequent time evolution of the state is given by
the Schrödinger equation (assuming $\hbar=1$)
$i\frac{d}{dt}\left|\Psi\left(t\right)\right\rangle=\mathcal{H}(t)\left|\Psi\left(t\right)\right\rangle.$
(8)
The time evolution creates a coherent superposition:
$\left|\Psi\left(t\right)\right\rangle=\sum_{j=1}^{N}c_{j}(t)\left|j\right\rangle,$
(9)
where $c_{j}(t)$ denotes the time-dependent probability amplitude for the
electron to be in the $j$-th QD that obeys the normalization condition
$\sum_{j=1}^{N}\left|c_{j}(t)\right|^{2}=1$. At time $\tau$ the fidelity of
the initial state transferring to the dot-$N$ is defined as
$F(\tau)=\left|\langle
N\left|\Psi(\tau)\right\rangle\right|^{2}=\left|c_{N}(\tau)\right|^{2}.$ (10)
A feasible proposal should be able to perform efficient high-fidelity QST in
the shortest possible time. In order to provide the most economical choice of
parameters for reaching high transfer efficiency, we used standard numerical
methods to integrate the Schrödinger equation for probability amplitudes.
Because the scheme relies on maintaining adiabatic conditions, we examine the
effect of system parameters on the target state population.
Figure 4: Probability of finding a single electron in basis states in the
ground state as a function of time for the values of $N=5$, $J=1.0$,
$\mu_{0}=20$, $\tau=500$ and (a) $\alpha=4/\tau$, (b) $\alpha=5/\tau$. (c)
Transfer fidelity $\left|c_{N}(t)\right|^{2}$ as a function of parameter
$\alpha$. As $\alpha$ is increased it is more able to obtain high fidelity
transfer. A smaller $\alpha$ introduces a population of excited states and the
transfer is no longer complete.
We show in Fig. 4 the probabilities as a function of time for different values
of $\alpha$ where we take a 5-dot structure for example. The time behavior of
$\mu_{A}(t)$ and $\mu_{B}(t)$ follow Gaussian functions with $\mu_{0}/J=20$
[see Fig. 2(a)] and have been performed in a finite time.
For the $\alpha=4/\tau$ case with $\tau=500$, which departs from the adiabatic
limit, we find the result in Fig. 4(a). Here the population transferred to the
target state is only about 11%. The reason is that the population is excited
to the upper energy states through nonadiabaticity. It is necessary to point
out that if one enlarged $\tau$ extremely in this case, adiabaticity would be
fulfilled, which would result in high transfer fidelity.
On the other hand, as shown in Fig. 3, enlarging $\alpha$ can increase the
level spacing between the first excited state and the ground state and hence
cause the adiabaticity of the system to become better. In Fig. 4(b) we show
the population evolution taking $\alpha=5/\tau$ as an example and the
populations of the states $\left|1\right\rangle$ and $\left|N\right\rangle$
are exchanged with a fidelity of $99.5\%$.
The choice of pulse modulation is therefore important with the maximum
transfer speed ultimately controlled by the adiabatic criteria for the
transfer. To see the emergence of the adiabatic limit, we plot in Fig. 4(c)
the transfer fidelity as a function of the adiabaticity parameter $\alpha$.
One can see that as $\alpha$ increases there is an exponential appearance of
the adiabaticity in the ideal limit. That means the smaller the overlapping of
two pulses is, the more ideal adiabatic transfer takes place. This result is
extendible to an arbitrary number of QDs. In the discussion that follows, we
choose $\alpha=5/\tau$.
Figure 5: Total evolution time $\tau$ (in units of $\mu_{0}/J^{2}$) as $N$ is
increased under the condition that $\left|c_{N}(\tau)\right|^{2}\geq 0.995$.
As $N$ is increased the energy gap is decreased, resulting in longer evolution
time across the array. The solid line is the quadratic curve fitting, which
indicates that the evolution time grows quadratically with the number of QDs.
In section II, we showed that the energy gap $\Delta$ also depends on the
system parameters, such as the number of QDs $N$ and coupling strength $J$. In
order to quantitatively determine the time needed to achieve high fidelity
QST, we solve the schrödinger equation for constants $\mu_{0}=20$, $J=1.0$. In
Fig. 5 we present results showing $\tau$ as a function of the QDs number $N$.
The quadratic curve fitting shows that the minimum possible transfer times are
proportional to $N^{2}$, giving a high-fidelity transfer of
$\left|c_{N}(\tau)\right|^{2}\geq 0.995$. On the other hand, the energy gap
$\Delta$ decreases when the peak value rises [see Fig. 3(a)]. Consequently,
the time scale $\tau$ is proportional to and of the order of $\mu_{0}/J^{2}$
for a given $N$. To sum up the above discussion, in practice the minimum
possible transfer timescale of this adiabatic passage will be of the same
order as $N^{2}\mu_{0}/J^{2}$.
For a long enough evolution time $\tau$, the maximum fidelity of this scheme
depends on the contrast ratio between peak values $\mu_{\kappa,max}$
$(\kappa=A,B)$ and coupling constants $J$. The reason is that small peak
values improve adiabaticity, but lead to a low fidelity because the initial
and final energy eigenstates are not the desired states $\left|1\right\rangle$
and $\left|N\right\rangle$, respectively.
Figure 6: (Color online) Plot of the transfer fidelity $|c_{N}(\tau)|^{2}$ as
a function of peak values (in units of $J$). (a)
$\mu_{A,max}=\mu_{B,max}=\mu_{0}$ varies from 10$J$ to 25$J$; (b)
$\mu_{A,max}$ and $\mu_{B,max}$ vary from 10$J$ to 25$J$ respectively. The
contour lines, labeled with the corresponding values of $|c_{N}(\tau)|^{2}$,
display the gradual increase of transfer fidelity as $\mu_{A,max}/J$ and
$\mu_{B,max}/J$ grow. The fidelity is close to one ($|c_{N}(\tau)|^{2}\geq
0.995$) when two peak values are achieved for $\mu_{\kappa,max}\geq 20J$
$(\kappa=A,B)$.
To determine the parameter range of $\mu_{\kappa,max}$ $(\kappa=A,B)$ needed
to achieve high fidelity transfer, we numerically integrate the density matrix
equations of motion, with varying peak values $\mu_{A,max/J}$ and
$\mu_{B,max}/J$ from 10 to 25. Fig. 6 shows the transfer fidelity
$|c_{N}(\tau)|^{2}$ plots as a function of $\mu_{A,max}/J$ and $\mu_{B,max}/J$
for $\tau=1000$ and $\alpha=5/\tau$ in a 5-dot system. The fidelity approaches
unity as $\mu_{A,max}$ and $\mu_{B,max}$ increase. The figure is nearly
symmetric with respect to the line $\mu_{A,max}=\mu_{B,max}$. Fig. 6(a) is
taken from Fig. 6(b) by slicing through the diagonal gray line. We can see
that to realize near-perfect fidelity transfer
($\left|c_{N}(\tau)\right|^{2}\geq 0.995$) one has to use peak values
satisfying $\mu_{\kappa,max}\geq 20J$ $(\kappa=A,B)$.
The other advantage of this scheme is defect tolerance of the system
parameters. We now assume that the tunnel coupling has a random but constant
offset $\delta\varepsilon_{j}$, i.e. $J_{j}=J(1-\delta\varepsilon_{j})$, where
$\varepsilon_{j}$ is drawn from the standard uniform distribution on the open
interval $(0,1)$ and all $\varepsilon_{j}$ are completely uncorrelated for all
sites along the chain. We show some examples in Fig. 7 for QST in the chain of
$N=5$ with maximum coupling offset bias $\delta=0.1,0.2$, and $0.3$. It shows
that weak fluctuations (up to $\delta=0.2$) in the coupling strengths do not
deteriorate the performance of our scheme. For $\delta=0.3$ we can see that
arbitrarily perfect transfer remains possible except for some rare
realizations of $0.3\varepsilon$. To realize high-fidelity QST transfer, the
price for unprecise couplings is thus a longer transmission time.
Figure 7: (Color online) The transfer fidelity $\left|c_{N}(\tau)\right|^{2}$
for a tight-binding chain of length $N=5$ with $\mu_{A,max}=\mu_{B,max}=20J$,
$\tau=500$ and $\alpha=5/\tau$. The coupling strengths are chosen randomly
from the interval $[(1-\delta)J,J]$ for $\delta=0.1$ (Square), $\delta=0.2$
(Circle) and $\delta=0.3$ (Triangle). The number of random samples is 20.
## 4 Summary
We have introduced a robust and coherent method of long-range coherent QST
through a tight-binding chain by adiabatic passage. This scheme is realized by
modulation of gate voltages applied on the two end QDs. Under suitable gate
pulses, the electron can be transported from one end of the chain to the
other, carrying along with it the quantum information encoded in its spin.
Different from the CTAPn Scheme [20], our method is to induce population
transfer through the tight-binding chain by maintaining the system in its
ground state and this is more operable in experiments. We have studied the
adiabatic QST through the system by theoretical analysis and numerical
simulations of the ground state evolution of the tight-binding model. The
result shows that it is an efficient high-fidelity process ($\geq 99.5\%$) for
a proper choice of standard deviation $\alpha\geq 5/\tau$ and peak values
$\mu_{0}\geq 20$ of gate voltages. For an increasing number of dots, we found
that the evolution time scale is $\tau\sim N^{2}\mu_{0}/J^{2}$. We also
consider the QST along the quantum chain if their coupling is changed by some
random amount. We further find that weak fluctuations in the coupling strength
still allow high fidelity QST.
## Acknowledgements
We acknowledge the support of the NSF of China (Grant No.10847150 and
No.11105086), the Shandong Provincial Natural Science Foundation (Grant No.
ZR2009AM026 and BS2011DX029), and the basic scientific research project of
Qingdao (Grant No.11-2-4-4-(6)-jch). Y. X. also thanks the Basic Scientific
Research Business Expenses of the Central University and Open Project of Key
Laboratory for Magnetism and Magnetic Materials of the Ministry of Education,
Lanzhou University (Grant No. LZUMMM2011001) for financial support.
## References
* [1] Bose S. Quantum communication through an unmodulated spin chain. Phys Rev lett, 2003, 91(20): 207901
* [2] Song Z, Sun C P. Quantum information storage and state transfer based on spin systems. Low Temperature Physics, 2005 31(8): 686
* [3] Christandl M, Datta N, Ekert A, et al. Perfect State Transfer in Quantum Spin Networks. Phys Rev Lett, 2004, 92(18): 187902
* [4] Eckert K, Lewenstein M, Corbal R, et al. Three-level atom optics via the tunneling interaction. Phys Rev A, 2004, 70(2): 023606
* [5] Greentree A D, Cole J H, Hamilton A R, et al. Coherent electronic transfer in quantum dot systems using adiabatic passage. Phys Rev B, 2004, 70(23): 235317
* [6] Vitanov N V, Halfmann T, Shore B W, et al. Laser-induced polulation transfer by adiabatic passage techniques. Annu Rev Phys Chem, 2001, 52(1):763-809
* [7] Eckert K, Mompart J, Corbalan R, et al. Three level atom optics in dipole traps and waveguides. Opt Commun, 2006, 264(2): 264-270
* [8] Opatrny T, Das K K. Conditions for vanishing central-well population in triple-well adiabatic transport. Phys Rev A, 2009, 79(1): 012113
* [9] Ohshima T, Ekert A, Oi D K L, et al. Robust state transfer and rotation through a spin chain via dark passage. e-print arXiv:quant-ph/0702019.
* [10] Zhang P, Xue Q K, Zhao X G, et al. Generation of spatially separated spin entanglement in a triple-quantum-dot system. Phys Rev A, 2004, 69(4): 042307
* [11] Fabian J, Hohenester U. Entanglement distillation by adiabatic passage in coupled quantum dots. Phys Rev B, 2005, 72(20): 201304(R)
* [12] Graefe E M, Korsch H J, Witthaut D. Mean-field dynamics of a Bose-Einstein condensate in a time-dependent triple-well trap: Nonlinear eigenstates, Landau-Zener models, and stimulated Raman adiabatic passage. Phys Rev A, 2006, 73(): 013617
* [13] Rab M, Cole J H, Parker N G, et al. Spatial coherent transport of interacting dilute Bose gases. Phys Rev A, 2008, 77(6): 061602(R)
* [14] Nesterenko V O, Nikonov A N, de Souza Cruz F F, et al. STIRAP transport of Bose-Einstein condensate in triple-well trap. Laser Phys, 2009, 19(4): 616-624
* [15] Hollenberg L C L, Greentree A D, Fowler A G, et al. Two-dimensional architectures for donor-based quantum computing. Phys Rev B, 2006, 74(4): 045311
* [16] Greentree A D, Devitt S J, Hollenberg L C L. Quantum-information transport to multiple receivers. Phys Rev A, 2006, 73(3): 032319
* [17] Chen B, Fan W, Xu Y. Adiabatic quantum state transfer in a nonuniform triple-quantum-dot system. Phys Rev A, 2011, 83(1): 014301
* [18] Cheng J, Zhou J Y. Ultrafast population transfer in three-level $\Lambda$ systems driven by few-cycle laser pulses. Phys Rev A, 2001 64(6): 065402
* [19] Goswami D. Optical pulse shaping approaches to coherent control. Phys Rep, 2003, 374(6): 385-481
* [20] Hollenberg L C L, Greentree A D, Fowler A G, et al. Two-dimensional architectures for donor-based quantum computing. Phys Rev B, 2006, 74(4): 045311
|
arxiv-papers
| 2012-06-28T14:51:20 |
2024-09-04T02:49:32.340834
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bing Chen, Q. H. Shen, Wei Fan, and Yan Xu",
"submitter": "Bing Chen",
"url": "https://arxiv.org/abs/1206.6716"
}
|
1206.6725
|
# A note on the norm and spectrum of a
Foguel operator
Mrinal Raghupathi Department of Mathematics, Vanderbilt University
Nashville, Tennessee, 37240, U.S.A. mrinal.raghupathi@vanderbilt.edu
http://www.math.vanderbilt.edu/ mrinalr
###### Abstract.
We present two ways to compute the norm of a Foguel operator. One of these is
algebraic and the other makes use of the Schur complement. This gives a two
simpler proof of a recent result of Garcia [2]. We also provide an extension
of these results.
###### Key words and phrases:
Foguel operator
###### 2000 Mathematics Subject Classification:
Primary 47A10; Secondary 47A30
The author was partially supported by a Young investigator award from Texas
A&M Univeristy in 2009.
## 1\. Introduction
Let $H$ be a Hilbert space. Given an isometry $V\in B(H)$ and an operator
$T\in B(H)$, the Foguel operator with symbol $T$ is defined by
$R_{T}=\begin{bmatrix}V^{\ast}&T\\\ 0&V\end{bmatrix}.$
In [2], a formula is given for the norm of a Foguel operator $R_{T}$. The
proof in [2] is based on an antilinear eigenvalue problem and certain
properties of complex symmetric matrices. In this note, we give a more direct
proof of this fact based on a direct computation of the inverse. We also show
how the formula for the norm can be obtained as an application of the Schur
complement.
Foguel operators have played a central role in counterexamples to similarity
conjectures. The most famous of these is Pisier’s counterexample [4] to the
Halmos conjecture [3].
## 2\. Spectrum and norm
In this section we derive a relationship between the spectrum of the operator
$R_{T}R_{T}^{*}$ and $TT^{*}$. The result is due to Garcia [2] but the proof
given here is more direct.
In the proof we make use of the fact that if $R$ is a selfadjoint operator on
a Hilbert space, then $R$ is invertible if and only if $R$ is right-
invertible. To see this, suppose that $RS=I$. Then $S^{*}R^{*}=S^{*}R=I$.
Hence, $R$ has a left-inverse, and is invertible.
###### Theorem 1 (Garcia).
If $\lambda>0$ and $\lambda\not=1$, then
$\lambda\in\operatorname{spec}(R_{T}R_{T}^{*})$ if and only if
$\lambda^{-1}(\lambda-1)^{2}\in\operatorname{spec}(TT^{*})$. The norm of
$R_{T}$ is given by
$\left\|R_{T}\right\|=\frac{\left\|T\right\|+\sqrt{\left\|T\right\|^{2}+4}}{2}.$
###### Proof.
Suppose that $\lambda\not\in\operatorname{spec}(R_{T}R_{T}^{\ast})$. Since
$R_{T}R_{T}^{*}-\lambda I$ is invertible there exists a selfadjoint operator
$S=\begin{bmatrix}A&X\\\ X^{\ast}&B\end{bmatrix}$
such that $R_{T}R_{T}^{\ast}S=\lambda S+I$. We have
$R_{T}R_{T}^{\ast}=\begin{bmatrix}V^{\ast}V+TT^{\ast}&TV^{\ast}\\\
VT^{\ast}&VV^{\ast}\end{bmatrix}=\begin{bmatrix}I+TT^{\ast}&TV^{\ast}\\\
VT^{*}&VV^{*}\end{bmatrix}.$
Writing out the entries of the operator matrix equation
$R_{T}R_{T}^{*}S=\lambda S+I$ we get
(1) $\displaystyle(I+TT^{*})A+TV^{*}X^{*}$ $\displaystyle=\lambda A+I$ (2)
$\displaystyle(I+TT^{\ast})X+TV^{*}B$ $\displaystyle=\lambda X$ (3)
$\displaystyle VT^{*}A+VV^{*}X^{*}$ $\displaystyle=\lambda X^{*}$ (4)
$\displaystyle VT^{*}X+VV^{*}B$ $\displaystyle=\lambda B+I.$
Using the fact that $V^{*}V=I$ and multiplying equation (3) on the left by
$V^{*}$ we get $T^{*}A+V^{*}X^{*}=\lambda V^{*}X^{*}$, which gives
(5) $T^{*}A=(\lambda-1)V^{*}X^{*}.$
Multiplying (1) by $(\lambda-1)$ and substituting the expression from (5) we
get
(6) $(\lambda-1)(I+TT^{*})A+TT^{\ast}A=(\lambda-1)(\lambda A+I).$
Rearranging and simplifying this last equation we get
$((\lambda-1)I+I)TT^{*}A+((\lambda-I)+\lambda(\lambda-1))A=(\lambda-1)I$
or
$(\lambda TT^{*}-(\lambda-I)^{2})A=(\lambda-1)I.$
Since we have assumed that $\lambda\not=0,1$ we get
(7)
$A=\frac{\lambda}{\lambda-1}\left(TT^{*}-\frac{(\lambda-1)^{2}}{\lambda}I\right)^{-1}.$
Hence $\lambda^{-1}(\lambda-1)^{2}\not\in\operatorname{spec}(TT^{*})$.
The operators $X$ and $B$ can also be computed explicitly. The choice of
$X=\frac{1}{\lambda-1}ATV^{\ast}$ satisfies (5). Multiplying (4) by $V^{*}$ we
get $T^{*}X+V^{*}B=\lambda V^{*}B+V^{*}$ and so
(8) $(\lambda-1)V^{*}B=T^{*}X-V^{*}.$
The choice
(9) $B=(\lambda-1)^{-1}(VT^{*}X-VV^{*})$
is a solution of (8).
Now consider the case where $\mu\not\in\operatorname{spec}(TT^{*})$ and assume
that $\mu\not=0$. The equation $\mu=(\lambda-1)^{2}/\lambda$ has two positive
solutions, choose $\lambda$ to be either of these solutions. We choose $A,X,B$
as in the equations above. It is a routine, but straightforward, calculation
to show that this choice of $A,B,X$ satisfies (1)–(4).
From the relationship between the spectrum of $R_{T}R_{T}^{*}$ and the
spectrum of $TT^{*}$ we get that
(10)
$\left\|T\right\|^{2}=\sup\\{\lambda^{-1}(\lambda-1)^{2}\,:\,\lambda\in\operatorname{spec}(R_{T}R_{T}^{*})\\}.$
The equation $\left\|T\right\|^{2}=\lambda^{-1}(\lambda-1)^{2}$ has two
solutions of the form $\lambda,\lambda^{-1}$. The function
$f(\lambda)=\lambda^{-1}(\lambda-1)^{2}$ is increasing for $\lambda\geq 1$ and
$\left\|T\right\|^{2}$ is the maximum value of the function $f$ on the set
$\operatorname{spec}(R_{T}R_{T}^{*})$. Since,
$\left\|R_{T}\right\|\geq\left\|V\right\|=1$ we see that
(11) $\left\|T\right\|^{2}=(\left\|R_{T}\right\|^{2}-1)/\left\|R_{T}\right\|.$
This gives,
(12)
$\left\|R_{T}\right\|=\frac{\left\|T\right\|+\sqrt{\left\|T\right\|^{2}+4}}{2}.$
∎
The proof of Theorem 1 actually gives us a little more information about the
invertibility of $R_{T}R_{T}^{*}$ and $TT^{*}$.
###### Proposition 2.
The operator $R_{T}R_{T}^{*}$ is invertible if and only if $V$ is unitary. The
operator $R_{T}R_{T}^{*}-I$ is invertible if and only if $T$ is invertible.
###### Proof.
From (4) we have, $V(T^{*}X+V^{*}B)=I$ and so $VY=I$ for some $Y$. Hence,
$Y=(VV^{*})Y=V^{\ast}(VY)=V^{\ast}$. Which proves that $VV^{\ast}=I$.
On the other hand, if $V$ is unitary, then $\begin{bmatrix}V&-VTV^{*}\\\
0&V^{*}\end{bmatrix}$ is the inverse of $R_{T}{R_{T}}^{*}$.
Now consider the case where $R_{T}R_{T}^{\ast}-I$ is invertible. In this case
from (1) we get $TT^{*}A+TV^{*}X^{*}=I$, which gives, $TY=I$ for some $Y$.
Multiply equation (4) by $V^{*}$ we get $T^{*}X=V^{*}$. Now multiply by $V$ to
get $T^{*}XV=I$. Hence, $ZT=I$ for some $Z$ and so $T$ is invertible. However,
we now get from (5) that $T^{\ast}A=0$ and so $A=0$. In this case, the inverse
of $R_{T}R_{T}^{\ast}-I$ is given by $\begin{bmatrix}0&X\\\
X^{\ast}&B\end{bmatrix}$, where $X=(T^{-1})^{*}V^{*}$ and $B=-I$.
Conversely, if $T$ is invertible, then there exists $Z$ such that $ZT=TZ=I$
and we can write down the inverse of $R_{T}R_{T}^{*}-I$ as above. ∎
As a final note in this section we point out that we can strengthen the power-
bounded result in [2] to the case of polynomials in $R_{T}$.
###### Proposition 3.
Let $A$ be a contraction and let $T$ be an operator on $H$. Let
$R=\begin{bmatrix}A^{*}&T\\\ 0&A\end{bmatrix}$. We have,
$\left\|R\right\|\leq\frac{\left\|T\right\|+\sqrt{\left\|T\right\|^{2}+4}}{2}$
###### Proof.
Let $V_{A}$ denote the usual isometric Sz.-Nagy dilation of $A$, that is,
$V_{A}=\begin{bmatrix}A&(I-AA^{*})^{1/2}\\\
(I-A^{*}A)^{1/2}&A^{*}\end{bmatrix}.$
Let $\tilde{T}$ denote the $2\times 2$ operator matrix that has $T$ in its
$(1,2)$ entry and is 0 otherwise. Let $W=\begin{bmatrix}V_{A^{*}}&\tilde{T}\\\
0&V_{A}\end{bmatrix}$. Note that $V_{A^{*}}=V_{A}^{*}$. Since $V_{A}$ is an
isometry, $W$ is a Foguel operator with symbol $\tilde{T}$ and so
$\left\|W\right\|=\frac{1}{2}\big{(}\left\|T\right\|+(\left\|T\right\|^{2}+4)^{1/2}\big{)}$,
since $\|\tilde{T}\|=\left\|T\right\|$. If we view $W$ as a $4\times 4$
operator matrix, then the operator $R$ is the compression of $W$ to the first
and fourth row and column. Hence, $\left\|R\right\|\leq\left\|W\right\|$. ∎
###### Corollary 4.
Let $p(z)=a_{0}+a_{1}z+\cdots+a_{m}z^{m}$ be a polynomial. Let
$\left\|p\right\|_{\infty}:=\sup_{z\in\mathbb{D}}\left\lvert p(z)\right\rvert$
and let $\tilde{p}(z)=\left\lvert a_{0}\right\rvert+\left\lvert
a_{1}\right\rvert z+\cdots+\left\lvert a_{m}\right\rvert z^{m}$. Let $A$ and
$T$ be operators on $H$ with $\left\|A\right\|\leq 1$ and let
$R=\begin{bmatrix}A^{*}&T\\\ 0&A\end{bmatrix}.$
We have,
$\left\|p(R)\right\|\leq\frac{\left\|\tilde{p}^{\prime}\right\|_{\infty}\left\|T\right\|+\sqrt{\left\|\tilde{p}^{\prime}\right\|_{\infty}^{2}\left\|T\right\|^{2}+4}}{2}$
###### Proof.
First note that we can assume that $\left\|p\right\|_{\infty}\leq 1$. A simple
computation shows that
$R^{n}=\begin{bmatrix}(A^{*})^{n}&\sum_{j=0}^{n-1}(A^{*})^{j}TA^{n-1-j}\\\
0&A^{n}\end{bmatrix}$
Let us denote the operator in the upper right corner of the above matrix by
$D_{n}(A,T)$ for $n\geq 1$.
We have,
$p(R)=\begin{bmatrix}p(A)^{*}&\sum_{j=1}^{m}a_{j}D_{j}(A,T)\\\
0&A^{n}\end{bmatrix}.$
Since $A$ is a contraction, von-Neumann’s inequality tells us that
$\left\|p(A)\right\|\leq\left\|p\right\|_{\infty}\leq 1$. Now,
$\displaystyle\left\|\sum_{j=1}^{m}a_{j}D_{j}(A,T)\right\|\leq$
$\displaystyle\sum_{j=1}^{m}\left\lvert
a_{j}\right\rvert\left\|D_{j}(A,T)\right\|\leq\sum_{j=1}^{m}\left\lvert
a_{j}\right\rvert\sum_{i=0}^{j-1}\left\|(A^{*})^{i}TA^{j-1-i}\right\|$
$\displaystyle\leq$ $\displaystyle\sum_{j=1}^{m}\left\lvert
a_{j}\right\rvert\sum_{i=0}^{j-1}\left\|A\right\|^{j-1}\left\|T\right\|\leq\left\|T\right\|\sum_{j=1}^{m}j\left\lvert
a_{j}\right\rvert\left\|A\right\|^{j-1}$ $\displaystyle=$
$\displaystyle\left\|T\right\|\tilde{p}^{\prime}(\left\|A\right\|)\leq\left\|T\right\|\left\|\tilde{p}^{\prime}\right\|_{\infty}$
The result now follows from Proposition 3. ∎
An application of the previous proposition with $p(z)=z^{n}$ gives the result
on the norm of $R_{T}^{n}$ obtained in [2]. We have $p=\tilde{p}$ and
$\left\|p^{\prime}\right\|_{\infty}=n$. Hence, we obtain the norm estimate
$\left\|R_{T}^{n}\right\|\leq\frac{n\left\|T\right\|+\sqrt{n^{2}\left\|T\right\|^{2}+4}}{2}.$
## 3\. A positivity proof of the norm
In this section we present a short proof of the norm equality using the Schur
complement.
We begin by giving a brief description of the Schur complement. Let
$R=\begin{bmatrix}P&X\\\ X^{*}&Q\end{bmatrix}$, with $P\geq 0,Q>0$. The
operator $R$ is positive if and only if the matrix $P-XQ^{-1}X^{*}\geq 0$. The
proof of this fact follows by first conjugating $R$ by the positive matrix
$I\oplus Q^{-1/2}$ which gives $R^{\prime}=\begin{bmatrix}P&XQ^{-1/2}\\\
Q^{-1/2}X^{\ast}&I\end{bmatrix}$, and then using the fact that
$\begin{bmatrix}P&Y\\\ Y^{*}&I\end{bmatrix}\geq 0$ if and only if
$P-YY^{*}\geq 0$.
For an operator $T\in B(H)$,
$\left\|T\right\|=\inf\\{M\,:\,M^{2}I-TT^{*}>0\\}$.
Consider the Foguel operator $R_{T}$. We know that $\left\|R_{T}\right\|\geq
1$. Let $M>1$. We have
(13) $\displaystyle M^{2}I-R_{T}R_{T}^{*}$
$\displaystyle=\begin{bmatrix}M^{2}I&0\\\
0&M^{2}I\end{bmatrix}-\begin{bmatrix}I+TT^{*}&TV^{*}\\\
VT^{*}&VV^{*}\end{bmatrix}$ (14)
$\displaystyle=\begin{bmatrix}(M^{2}-1)I-TT^{*}&-TV^{*}\\\
-VT^{*}&M^{2}I-VV^{*}\end{bmatrix}$
By applying the Schur complement we see that this operator is positive if and
only if
$(M^{2}-1)I-TT^{*}-TV^{*}(M^{2}I-VV^{*})^{-1}VT^{*}\geq 0.$
Since $M>1$ and $V$ is a contraction we can expand $(M^{2}I-VV^{*})^{-1}$ in
its Neumann series as $M^{-2}\sum_{j=0}^{\infty}(VV^{*})^{j}/(M^{2j})$. Hence,
(15) $\displaystyle TV^{*}(M^{2}I-VV*)^{-1}VT^{*}$
$\displaystyle=T\left(M^{-2}\sum_{j=0}^{\infty}\frac{V(VV^{*})^{j}V^{*}}{M^{2j}}\right)T^{*}$
(16)
$\displaystyle=M^{-2}TT^{*}\sum_{j=0}^{\infty}M^{-2j}=(M^{2}-1)^{-1}TT^{*}.$
Hence, the positivity condition is
$(M^{2}-1)I-TT^{*}-(M^{2}-1)^{-1}TT^{*}=(M^{2}-1)I-M^{2}(M^{2}-1)^{-1}TT^{*}\geq
0.$
This happens if and only if
$TT^{*}\leq M^{-2}(M^{2}-1)^{2},$
which happens if and only if
$\left\|T\right\|\leq M^{-1}(M^{2}-1).$
Hence,
$\left\|T\right\|=\left\|R_{T}\right\|^{-1}(\left\|R_{T}\right\|^{2}-1)$,
which is the same relation obtained in (11).
## References
* [1] Jon F. Carlson, Douglas N. Clark, Ciprian Foias, and J. P. Williams, Projective Hilbert $A(\mathbb{D})$-modules, New York J. Math, 1 (1994/95), 26–38.
* [2] Stephan Ramon Garcia, The norm and modulus of a Foguel operator, Indiana Univ. Math. J., 58 No. 5 (2009), 2305–2316.
* [3] Paul R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933.
* [4] Gilles Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc., 10 (1997), no. 2, 351 – 369.
|
arxiv-papers
| 2012-06-28T15:23:13 |
2024-09-04T02:49:32.348243
|
{
"license": "Public Domain",
"authors": "Mrinal Raghupathi",
"submitter": "Mrinal Raghupathi",
"url": "https://arxiv.org/abs/1206.6725"
}
|
1206.6927
|
# Consistent Biclustering
Cheryl J. Flynnlabel=e1]cflynn@stern.nyu.edu [ Patrick O.
Perrylabel=e2]pperry@stern.nyu.edu [ New York University Leonard N. Stern
School of Business
New York University
44 West 4th Street
New York, NY 10012-1126
E-mail: e2
###### Abstract
Biclustering, the process of simultaneously clustering observations and
variables, is a popular and effective tool for finding structure in a high-
dimensional dataset. A variety of biclustering algorithms exist, and they have
been applied successfully to data sources ranging from gene expression arrays
to review-website data. Currently, while biclustering appears to work well in
practice, there have been no theoretical guarantees about its performance. We
address this shortcoming with a theorem providing sufficient conditions for
asymptotic consistency when both the number of observations and the number of
variables in the dataset tend to infinity. This theorem applies to a broad
range of data distributions, including Gaussian, Poisson, and Bernoulli. We
demonstrate our results through a simulation study and with examples drawn
from microarray analysis and collaborative filtering.
62H30,
62P10,
62P25,
Biclustering,
block model,
profile likelihood,
consistency,
microarray data,
collaborative filtering,
###### keywords:
[class=AMS]
###### keywords:
name
and
## 1 Introduction
Suppose we are given a data matrix $\boldsymbol{X}=[X_{ij}]$, and our goal is
to cluster the rows and columns of $\boldsymbol{X}$ into meaningful groups.
For example, $X_{ij}$ could be the log activation level of gene $j$ in patient
$i$; our goal is to seek groups of patients with similar genetic profiles,
while at the same time finding groups of genes with similar activation levels.
Alternatively, $X_{ij}$ can indicate whether or not user $i$ reviewed movie
$j$; our goal is to simultaneously cluster the users and the movies. Mirkin
(1996) termed the general clustering process “biclustering,” but it is also
known as direct clustering (Hartigan, 1972), block modeling (Arabie, Boorman
and Levitt, 1978), and co-clustering (Dhillon, 2001).
Empirical results from a broad range of disciplines indicate that biclustering
data is useful in practice. For example, Ungar and Foster (1998) and Hofmann
(1999) found that biclustering helps identify structure in latent class models
for collaborative filtering problems where the data is sparse and diverse
tastes make it difficult to cluster purely based on purchasing or review
habits.
Several biclustering applications exist in the biological sciences. Eisen et
al. (1998) was one of the first papers to note the benefits of clustering
genes and conditions in microarray data, finding that genes with similar
functions cluster together. Recently, Harpaz et al. (2010) applied
biclustering methods to a Food and Drug Adminstration report database,
identifying associations between certain active ingredients and adverse
medical reactions. Several other applications of biclustering exist; see Cheng
and Church (2000),Getz, Levine and Domany (2000), Lazzeroni and Owen (2002),
and Kluger et al. (2003) as well as Madeira and Oliveira (2004) for a
comprehensive survey.
Although their goals are the same, the references above use a variety of
different biclustering algorithms. Clearly, many of these algorithms work well
in practice, but they are often ad-hoc, and there are no rigorous guarantees
as to their performance. In particular, the lack of any notion of consistency
means that practitioners cannot be assured that their discoveries from
biclustering will generalize or be reproducible; collecting more data may lead
to completely different biclusters.
Our first objective in this report is to establish a probabilistic model for
the data matrix, so that biclustering can be formalized as an estimation
problem. Once we have done this, we study a class of biclustering algorithms
based on profile-likelihood, as proposed by Ungar and Foster (1998). We show
that these profile-based procedures are asymptotically consistent as the
dimensions of the matrix tend to infinity, under weak assumptions on the
elements of $\boldsymbol{X}$. Notably, our methods can handle both dense and
sparse data matrices. To our knowledge, this is the first general consistency
result for a biclustering algorithm.
The present treatment was inspired by recent developments in clustering
methods for symmetric binary networks. In that context, $\boldsymbol{X}$ is an
$n$-by-$n$ symmetric binary matrix, and the clusters for the rows of
$\boldsymbol{X}$ are the same as the clusters for the columns of
$\boldsymbol{X}$. Bickel and Chen (2009) used methods similar to those used
for proving consistency of M-estimators to derive results for network
clustering when $n$ tends to infinity. This work was later extended by Choi,
Wolfe and Airoldi (2012), who allow the number of clusters to increase with
$n$; Zhao, Levina and Zhu (2011), who allow for nodes not belonging to any
cluster; and Zhao, Levina and Zhu (2012), who incorporate individual-specific
effects. In parallel to this work, Rohe, Chatterjee and Yu (2011) study the
performance of spectral clustering for symmetric binary networks; and Rohe and
Yu (2012) study spectral methods for unsymmetric binary networks.
The main contribution of our work is recognizing that methods originally
developed for an extremely specialized context (symmetric binary networks) can
be generalized to handle clustering for arbitrary data matrices. Using a
Lindeberg-type condition, we are able to generalize the Bickel and Chen (2009)
results beyond Bernoulli random variables. We have also managed to simplify
the presentation of the main proof. To our knowledge, this is the first time
methodologies for binary networks have been used to study general biclustering
methods.
The main text of the paper is organized as follows. First Section 2 describes
the theoretical setup and Section 3 presents the consistency theorem. Section
4 presents the main steps of the proof. Next, Section 5 corroborates the
theoretical findings through a simulation study, and Section 6 presents an
application to a microarray dataset. Finally, Section 7 presents some
concluding remarks. The appendices include additional technical and empirical
results.
## 2 Block models and profile likelihood
As above, let $\boldsymbol{X}=[X_{ij}]\in\mathbb{R}^{m\times n}$ be a data
matrix. One way to formalize the biclustering problem is to follow the network
clustering literature and posit existence of $K$ _row classes_ and $L$ _column
classes_ , such that the mean value of entry $X_{ij}$ is determined solely by
the classes of row $i$ and column $j$. That is, there is an unknown row class
membership vector $\boldsymbol{c}\in K^{m}$, an unknown column class
membership vector $\boldsymbol{d}\in L^{n}$, and an unknown mean matrix
$\boldsymbol{M}=[\mu_{kl}]\in\mathbb{R}^{K\times L}$ such that
$\operatorname{E}X_{ij}=\mu_{c_{i}d_{j}};$
we refer to this model as a _block model_ , after the related model for
undirected networks proposed by Holland, Laskey and Leinhardt (1983). With a
block model, our goal is to estimate $\boldsymbol{c}$ and $\boldsymbol{d}$. We
do this by assigning labels to the rows and columns of $\boldsymbol{X}$,
codified in vectors $\boldsymbol{g}\in K^{m}$ and $\boldsymbol{h}\in L^{n}$.
Ideally, $\boldsymbol{g}$ and $\boldsymbol{h}$ match $\boldsymbol{c}$ and
$\boldsymbol{d}$. Note that we are assuming that the true numbers of row and
column clusters, $K$ and $L$, are fixed and known.
Analogously to Bickel and Chen (2009), we can employ profile-likelihood
(Murphy and van der Vaart, 2000) for the biclustering task. However, we cannot
assume that the elements of $\boldsymbol{X}$ are Bernoulli random variables.
Instead, we proceed by initially proposing a simple distribution for
$\boldsymbol{X}$ and we deriving the maximum profile likelihood estimator for
$\boldsymbol{c}$ and $\boldsymbol{d}$ under this model. Later, we consider a
far more general distribution for $\boldsymbol{X}$, and we show that the
simple profile-likelihood based estimator is still consistent, even under
model misspecification.
Suppose that the elements of $\boldsymbol{X}$ are independent draws whose
distribution is specified by a single-paremeter exponential family.
Conditional on $\boldsymbol{c}$ and $\boldsymbol{d}$, entry $X_{ij}$ has
density $g(x;\eta_{c_{i}d_{j}})$ with respect to some $\sigma$-finite measure
$\nu$, where
$g(x;\eta)=\exp\\{x\eta-\psi(\eta)\\};$
$\psi(\eta)$ is the cumulant function, and
$\eta_{kl}=(\psi^{\prime})^{-1}(\mu_{kl})$. With labels $\boldsymbol{g}$ and
$\boldsymbol{h}$, the complete data log-likelihood is
$\operatorname{l}(\boldsymbol{g},\boldsymbol{h},\boldsymbol{M})=mn\sum_{k,l}\hat{p}_{k}\,\hat{q}_{l}\,\\{\bar{X}_{kl}\,\eta_{kl}-\psi(\eta_{kl})\\},$
where
$\hat{p}_{k}=\frac{1}{m}\sum_{i}\operatorname{I}(g_{i}=k),\qquad\hat{q}_{l}=\frac{1}{n}\sum_{j}\operatorname{I}(h_{j}=l),$
and
$\bar{X}_{kl}=\frac{\sum_{i,j}X_{ij}\operatorname{I}(g_{i}=k,h_{j}=l)}{\sum_{i,j}\operatorname{I}(g_{i}=k,h_{j}=l)}.$
The profile log-likelihood is
$\operatorname{pl}(\boldsymbol{g},\boldsymbol{h})=\sup_{\boldsymbol{M}}\operatorname{l}(\boldsymbol{g},\boldsymbol{h},\boldsymbol{M})=mn\sum_{k,l}\hat{p}_{k}\,\hat{q}_{l}\,\psi^{\ast}(\bar{X}_{kl}),$
where $\psi^{\ast}(x)=\sup_{\eta}\\{x\eta-\psi(\eta)\\}$ is the convex
conjugate of $\psi$, known as the _rate function_ in the large deviations
literature (Dembo and Zeitouni, 1998). Following the above derivation, it is
natural to estimate $\boldsymbol{c}$ and $\boldsymbol{d}$ by the label vectors
which maximize $\operatorname{pl}(\boldsymbol{g},\boldsymbol{h})$.
In the sequel, we consider a far more general setting. We consider criterion
functions of the form
$F(\boldsymbol{g},\boldsymbol{h})=mn\sum_{kl}\hat{p}_{k}\,\hat{q}_{l}\,f(\bar{X}_{kl}/\rho),$
(2.1)
where $f$ is any smooth convex function and $\rho$ is a scale parameter.
Borrowing terminology from the large deviations literature, we refer to $f$ as
the rate function of the criterion; though, since we allow $f$ to take
negative values, we do not require that $f$ be a rate function in the
strictest sense. We permit the elements of $\boldsymbol{X}$ to have different
distributions, allowing for heteroscedasticity and model misspecification. We
show that under mild technical conditions, the maximizer of $F$ is a
consistent estimator of the true row and column classes.
## 3 Consistency results
To prove consistency, we need to work with a sequence $\boldsymbol{X}_{n}$ of
data matrices. Suppose that $\boldsymbol{X}_{n}\in\mathbb{R}^{m\times n}$ and
$m=m(n)$ with $n/m\to\gamma$ for some finite constant $\gamma>0$. Suppose that
for each $n$ there exists a row class membership vector $\boldsymbol{c}_{n}\in
K^{m}$ and a column class membership vector $\boldsymbol{d}_{n}\in L^{n}$. We
assume that there exist vectors $\boldsymbol{p}\in\mathbb{R}^{K}$ and
$\boldsymbol{q}\in\mathbb{R}^{L}$ such that $\hat{p}_{k}(\boldsymbol{c})\to
p_{k}$ and $\hat{q}_{l}(\boldsymbol{d})\to q_{l}$ as $n\to\infty$ almost
surely for all k and l. This assumption is satisfied, for example, if the
class labels are independently drawn from a multinomial distribution. When
there is no ambiguity, we omit the subscript $n$.
Assume that the mean of element $X_{ij}$ depends only on the row and column
memberships $c_{i}$ and $d_{j}$, so that
$\operatorname{E}(X_{ij}\mid\boldsymbol{c},\boldsymbol{d})=\mu_{c_{i}d_{j}}$
for some matrix $\boldsymbol{M}=[\mu_{kl}]\in\mathbb{R}^{K\times L}$, possibly
varying with $n$. To model sparsity in $\boldsymbol{X}$, we allow
$\boldsymbol{M}$ to tend to $\mathbf{0}$. To avoid degeneracy, we suppose that
there exists a sequence $\rho$ and a fixed matrix
$\boldsymbol{M}_{0}\in\mathbb{R}^{K\times L}$ such that
$\rho^{-1}\boldsymbol{M}\to\boldsymbol{M}_{0}$.
Define the normalized confusion matrices
$\boldsymbol{C}(\boldsymbol{g})\in\mathbb{R}^{K\times K}$ and
$\boldsymbol{D}(\boldsymbol{h})\in\mathbb{R}^{L\times L}$ by
$C_{ak}(\boldsymbol{g})=\frac{1}{m}\sum_{i}I(c_{i}=a,g_{i}=k),\qquad
D_{bl}(\boldsymbol{h})=\frac{1}{n}\sum_{j}I(d_{j}=b,h_{j}=l).$
Entry $C_{ak}$ is the proportion of nodes with class $a$ and label $k$; entry
$D_{bl}$ is defined similarly. These matrices are normalized so that
$\boldsymbol{C}^{T}\boldsymbol{1}=\hat{\boldsymbol{p}}$ and
$\boldsymbol{D}^{T}\boldsymbol{1}=\hat{\boldsymbol{q}}$.
We only consider nontrivial partitions; to this end, for $\varepsilon>0$,
define
$\mathcal{J}_{\varepsilon}=\\{\boldsymbol{g},\boldsymbol{h}:\hat{p}_{k}(\boldsymbol{g})>\varepsilon,\hat{q}_{l}(\boldsymbol{h})>\varepsilon\\}.$
For fixed convex rate function $f$, we let $F$ be a criterion function as in
(2.1).
### 3.1 Assumptions
Denote by $\mathcal{M}_{0}\in\mathbb{R}$ the convex hull of the entries of
$\boldsymbol{M}_{0}$. Let $\mathcal{M}$ be a neighborhood of
$\mathcal{M}_{0}$. We require the following assumptions:
1. (A1)
The biclusters are identifiable. No two rows of $\boldsymbol{M}_{0}$ are
equal, and no two columns of $\boldsymbol{M}_{0}$ are equal.
2. (A2)
The rate function is locally strictly convex. That is,
$f^{\prime\prime}(\mu)>0$ when $\mu\in\mathcal{M}$.
3. (A3)
The third derivative of the rate function is locally bounded. That is, $\lvert
f^{\prime\prime\prime}(\mu)\rvert$ is bounded when $\mu\in\mathcal{M}$.
4. (A4)
The average variance of the elements is of the same order as $\rho$.
$\limsup_{n\to\infty}\frac{1}{\rho
mn}\sum_{i,j}\operatorname{E}[(X_{ij}-\mu_{c_{i}d_{j}})^{2}\mid\boldsymbol{c},\boldsymbol{d}]<\infty.$
5. (A5)
The mean matrix does not converge to zero too quickly. That is,
$\limsup_{n\to\infty}\rho n=\infty$.
6. (A6)
The elements satisfy a Lindeberg condition. For all $\varepsilon>0$,
$\lim_{n\to\infty}\frac{1}{\rho^{2}mn}\sum_{i,j}\operatorname{E}[(X_{ij}-\mu_{c_{i}d_{j}})^{2}\operatorname{I}(\lvert
X_{ij}-\mu_{c_{i}d_{j}}\rvert>\varepsilon\sqrt{mn}\rho)\mid\boldsymbol{c},\boldsymbol{d}]=0.$
Assumption (A4) is trivially satisfied for dense data and is satisfied for
Binomial and Poisson data so long as
$\rho^{-1}\boldsymbol{M}\to\boldsymbol{M}_{0}$. However, this assumption
cannot handle arbitrary sparsity. For example, if the elements of
$\boldsymbol{X}$ are distributed as Negative Binomial random variables, then
assumption (A4) requires that the product of the mean and the dispersion
parameter does not tend to infinity. In other words, for sparse count data,
the counts cannot be too heterogeneous.
Assumption (A5) places a sparsity restriction on the mean matrix. Zhao, Levina
and Zhu (2012) used the same assumption to establish weak consistency for
network clustering. A variant Lyaponuv’s condition (Varadhan, 2001) implies
(A6). That is, if
$\lim_{n\to\infty}\frac{1}{(\rho\sqrt{mn})^{2+\delta}}\sum_{i,j}\operatorname{E}|X_{ij}-\mu_{c_{i}d_{j}}|^{2+\delta}=0$
for some $\delta>0$, then (A6) follows. In particular, for dense data ($\rho$
bounded away from zero), uniformly bounded $(2+\delta)$ absolute central
moments for any $\delta>0$ is sufficient. For certain types of sparse data
(Bernoulli or Poisson data with $\rho$ converging to zero), (A5) is a
sufficient condition for (A6).
### 3.2 Main results
Given the above setup and assumptions, it follows that the maximizer of
$F(\boldsymbol{g},\boldsymbol{h})$ is a consistent estimator of the true row
and column labels.
###### Theorem 3.1.
Fix any $\varepsilon>0$ with $\varepsilon<\min_{a}\\{p_{a}\\}$ and
$\varepsilon<\min_{b}\\{q_{b}\\}$. Let
$(\boldsymbol{\hat{g}},\boldsymbol{\hat{h}})$ satisfy
$F(\boldsymbol{\hat{g}},\boldsymbol{\hat{h}})=\max_{\mathcal{J}_{\varepsilon}}F(\boldsymbol{g},\boldsymbol{h})$.
If assumptions (A1)–(A6) hold, then all limit points of
$\boldsymbol{C}(\boldsymbol{\hat{g}})$ and
$\boldsymbol{D}(\boldsymbol{\hat{h}})$ are permutations of diagonal matrices,
i.e. the proportions of mislabeled rows and columns converge to zero in
probability.
The statement of the theorem is somewhat awkward, because we can permute the
row or column labels and get the same value of the criterion function, $F$.
Thus, $F$ has multiple maximizers, and it is only possible to recover the true
classes up to a permutation of labels.
From the discussion in Section 2, it follows that maximizing the profile log-
likelihood associated with a single-parameter exponential family may be a
reasonable biclustering procedure. Furthermore, the proof of this result does
not require the distribution of the data to be correctly specified and allows
for possible model misspecification so long as (A1)-(A6) are satisfied. This
implies that this result can be applied to binary matrices, count data, and
continuous data which could be reasonably modeled by Binomial, Poisson, and
Gaussian distributed data, respectively, making this a suitable result for our
motivating examples.
Under stronger distributional assumptions, we can use the methods of the proof
to establish finite-sample results. For example, if we assume that the
elements of $\boldsymbol{X}$ are Gaussian, then the following result holds.
The proof of the finite-sample result follows the same outline as the
asymptotic result; see Appendix B.
###### Theorem 3.2.
Fix any $\varepsilon>0$. Let $(\hat{\boldsymbol{g}},\hat{\boldsymbol{h}})$
satisfy
$F(\hat{\boldsymbol{g}},\hat{\boldsymbol{h}})=\max_{\mathcal{J}_{\varepsilon}}F(\boldsymbol{g},\boldsymbol{h})$.
If the elements of $\boldsymbol{X}$ are independent Gaussian random variables
with constant variance $\sigma^{2}$ and assumptions (A1)-(A3) hold, then for
any
$0<\delta<\min\Big{\\{}1,\frac{8c\sigma\min\\{K^{2},L^{2}\\}}{\tau\varepsilon^{2}}\Big{\\}}$,
$\Pr\Big{(}\big{(}\boldsymbol{C}(\boldsymbol{\hat{g}}),\boldsymbol{D}(\boldsymbol{\hat{h}})\big{)}\notin\mathcal{P}_{\delta}\cap\mathcal{Q}_{\delta}\Big{)}\leq
2K^{m+1}L^{n+1}\exp\Big{\\{}-\frac{T_{n}\tau^{2}\varepsilon^{4}\delta^{2}}{256c^{2}\sigma^{2}\min\\{K^{4},L^{4}\\}}\Big{\\}},$
where $c=\sup|f^{\prime}(\mu)|$ for $\mu$ in $\mathcal{M}$,
$T_{n}=\inf_{k,l}\Big{\\{}\sum_{i}\sum_{j}\operatorname{I}(g_{i}=k,h_{j}=l)|(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}\Big{\\}}.$
and $\mathcal{P}_{\delta}$ and $\mathcal{Q}_{\delta}$ are neighborhoods of the
optimal confusion matrices defined in the sequel in equation (4.1).
To prove the main theorem, we first establish that in the limit, $F$ is close
to a nonrandom “population version,” $G$. Then, we establish that $G$ is
maximized at the true class labels. Finally, we show that outside any
neighborhood around the true class labels, $G$ is smaller than at the true
values.
As alluded to in the Introduction, the main thrust of the proof is similar to
that used in the literature on clustering for symmetric binary networks
initiated by Bickel and Chen (2009) and extended by Choi, Wolfe and Airoldi
(2012), Zhao, Levina and Zhu (2011) and Zhao, Levina and Zhu (2012). There are
three main points of departure from this previous work. First, the literature
on clustering for symmetric binary networks works only with Bernoulli random
variables, whereas we consider arbitrary distributions that satisfy a
Lindeberg-type condition. Second, these papers work with uniformly Lipschitz
criterion functions, while we consider potentially unbounded criterion
functions. Lastly, the results for symmetric binary networks require an
additional assumption: that the criterion function is maximized at the
population parameters. We show that this condition is satisfied as long as the
parameters are identifiable and the rate function is strictly convex.
## 4 Proof of consistency theorem
In this section, we continue with the setup and notation of Section 3. To
prove Theorem 3.1, we need to introduce some additional notation.
Define expectation matrix
$\boldsymbol{E}(\boldsymbol{g},\boldsymbol{h})\in\mathbb{R}^{K\times L}$ with
$E_{kl}(\boldsymbol{g},\boldsymbol{h})=\operatorname{E}(\bar{X}_{kl}(\boldsymbol{g},\boldsymbol{h})\mid\boldsymbol{c},\boldsymbol{d})=\frac{[\boldsymbol{C}^{T}\,\boldsymbol{M}\,\boldsymbol{D}]_{kl}}{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}},$
where $\boldsymbol{C}=\boldsymbol{C}(\boldsymbol{g})$ and
$\boldsymbol{D}=\boldsymbol{D}(\boldsymbol{h})$. Also, define normalized
residual matrix
$\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\in\mathbb{R}^{K\times L}$ by
$\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})=\rho^{-1}\\{\bar{\boldsymbol{X}}(\boldsymbol{g},\boldsymbol{h})-\boldsymbol{E}(\boldsymbol{g},\boldsymbol{h})\\}.$
The weak law of large numbers establishes that for fixed $\boldsymbol{g}$ and
$\boldsymbol{h}$, the convergence
$R_{kl}(\boldsymbol{g},\boldsymbol{h})\overset{\mathit{P}}{\to}0$ holds. We
can prove a stronger result, that this convergence is uniform over all
$\boldsymbol{g}$ and $\boldsymbol{h}$.
###### Lemma 4.1.
Under assumptions (A1)–(A6), for all $\varepsilon>0$,
$\sup_{\mathcal{J}_{\varepsilon}}\|\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\|_{\infty}\overset{\mathit{P}}{\to}0,$
where $\|\mathbf{A}\|_{\infty}=\max_{k,l}|A_{kl}|$ for any matrix
$\mathbf{A}$.
With Lemma 4.1 (proved in Appendix A), we can establish that in the limit,
$F(\boldsymbol{g},\boldsymbol{h})$ is close to its “population version,” which
depends only on $\boldsymbol{C}$ and $\boldsymbol{D}$. To define this
population version, first, for each $M$, define $\mathcal{S}_{M}$ to be the
set of $M\times M$ matrices with nonnegative entries summing to one:
$\mathcal{S}_{M}=\\{\boldsymbol{X}\in\mathbb{R}_{+}^{M\times
M}:\mathbf{1}^{T}\boldsymbol{X}\boldsymbol{1}=1\\}.$ Next, define function
$G_{\boldsymbol{M}_{0}}:\mathcal{S}_{K}\times\mathcal{S}_{L}\to\mathbb{R}$ to
be the population version of $F$:
$\displaystyle G_{\boldsymbol{M}_{0}}(\boldsymbol{C},\boldsymbol{D})$
$\displaystyle=\sum_{k,l}[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}\,[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}\,f\Big{(}\frac{[\boldsymbol{C}^{T}\boldsymbol{M}_{0}\boldsymbol{D}]_{kl}}{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}\,[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}}\Big{)}.$
###### Lemma 4.2.
$F$ is close to its population version in the sense that, for all
$\varepsilon>0$,
$\sup_{\mathcal{J}_{\varepsilon}}|F(\boldsymbol{g},\boldsymbol{h})-G_{\boldsymbol{M}_{0}}(\boldsymbol{C},\boldsymbol{D})|\overset{\mathit{P}}{\to}0.$
This is a direct consequence of Lemma 4.1 and assumption (A3), which implies
that $f$ is locally Lipschitz; see Appendix A for details.
Once we have established that $F$ is close to its population version, our next
task is to show that the population version is maximized at the true class
labels. To do this, for $\delta>0$, first define
$\displaystyle\mathcal{P}_{\delta}$
$\displaystyle=\\{\boldsymbol{C}\in\mathcal{S}_{K}:\max_{a\neq
a^{\prime}}C_{ak}C_{a^{\prime}k}<\delta\\}$ (4.1a) and
$\displaystyle\mathcal{Q}_{\delta}$
$\displaystyle=\\{\boldsymbol{D}\in\mathcal{S}_{L}:\max_{b\neq
b^{\prime}}D_{bl}D_{b^{\prime}l}<\delta\\}.$ (4.1b)
A permutation of a diagonal matrix has only one non-zero entry in each column,
so taking $\delta$ close to zero restricts the confusion matrices to be close
to permutations of diagonal matrices.
###### Lemma 4.3.
If $\min_{a}\\{[\boldsymbol{C}\boldsymbol{1}]_{a}\\}>\eta,$
$\min_{b}\\{[\boldsymbol{D}\boldsymbol{1}]_{b}\\}>\eta,$ and
$(\boldsymbol{C},\boldsymbol{D})\notin\mathcal{P}_{\delta}\times\mathcal{Q}_{\delta}$,
then $G_{\boldsymbol{M}_{0}}(\boldsymbol{C},\boldsymbol{D})$ is small, in the
sense that
$G_{\boldsymbol{M}_{0}}(\boldsymbol{C},\boldsymbol{D})-\sum_{a,b}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f([\boldsymbol{M}_{0}]_{ab})\leq-\kappa\eta^{2}\delta,$
where $\kappa$ is a constant independent of $\delta$ and $\eta$.
We prove Lemma 4.3 in Appendix A.
Next, we prove the consistency theorem.
###### Proof of Theorem 3.1.
Fix $\delta>0$ and define $\mathcal{P}_{\delta}$ and $\mathcal{Q}_{\delta}$ as
in (4.1). We will show that if
$(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}$ and if
$(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h}))\notin(\mathcal{P}_{\delta},\mathcal{Q}_{\delta})$,
then $F(\boldsymbol{g},\boldsymbol{h})<F(\boldsymbol{c},\boldsymbol{d})$ with
probability approaching one. Moreover, this inequality holds uniformly over
all such choices of $(\boldsymbol{g},\boldsymbol{h})$. Since $\delta$ is
arbitrary, this implies that $\boldsymbol{C}(\boldsymbol{\hat{g}})$ and
$\boldsymbol{D}(\boldsymbol{\hat{h}})$ converge to permutations of diagonal
matrices, i.e. the proportions of misclassified rows and columns converge to
zero.
Set
$r_{n}=\sup_{\mathcal{J}_{\varepsilon}}|F(\boldsymbol{g},\boldsymbol{h})-G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h})|$.
Suppose $(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}$. In this
case,
$\displaystyle
F(\boldsymbol{g},\boldsymbol{h})-F(\boldsymbol{c},\boldsymbol{d})$
$\displaystyle\leq
2r_{n}+\\{G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h}))-G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{c}),\boldsymbol{D}(\boldsymbol{d}))\\}$
$\displaystyle=2r_{n}+\big{\\{}G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h}))-\sum_{a,b}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f([\boldsymbol{M}_{0}]_{ab})\big{\\}}.$
Pick $\eta>0$ smaller than $\min_{a}\\{p_{a}\\}$ and $\min_{b}\\{q_{b}\\}$. By
assumption, the true row and column class proportions converge to
$\boldsymbol{p}$ and $\boldsymbol{q}$. Thus, for all $\boldsymbol{g}\in K^{m}$
and $\boldsymbol{h}\in L^{n}$, for $n$ large enough,
$[\boldsymbol{C}(\boldsymbol{g})]_{a}\geq\eta$ and
$[\boldsymbol{D}(\boldsymbol{h})]_{b}\geq\eta$; this holds uniformly over all
choices of $(\boldsymbol{g},\boldsymbol{h})$.
Applying Lemma 4.3, to the second term in the inequality, we get that with
probability approaching one,
$F(\boldsymbol{g},\boldsymbol{h})-F(\boldsymbol{c},\boldsymbol{d})\leq
2r_{n}-\kappa\eta^{2}\delta$
for all $(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}$ such
that
$(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h}))\notin\mathcal{P}_{\delta}\times\mathcal{Q}_{\delta}$.
By Lemma 4.2, $r_{n}\overset{\mathit{P}}{\to}0$. Thus, with probability
approaching one,
$(\boldsymbol{C}(\boldsymbol{\hat{g}}),\boldsymbol{D}(\boldsymbol{\hat{h}}))\in\mathcal{P}_{\delta}\times\mathcal{Q}_{\delta}$.
Since this result holds for all $\delta$, all limit points of
$\boldsymbol{C}(\boldsymbol{\hat{g}})$ and
$\boldsymbol{D}(\boldsymbol{\hat{h}})$ must be permutations of diagonal
matrices. ∎
## 5 Empirical evaluation
We study the performance of profile log-likelihood for biclustering Bernoulli,
Poisson, Gaussian, and Non-standardized Student’s t data. We consider the
following three rate functions:
$\displaystyle f_{\text{Bernoulli}}(\mu)$
$\displaystyle=\mu\log\mu+(1-\mu)\log(1-\mu),$ (5.1a) $\displaystyle
f_{\text{Poisson}}(\mu)$ $\displaystyle=\mu\log\mu-\mu,$ (5.1b) $\displaystyle
f_{\text{Gaussian}}(\mu)$ $\displaystyle=\mu^{2}/2.$ (5.1c)
We use the appropriate rate function for the Bernoulli and Gaussian
simulations. To study performance under model misspecification, we compare the
performance of the Poisson and Gaussian rate functions for the Poisson
simulations and use the Gaussian rate function for the t simulations.
For the Poisson and Gaussian rate functions (5.1b) and (5.1c), the maximizer
of the criterion function (2.1) does not depend on the scale factor $\rho$.
This is immediately obvious in the Gaussian case. For the Poisson case, the
function
$f_{Poisson}(\mu/\rho)=\frac{1}{\rho}\mu\log(\mu)-\frac{1}{\rho}\mu(1+\log(\rho))$.
When summed over all biclusters, the second term in this sum is equal to a
constant so the value of $\mu$ which maximizes $f_{Poisson}(\mu/\rho)$ does
not depend on the value of $\rho$.
This is not the case for the Binomial rate function (5.1a), but the parameter
$\rho$ is not identifiable in practice so it does not make sense to try to
estimate it. For our simulations we use which maximizes $\rho=1$ in the
fitting procedure, regardless of the true scale factor for the mean matrix
$\boldsymbol{M}$. Even though in the simulations assumption (A1) doesn’t hold
for this choice of $\rho$, we still get consistency, because the maximizer of
the criterion with $f_{\text{Bernoulli}}(\mu)$ is close to the maximizer with
$f_{\text{Poisson}}(\mu)$. See Perry and Wolfe (2012) for discussion of a
related phenomenon.
To maximize the profile log-likelihood, we compute initial partitions for the
rows and columns. Then, we iteratively update the cluster assignments in a
greedy manner, based on the Kernighan-Lin heuristic (Kernighan and Lin, 1970)
employed by Newman (2006):
1. 1.
For each row and column, we compute the optimal label assignment while keeping
the labels of all other rows and columns fixed; we also record the improvement
made by making this assignment.
2. 2.
In order of the local improvements recorded in step 1, we perform the label
reassignments determined in step 1. Note that these assignments are no longer
locally optimal since the labels of many of the rows and columns change during
this step.
3. 3.
Out of those labels considered in step 2, we choose the one which has the
highest profile likelihood.
We iteratively perform steps 1–3 until the profile likelihood converges. The
algorithm is not guaranteed to converge to a global optimum, but it seems to
perform well in practice. If implemented efficiently, each update to the log-
likelihood can be done in a constant number of operations. Therefore, the most
complex part of this algorithm is the ordering of the local improvements. On
average, using a conventional sorting algorithm this requires $O(n\log(n))$
operations. For comparison, the algorithmic costs for spectral-based
biclustering algorithms are at least $O(n^{2})$, the cost of computing the top
singular vectors of the data matrix using an indirect method (Golub and Loan,
1996).
In our simulations, we report the proportion of misclassified rows and columns
by the profile-likelihood-based method (PL), which Theorem 3.1 guarantees to
be consistent. For the Poisson simulations we report the profile-liklihood-
based method with the Poisson rate function (PL-Pois) and with the Gaussian
rate function (PL-Gaus). We also report misclassification errors for $k$-means
applied separately to the rows and columns (KM) and for the DI-SIM
biclustering algorithm (DS) of Rohe and Yu (2012). For the profile-likelihood-
based method, we compute initial partitions for the rows and columns by
applying $k$-means separately to the rows and the columns. In principle any
initialization can be used. The algorithm is run once for each simulation.
For the Poisson simulation, we simulate from a block model with $K=2$ row
clusters and $L=3$ column clusters. We vary the number of columns, $n$,
between 200 to 1500 and we take the number of rows as $m=\gamma n$ where
$\gamma\in\\{0.5,1,2\\}$.
We set the row and column class memberships as independent multinomials with
probabilities $\boldsymbol{p}=(0.3,0.7)$ and $\boldsymbol{q}=(0.2,0.3,0.5)$.
We choose the matrix of block parameters to be
$\boldsymbol{M}=[\mu_{ab}]=\frac{b}{\sqrt{n}}\begin{pmatrix}0.92&0.77&1.66\\\
0.17&1.41&1.45\end{pmatrix},$
where $b$ is chosen between 5 and 20; the entries of the matrix were chosen
randomly, uniformly on the interval $[0,2]$. Note that
$\boldsymbol{M}\to\boldsymbol{0}$, so the data matrix is sparse. We generate
the data conditional on the row and column classes as
$X_{ij}\mid\boldsymbol{c},\boldsymbol{d}\sim\mathrm{Poisson}(\mu_{c_{i}d_{j}}).$
Figure 1 presents the average bicluster misclassification rates for each
sample size and Tables 1 and 2 report the standard deviations. In all of the
scenarios considered, the biclustering based on the profile log-likelihood
criterion performs at least as well as the other methods and shows signs of
convergence even if the rate function is misspecified. Although the $k$-means
algorithm and the DI-SIM algorithm perform well in some settings, in other
settings the performance is not well-behaved. We also see that the standard
deviation of the misclassification rate tends to zero for the PL bicustering,
but not for the other two algorithms. The results in McSherry (2001) and Rohe
and Yu (2012) suggest that the DI-SIM algorithm may be a consistent
biclustering algorithm, but require strong assumptions that do not hold in our
simulations. Specifically, the former requires that the variance of the
entries in $\boldsymbol{M}$ to be $\gg\log^{6}(n)/n$ and the later requires
that the minimum expected degree eventually exceeds
$\sqrt{2}n/\sqrt{\log(n)}$. Therefore it is not clear if the convergence of
the DI-SIM algorithm is slow or if the results do not apply in this setting.
Appendix C describes in detail the simulations for Bernoulli, Gaussian, and t
data. For the Bernoulli simulation, the profile-likelihood-based method
outperforms the other procedures. For the Gaussian and t-distributed data, the
DI-SIM algorithm is much more competitive, but our algorithm still beats it in
almost all cases.
Overall, the simulation results corroborate the conclusions of Theorem 3.1 and
support the use of biclustering based on the profile log-likelihood criterion.
Figure 1: Average misclassification rates for Poisson example over 100
simulations. Table 1: Standard deviations for Poisson example over 100
simulations for the profile likelihood methods based on Poisson and Gaussian
criteria. $m=0.5n$
---
| PL-Pois | PL-Norm
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.0683 | 0.0177 | 0.0554 | 0.1376 | 0.0254 | 0.0566
500 | 0.0112 | 0.0046 | 0.0392 | 0.0161 | 0.0069 | 0.0391
1000 | 0.0050 | 0.0013 | 0.0002 | 0.0072 | 0.0022 | 0.0004
1500 | 0.0034 | 0.0007 | 0.0001 | 0.0046 | 0.0012 | 0.0002
$m=n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.0158 | 0.0041 | 0.0777 | 0.0481 | 0.0061 | 0.0771
500 | 0.0039 | 0.0008 | 0.0000 | 0.0064 | 0.0014 | 0.0000
1000 | 0.0015 | 0.0002 | 0.0000 | 0.0019 | 0.0004 | 0.0000
1500 | 0.0007 | 0.0000 | 0.0000 | 0.0012 | 0.0001 | 0.0000
$m=2n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.0063 | 0.0009 | 0.1259 | 0.0010 | 0.0002 | 0.0124
500 | 0.0012 | 0.0000 | 0.0552 | 0.0002 | 0.0000 | 0.0054
1000 | 0.0002 | 0.0001 | 0.0000 | 0.0001 | 0.0000 | 0.0000
1500 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Table 2: Standard deviations for Poisson example over 100 simulations for the
K-means and DI-SIM algorithms. $m=0.5n$
---
| KM | DS
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.1325 | 0.0635 | 0.0487 | 0.0488 | 0.0355 | 0.0385
500 | 0.0666 | 0.0094 | 0.0357 | 0.0209 | 0.0090 | 0.0384
1000 | 0.0134 | 0.0029 | 0.0005 | 0.0090 | 0.0394 | 0.1069
1500 | 0.0061 | 0.0015 | 0.0445 | 0.0065 | 0.0387 | 0.1113
$m=n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.1205 | 0.0102 | 0.0811 | 0.0298 | 0.0521 | 0.0798
500 | 0.0124 | 0.0021 | 0.0000 | 0.0121 | 0.0534 | 0.0926
1000 | 0.0039 | 0.0401 | 0.0405 | 0.0043 | 0.1162 | 0.1303
1500 | 0.0017 | 0.0002 | 0.0561 | 0.0018 | 0.0761 | 0.1050
$m=2n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.0104 | 0.0009 | 0.1100 | 0.0056 | 0.0007 | 0.0858
500 | 0.0017 | 0.0000 | 0.0668 | 0.0017 | 0.0002 | 0.0844
1000 | 0.0003 | 0.0000 | 0.1020 | 0.0002 | 0.0000 | 0.1174
1500 | 0.0001 | 0.0000 | 0.0934 | 0.0001 | 0.0000 | 0.1077
## 6 Application
In this section we use profile-likelihood-based biclustering to reveal
structure in two high-dimensional datasets. For each example, we maximize the
profile log-likelihood using the algorithm described in Section 5.
### 6.1 AGEMAP
Biclustering is commonly used for microarray data to visualize the activation
patterns of thousands of genes simultaneously. It is used to identify patterns
and discover distinguishing properties between genes and individuals. We use
the AGEMAP dataset (Zahn et al., 2007) to illustrate this process.
AGEMAP is a large microarray data set containing the log expression levels for
40 mice across 8932 genes measured on 16 different tissue types. For this
analysis, we restrict attention to two tissue types: cerebellum and cerebrum.
The 40 mice in the dataset belong to four age groups, with five males and five
females in each group. One of the mice is missing data for the cerebrum tissue
so it has been removed from the dataset.
To study the relationship between mice and gene expression levels, we
bicluster the 39 $\times$ 17864 residual matrix computed from the least
squares fit to the multiple linear regression model
$Y_{ij}=\beta_{0j}+\beta_{1j}A_{i}+\beta_{2j}S_{i}+\varepsilon_{ij},$
where $Y_{ij}$ is the log-activation of gene $j$ in mouse $i$, $A_{i}$ is the
age of mouse $i$, $S_{i}$ indicates if mouse $i$ is male, $\varepsilon_{ij}$
is a random error, and $(\beta_{0j},\beta_{1j},\beta_{2j})$ is a gene-specific
coefficient vector.
The entries of the residual matrix are not independent (for example, the sum
of each column is zero). Also, the responses of many genes are likely
correlated with each other. Thus, the block model required by Theorem 3.1 is
not in force, so its conclusion will not hold unless the dependence between
the residuals is negligible. In light of this caveat, the example should be
considered as exploratory data analysis.
We perform biclusterring using profile log-likelihood based on the Gaussian
criterion (5.1c). Based on the preliminary analysis in Perry and Owen (2010),
it appears that there are three mice groups. To determine an appropriate
number of gene groups we experiment with values between two and five. Using
four gene groups appears to give a reasonable representation of the data. The
heatmap presented in Figure 2 shows the results.
Figure 2: Heatmap generated from AGEMAP residual data reflecting the varying
expression patterns in the different biclusters. The colors for the matrix
entries correspond encode to the first quartile, the middle two quartiles, and
the upper quartile.
Although the expression levels for the fourth gene group appear to be fairly
neutral across the three mouse groups, the first three gene groups each have a
visually apparent pattern. It appears that a mouse can have high expression
levels at most two of the first three gene groups. Mouse group 1 has high
expression for gene groups 2 and 3; mouse group 2 has high expression for gene
group 1; and mouse group 3 has high expression for gene groups 1 and 3.
The three clusters of mice agree with those found by Perry and Owen (2010).
That analysis identified the mouse clusters, but could not attribute meaning
to them. The bicluster-based analysis has deepend our understanding of the
three mouse clusters while suggesting some interesting interactions between
the genes.
### 6.2 MovieLens
Collaborative filtering uses the behavior of similar consumers to provide
better recommendations of products to new individuals. Since consumer habits
likely vary depending on products, biclustering can help identify structure in
the data and identify groups of consumers and groups of products with similar
patterns. As an application of this we apply biclustering to the MovieLens
data set (GroupLens Research Project, 1998). Appendix D describes the results
from this application which splits individuals who rate movies from all time
periods from individuals who only rate recent movies.
## 7 Discussion
In this report we developed a statistical setting for studying the performance
of biclustering algorithms. Under the assumption that the data follows a
stochastic block model, we derived sufficient conditions for an algorithm
based on maximizing the profile-likelihood to be consistent. This is the first
theoretical guarantee for any biclustering algorithm which can be applied to a
broad range of data distributions and can handle both sparse and dense data
matrices. These results can easily be extended to symmetric matrices. Our
empirical comparisons demonstrated that the method performs well in a variety
of situations and can outperform existing procedures.
It is important to note that the theoretical results operate under the
assumption that the true number of row and column classes are known. In
practice, this is a simplifying assumption and the optimal number of
biclusters needs to be inferred from the data. While this is an open problem
in the biclustering literature, the formalization of biclustering as an
estimation process may help identify parallels that can be drawn from
classical model selection procedures to this setting and is an interesting
topic for future research.
Applying the profile-likelihood based biclustering algorithm to real data
revealed several interesting findings. Biclustering the genes and mice in the
AGEMAP data exposed an interesting pattern in the expression of certain genes
and we found that at most two gene groups can have high expression levels for
any one mouse. Our results from the MovieLens dataset identified a distinct
difference in movie review behavior and split individuals who only rate recent
movies from individuals who rate movies from all time periods. The consistency
theorem proved in this report gives conditions under which we can have
confidence in the robustness of these findings.
## Appendix A Additional technical results
###### Lemma A.1.
For each $n$, let $X_{n,m}$, $1\leq m\leq n$, be independent random variables
with $\operatorname{E}X_{n,m}=0$. Let $\rho_{n}$ be a sequence of positive
numbers. Let $\mathcal{I}_{n}$ be a subset of the powerset $2^{[n]}$, with
$\inf\\{|I|:I\in\mathcal{I}_{n}\\}\geq L_{n}.$ Suppose
1. (i)
$\frac{1}{n\rho_{n}}\sum_{m=1}^{n}\operatorname{E}\lvert X_{n,m}\rvert^{2}$ is
uniformly bounded in $n$;
2. (ii)
For all $\varepsilon>0$,
$\frac{1}{n\rho_{n}^{2}}\sum_{m=1}^{n}\operatorname{E}(\lvert
X_{n,m}\rvert^{2};\lvert X_{n,m}\rvert>\varepsilon\sqrt{n}\rho_{n})\to 0;$
3. (iii)
$\varlimsup_{n\to\infty}\frac{n}{L_{n}}<\infty;$
4. (iv)
$\varlimsup_{n\to\infty}\frac{\log\lvert\mathcal{I}_{n}\rvert}{\sqrt{n}}<\infty.$
5. (v)
$\varlimsup_{n\to\infty}\rho_{n}\sqrt{n}=\infty.$
Then
$\sup_{I\in\mathcal{I}_{n}}\Big{|}\frac{1}{\rho_{n}\lvert I\rvert}\sum_{m\in
I}X_{n,m}\Big{|}\overset{\mathit{P}}{\to}0.$
###### Proof.
Let $\varepsilon>0$ be arbitrary. Define
$Y_{n,m}=\rho_{n}^{-1}X_{n,m}\operatorname{I}(\lvert
X_{n,m}\rvert\leq\varepsilon\sqrt{n}\rho_{n}),$ and note that
$\Pr(Y_{n,m}\neq\rho_{n}^{-1}X_{n,m}\text{ for some $1\leq m\leq
n$})\leq\sum_{m=1}^{n}\Pr(\lvert X_{n,m}\rvert>\varepsilon\sqrt{n}\rho_{n})\\\
\leq\frac{1}{\varepsilon^{2}n\rho_{n}^{2}}\sum_{m=1}^{n}\operatorname{E}(\lvert
X_{n,m}\rvert^{2};\lvert X_{n,m}\rvert>\varepsilon\sqrt{n}\rho_{n}).$
Fix an arbitrary $t>0$. Set $\mu_{n,m}=\operatorname{E}Y_{n,m}$ and for
$I\in\mathcal{I}_{n}$ define
$\mu_{n}(I)=\frac{1}{\lvert I\rvert}\sum_{m\in I}\mu_{n,m}.$
For $I\in\mathcal{I}_{n}$, write
$\Pr\Big{(}\sum_{m\in I}Y_{n,m}>t\,\lvert I\rvert\Big{)}=\Pr\Big{(}\sum_{m\in
I}(Y_{n,m}-\mu_{n,m})>\lvert I\rvert\big{(}t-\mu_{n}(I)\big{)}\Big{)}.$
Note that since $\operatorname{E}X_{n,m}=0$, it follows that
$\lvert\mu_{n,m}\rvert=\lvert-\operatorname{E}(\rho_{n}^{-1}X_{n,m};\lvert
X_{n,m}\rvert>\varepsilon\sqrt{n}\rho_{n})\rvert\leq\frac{1}{\varepsilon\sqrt{n}\rho^{2}_{n}}\operatorname{E}(\lvert
X_{n,m}\rvert^{2};\lvert X_{n,m}\rvert>\varepsilon\sqrt{n}\rho_{n}).$
Thus, by (ii) and (iii) we have that
$\sup_{I\in\mathcal{I}_{n}}\\{\lvert\mu_{n}(I)\rvert\\}\to 0;$ in particular,
for $n$ large enough,
$\sup_{I\in\mathcal{I}_{n}}\\{\lvert\mu_{n}(I)\rvert\\}<\frac{t}{2}.$
Consequently, for $n$ large enough, uniformly for all $I$,
$\Pr\Big{(}\sum_{m\in I}Y_{n,m}>t\,\lvert
I\rvert\Big{)}\leq\Pr\Big{(}\sum_{m\in I}(Y_{n,m}-\mu_{n,m})>t\,\lvert
I\rvert/2\Big{)}.$
Similarly,
$\Pr\Big{(}\sum_{m\in I}Y_{n,m}<-t\,\lvert
I\rvert\Big{)}\leq\Pr\Big{(}\sum_{m\in I}(Y_{n,m}-\mu_{n,m})<-t\,\lvert
I\rvert/2\Big{)}.$
We apply Bernstein’s inequality to the bound. Define
$\sigma_{n,m}^{2}=\operatorname{E}(Y_{n,m}-\mu_{n,m})^{2}$ and
$v_{n}(I)=\sum_{m\in I}\sigma_{n,m}^{2}$. Note that $\lvert
Y_{n,m}-\mu_{n,m}\rvert\leq 2\varepsilon\sqrt{n}$. By Bernstein’s inequality,
$\Pr\Big{(}\Big{|}\sum_{m\in I}(Y_{n,m}-\mu_{n,m})\Big{|}>t\,\lvert
I\rvert/2\Big{)}\leq 2\exp\Big{\\{}-\frac{t^{2}\lvert
I\rvert^{2}/8}{v_{n}(I)+\varepsilon t\lvert I\rvert\sqrt{n}/3}\Big{\\}}.$
By (i), (iv), and (v), it follows that for $n$ large enough,
$v_{n}(I)<\varepsilon t\lvert I\rvert\sqrt{n}/3,$ so
$\Pr\Big{(}\Big{|}\sum_{m\in I_{n}}(Y_{n,m}-\mu_{n,m})\Big{|}>t\lvert
I\rvert/2\Big{)}\leq 2\exp\Big{\\{}-\frac{t\lvert
I\rvert}{8\varepsilon\sqrt{n}}\Big{\\}}.$
We use this bound for each $I$ to get the union bound:
$\Pr\Big{(}\sup_{I\in\mathcal{I}_{n}}\Big{|}\frac{1}{\lvert I\rvert}\sum_{m\in
I}Y_{n,m}\Big{|}>t\Big{)}\leq
2\lvert\mathcal{I}_{n}\rvert\exp\Big{\\{}-\frac{tL_{n}}{8\varepsilon\sqrt{n}}\Big{\\}}=2\exp\Big{\\{}\log\lvert\mathcal{I}_{n}\rvert-\frac{tL_{n}}{8\varepsilon\sqrt{n}}\Big{\\}}.$
By (iii) and (iv), it is possible to choose $\varepsilon$ such that the right
hand side goes to zero. It follows then that
$\Pr\Big{(}\sup_{I\in\mathcal{I}_{n}}\Big{|}\frac{1}{\rho_{n}\lvert
I\rvert}\sum_{m\in
I}X_{n,m}\Big{|}>t\Big{)}\leq\Pr(Y_{n,m}\neq\rho_{n}^{-1}X_{n,m}\text{ for
some $1\leq m\leq n$})\\\
+\Pr\Big{(}\sup_{I\in\mathcal{I}_{n}}\Big{|}\frac{1}{\lvert I\rvert}\sum_{m\in
I}Y_{n,m}\Big{|}>t\Big{)}\to 0.$
∎
###### Proof of Lemma 4.1.
For all $t>0$,
$\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}\lVert\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\rVert_{\infty}>t\Big{)}\\\
\leq KL\Pr\Big{(}\sup_{I\in\mathcal{I}_{n}}\rho^{-1}\Big{|}\sum_{\\{i,j\\}\in
I}(X_{ij}-\mu_{c_{i}d_{j}})\Big{|}>t\lvert I\rvert\Big{)},$
where $\mathcal{I}_{n}\subset 2^{[n]}\times 2^{[m]}$ is the set of all
biclusters such that $\hat{p}_{k}>\varepsilon$ for all $k$ and
$\hat{q}_{l}>\varepsilon$ for all $l$. Since $\mathcal{I}_{n}$ is a subset of
the powerset $2^{[nm]}$, by Lemma A.1, it follows that
$\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}\lVert\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\rVert_{\infty}>t\Big{)}\rightarrow
0.$
∎
###### Proof of Lemma 4.2.
The technical assumptions of $f$ imply that its first derivative is bounded.
Therefore, $f$ is locally Lipschitz continuous with Lipschitz constant
$c=\sup|f^{\prime}(\mu)|$ for $\mu$ in a neighborhood of $\mathcal{M}$ and
$\left|F(\rho^{-1}\bar{\boldsymbol{X}}(\boldsymbol{g},\boldsymbol{h}),\hat{\boldsymbol{p}}(\boldsymbol{g}),\hat{\boldsymbol{q}}(\boldsymbol{h}))-G_{\boldsymbol{M}_{0}}(\boldsymbol{C},\boldsymbol{D})\right|\leq
c\|\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\|_{\infty}.$
From Lemma 4.1, the righthand side converges to zero almost surely and the
result follows. ∎
###### Lemma A.2.
Let $\boldsymbol{C}=[C_{ak}]\in\mathbb{R}^{A\times K}$ and
$\boldsymbol{D}=[D_{bl}]\in\mathbb{R}^{B\times L}$ be nonnegative matrices
whose entries sum to one. Let $\boldsymbol{M}=[\mu_{ab}]\in\mathbb{R}^{A\times
B}$ be a matrix with no two identical columns. Define $\mathcal{M}$ to be the
convex hull of the entries of $\boldsymbol{M}$ and let
$f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable convex function with
$f^{\prime\prime}$ bounded away from zero in $\mathcal{M}$. Suppose that
$[\boldsymbol{C}\boldsymbol{1}]_{a}\geq\eta$ for all $a$ and that
$D_{bl}D_{b^{\prime}l}>\delta$ for some $l$ and $b\neq b^{\prime}$. There
exists a positive constant $C$ depending on $\boldsymbol{M}$ and $f$ such that
$\sum_{k=1}^{K}\sum_{l=1}^{L}[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}f\bigg{(}\frac{[\boldsymbol{C}^{T}\boldsymbol{M}\boldsymbol{D}]_{kl}}{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}}\bigg{)}\\\
\leq\sum_{a=1}^{A}\sum_{b=1}^{B}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f(\mu_{ab})-\frac{C\eta^{2}\delta}{K^{2}}.$
###### Proof.
Let $l$, and $b\neq b^{\prime}$ be such that $D_{bl}D_{b^{\prime}l}>\delta$.
Since no two columns of $\boldsymbol{M}$ are identical, there exists an $a$
such that $\mu_{ab}\neq\mu_{ab^{\prime}}$. Let $k$ be the index of the largest
element in row $a$ of matrix $\boldsymbol{C}$; this element must be at least
as large as the mean, i.e.
$C_{ak}\geq\frac{[\boldsymbol{C}\boldsymbol{1}]_{a}}{K}\geq\frac{\eta}{K}.$
Let
$W=[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}$;
this is nonzero. Now, there exists $\mu_{\ast}\in\mathcal{M}$ such that
$[\boldsymbol{C}^{T}\boldsymbol{M}\boldsymbol{D}]_{kl}=C_{ak}D_{bl}\mu_{ab}+C_{ak}D_{b^{\prime}l}\mu_{ab^{\prime}}+(W-C_{ak}D_{bl}-C_{ak}D_{b^{\prime}l})\mu_{\ast}.$
Let $z=\frac{[\boldsymbol{C}^{T}\boldsymbol{M}\boldsymbol{D}]_{kl}}{W}$. Set
$\kappa=\inf_{\mu\in\mathcal{M}}f^{\prime\prime}(\mu)$ and define
$\boldsymbol{N}=[\nu_{ab}]\in\mathbb{R}^{A\times B}$ with
$\nu_{ab}=f(\mu_{ab})$. By a second-order Taylor expansion,
$\displaystyle\frac{[\boldsymbol{C}^{T}\boldsymbol{N}\boldsymbol{D}]_{kl}}{W}-f(z)$
$\displaystyle\geq\frac{1}{2}\kappa\Big{(}\frac{C_{ak}D_{bl}}{W}(\mu_{ab}-z)^{2}+\frac{C_{ak}D_{b^{\prime}l}}{W}(\mu_{ab^{\prime}}-z)^{2}\Big{)}$
$\displaystyle\geq\kappa\frac{C^{2}_{ak}D_{bl}D_{b^{\prime}l}}{W^{2}}\Big{(}\frac{1}{2}(\mu_{ab}-z)^{2}+\frac{1}{2}(z-\mu_{ab^{\prime}})^{2}\Big{)}$
$\displaystyle\geq\kappa\frac{C^{2}_{ak}D_{bl}D_{b^{\prime}l}}{W^{2}}\Big{(}\frac{1}{2}(\mu_{ab}-z)+\frac{1}{2}(z-\mu_{ab^{\prime}})\Big{)}^{2}$
$\displaystyle=\kappa\frac{C^{2}_{ak}D_{bl}D_{b^{\prime}l}}{4W^{2}}(\mu_{ab}-\mu_{ab^{\prime}})^{2}.$
It follows that
$\displaystyle[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}f\bigg{(}\frac{[\boldsymbol{C}^{T}\boldsymbol{M}\boldsymbol{D}]_{kl}}{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}}\bigg{)}-[\boldsymbol{C}^{T}\boldsymbol{N}\boldsymbol{D}]_{kl}$
$\displaystyle\leq-\frac{\kappa}{2}(\mu_{ab}-\mu_{ab^{\prime}})^{2}\frac{C_{ak}^{2}D_{bl}D_{b^{\prime}l}}{W}$
$\displaystyle\leq-\frac{\kappa\eta^{2}\delta}{4K^{2}}(\mu_{ab}-\mu_{ab^{\prime}})^{2}.$
This inequality only holds for one particular choice of $k$ and $l$; for other
choices, the left hand side is nonpositive by Jensen’s inequality. The result
of the theorem follows, with $C$ defined by
$C=\frac{\kappa}{4}\min_{a,b\neq b^{\prime}}(\mu_{ab}-\mu_{ab^{\prime}})^{2}.$
∎
###### Proof of Lemma 4.3.
If $\boldsymbol{D}\notin\mathcal{Q}_{\delta}$, then for some $l$ and some
$b\neq b^{\prime}$, $D_{bl}D_{b^{\prime}l}\geq\delta$. By Lemma A.2,
$\sum_{k,l}[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}\,[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}\,f\Big{(}\frac{[\boldsymbol{C}^{T}\boldsymbol{M}_{0}\boldsymbol{D}]_{kl}}{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}\,[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}}\Big{)}-\sum_{a,b}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f([\boldsymbol{M}_{0}]_{a,b})\leq-\frac{\kappa_{0}\,\eta^{2}\delta}{K^{2}},$
where $\kappa_{0}$ depends only on $\boldsymbol{M}_{0}$ and $f$. Similarly, if
$\boldsymbol{C}\notin\mathcal{P}_{\delta}$, then the right hand side is
bounded by $-\kappa_{0}\,\eta^{2}\delta/L^{2}.$ The result of the lemma
follows with $\kappa=\kappa_{0}/\min\\{K^{2},L^{2}\\}$. ∎
## Appendix B Finite Sample Results
In this appendix we derive a finite sample tail bound for the probability that
the class assignments that maximize the profile likelihood are close to the
true class labels. To proceed in this setting, we make stronger distributional
assumptions than in the asymptotic case. Specifically, we assume here that the
entries $X_{ij}|\boldsymbol{c},\boldsymbol{d}$ follow a Gaussian distribution
with mean $\mu_{c_{i}d_{j}}$ and finite variance $\sigma^{2}$. We proceed with
the notation from the main text.
###### Lemma B.1.
For all $\varepsilon>0$, if $t<\sigma$ then
$\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}\|\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\|_{\infty}>t\Big{)}\leq
2K^{m+1}L^{n+1}\exp\Big{(}-\frac{L_{n}t^{2}}{4\sigma^{2}}\Big{)},$
and if $t\geq\sigma$ then
$\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}\|\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\|_{\infty}>t\Big{)}\leq
2K^{m+1}L^{n+1}\exp\Big{(}-\frac{L_{n}t}{4\sigma}\Big{)}$
where $\|\boldsymbol{A}\|_{\infty}=\max_{k,l}|A_{kl}|$ for any matrix
$\boldsymbol{A}$.
###### Proof.
If the entries $X_{ij}$ follow a Gaussian distribution with mean
$\mu_{c_{i}d_{j}}$ and variance $\sigma^{2}$ then
$\operatorname{E}\Big{(}|X_{ij}-\mu_{c_{i}d_{j}}|^{l}\Big{)}\leq\frac{\sigma^{2}}{2}\sigma^{l-2}l!$
so the conditions of Bernstein’s inequality hold. It follows that for any
bicluster $I$, for all $t>0$,
$\displaystyle\Pr\Big{(}\lvert\sum_{i,j\in
I}X_{ij}-\mu_{c_{i}d_{j}}\rvert>t|I|\Big{)}$ $\displaystyle\leq
2\exp\Big{\\{}-\frac{|I|^{2}t^{2}}{2(\sigma^{2}|I|+\sigma|I|t)}\Big{\\}}$
$\displaystyle\leq 2\exp\Big{\\{}-\frac{L_{n}t^{2}}{4\max\\{\sigma^{2},\sigma
t\\}}\Big{\\}}.$
Applying a union bound,
$\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}\|\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\|_{\infty}>t\Big{)}\leq
2K^{m+1}L^{n+1}\exp\Big{\\{}-\frac{L_{n}t^{2}}{4\max\\{\sigma^{2},\sigma
t\\}}\Big{\\}}.$
∎
Lemma B.1 is used to establish a finite sample bound on the difference between
$F(\boldsymbol{g},\boldsymbol{h})$ and its population version.
###### Lemma B.2.
Under assumptions (A2) and (A3), for any $t>0$,
$\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}|F(\boldsymbol{g},\boldsymbol{h})-G_{M_{0}}(\boldsymbol{c},\boldsymbol{d})|>t\Big{)}\leq\Pr\Big{(}\sup_{\mathcal{J}_{\varepsilon}}\|\boldsymbol{R}(\boldsymbol{g},\boldsymbol{h})\|_{\infty}>\frac{t}{c}\Big{)}$
where $c=\sup|f^{\prime}(\mu)|$ for $\mu$ in $\mathcal{M}$.
Lemma B.2 is a direct consequence of the fact that $f$ is locally Lipschitz
continuous under assumptions (A2) and (A3). The details are similar to proof
of Lemma 4.2.
The next step is to show that population version is maximized at the true
class labels.
###### Lemma B.3.
Choose $\tau>0$ such that $\min_{a\neq
a^{\prime},b}(\mu_{ab}-\mu_{a^{\prime}b})^{2}\geq\tau$ and $\min_{a,b\neq
b^{\prime}}(\mu_{ab}-\mu_{ab^{\prime}})^{2}\geq\tau$. Then for all
$\varepsilon>0$, for
$(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}$ and
$(\boldsymbol{C},\boldsymbol{D})\notin\mathcal{P}_{\delta}\cap\mathcal{Q}_{\delta}$,
$G_{M_{0}}(\boldsymbol{C},\boldsymbol{D})$ is small in the sense that
$G_{M_{0}}(\boldsymbol{C},\boldsymbol{D})-\sum_{a,b}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f([\boldsymbol{M}_{0}]_{ab})\leq-\frac{\tau\varepsilon^{2}\delta}{4\min\\{K^{2},L^{2}\\}}.$
The proof of Lemma B.3 is similar to the proof of Lemma 4.3 except that the
bound on the difference uses $\varepsilon$ in place of the random value
$\eta$. Some details follow.
###### Proof of Lemma B.3.
First note that in Lemma A.2, we can let $a$ be the index of the largest
element in column $k$ of matrix $\boldsymbol{C}$; then, since we are
restricted to the set $\mathcal{J}_{\varepsilon}$, this element must be at
least as large as
$C_{ak}\geq\frac{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}}{K}\geq\frac{\varepsilon}{K}.$
Therefore the bound in Lemma A.2 can be rewritten as
$\sum_{k=1}^{K}\sum_{l=1}^{L}[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}f\bigg{(}\frac{[\boldsymbol{C}^{T}\boldsymbol{M}\boldsymbol{D}]_{kl}}{[\boldsymbol{C}^{T}\boldsymbol{1}]_{k}[\boldsymbol{D}^{T}\boldsymbol{1}]_{l}}\bigg{)}\\\
\leq\sum_{a=1}^{A}\sum_{b=1}^{B}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f(\mu_{ab})-\frac{C\varepsilon^{2}\delta}{K^{2}}.$
Noting that for the Gaussian rate function $f^{\prime\prime}(\mu)=1$ for all
$\mu\in\mathcal{M}$, the remainder of the proof is similar to the proof of
Lemma 4.3. ∎
We establish a finite sample bound by combining these results.
###### Proof of Theorem 3.2.
Fix $\delta>0$ and define $\mathcal{P}_{\delta}$ and $\mathcal{Q}_{\delta}$ as
in Lemma 4.3. Set
$r_{n}=\sup_{\mathcal{J}_{\varepsilon}}|F(\boldsymbol{g},\boldsymbol{h})-G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h})|$.
Suppose $(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}$. In this
case,
$\displaystyle
F(\boldsymbol{g},\boldsymbol{h})-F(\boldsymbol{c},\boldsymbol{d})$
$\displaystyle\leq
2r_{n}+\\{G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h}))-G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{c}),\boldsymbol{D}(\boldsymbol{d}))\\}$
$\displaystyle=2r_{n}+\big{\\{}G_{\boldsymbol{M}_{0}}(\boldsymbol{C}(\boldsymbol{g}),\boldsymbol{D}(\boldsymbol{h}))-\sum_{a,b}[\boldsymbol{C}\boldsymbol{1}]_{a}[\boldsymbol{D}\boldsymbol{1}]_{b}f([\boldsymbol{M}_{0}]_{ab})\big{\\}}.$
Applying Lemma B.3 to the second term in the inequality, we get that
$F(\boldsymbol{g},\boldsymbol{h})-F(\boldsymbol{c},\boldsymbol{d})\leq
2r_{n}-\frac{\tau\varepsilon^{2}\delta}{4\min\\{K^{2},L^{2}\\}}$
for all $(\boldsymbol{g},\boldsymbol{h})\in\mathcal{J}_{\varepsilon}$ such
that
$(\boldsymbol{C},\boldsymbol{D})\notin\mathcal{P}_{\delta}\cap\mathcal{Q}_{\delta}$.
The result follows by applying Lemma B.2. ∎
## Appendix C Additional empirical results
This appendix reports additional empirical results for Bernoulli, Gaussian,
and non-standardized Student’s t distributed data. Figures 3-5 present the
average bicluster misclassification rates for each sample size and Tables 3-5
report the standard deviations for the Bernoulli, Gaussian, and t simulations,
respectively. Since the normalization for the DI-SIM algorithm is only
sbpecified for non-negative data, the algorithm is run on the un-normalized
matrix for the Gaussian and non-standardized Student’s t examples.
For the Bernoulli simulation, we simulate from a block model with $K=2$ row
clusters and $L=3$ column clusters. We vary the number of columns, $n$,
between 200 to 1500 and we take the number of rows as $m=\gamma n$ where
$\gamma\in\\{0.5,1,2\\}$.
We set the row and column class membership probabilities as
$\boldsymbol{p}=(0.3,0.7)$ and $\boldsymbol{q}=(0.2,0.3,0.5)$. We choose the
matrix of block parameters to be
$\boldsymbol{M}=\frac{b}{\sqrt{n}}\begin{pmatrix}0.43&0.06&0.13\\\
0.10&0.34&0.17\end{pmatrix}.$
where the entries were selected to be on the same scale as Bickel and Chen
(2009). We vary $b$ between 5 and 20. We generate the data conditional on the
row and column classes as
$X_{ij}\mid\boldsymbol{c},\boldsymbol{d}\sim\mathrm{Bernoulli}(\mu_{c_{i}d_{j}}).$
Figure 3: Average misclassification rates for Bernoulli example over 100
simulations. Table 3: Standard deviations for Bernoulli example over 100
simulations. $m=0.5n$
---
| PL | KM | DS
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.1474 | 0.0281 | 0.0344 | 0.1230 | 0.1490 | 0.0982 | 0.0614 | 0.0503 | 0.0408
500 | 0.0471 | 0.0096 | 0.0029 | 0.1217 | 0.1490 | 0.1399 | 0.0321 | 0.0205 | 0.0494
1000 | 0.0111 | 0.0052 | 0.0010 | 0.1573 | 0.1426 | 0.1490 | 0.0223 | 0.0123 | 0.1075
1500 | 0.0073 | 0.0030 | 0.0005 | 0.1529 | 0.1720 | 0.1688 | 0.0189 | 0.0087 | 0.1229
$m=n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.0889 | 0.0113 | 0.0285 | 0.1234 | 0.1357 | 0.1112 | 0.0526 | 0.0248 | 0.0657
500 | 0.0121 | 0.0376 | 0.0003 | 0.1466 | 0.1483 | 0.1544 | 0.0233 | 0.0391 | 0.0905
1000 | 0.0056 | 0.0013 | 0.0001 | 0.1659 | 0.1687 | 0.1666 | 0.0132 | 0.0956 | 0.1267
1500 | 0.0032 | 0.0403 | 0.0000 | 0.1675 | 0.1720 | 0.1641 | 0.0090 | 0.0637 | 0.0923
$m=2n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
200 | 0.0275 | 0.0321 | 0.0000 | 0.1336 | 0.1611 | 0.1397 | 0.0373 | 0.0756 | 0.0998
500 | 0.0048 | 0.0009 | 0.0000 | 0.1598 | 0.1668 | 0.1481 | 0.0120 | 0.1112 | 0.1166
1000 | 0.0015 | 0.0001 | 0.0000 | 0.1590 | 0.1652 | 0.1485 | 0.0543 | 0.1230 | 0.1232
1500 | 0.0009 | 0.0000 | 0.0000 | 0.1591 | 0.1679 | 0.1617 | 0.0746 | 0.0859 | 0.0854
For the Gaussian simulation, we simulate from a block model with $K=2$ row
clusters and $L=3$ column clusters. We vary the number of columns, $n$,
between 50 to 400 and we take the number of rows as $m=\gamma n$ where
$\gamma\in\\{0.5,1,2\\}$.
We set the row and column class membership probabilities as
$\boldsymbol{p}=(0.3,0.7)$ and $\boldsymbol{q}=(0.2,0.3,0.5)$. We choose the
matrix of block parameters to be
$\boldsymbol{M}=b\begin{pmatrix}\phantom{-}0.47&0.15&-0.60\\\
-0.26&0.82&\phantom{-}0.80\end{pmatrix}$
where the entries were simulated from a uniform distribution on $[-1,1]$. We
vary $b$ between 0.5 and 2. We generate the data conditional on the row and
column classes as
$X_{ij}\mid\boldsymbol{c},\boldsymbol{d}\sim\mathrm{Gaussian}(\mu_{c_{i}d_{j}},\sigma=1).$
Figure 4: Average misclassification rates for Gaussian example over 500
simulations. Table 4: Standard deviations for Gaussian example over 500
simulations. $m=0.5n$
---
| PL | KM | DS
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
50 | 0.1300 | 0.0874 | 0.1136 | 0.1019 | 0.1068 | 0.1247 | 0.0961 | 0.1890 | 0.1781
100 | 0.0681 | 0.0808 | 0.1094 | 0.0737 | 0.1074 | 0.1116 | 0.1264 | 0.1750 | 0.1975
200 | 0.0406 | 0.0958 | 0.1077 | 0.0700 | 0.0966 | 0.1122 | 0.1699 | 0.1846 | 0.2108
400 | 0.0459 | 0.1074 | 0.1064 | 0.0754 | 0.1071 | 0.1106 | 0.1274 | 0.2111 | 0.2151
$m=n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
50 | 0.1121 | 0.0830 | 0.1181 | 0.1010 | 0.1156 | 0.1234 | 0.1076 | 0.1940 | 0.1865
100 | 0.0576 | 0.0964 | 0.1047 | 0.0690 | 0.1002 | 0.1121 | 0.1576 | 0.1727 | 0.2060
200 | 0.0576 | 0.1120 | 0.1137 | 0.0947 | 0.1119 | 0.1204 | 0.1516 | 0.2033 | 0.2106
400 | 0.0843 | 0.1102 | 0.1124 | 0.0819 | 0.1096 | 0.1281 | 0.1712 | 0.2144 | 0.2147
$m=2n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
50 | 0.0829 | 0.1014 | 0.1171 | 0.0858 | 0.1290 | 0.1305 | 0.1194 | 0.1604 | 0.1893
100 | 0.0527 | 0.1215 | 0.1255 | 0.0851 | 0.1212 | 0.1397 | 0.1696 | 0.1956 | 0.2050
200 | 0.0786 | 0.1126 | 0.1126 | 0.0911 | 0.1146 | 0.1364 | 0.1578 | 0.2084 | 0.2106
400 | 0.0848 | 0.0966 | 0.0995 | 0.0837 | 0.0992 | 0.1410 | 0.2010 | 0.2118 | 0.2139
For the non-standardized Student’s t simulation, we use the same parameters as
in the Gaussian simulation and we generate the data conditional on the row and
column classes as
$X_{ij}\mid\boldsymbol{c},\boldsymbol{d}\sim\mathrm{t}(\mu_{c_{i}d_{j}},\sigma=1)$
with four degrees of freedom.
Figure 5: Average misclassification rates for t example over 500 simulations.
Table 5: Standard deviations for t example over 500 simulations. $m=0.5n$
---
| PL | KM | DS
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
50 | 0.0039 | 0.0059 | 0.0042 | 0.0033 | 0.0050 | 0.0059 | 0.0034 | 0.0061 | 0.0080
100 | 0.0056 | 0.0029 | 0.0042 | 0.0040 | 0.0041 | 0.0051 | 0.0043 | 0.0065 | 0.0078
200 | 0.0031 | 0.0029 | 0.0054 | 0.0028 | 0.0047 | 0.0055 | 0.0055 | 0.0060 | 0.0088
400 | 0.0022 | 0.0040 | 0.0049 | 0.0032 | 0.0046 | 0.0050 | 0.0055 | 0.0080 | 0.0096
$m=n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
50 | 0.0047 | 0.0045 | 0.0041 | 0.0036 | 0.0044 | 0.0059 | 0.0037 | 0.0064 | 0.0074
100 | 0.0044 | 0.0034 | 0.0049 | 0.0039 | 0.0049 | 0.0054 | 0.0052 | 0.0066 | 0.0084
200 | 0.0025 | 0.0044 | 0.0054 | 0.0027 | 0.0053 | 0.0052 | 0.0062 | 0.0068 | 0.0092
400 | 0.0026 | 0.0049 | 0.0050 | 0.0045 | 0.0055 | 0.0050 | 0.0048 | 0.0092 | 0.0096
$m=2n$
n | b=5 | b=10 | b=20 | b=5 | b=10 | b=20 | b=5 | b=10 | b=20
50 | 0.0057 | 0.0037 | 0.0047 | 0.0043 | 0.0072 | 0.0058 | 0.0045 | 0.0101 | 0.0081
100 | 0.0034 | 0.0032 | 0.0048 | 0.0032 | 0.0057 | 0.0052 | 0.0060 | 0.0068 | 0.0090
200 | 0.0034 | 0.0039 | 0.0055 | 0.0034 | 0.0047 | 0.0054 | 0.0063 | 0.0073 | 0.0093
400 | 0.0041 | 0.0039 | 0.0050 | 0.0050 | 0.0045 | 0.0052 | 0.0060 | 0.0087 | 0.0095
Similar to the Poisson simulation, biclustering based on the profile log-
likelihood criterion performs at least as well as the other methods and shows
signs of convergence in all three examples. These results provide further
verification of the theoretical findings and support the use of biclustering
based on the profile log-likelihood criterion.
## Appendix D Additional Application - Movielens
The MovieLens data set consists of 100,000 movie reviews on 1682 movies by 943
users. Each user has rated at least 20 movies and each movie is rated on a
scale from one to five. In addition to the review rating, the release date and
genre of each movie is available as well as some demographic information about
each user including gender, age, occupation and zip code.
In order to retain customers, movie-renting services strive to recommend new
movies to individuals who are likely to view them. Since most users only
review movies that they have already seen, we can use the structure of the
user-movie review matrix to identify associations between users and viewing
habits of movies. Specifically, we consider the 943$\times$1682 binary matrix
$\boldsymbol{X}$ where $X_{ij}=1$ if user $i$ has rated movie $j$ and
$X_{ij}=0$ otherwise. To find structure in $\boldsymbol{X}$, we biclustered
the rows and columns of $\boldsymbol{X}$ using the profile log-likelihood
based on the Bernoulli criterion (5.1a).
To determine a reasonable selection for the number of biclusters we varied the
number of user groups from two to three and the number of movie groups from
two to seven. Qualitatively, the model with three user groups and six movie
groups seems to provide a parsimonious description of the data.
Figure 6 presents the heatmap of the data based on the resulting bicluster
assignments, with the ordering of the clusters determined by the total number
of a reviews in each cluster. Table 6 reports the top ten movies in each
group. The eclectic mix of genres within each movie group suggests that the
rating behavior of users is not explained by genre alone. Figure 7 presents a
boxplot comparing the distributions of the movie release years for each group.
We can see a clear ordering of the movie groups by median release date.
Figure 6: Heatmap generated from MovieLens data reflecting the varying review
patterns in the different biclusters. Blue identfies movies with no review and
white identifies rated movies.
The median ages within the user group were 32, 31, and 30.5, and the
percentages of male users within each group were 67.6%, 72.8%, and 77.8%.
These statistics suggest that there is some age and gender effect on the
reviewing habits of the users.
Table 6: The top ten movies in each cluster based on the total number of reviews. Group 1 | Group 2
---|---
Jade (1995) | She’s the One (1996)
When the Cats Away (1996) | Jack (1996)
Jaws 3-D (1983) | The Preacher’s Wife (1996)
Bastard Out of Carolina (1996) | Striptease (1996)
Exit to Eden (1994) | Mirror Has Two Faces, The (1996)
The Ruling Class (1972) | Hercules (1997)
The Air Up There (1994) | Kids in the Hall: Brain Candy (1996)
Bad Taste (1987) | Jean de Florette (1986)
Stuart Saves His Family (1995) | The Fan (1996)
Cabin Boy (1994) | Extreme Measures (1996)
Group 3 | Group 4
The Firm (1993) | Courage Under Fire (1996)
The Abyss (1989) | Volcano (1997)
Die Hard: With a Vengeance (1995) | Murder at 1600 (1997)
Remains of the Day, The (1993) | Mars Attacks! (1996)
Sneakers (1992) | The People vs. Larry Flynt (1996)
The Professional (1994) | Starship Troopers (1997)
Clerks (1994) | Eraser (1996)
Reservoir Dogs (1992) | Das Boot (1981)
Like Water For Chocolate (1992) | Good Will Hunting (1997)
Chinatown (1974) | The Fifth Element (1997)
Group 5 | Group 6
Raiders of the Lost Ark (1981) | Star Wars (1977)
Pulp Fiction (1994) | Contact (1997)
The Silence of the Lambs (1991) | Fargo (1996)
The Empire Strikes Back (1980) | Return of the Jedi (1983)
Back to the Future (1985) | Liar Liar (1997)
The Fugitive (1993) | The English Patient (1996)
Indiana Jones and the Last Crusade (1989) | Scream (1996)
The Princess Bride (1987) | Toy Story (1995)
Forrest Gump (1994) | Air Force One (1997)
Monty Python and the Holy Grail (1974) | Independence Day (1996)
Figure 7: Boxplot comparing the different clusters based on movie release
dates.
Roughly speaking, user group 3 is consistently active across all movie groups
with increasing activity as the popularity of the movie increases. The
reviewing habits of user group 2 follow a similar pattern but to a lesser
extent. In contrast, user group 1 is consistently inactive with the only
exceptions being movie groups 4 and 6. From Figure 7, it appears that the
users in group 1 only rate recent movies whereas users in groups 2 and 3 rate
movies from all time periods.
The biclusters discovered here suggest that a movie-renting service should
recommend under-reviewed movies to individuals in user group 3, and it should
recommend new releases to users in group 1. This information can be used in an
ensemble-based recommendation engine like that of Töscher, Jahrer and Bell
(2009).
## References
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|
arxiv-papers
| 2012-06-29T01:19:35 |
2024-09-04T02:49:32.359118
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Cheryl J. Flynn, Patrick O. Perry",
"submitter": "Cheryl Brooks",
"url": "https://arxiv.org/abs/1206.6927"
}
|
1206.7020
|
# The thermodynamics of urban population flows
A. Hernando1, A. Plastino2, 3 1 Laboratoire Collisions, Agrégats, Réactivité,
IRSAMC, Université Paul Sabatier 118 Route de Narbonne 31062 - Toulouse CEDEX
09, France
2 National University La Plata, Physics Institute (IFLP-CCT-CONICET) C.C. 727,
1900 La Plata, Argentina
3 Universitat de les Illes Balears and IFISC-CSIC, 07122 Palma de Mallorca,
Spain
###### Abstract
Orderliness, reflected via mathematical laws, is encountered in different
frameworks involving social groups. Here we show that a thermodynamics can be
constructed that macroscopically describes urban population flows. Microscopic
dynamic equations and simulations with random walkers underlie the macroscopic
approach. Our results might be regarded, via suitable analogies, as a step
towards building an explicit social thermodynamics.
## I Introduction
The application of mathematical models to social sciences has a long and
distinguished history 1st . One may speak of empirical data from scientific
collaboration networks cites , cites of physics journals nosotrosZ , the
Internet traffic net1 , Linux packages links linux , popularity of chess
openings chess , as well as electoral results elec1 ; ccg , urban
agglomerations ciudad ; ciudad2 and firm sizes all over the world firms . A
specially relevant issue is that of universality classes defined by to the so-
called Zipf’s law (ZL) in the cumulative distribution or rank-size
distributions zipf ; nosotrosZ ; net1 ; linux ; chess ; ciudad ; ciudad2 ;
firms ; chin ; upf ; citis . Maillart et al. linux have found that links’
distributions follow ZL as a consequence of stochastic proportional growth.
Such kind of growth assumes that an element of the system becomes enlarged
proportionally to its size $k$, being governed by a Wiener process. The class
emerges from a condition of stationarity (dynamic equilibrium) citis . ZL also
applies for processes involving either self-similarity chess or fractal
hierarchy chin , all of them mere examples amongst very general stochastic
ones upf . A second universality-class was found by Costa Filho et al. elec1 ,
who studied vote-distributions in Brazil’s electoral results. Therefrom emerge
multiplicative processes in complex networks ccg . Such behavior ensues as
well in i) city-population rank distributions nosotros , ii) Spanish electoral
results nosotros , and iii) the degree distribution of social networks
nosotros2 . As shown in Ref. benford2 , this universality class encompasses
Benford’s Law benford1 . In the present vein, still another kind of
idiosyncratic distribution is often reported: the log-normal one lnwiki , that
has been observed in biology (length and sizes of living tissue bio ), finance
(in particular, the Black and Scholes model black ), and firms-sizes. The
latter instance obeys Gibrat’s rule of proportionate growth gibrat , that also
applies to cities’ sizes.
Together with geometric Brownian motion, there is a variety of models arising
in different fields that yield Zipf’s law and other power laws on a case-by-
case basis ciudad ; ciudad2 ; citis ; mod1 ; exp ; renorm , as preferential
attachment net1 and competitive cluster growth ccg ; nosotros2 in complex
networks, used to explain many of the scale-free properties of social
networks. For instance, we may mention detailed realistic approaches in urban
modelling ud ; otros , opinion dynamics oppi , and electoral results elec1 ;
elec2 . Of course, the renormalization group is intimately related to scale
invariance and associated techniques have been fruitfully exploited in these
matters (as a small sample see renorm, ; voteGalam, ).
It has been recently shown, in Ref. epjb2, , that a variational principle
based on MaxEnt can be successfully applied to scale-invariant social systems.
Used in the present context, it allows for a classification of the above cited
behaviors on the basis of inferences drawn from objective observables of the
system. We had also shown emp that including some dynamical information in
the variational scheme epjb1 one is able to reproduce the shape of empirical
city-population distributions, going beyond the customary universality classes
conventionally used in such regards. Indeed, a connection between explicit
microscopic growth equations and the macroscopic characterization exists,
illustrated for logistic-growth in Ref. logistic, . We will here describe the
manner in which the methods of that paper can be generalized to first-
principles theoretical framwework describing population flows in terms of
thermodynamic concepts.
### I.1 Motivation, statement of the problem and goal
We are looking here for more that models: what we aim for is to discover
physical principles that may underlie some social phenomena. Our system is a
specific geographical area whose population is distributed amongst several
population-nuclei (cities, villages, towns, etc.) Each nucleus’ population is
time-dependent due to migration, birth, death, etc. Our aim is to
quantitatively describe the population-nuclei’s variation. Microscopic
variables are plentiful, but our main goal is to be able to identify
macroscopic variables that can give a reasonable account of urban population-
variations.
We will proceed in seven steps, as indicated in the scheme below:
1. Introduce the basic observables and the empirical data sets. 2. Identify the stochastic nature of the city-population growth rates. 3. Postulate dynamic microscopic equations and empirically validate them. 4. Perform numerical simulations with random walkers following these dynamical equations and parametrize the macroscopic evolution. 5. Show that equilibrium configurations of such evolutions can be predicted by MaxEnt using few macroscopic parameters. 6. Derive thermodynamic-like relations between these macro-parameters. 7. Show the applicability of our thermodynamic description by modeling empirical urban flows as an scale invariant ideal gas.
The paper is organized as follows. Step 1 is addressed in the next Section II.
Section III deals with step 2, Section IV with step 3, Section V with step 4,
and Section VI with step 5 and 6. Finally, the application is dealt with in
Section VII, and some conclusions are drawn in Section VIII.
## II Preliminary matters
The basic ingredients we need in our approach, following Refs. emp, ; epjb1, ,
are
* i)
$n$, the total number of “population-nuclei”;
* ii)
$x_{i}(t)$, the population of the $i$-th nucleus at time $t$ (and
$\mathbf{x}(t)=\\{x_{i}(t)\\}_{i=1}^{n}$ a vector with all the populations);
* iii)
$x_{0}$ and $x_{M}$, the minimum and maximum allowed nucleus’ population (in
general $x_{0}=1$ and $x_{M}=\infty$);
* iv)
$N_{T}$, the total area’s population ($N_{T}=\sum_{i=1}^{n}x_{i}(t)$);
* v)
$\dot{x}_{i}(t)$, the time-derivative of $x_{i}(t)$ (thus the pairs
$\\{(x_{i},\dot{x}_{i})\\}_{i=1}^{n}$ compose the “urban phase space”); and
* v)
some a priori knowledge of the dynamics at hand, written as
$\dot{x}_{i}(t)=k_{i}(t)g_{i}[\mathbf{x}(t)]$ (1)
where $g_{i}$ are population-functions to be determined and $k_{i}(t)$ growth
rates independent of the $g_{i}$.
The raw data used in our analysis is obtained from the Spanish state institute
INEine and cover annually the period 1996-2010 (with the exception of 1997).
It encompasses up to 8000 municipalities (the smallest Spanish administrative
unit) distributed within 50 provinces (the building blocks of the autonomous
communities). We use provinces and municipalities as the closest
representatives of the ideal of a closed system’s fundamental elements. Also
other regions of the world are used as examples along the text. In this
tableau, the total population $N_{T}$ of a province is apportioned in $n$
nucleus. The $i$-th nucleus account a population of $x_{0}\leq x_{i}(t)\leq
x_{M}$ at time $t$, which time-evolution obeys Eq. (1).
## III The stochastic nature of population growth rates
We begin dealing with step 2 of our Scheme, saying something meaningful
concerning the form of the growth rates $k_{i}(t)$ in Eq. (1). The value of
$k_{i}$ above depends upon millions of individual decisions, so it is expected
some stochastic behavior. We should know both the average
$m_{i}=\langle\dot{x}_{i}(t)\rangle_{\delta t}$ and the standard deviation
$s_{i}=\langle(\dot{x}_{i}(t)-m_{i})^{2}\rangle_{\delta t}^{1/2}$ (for each
$i$) in a time-window ${\delta t}$ around $t$. Trying then to study the
distribution of $\xi_{i}(t)=(\dot{x}_{i}(t)-m_{i})/s_{i}$ one immediately
finds
$\displaystyle m_{i}$ $\displaystyle=$ $\displaystyle\langle
k_{i}(t)g_{i}(t)\rangle_{\delta t}$ (2) $\displaystyle=$ $\displaystyle\langle
k_{i}(t)\rangle_{\delta t}\times\langle g_{i}(t)\rangle_{\delta t},$
$\displaystyle s_{i}^{2}$ $\displaystyle=$
$\displaystyle\left\langle(k_{i}(t)g_{i}(t)-\langle
k_{i}(t)g_{i}(t)\rangle_{\delta t})^{2}\right\rangle_{\delta t}$ (3)
$\displaystyle=$ $\displaystyle\sigma_{k_{i}}^{2}\langle
g_{i}(t)\rangle_{\delta t}^{2}-\langle k_{i}(t)\rangle_{\delta
t}^{2}~{}\sigma^{2}_{g_{i}}$
with $\sigma_{k_{i}}$, $\sigma_{g_{i}}$ being the standard deviations of
$k_{i}(t)$ and $g_{i}(t)\equiv g_{i}[\mathbf{x}(t)]$, respectively. Assuming
now that the function $g_{i}$’s variation in the time-window for which one
evaluates the pair $m_{i}$ \- $s_{i}$ is negligible (i.e.,
$\sigma_{g_{i}}^{2}\ll\sigma_{k_{i}}^{2}$), to a good approximation one has
$\xi_{i}(t)=\frac{k_{i}(t)-\langle k_{i}(t)\rangle_{\delta
t}}{\sigma_{k_{i}}},$ (4)
entailing that $\xi_{i}(t)$ has null average and unit standard deviation. If
this assumption is correct the shape of the $p_{\Xi}-$distribution of the
variable $\xi_{i}(t)$ should not depend upon $x_{i}(t)$. We have verified the
hypothesis, as our first result here, with reference to all (8116) Spain’s
municipalities. Fig. 1 displays the $(x_{i},\xi_{i})-$pairs for every township
in the time-window $\delta t=15$ years. From them we evaluate appropriate
points taken at regular intervals from the cumulative distribution function of
our random variable (quantiles) as a function of the population $x$. No
apparent $x-$dependence can be detected. The overall distribution
$p_{\Xi}(\xi)$ shape looks like a normal one
$p_{\Xi}(\xi)=\frac{e^{-\xi^{2}/2}}{\sqrt{2\pi}},$ (5)
with cumulative distributions of the form
$P_{\Xi}(\xi)=\frac{1}{2}\left[1+\mathrm{erf}\left(\frac{\xi}{2}\right)\right],$
(6)
shown in Fig. 1. Save for some fluctuations, we have not found any dependence
on the shape of $p_{\Xi}(\xi)$ for the different provinces (same Fig. 1).
Accordingly, but with a grain of salt, one may speak of “universality”.
Consequently, we will consider herefrom that our variable $\xi$ can be
regarded as belonging to a Wiener process, our second result here.
## IV Introducing microscopic equations of motion
### IV.1 Proportional growth
We are now at step 3. For the $g_{i}$’s shape we will assume that it depends
only on its own $x_{i}$’s population, i.e., $g_{i}[\mathbf{x}(t)]\simeq
g_{i}[x_{i}(t)]$. In order to guess the explicit analytical form we appeal to
a cluster-growth model in networks,ccg ; nosotros2 used successfully before
describing city-population distributions. We firstly consider a network of
nodes (that eventually represents the social network) and a single node as
seed of a cluster. Initially, the first neighbors of the seed will belong to
the cluster with a given probability $P(t=0)$. At a subsequent time $t$, the
first neighbors of the members of the cluster become also members with
probability $P(t)$. Proceeding in this vein, it is reasonable to conjecture
that the time-variation of the cluster-size $\dot{x}$ at time $t$ acquire the
form
$\dot{x}(t)=\sum_{j=1}^{x(t)}P(t)c_{j}(t),$ (7)
where $c_{j}(t)$ is the first-neighbors-number of node $i$-th at time $t$. We
appeal now to the _central limit theorem_ to write
$\dot{x}(t)=P(t)\left(\overline{c}(t)x(t)+\sigma_{c}(t)\sqrt{x(t)}\xi(t)\right).$
(8)
Here $\overline{c}(t)$ is the mean neighbor’s number at time $t$,
$\sigma_{c}(t)$ its standard deviation, and $\xi(t)$ an independent normally-
distributed number. This last summand, usually neglected for very large sizes,
is associated to finite-size effects. The first term, size-proportional,
generates proportional (or multiplicative) growth. In view of this result, we
consider the form
$g_{i}(x_{i})=[x_{i}]^{\alpha}$ (9)
with $\alpha=1$ or $1/2$. Considering then both terms in the microscopic
dynamics we write
$\dot{x}_{i}(t)=k_{i1}(t)x_{i}(t)+k_{i1/2}(t)\sqrt{x_{i}(t)},$ (10)
with the $k_{i1}(t)$ and $k_{i1/2}(t)$ two (a priori) independent Wiener
coefficients. This dependence is checked out by comparison of the previously
employed $s_{i}-$numbers with a functional form of the type
$\displaystyle s_{i}^{2}(x_{i})$ $\displaystyle=$
$\displaystyle\langle[\dot{x}_{i}]^{2}-\langle\dot{x}_{i}\rangle_{\delta
t}^{2}\rangle_{\delta t}$ (11) $\displaystyle=$
$\displaystyle\sigma_{i1}^{2}x_{i}^{2}+\sigma_{i1/2}^{2}x_{i},$
where $\sigma_{i1}$ and $\sigma_{i1/2}$ are the associated deviations of
$k_{i1/2}$ and $k_{i1}$, respectively. Rewriting (11) in a more convenient way
we have
$s_{i}^{2}(x_{i})/x_{i}=\sigma_{i1}^{2}x_{i}+\sigma_{i1/2}^{2},$ (12)
that, for sizes small enough reduces to
$s_{i}^{2}(x_{i})/x_{i}\approx\sigma_{i1/2}^{2},$ (13)
while for very large sizes one has
$s_{i}^{2}(x_{i})/x_{i}\approx\sigma_{i1}^{2}x_{i}.$ (14)
The transition between these two regimes should take place at a value
$x_{T}=\sigma_{1/2}^{2}/\sigma_{1}^{2}$. Fig. 2 displays, as our third result,
the $(x_{i},s_{i}^{2}(x)/x_{i})-$pairs for all the Spanish municipalities,
together with appropriate quantiles. The median
$\mathrm{med}(s_{i}(x_{i})/x_{i})$ nicely fits things with
$\sigma_{i1}=0.0119$ and $\sigma_{i1/2}=0.47$. We appreciate the fact that
finite size fluctuations are larger than multiplicative ones, the later
dominating, of course, for large sizes. Our transition occurs at population-
values of the order of $1500$ inhabitants. Surprisingly enough, the
distribution of the variable
$s^{\prime}_{i}=\log[s_{i}(x_{i})/\sqrt{x_{i}}]-\log[\mathrm{med}(s_{i}(x_{i})/\sqrt{x_{i}})]$
becomes independent of $x_{i}$, being of a Gaussian nature.
At this point, we need still to address a further question. The finite-size
term average is $\langle k_{i1/2}(t)\rangle_{\delta t}=0$ (by definition), but
this is not so for the multiplicative one $\langle k_{i1}(t)\rangle_{\delta
t}\neq 0$, that is _a priori_ regarded as constant and size-independent. This
is indeed empirically true on occasions, but not always. For instance, such
assumption cannot account for the migration from the countryside to big
cities, where the mean growth rate correlates with the city-population.
### IV.2 Taking into account internal flow
It is a fact that small populations tend to diminish while large towns tend to
increase their population. We encounter this scenario for most of the 50
provinces of Spain. We intend to tackle this issue below.
We can show that the effect can be described by recourse to a smooth
dependence of the mean relative growth $\langle\dot{x}_{i}/x_{i}\rangle$ on
$\log(\langle x\rangle)$ that generates what we will call internal flow. A
second order expansion in $\log(\langle x\rangle)$ reads
$\langle\dot{x}/x\rangle\simeq a+b\log(\langle x\rangle)+c\log(\langle
x\rangle)^{2}$ (15)
where the values of $a$, $b$ and $c$ come from the corresponding Taylor
coefficients. Assuming $b\gg c$ we can safely write it as
$\langle\dot{x}/x\rangle\simeq\langle k_{1}\rangle+\langle
k_{q}\rangle[\langle x\rangle]^{q-1},$ (16)
where we have defined for convenience $\langle k_{1}\rangle=a-b^{2}/2c$,
$\langle k_{q}\rangle=b^{2}/2c$ and $q-1=2c/b$. To validate our assumptions,
we fitted the empirical provincial data to Eq. (16) via $\langle
k_{1}\rangle$, $\langle k_{q}\rangle$ and $q$, when possible (in some cases a
quasi-linear relation is found, generating large uncertain in the optimal
values). We have found for the exponent $q$ a mean value of 1.2 and a standard
deviation of 0.45, with $|q-1|<1$ in all cases. This result confirms the
assumption $b\gg c$ validating the second-order expansion of
$\langle\dot{x}_{i}/x_{i}\rangle$. Moreover, as seen in Fig. 3 (our fourth
result), nice fits are found in general with very few exceptions.
With this new hypothesis our complete dynamic equation turns out to be
$\dot{x}_{i}(t)=k_{iq}(t)[x_{i}(t)]^{q}+k_{i1}(t)x_{i}(t)+k_{i1/2}(t)\sqrt{x_{i}(t)},$
(17)
with $k_{iq}(t)$, $k_{i1}(t)$ and $k_{i1/2}(t)$ independent (a priori) Wienner
processes. Summing up, we have assumed
* •
a finite size term that dominates things for low population levels ($<1500$),
* •
a multiplicative term that accounts for population’s growth/diminution
(births, death or o external migration, and
* •
a power-law (exponent $q\sim 1$) accounting for internal migration.
Since for most of the population range only one term dominates, we will
include only one term in the considerations what follow below.
## V From microscopic to macroscopic descriptions
We arrive to stage 4, having discussed above a microscopic population
dynamics. We will tray now to ascertain whether a macroscopic description is
also feasible. Our goal is to reduce the $2n$ microscopic degrees of freedom
to a few macroscopic ones. We will separately consider each of the three terms
of the dynamic equation. The ensuing results will be valid in the domains in
which each term dominates.
Consider $n$ random walkers characterized by a dynamic coordinate $x_{i}(t)$
obeying
$\dot{x}_{i}(t)=k_{i}(t)[x_{i}(t)]^{q},$ (18)
with
$\langle(k_{i}(t)-\overline{k})(k_{j}(t)-\overline{k})\rangle=\sigma_{k}\delta_{ij}\delta(t-t^{\prime})$.
Parameter $q$ will take as special possible values $1/2$ or $1$, or in
general, $0\leq q$.
### V.1 Brownian motion and diffusion equation
We start with $q=0$ as control case. One has $\dot{x}_{i}(t)=k_{i}(t)$ so that
we deal with the well-known brownian random walkers. Consider this numerical
procedure: initially, the $n$ walkers are located at, say, $x=x_{0}$. By
$\rho(x,t)dx$ we will refer to the walker’s normalized histogram, at time $t$,
that indicates the walker’s relative number positioned in the interval $dx$
around $x$. The associated initial density would read
$\rho(x,0)=\delta(x-x_{0})$. A discrete version of the pertinent dynamic
equation is
$x_{i}(t+\Delta t)=x_{i}(t)+\Delta tk_{i}(t),$ (19)
that forces the walkers to “move” during the period $\Delta t$ in a amount
given by $\Delta tk_{i}(t)$, with $k_{i}(t)$ a random number generated from a
Gaussian distribution determined by an standard deviation $\sigma_{k}$ and
mean $\overline{k}$, as defined above. We have
$x_{i}(t=M\Delta t)=x_{0}+\Delta t\sum_{m=1}^{M}k_{i}[(M-1)\Delta t],$ (20)
so that after $M$ iterations the walkers-distributions coincides with that of
a random number generated by summing up $M$ Gaussian numbers characterized by
$\Delta t\sigma_{k}$ and $\Delta t\overline{k}$. Remind that a distribution
that follows a random number composed of two other numbers of that character
is the convolution of the distributions associated to these later numbers.
Thus, $x(t)$ is described by the $M-$th convolution of the $k$’s Gaussian
distribution. By recourse to a Fourier transform $\mathcal{F}$ for
convolutions we have
$\displaystyle\mathcal{F}[\rho(x,t)]$ $\displaystyle=$
$\displaystyle\left(\mathcal{F}\left[\frac{e^{-(k-\Delta
t\overline{k})^{2}/2(\Delta t\sigma_{k})^{2}}}{\sqrt{2\pi}\Delta
t\sigma_{k}}\right]\right)^{M}$ (21) $\displaystyle=$
$\displaystyle\left(e^{-\Delta t^{2}\sigma_{k}^{2}\omega^{2}/2+i\Delta
t\overline{k}\omega}\right)^{M}$ $\displaystyle=$ $\displaystyle e^{-M\Delta
t^{2}\sigma_{k}^{2}\omega^{2}/2+iM\Delta t\overline{k}\omega},$
and, appealing to the inverse transformation,
$\displaystyle\rho(x,t)$ $\displaystyle=$ $\displaystyle\frac{e^{-(k-M\Delta
t\overline{k})^{2}/(2M(\Delta t\sigma_{k})^{2})}}{\sqrt{2\pi M}\Delta
t\sigma_{k}}$ (22) $\displaystyle=$
$\displaystyle\frac{e^{-(k-t\overline{k})^{2}/(4Dt)}}{\sqrt{4\pi Dt}},$
where we have introduced for convenience $2D=\Delta t\sigma_{k}^{2}$. An
arbitrary density $\rho(x,t)$ will evolve in $\Delta t$, via the convolution
of that density with a Gaussian of deviation $\Delta t\sigma_{k}=\sqrt{2\Delta
tD}$ and mean $\Delta t\overline{k}$, as
$\displaystyle\mathcal{F}[\rho(x,t+\Delta t)]$ $\displaystyle=$
$\displaystyle\mathcal{F}[\rho(x,t)]\times e^{-\Delta tD\omega^{2}+i\Delta
t\overline{k}\omega}$ $\displaystyle\simeq$
$\displaystyle\mathcal{F}[\rho(x,t)]\left(1-\Delta tD\omega^{2}+i\Delta
t\overline{k}\omega\right),$
where we take $\Delta t$ arbitrarily small. A simple manipulation involving
division by $\Delta t$ leads now to
$\frac{\mathcal{F}[\rho(x,t+\Delta t)]-\mathcal{F}[\rho(x,t)]}{\Delta
t}=\left(-D\omega^{2}+i\overline{k}\omega\right)\mathcal{F}[\rho(x,t)].$ (24)
By recourse to the inverse transformation and taking the limit $\Delta
t\rightarrow 0$ we get
$\partial_{t}\rho(x,t)=D\partial^{2}_{x}\rho(x,t)-\overline{k}\partial_{x}\rho(x,t),$
(25)
which is a diffusion equation. Accordingly, we reach an important result here
(our fifth one):
Our original $2n$ degrees of freedom-problem can now be tackled via just a few
macroscopic parameters.
### V.2 $q$-metric Brownian motion
In the general instance $q\neq 0$ we introduce a variable
$u_{i}=\log_{q}(x_{i})$, where $\log_{q}$ is Tsallis’ $q$-logarithm tbook .
The Jacobian for the transform is $du/dx=1/x^{q}$ so that
$\dot{u}=\dot{x}/x^{q}$ and the associated dynamical equation becomes
$\dot{u}_{i}(t)=k_{i}(t).$ (26)
In the set $\\{(u_{i},\dot{u}_{i})\\}_{i=1}^{n}$, the variables $u_{i}$ and
$\dot{u}_{i}$ are independent of each other. We regard them, of course, as our
dynamical variables. Note that one recovers Brownian motion for $u$. Indeed,
$u_{i}(t=M\Delta t)=u_{i}(0)+\Delta t\sum_{m=1}^{M}k_{i}[(M-1)\Delta t],$ (27)
and then the demonstration of the preceding subsection becomes valid, now for
$u$ and $\rho(u,t)du$. Our new diffusion equation reads
$\partial_{t}\rho(u,t)=D\partial^{2}_{u}\rho(u,t)-\overline{k}\partial_{u}\rho(u,t),$
(28)
and, starting from a density $\rho(u,0)=\delta(u-u_{0})$ we end up with
$\rho(u,t)du=\frac{du}{4\pi
Dt}\exp\left[-\frac{(u-u_{0}-\overline{k}t)^{2}}{4Dt}\right].$ (29)
The $x-$density is governed accordingly by a $q$log-normal distribution
$\displaystyle\rho_{X}(x,t)dx$ $\displaystyle=$
$\displaystyle\rho[u(x),t]\frac{dx}{du}du$ $\displaystyle=$
$\displaystyle\frac{dx}{\sqrt{4\pi
Dt}x^{q}}\exp\left[-\frac{(\log_{q}(x)-u_{0}-\overline{k}t)^{2}}{4Dt}\right].$
In particular, for $q=1/2$ one has
$\rho_{X}(x,t)dx=\frac{dx}{\sqrt{4\pi
Dtx}}\exp\left[-\frac{(2(\sqrt{x}-1)-u_{0}-\overline{k}t)^{2}}{4Dt}\right],$
(31)
and, for $q=1$ the well known log-normal
$\rho_{X}(x,t)dx=\frac{dx}{\sqrt{4\pi
Dt}x}\exp\left[-\frac{(\log(x)-u_{0}-\overline{k}t)^{2}}{4Dt}\right].$ (32)
We have again reduced the microscopic number of degrees of freedom to just a
few macroscopic parameters.
### V.3 Examples of diffusion
Numerical experiments confirm our findings above. We start with our dynamical
equation in discrete form
$x_{i}(t+\Delta t)=x_{i}(t)+\Delta tk_{i}(t)[x_{i}(t)]^{q}$ (33)
using $k_{i}(t)=\sqrt{2D/\Delta t}\xi_{i}(t)+\overline{k},$ where the random
numbers $\xi$ follow a normal distribution such that
$\langle\xi_{i}(t)\xi_{j}(t)\rangle=\delta_{ij}\delta(t-t^{\prime})$. We have
taken $q=1/2$ and $1$ for our examples, and find that the associated
distributions exactly follow the diffusion equation’s predictions. We have
used in the former case $u_{0}=\log_{1/2}(220)$, $\overline{k}=0$, and
$\sigma_{k}^{2}=10$, in intervals of $\Delta t=0.01$. In the later instance we
had $u_{0}=\log(4400)$ instead. Indeed, the walkers’ histograms’ evolution
follow Eq. (31) and Eq. (32), respectively, with $D=\Delta t\sigma_{k}^{2}/2$
as defined above (see Fig. 4 for the cumulative distributions).
As empirical examples we discovered that for small populations $<1500$
inhabitants the finite-size noise dominates. Provinces for which most towns
are scarcely populated will obey the dynamical equation with $q=1/2$. Such is
the case for the province of, i.e., Salamanca, as shown in top panel of Fig.
4. The ensuing dynamics confirms this assertion. The relative growth of most
of the towns follows a dynamics with a variance $s^{2}\propto\sqrt{x}$ (red
line of the inset). The ensuing distribution fits the final state predicted by
the diffusion equation for that dynamics, Eq. (31), with
$u_{0}+\overline{k}t=\log_{1/2}(216.3)$ and $2Dt=95.6$ for year 2010 (see Fig.
4). Remark that the 1/2-log-normal can be easily confused with the usual log-
normal, although the former exhibits asymmetries in log-scale. As a
$q=1-$example we mention Florida State in the US usa (see also bottom Fig.
4). Using data from 1990, 2000, and 2010, we have verified that the
microscopic dynamics confirms the proportional growth assumption (with a
variance of the relative growth independent of the size, as illustrated in the
inset). The city-populations distribution follows a log-normal distribution,
that of Eq. (32), which can be the one pertaining to geometrical random-
walkers’ diffusion, with $u_{0}+\overline{k}t=\log(4380)$ and $2Dt=2.96$.
### V.4 Constrained diffusion
$q$-log-normal distributions do not set any limits to population-sizes.
However, it is reasonable to assume that physical space does pose limits to a
city’s population-growth. Unlimited growth is unrealistic since in the case of
internal migrations the total population $N_{T}$ should remain constant and a
free-diffusion model is, again, unrealistic. Constrained diffusion must be
contemplated instead.
We pass now to consider numerical experiments with random walkers that fix
lower and upper bounds for population. These are denoted by $x_{0}$ and
$x_{M}$, respectively. Now, walkers “moves” leading to values outside the
range $x_{0}<x<x_{M}$ are to be rejected in our simulations. Fig. 5 shows that
a $q$-metric walkers’ evolution begins by faithfully following the diffusion
equation Eq. (28) till they bump off these extreme values. Now their density
deviates from that of “free” evolution. After some time has elapsed, an
equilibrium $x-$distribution is reached that follows a power-law with exponent
$q$, independently of the initial state. _The origin of this systematic result
can not be unraveled by the simulations, so a higher-level of theory is
needed._
Now we use a total population constraint. This is equivalent to make the
walkers move under the rule of a _$q$ -generalized multi-component logistic
equation_
$\dot{x}_{i}(t)=[x_{i}(t)]^{q}\left[k_{i}(t)-\frac{\sum_{i=1}^{n}k_{i}(t)[x_{i}(t)]^{q}}{\sum_{i=1}^{n}[x_{i}(t)]^{q}}\right].$
(34)
Indeed, it is easy to check that
$\partial_{t}N_{T}=\sum_{i=0}^{n}\dot{x}_{i}(t)=0$, thus preserving the value
of $N_{T}$ in time. Also the original $q$-symmetry of the dynamics is
preserved. This equation is the $q$-generalization of the scale-invariant
multi-component logistic equation presented in logistic, . Results are
displayed in Fig. 6 for $q=1$, 1.5 and 2, using $n=100000$ walkers and a total
population of $N=250000$ inhabitants (with $x_{0}=1$). Remarkably enough,
equilibrium is always reached, to a density that does not depend upon the
initial state or the $k$-parameters. The shape of the distributions resembles
$x$ power-laws with exponential cut-off. Again, the simulation can not unravel
the origin of this form. _Finding the properties and the exact analytical form
of those macroscopic equilibrium distributions is our goal in the sext
Section._
## VI The macroscopic conundrum
We tread now step 5. Our simulations with random walkers suggest that it is
indeed possible to pass from a description that uses $2n$ microscopic
variables to a description involving just a few macroscopic parameters. The
big question is: do they behave in thermodynamic fashion, satisfying the
pertinent partial derivatives-relationships? We wish to tackle this issue now
looking for a way to reduce the number of microscopic degrees of freedom to a
few manageable macroscopic ones while keeping a coherent, reasonable
description of our system, mimicking the kind of scenario that links
statistical mechanics to thermodynamics. This requires appropriate
constraints, a topic to be addressed below by enumerating the appropriate
“social” constraints we need.
### VI.1 Macroscopic constraints
$\bullet$ Total number of cities $n$. Since there is some confusion in the
available data about what the administrative meaning of city is, we wish to
ascertain that this issue is of no importance. Consider
$x_{i}=\sum_{j}^{n_{i}}x_{ij},$ where $n_{i}$ is the number of sub-
administrative units included in the administrative unit $i$, with $x_{ij}$
their sub-administrative populations. Considering proportional growth, we
write for the time-evolution
$\displaystyle\dot{x}_{i}(t)$ $\displaystyle=$
$\displaystyle\sum_{j}^{n_{i}}\dot{x}_{ij}(t)$ (35) $\displaystyle=$
$\displaystyle\sum_{j}^{n_{i}}k_{ij}(t)x_{ij}(t)$ $\displaystyle=$
$\displaystyle\frac{\sum_{j}^{n_{i}}k_{ij}(t)x_{ij}(t)}{\sum_{j}^{n_{i}}x_{ij}(t)}\sum_{j}^{n_{i}}x_{ij}(t)$
$\displaystyle=$ $\displaystyle k^{\prime}_{i}(t)x_{i}(t),$
where we have defined $k^{\prime}_{i}(t)$ as a new variable defined as an
average weighted by the populations $x_{ij}$. If the growth rates $k_{ij}$ are
random variables with approximately the same mean and variance, it is easy to
check that $k^{\prime}_{i}(t)$ is in turn a random variable of the same mean
and variance. The dynamical behavior of the ensemble of administrative units
$\mathbf{x}$ is thus equivalent of that of the sub-units, and the procedure
described in this work is still applicable.
$\bullet$ Maximum/minimum population $x_{M}/x_{0}$. It is well-known that a
typical minimum population size equals the Dunbar numberdum ($\sim 150$),
heuristically associated to the maximum (allowable by our neo-cortex) number
of stable human relationships. Thus, it is reasonable to think of a minimum
size $x_{0}\sim 150$. In many cases a maximum number for a city population
$x_{M}$ can be established via consideration of geographical peculiarities as
mountains aostamap or oceansmarmap (See Fig. 7 for an example). In such
cases it is convenient to employ the transform $u=\log_{q}(x/x_{0})$. An
associated, valuable macroscopic parameter is $u_{M}=\log_{q}(x_{M}/x_{0})$.
We will be dealing then with a “volume” $0<u<u_{M}$.
$\bullet$ Total population $N_{T}$. We have $N_{T}=\sum_{i=1}^{n}\,x_{i}$ that
gets transformed into $N_{T}=x_{0}\sum_{i=1}^{n}\,e_{q}^{u_{i}}$. A useful
quantity becomes then $N=N_{T}/x_{0}$.
$\bullet$ Total variance of $\dot{u}.$ With reference to the dynamics, a
useful observable is the total variance for relative growth
$\sigma^{2}=\sum_{i=1}^{n}\langle(\dot{u}_{i}-\langle\dot{u}_{i}\rangle_{t})^{2}\rangle/n$.
For a Gaussian form (see Fig. 1) this quantity measures fluctuation-
intensities. Generalizing, this quantity can be defined by the covariance
matrix with elements
$Q_{ij}=\langle(\dot{u}_{i}-\langle\dot{u}_{i}\rangle)(\dot{u}_{j}-\langle\dot{u}_{j}\rangle)\rangle$,
using its trace as a thermodynamical variable
$\mathrm{Tr}(Q)=\sum_{i=1}^{n}Q_{ii}$:
$\displaystyle U$ $\displaystyle=$ $\displaystyle\frac{\tau}{2}\mathrm{Tr}(Q)$
(36) $\displaystyle=$
$\displaystyle\frac{\tau}{2}\sum_{i=1}^{n}\langle(\dot{u}_{i}-\langle\dot{u}_{i}\rangle)^{2}\rangle$
$\displaystyle=$ $\displaystyle\frac{\tau}{2}n\sigma^{2},$
where we add for dimensional convenience a factor $\tau/2$.
### VI.2 Fundamental hypothesis for urban-thermodynamics
Let us discuss the pertinent three hypothesis that we need in our Scheme:
$\bullet$ _H-I. Microscopic hypothesis._
We adopt as fundamental dynamical equation Eq. (18) [$\dot{x}=kx^{q}$] for the
population of a center, linearized via the variable $u=\log_{q}(x/x_{0})$. We
will think of the pair $(u,\dot{u})$ as constituting our social phase space
coordinates. We can speak of an:
$\bullet$ _H-II. A priori phase space equiprobability in $(u,\dot{u})$.katz _
The probability density distribution for the $i-$th phase space cell centered
at $(u_{i},\dot{u}_{i})$ of size $dud\dot{u}$ is defined as
$\rho[\\{(u_{i},\dot{u}_{i})\\}_{i=1}^{n}]d^{n}ud^{n}\dot{u}$. Accordingly to
H-II, the system’s entropy is written as
$S[\rho]=-\int
d^{n}ud^{n}\dot{u}~{}\rho[\\{(u_{i},\dot{u}_{i})\\}_{i=1}^{n}]\log\left[\rho[\\{(u_{i},\dot{u}_{i})\\}_{i=1}^{n}]\right].$
(37)
Since none of our macroscopic observables is able to distinguish amongst
population nuclei, towns are thus indistinguishable. In this case, the useful
distribution is the one-body density $\rho(u,\dot{u})$ defined as
$\rho(u,\dot{u})=\int
d^{n-1}ud^{n-1}\dot{u}~{}\rho[\\{(u_{i},\dot{u}_{i})\\}_{i=1}^{n}],$ (38)
and thus,
$S[\rho]=-\int dud\dot{u}~{}\rho(u,\dot{u})\log\left[\rho(u,\dot{u})\right].$
(39)
Macroscopic observables are written in terms of the one-body density as
$\displaystyle n$ $\displaystyle=$ $\displaystyle\int
dud\dot{u}~{}\rho(u,\dot{u}),$ (40) $\displaystyle N$ $\displaystyle=$
$\displaystyle\int dud\dot{u}~{}\rho(u,\dot{u})e_{q}(u),$ (41) $\displaystyle
U$ $\displaystyle=$ $\displaystyle\frac{\tau}{2}\int
dud\dot{u}~{}\rho(u,\dot{u})\dot{u}^{2}.$ (42)
$\bullet$ _H-III. Maximum entropy principle (MaxEnt).katz _
Equilibrium is determined via constrained entropic maximization using $n$,
$u_{M}$, $N$ y $U$. This determines the equilibrium density $\rho(u,\dot{u})$
that is a solution of the entropic variational problem
$\delta\left\\{S[\rho]-\beta A[\rho]\right\\}=0,$ (43)
with
$A=U-\mu n+pu_{M}+\Lambda N,$ (44)
where $\beta$, $\mu$, $p$ and $\Lambda$ stand for the pertinent Lagrange
multipliers, that will be seen below to acquire the character of intensive
thermal-quantities.
### VI.3 Thermodynamical relations
We enter step 6 by considering the _Lagrangian_ $A[\rho]$ [and Lagrangian
density $a(u,\dot{u})$]. It reads
$\displaystyle A[\rho]$ $\displaystyle=$ $\displaystyle\int
dud\dot{u}~{}\rho(u,\dot{u})a(u,\dot{u})$ $\displaystyle=$ $\displaystyle\int
dud\dot{u}~{}\rho(u,\dot{u})\left\\{\frac{\tau}{2}\dot{u}^{2}-\mu+pv(u)+\Lambda
e_{q}(u)\right\\},$
where the volume condition is enforced by an infinite-well potential
$v(u)=\left\\{\begin{array}[]{ll}u_{M}/n&for\,\,\,0<u<u_{M};\\\
\infty&otherwise\end{array}\right.$ (46)
The well-known general solution to the entropic problem Eq. (43) iskatz
$\rho(u,\dot{u})=\exp[-\beta a(u,\dot{u})]$, so that
$\rho(u,\dot{u})=\frac{n}{Z}e^{-\frac{\beta\tau}{2}\dot{u}^{2}-\beta\Lambda
e_{q}(u)}~{}~{}(0<u<u_{M}),$ (47)
where the normalization factor $Z$ (partition function) becomes
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}d\dot{u}\int_{0}^{u_{M}}du~{}e^{-\frac{\beta\tau}{2}\dot{u}^{2}-\beta\Lambda
e_{q}(u)}$ (48) $\displaystyle=$
$\displaystyle\sqrt{\frac{2\pi}{\beta\tau}}E_{q}(\beta\Lambda,u_{M}),$
with $E_{q}(l,m)$ the generalized exponential function of order $q$
$E_{q}(l,m)=E_{q}(l)-e^{(1-q)m}E_{q}(le^{m}).$ (49)
Our constraints in $U$ and $N$ determine the multipliers $\beta$ and
$\Lambda$-values. On the one hand, we have the $\dot{u}-$variance
$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{\tau}{2}n\int_{-\infty}^{\infty}d\dot{u}~{}\sqrt{\frac{\beta\tau}{2\pi}}e^{-\frac{\beta\tau}{2}\dot{u}^{2}}\dot{u}^{2}$
(50) $\displaystyle=$ $\displaystyle\frac{n}{2\beta},$
and on the other one, we have to deal with the total population
$\displaystyle N$ $\displaystyle=$ $\displaystyle
n\int_{0}^{u_{M}}du~{}\frac{e_{q}(u)e^{-\beta\Lambda
e_{q}(u)}}{E_{q}(\beta\Lambda,u_{M})}$ (51) $\displaystyle=$ $\displaystyle
n\frac{E_{q-1}(\beta\Lambda,u_{M})}{E_{q}(\beta\Lambda,u_{M})},$
$\displaystyle=$ $\displaystyle nF_{q}(\beta\Lambda,u_{M}),$
where we use the function $F_{q}(l,m)=\partial_{l}\log[E_{q}(l,m)]$. We obtain
from the former the direct result $\beta=\frac{n}{2U},$ (52) and, via
inversion of the latter equation [defining first $L_{q}(f,m)=F_{q}^{-1}(f,m)$
and thus $F_{q}[L_{q}(f,m),m]=f$], we finally obtain the relation between the
system variables _(equation of state)_ $\beta\Lambda=L_{q}(N/n,u_{M}).$ (53)
Note that we have intensive quantities on the left hand side, while extensive
ones appear in the r.h.s.. The entropy becomes
$\displaystyle S$ $\displaystyle=$ $\displaystyle
n\log\left[\frac{1}{n}\sqrt{\frac{2\pi}{\beta\tau}}E_{q}(\beta\Lambda,u_{M})\right]$
(54) $\displaystyle+n\left[\frac{1}{2}+\beta\Lambda
F_{q}(\beta\Lambda,u_{M})\right].$
Using now Eq. (53) we can recast things in term of the natural variables as
$\displaystyle S(U,n,u_{M},N)$ $\displaystyle=$ $\displaystyle
n\log\left[\frac{2}{n}\sqrt{\frac{\pi
U}{n\tau}}E_{q}[L_{q}(N/n,u_{M}),u_{M}]\right]$ (55)
$\displaystyle+\frac{n}{2}+L_{q}(N/n,u_{M})N.$
It is easy to verify, but crucial to our present goals, that macroscopic
observables and Lagrange multipliers become linked entropic-wise via
$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\left.\frac{\partial
S}{\partial U}\right|_{n,u_{M},N},$ (56) $\displaystyle\mu$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\left.\frac{\partial S}{\partial
n}\right|_{U,u_{M},N},$ (57) $\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\left.\frac{\partial S}{\partial
u_{M}}\right|_{U,n,N},$ (58) $\displaystyle\Lambda$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\left.\frac{\partial S}{\partial
N}\right|_{U,u_{M},n}.$ (59)
The first relation leads to Eq. (52), also showing that the $\beta-$multiplier
is the inverse of the variance $\beta=1/\tau\sigma^{2}$. The last relation
takes us to Eq. (53) and is indeed one of our equations of state. The other
two are $\displaystyle\beta\mu$ $\displaystyle=$
$\displaystyle\log\left[\frac{2}{n}\sqrt{\frac{\pi
U}{n\tau}}E_{q}[L_{q}(N/n,u_{M}),u_{M}]\right]-1$ (60)
$\displaystyle-2L_{q}^{(1,0)}(N/n,u_{M})\left(\frac{N}{n}\right)^{2}$ and
$\displaystyle\beta p$ $\displaystyle=$ $\displaystyle
n\frac{\exp[-L_{q}(N/n,u_{M})e_{q}(u_{M})]}{E_{q}[L_{q}(N/n,u_{M}),u_{M}]}$
(61) $\displaystyle+2L_{q}^{(0,1)}(N/n,u_{M})N.$
At this stage the reader will agree that it is fair to assert that our goal
has been successfully reached. _We have indeed constructed a social
thermodynamics for urban population flows_. The following equivalence may be
established vis-a vis the thermodynamics of chemical species:
• Temperature $\leftrightarrow$ Variance of relative growth. • Number of
inhabitants $\leftrightarrow$ Number of particles. • Number of towns
$\leftrightarrow$ Number of chemical species. • Volume $\leftrightarrow$
Maximum possible town’s population.
## VII Application: the scale-free ideal gas (SFIG)
This is our final step 7. We envision two main regimes, according to the
$\Lambda-$value: $\Lambda=0$ and $\Lambda>0$.
### VII.1 The SFIG-in-a-box
We will consider in some detail the first case here. Different scenarios can
be associated to $\Lambda\rightarrow 0$: i) the system is not isolated and
exchanges population with its surroundings, with a maximum-size constraint,
ii) the triplet $n$, $N$, $u_{M}$ is such that the equation of state yields
$\Lambda=0$, i.e., $N/n=\log_{q-1}[e_{q}(u_{M})]/u_{M}$, or iii) no size-
limitation exists ($u_{M}\rightarrow\infty$) but $N/n$ is large enough to
consider $\Lambda\sim 0$. In the latter case one can obtain an effective
$u_{M}-$value from normalization such that $u_{M}=E_{q}(\beta\Lambda)$. When
$\Lambda=0$ the Lagrangian $A$ is written as
$A=U-\mu n+pu_{M},$ (62)
so that we do not need knowledge of $N$. The equilibrium density is
$\rho(u,\dot{u})dud\dot{u}=\frac{n}{u_{M}}\sqrt{\frac{\beta\tau}{2\pi}}~{}e^{-\frac{\beta\tau}{2}\dot{u}^{2}}dud\dot{u}~{}~{}(0<u<u_{M}).$
(63)
The partial density $\rho(u)=\int d\dot{u}\rho(u,\dot{u})=n/u_{M}$ is constant
in $u$ so that $x$ is given by a power-law
$\rho_{X}(x)dx=\rho[u(x)]\frac{du}{dx}dx=n\frac{x_{0}^{q-1}}{u_{M}}\frac{dx}{x^{q}},$
(64)
with an associated rank-plot given by
$x(r)=x_{0}e_{q}[u_{M}(1-r/n)],$ (65)
where $r$ is the rank from 1 to $n$. Comparing with the equilibrium densities
found above in our numerical experiments with random walkers, a very nice fit
ensues as seen in Fig. 5, which _validates our methodology_. The entropy
becomes
$S(U,n,u_{M})=n\log\left[\frac{2}{n}\sqrt{\frac{\pi
U}{n\tau}}u_{M}\right]+\frac{n}{2},$ (66)
resembling that of the one-dimensional ideal gas. The state-equations are
$\beta\mu=\log\left[\frac{2}{n}\sqrt{\frac{\pi U}{n\tau}}u_{M}\right]-1,$ (67)
and
$\beta p=\frac{n}{u_{M}},$ (68)
in exact agreement with the ideal gas scenario.
As empirical $q=1$-examples we show the cases of i) Marshall Islands,mar ii)
d’Agosta Valley (Italy) ita and iii) Huelva-province (spain)ine . In all
instances the relative growth is nearly independent of the population (with
some finite-size noise) as in the case of Huelva, or with a secondary constant
trend for low-populated cities, as in d’Agosta Valley’s intance. Thus we
consider that the microscopic dynamics fits the proportional growth
hypothesis, with densities $\rho(u,\dot{u})$ nicely adapted to the ensuing
thermodynamic predictions. In all cases, geographical conditions set strong
limits to the city-sizes. The values for the macroscopic parameters are shown
in Table 1. Remarkably enough, the pressure $p$ due to the limited space is
highest for the Marshall Islands. Indeed, this system exhibits the lowest
volume $u_{M}$ and the lowest $\beta$ (highest “temperature”) for a large
number of units $n$.
| Marshall Islands | Agosta Valley | Huelva
---|---|---|---
$n$ | $160$ | $74$ | $79$
$x_{0}$ | $2.14$ | $126$ | $206$
$u_{M}$ | $0.038$ | $0.048$ | $0.059$
$\beta$ | $0.504$ | $2.05$ | $11.7$
$p$ | $0.829$ | $0.0133$ | $0.0115$
Table 1: Macroscopic parameters for the four SFIG-in-a-box examples
($\tau=100$ years2).
### VII.2 The SFIG under total population-constraint
We now consider $\Lambda>0$, with $u_{M}\rightarrow\infty$ for simplicity.
This case describes regions where internal migration dominates the microscopic
dynamics, and no upper limit is found for the city-size. According to the
equation of state Eq. (61), $p=0$ in this limit, so that we deal with the
lagrangian
$A=U-\mu n+\Lambda N.$ (69)
The equilibrium density is
$\rho(u,\dot{u})dud\dot{u}=\frac{n}{E_{q}(\beta\Lambda)}\sqrt{\frac{\beta\tau}{2\pi}}e^{-\frac{\beta\tau}{2}\dot{u}^{2}-\beta\Lambda
e_{q}(u)}dud\dot{u}~{}~{}(0<u),$ (70)
and the ecuation of state can be written in the form
$N=n\frac{E_{q-1}(\beta\Lambda)}{E_{q}(\beta\Lambda)}.$ (71)
The partial density for $x$ is given by a power-law with exponential cut-off
$\rho_{X}[x]dx=n\frac{x_{0}^{q-1}}{E_{q}(\beta\Lambda)}\frac{e^{-\beta\Lambda
x}}{x^{q}}dx,$ (72)
with an associated rank-plot
$x(r)=\frac{x_{0}}{\Lambda}E^{-1}_{q}\left[E_{q}(\Lambda)r/n\right].$ (73)
Again, this result fits the numerical equilibrium densities found above in our
numerical simulations (Fig. 6), _validating again our methodology_.
This is the most common situation in the Spanish provinces (for more details
see Ref. emp ). We have found nice agreement between the $q$ value obtained
from a fit to the microscopic dynamics and the $q-$value obtained from the fit
of the rank-plot to Eq. (73). We show some examples in Fig. 9, and the
associated macroscopic numerical results in Table 2. In the examples presented
below, using the parameter $\Lambda$ as a measure of the pressure generated by
the total population constraint, it turns out that Alicante is the province
with the highest pressure and Girona that with the lowest one, correlated with
a highest and a lowest ‘temperature’, respectively.
| Alicante | Almería | Girona | Lleida
---|---|---|---|---
$n$ | $140$ | $101$ | $220$ | $230$
$x_{0}$ | $83.9$ | $147$ | $141$ | $105$
$N$ | $18355.7$ | $3245.53$ | $4597.37$ | $2818.12$
$\log(\beta\Lambda)$ | $-6.26$ | $-5.35$ | $-5.18$ | $-4.02$
$\Lambda$ | $4.03~{}10^{-3}$ | $1.83~{}10^{-4}$ | $7.89~{}10^{-5}$ | $4.1~{}10^{-4}$
$\beta$ | $0.47$ | $25.9$ | $71.1$ | $43.8$
$q$ | $0.862$ | $1.135$ | $1.27$ | $1.16$
| Navarra | Vizcaya | Zaragoza | Granada
$n$ | $271$ | $111$ | $292$ | $167$
$x_{0}$ | $47.6$ | $219$ | $40.7$ | $263$
$N$ | $8976.55$ | $3571.01$ | $6906.03$ | $2539.5$
$\log(\beta\Lambda)$ | $-5.16$ | $-5.18$ | $-5.06$ | $-4.01$
$\Lambda$ | $6.69~{}10^{-4}$ | $1.18~{}10^{-4}$ | $6.22~{}10^{-4}$ | $2.03~{}10^{-3}$
$\beta$ | $8.57$ | $47.4$ | $10.2$ | $8.91$
$q$ | $1.06$ | $1.08$ | $1.18$ | $1.02$
Table 2: Macroscopic parameters for the SFIG-under-pop-constraint examples.
## VIII Conclusions
After initially introducing some useful social-macroscopic and social-
stochastic quantities we have
1. 1.
Postulated social, dynamic microscopic equations.
2. 2.
Validated them using urban population data.
3. 3.
Performed numerical simulations with random walkers that conclusively
demonstrated that a description using many microscopic variables has as a
counterpart a macroscopic one with few parameters.
4. 4.
Showed that such macroscopic description can be given an appropriate MaxEnt
form after constructing a “social” phase space, that allows one to derive
thermodynamic-like relations amongst our macro-parameters.
5. 5.
Finally, as an application, we successfully analyzed urban flows as modelled
by a scale invariant ideal gas.
## References
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* (9) L. C. Malacarne, R. S. Mendes, and E. K. Lenzi, Phys. Rev. E 65, 017106 (2001).
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* (14) B. C. Murtra, R.V. Solé, _Phys. Rev. E_ 82, 011102 (2010).
* (15) X. Gabaix, Quart. J. Econom. 114, 739 (1999).
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* (18) Weisstein, Eric W., _Benford’s Law_ from MathWorld.
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* (26) S. Galam, J. Stat. Phys. 61, 943 (1990).
* (27) UrbanSim: http://www.urbansim.org.
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* (41) Demis, edited by Hanno Sandvik, Wikimedia Commons.
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* (44) National Statistical Institute, Italy (web).
Figure 1: Top panel: quantiles from 0.1 to 0.9 each 0.1 for $p_{\Xi}(\xi)$ as
a function of the population $x$ (the median in shown in black). Bottom panel:
$P_{\Xi}(\xi)$’s cumulative distribution for each of Spain’s provinces
.
Figure 2: Variance $s^{2}/x$ vs. $x$. Red: quantiles from 0.1 to 0.9 each 0.1.
Solid black: fit to the median value. Dashed black lines: Finite-size’s
fluctuations are constant, while the multiplicative regime is given by a
straight line. Figure 3: Fit of $\langle\dot{x}/x\rangle$ to an expression of
the type represented by Eq. (16) for 12 Spanish provinces: Asturias, Almería,
Cáceres, Cuenca, Baleares, Lleida, Badajoz, Ávila, Guipúzcoa, Castellón,
Valladolid and Guadalajara.
Figure 4: Top panel: $q=21/2$-metric diffusion at times $t=20$ (green dotted
line), 70 (red dashed) and 110 (blue solid) using the text-parameters compared
with the distribution of Salamanca towns’ population in 2010, fitted to a
$1/2$-log-normal distribution (solid line). Inset: variance of the relative
growth vs. log-population (dots), confirming the $\sqrt{x}$ dependence for
$q=1/2$ dynamics (red line). Bottom panel: geometric diffusion $(q=1)$ at
times $t=4$ (green dotted), 9 (red dashed) and 29 (blue solid) compared with
the population distribution of Florida State (US) in 2010, fitted to a log-
normal distribution (solid line). Inset: same as top panel’s inset, confirming
size independence and thus proportional dynamics. Figure 5: $q$-metric
diffusion with maximum size constraint for (from left to right panels)
$q=1/2$, $1$ and $1.5$. Rank-distributions and evolution (insets). Figure 6:
$N$-constrained $q$-metric diffusion for $q=1$, $1.5$, and $2$. Figure 7: An
example of population restriction arising out of geographical reasons. Left
Panel: d’Aosta valley (Italy)aostamap . Right Panel: Marshall islandsmarmap .
Figure 8: SFIG-in-a-box examples, from top to bottom: RD of Huelva-province
(Spain), D’Aosta Valley (Italy), and Marshall Islands (dots), compared with
distribution Eq. (65) (lines). In the insets, the relative growth $\dot{u}$
vs. the logarithmic population $u$. Figure 9: SFIG under total-population
constraint examples, in the reading order: RD of Navarra, Lleida, Zaragoza,
Girona, Granada, Almería, Alicante, and Vizcaya (dots) and RD of Eq. (73)
(lines). Insets: relative growth $\dot{x}/x$ vs. log-population $\log x$
fitted to Eq. (16) with the same value of $q$ as in the RD.
|
arxiv-papers
| 2012-06-29T13:35:50 |
2024-09-04T02:49:32.372768
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Hernando, A. Plastino",
"submitter": "Alberto Hernando",
"url": "https://arxiv.org/abs/1206.7020"
}
|
1206.7024
|
11institutetext: Physikalisches Institut, University of Freiburg, Germany
# Physics at the LHC
-From Standard Model measurements to Searches for New Physics-
Karl Jakobs
###### Abstract
The successful operation of the Large Hadron Collider (LHC) during the past
two years allowed to explore particle interaction in a new energy regime.
Measurements of important Standard Model processes like the production of
high-$p_{\mathrm{T}}$ jets, $W$ and $Z$ bosons and top and $b$-quarks were
performed by the LHC experiments. In addition, the high collision energy
allowed to search for new particles in so far unexplored mass regions.
Important constraints on the existence of new particles predicted in many
models of physics beyond the Standard Model could be established. With
integrated luminosities reaching values around 5 fb-1 in 2011, the experiments
reached as well sensitivity to probe the existence of the Standard Model Higgs
boson over a large mass range. In the present report the major physics results
obtained by the two general-purpose experiments ATLAS and CMS are summarized.
## 0.1 Introduction
In March 2010 the Large Hadron Collider started its operation at the highest
centre-of-mass energy ever reached and delivered first proton-proton
collisions at 7 TeV. The years 2010 and 2011 showed a very successful
operation of both the collider and the associated experiments. During the
start-up year 2010 data corresponding to an integrated luminosity of about 48
pb-1 could be delivered. This successful operation was followed by an even
more successful year 2011 where the collider delivered data corresponding to
an integrated luminosity of 5.5 fb-1 and exceeded the original design goal of
1 fb-1 by far. In April 2011 the world record on the instantaneous luminosity
was reached with a luminosity of 4.7 $\cdot$ 1032 cm-2sec-1. Meanwhile
luminosities beyond 3$\cdot$1033 cm-2sec-1 have been reached.
However, not only the accelerator but also the experiments showed an extremely
successful operation. They were able to record the delivered luminosity with
efficiencies of the order of 94%. All detector subsystems worked well with a
high number of functioning channels, typically exceeding 99%.
The data were used to test the Standard Model [1, *SM2, *SM3, 4, *qcd2, *qcd3]
of particle physics in the new energy regime. At the LHC as a hadron collider
the production of particles via the strong interaction is dominating.
Therefore, the test of Quantum Chromodynamics (QCD) [4, *qcd2, *qcd3], the
theory of strong interactions, was in the focus during the early phase. Tests
of QCD can be performed at small distances or for processes with large
momentum transfer. Among them the production of jets with large transverse
momenta ($p_{\mathrm{T}}$) has the largest cross section. The investigation of
the production of $W$ and $Z$ bosons, their associated production with jets
and the production of top quarks constitute other important tests of QCD in
the new energy regime.
Due to the high centre-of-mass energy of 7 TeV the LHC has a large discovery
potential for new particles with masses beyond the limits set by the Tevatron
experiments, even already in the initial phase of operation. Due to the
dominating strong production, this holds in particular for particles that
carry colour charge, like e.g. the supersymmetric partners of quarks and
gluons. Due to the excellent luminosity performance of the LHC in 2011 the
sensitivity for many models of new physics were pushed far beyond the mass
range explored so far.
In this review the physics motivation for the LHC is briefly recalled in
Section 2. The phenomenology of proton-proton collisions and the calculation
of cross sections is briefly reviewed in Section 3. The measurement of
important Standard Model processes at the LHC is discussed in Section 4. The
status of the search for the Standard Model Higgs boson is summarized in
Section 5. It should be noted that in this paper the status of the Higgs boson
search as of March 2012 is presented. Given the large increase in the
integrated luminosity during the second half of 2011, these results supersede
by far those presented at the school in September 2011. In the remaining
sections of the paper the searches for physics beyond the Standard Model are
discussed. The search for supersymmetric particles is described in Section 6,
the search for other scenarios is summarized in Section 7.
## 0.2 The Physics Questions at the LHC
The Standard Model is a very successful description of the interactions of
particles at the smallest scales (10-18m) and highest energies accessible to
current experiments. It is a quantum field theory which describes the
interactions of spin-$\nicefrac{{1}}{{2}}$ pointlike fermions whose
interactions are mediated by spin-1 gauge bosons. The bosons are a consequence
of local gauge invariance of the underlying Lagrangian under the symmetry
group $SU(3)$x$SU(2)$x$U(1)$ [1, *SM2, *SM3].
The $SU(2)$ x $U(1)$ symmetry group, which describes the electroweak
interactions, is spontaneously broken by the existence of a postulated scalar
field, the so-called Higgs field, with a non-zero vacuum expectation value [7,
8, 9, 10, 11, 12]. This leads to the emergence of massive vector bosons, the
$W$ and $Z$ bosons, which mediate the weak interaction, while the photon of
electromagnetism remains massless. One physical degree of freedom remains in
the Higgs sector which should manifest as a neutral scalar boson $H$ which is
so far unobserved. The description of the strong interaction (Quantum
Chromodynamics or QCD) is based on the gauge group $SU(3)$ [4, *qcd2, *qcd3].
Eight massless gluons mediate this interaction.
All experimental particle physics measurements performed up to date are in
excellent agreement with the predictions of the Standard Model. The only
noticeable exception is the evidence for non-zero neutrino masses observed in
neutrino-oscillation experiments [13]. There remain, however, many key
questions open and it is generally believed that the Standard Model can only
be a low energy effective theory of a more fundamental underlying theory. One
of the strongest arguments for an extension of the Standard Model is the
existence of Dark Matter [14] in the universe. There is no explanation for
such a type of matter in the Standard Model.
The key questions can be classified to be linked to mass, unification and
flavour:
* •
Mass: What is the origin of mass?
How is the electroweak symmetry broken? Is the solution, as implemented in the
Standard Model, realized in Nature, and linked to this, does the Higgs boson
exist?
* •
Unification: What is the underlying fundamental theory?
Can the three interactions which are relevant for particle physics be unified
at larger energy and are there new symmetries found towards unification? Are
there new particles, e.g. supersymmetric particles, at higher energy scales?
And finally, it must also be answered how gravity can eventually be
incorporated.
* •
Flavour: Why are there three generations of matter particles? What is the
origin of CP violation in the weak interaction? What is the origin of neutrino
masses and mixings?
The high energy and luminosity of the LHC offers a large range of physics
opportunities. The primary role of the LHC is to explore the TeV-energy range
where answers to at least some of the aforementioned questions are expected to
be found. In the focus is certainly the search for the Higgs boson. The LHC
experiments have the potential to explore the full relevant mass range and
either to discover the Standard Model Higgs boson or to exclude its existence.
Another focus area constitutes the search for supersymmetric particles which
can be carried out at the LHC up the masses of a few TeV. If such particles
are discovered, their link to the dark matter in the universe must be
investigated. This can only be done in conjunction with experiments on direct
dark matter detection [15].
However, it is important to stress that the remit of the LHC is not only to
look for the expected or theoretically favoured models, but to carry out a
thorough investigation of as many final states as possible. It is important to
search for any deviation from the Standard Model predictions. This implies
that the Standard Model predictions must be reliably tested in the new energy
domain. In particular during the early phase of experimentation at the LHC,
detailed measurements of Standard Model processes must be carried out. Some of
these processes can as well be used for the understanding of the detector
response and its calibration.
Finally, with increasing precision of the Standard Model measurements, it is
also important to test the consistency of the model via quantum corrections.
Important contributions from the LHC in this area will be precise measurements
of the $W$ mass and of the top-quark mass, which can be used to constrain the
Higgs boson mass. A direct confrontation of this prediction to a direct Higgs
boson mass measurement may constitute the ultimate test of the Standard Model
at the LHC.
## 0.3 Phenomenology of proton-proton collisions
Scattering processes at high-energy hadron colliders can be classified as
either hard or soft. Quantum Chromodynamics is the underlying theory for all
such processes, but the approach and level of understanding is very different
for the two cases. For hard processes, e.g. high-$p_{\mathrm{T}}$ jet
production or $W$ and $Z$ production, the rates and event properties can be
predicted with good precision using perturbation theory. For soft processes,
e.g. the total cross section, the underlying event etc., the rates and
properties are dominated by non-perturbative QCD effects, which are less well
understood. An understanding of the rates and characteristics of predictions
for hard processes, both signals and backgrounds, using perturbative QCD
(pQCD) is crucial for tests of the theory and for searches for new physics.
Figure 1: Diagrammatic structure of a generic hard-scattering process (from
Ref. [16]).
The calculation of a hard-scattering process for two hadrons $A$ and $B$ can
be illustrated as displayed in Fig. 1. Two partons of the incoming hadrons
undergo a hard scattering process characterized by the cross section
$\hat{\sigma}$. The structure of the incoming hadrons is described by the
parton density functions (PDFs) $f_{a/A}(x_{a},\mu_{F}^{2})$ (see Section
0.3.1), i.e. the probability to find a parton $a$ in hadron $A$ with a
momentum fraction $x_{a}$ at the energy scale $\mu_{F}^{2}$. To obtain the
hadron-hadron cross section, a summation over all possible parton-parton
scattering processes and an integration over the momentum fractions has to be
performed [16]:
$\sigma_{AB}=\sum\limits_{a,b}\int\rm{d}x_{a}\cdot\rm{d}x_{b}\
f_{a/A}(x_{a},\mu_{F}^{2})\ f_{b/B}(x_{b},\mu_{F}^{2})\
\hat{\sigma}_{ab}(x_{a},x_{b},\alpha_{s}(\mu_{R}^{2}))\leavevmode\nobreak\ .$
(1)
The calculations of the hard scattering process $\hat{\sigma}_{ab}$ are
performed in perturbative QCD and the results depend on the strong coupling
constant $\alpha_{s}$ and its renormalization scale $\mu_{R}$. The scale
$\mu_{F}$ that appears in the parton density functions is the so-called
factorization scale, which can be thought of as the scale that separates long-
and short-distance physics [16]. Large logarithms related to gluons emitted
collinear with incoming quarks can be absorbed in the definition of the parton
densities, giving rise to logarithmic scaling violations which can be
described via the DGLAP111Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution
equations [17]. The perturbative calculation can be written as
$\hat{\sigma}_{ab}^{[n]}=\hat{\sigma}_{ab}^{[0]}+\sum\limits_{j=k+1}^{k+n}c_{j}\cdot\alpha_{s}^{j}\leavevmode\nobreak\
,$ (2)
where $\hat{\sigma}_{ab}^{[0]}$ denotes the leading order (LO) cross section
and $n$ denotes the perturbative order of the calculation. The index $k$
denotes the order of $\alpha_{s}$ appearing in the leading order calculation,
which might as well be 0, like for the Drell-Yan production of $W$ and $Z$
bosons, as discussed below. The cross sections at higher orders, which are
usually denoted as next-to-leading order (NLO) and next-to-next-to-leading
order (NNLO) etc., are often parametrized in terms of total $K$ factors,
defined at each perturbative order [$n$] as the ratio of the cross section
computed to that order normalized to the Born level cross section:
$\sigma^{[n]}\ =\ \sigma^{[0]}\cdot K_{\rm tot}^{[n]}.$ (3)
As discussed above, the scale $\mu$ is an arbitrary parameter, which in
general is, however, chosen to be of the order of the energy characterizing
the parton-parton interaction, like, for example, the mass of the vector
bosons or the transverse momenta of outgoing jets. The more orders are
included in the perturbative expansion, the weaker the dependence on $\mu$. As
an example, the production of $W$ and $Z$ bosons is discussed in Section
0.3.3.
Those partons which do not take part in the hard scattering process will
produce what is generally called the ‘underlying event’. Finally, it should be
stressed that Eq. 2 does not describe the bulk of the events which occur at a
hadron collider. It can only be used to describe the most interesting classes
of events which involve a hard interaction. Most events result from elastic
and soft inelastic interactions generally called ‘minimum bias’ events. In the
following a few specific examples of hard scattering processes are discussed.
### 0.3.1 Parton Distribution Functions
The parton distribution functions (PDFs) $f(x,Q^{2})$ for a hadron provide the
probability density of finding a parton with momentum fraction $x$ at momentum
transfer $Q^{2}$ which defines the energy scale of the process. The $Q^{2}$
dependence is induced by the usage of perturbation theory and the resulting
higher order corrections. It is described by the DGLAP evolution equations
[17]. However, the functional form of the PDFs is not predicted by
perturbative QCD and has to be measured experimentally.
Various classes of experiments are sensitive to the proton PDFs, such as deep
inelastic scattering at fixed target experiments with electron, muon or
neutrino beams, and electron-proton scattering at the HERA collider. Also
experiments at pure hadronic colliders such as the Tevatron ($p\bar{p}$) and
the LHC ($pp$) can yield valuable information.
In order to determine the parton distributions from the measurements, a
parametrization is assumed to be valid at some starting value
$Q^{2}=Q^{2}_{0}$. The DGLAP evolution functions are used to evolve the PDFs
to a different $Q^{2}$ where predictions of the measured quantities (e.g.
structure functions) are obtained. The predictions are then fitted to the
measured datasets, thus constraining the parameters (typically $10$ to $20$)
of the parametrisation [18]. Various collaborations performed fits to the
available datasets and provided PDF sets for the proton, for instance the
groups ABKM [19], CTEQ [20], CT10 [21], HERAPDF [22, 23], JR [24], MSTW [25]
and NNPDF [26, 27]. The NNPDF collaboration has already included the first
lepton charge asymmetry measurements in the $W$ boson production by the ATLAS
[28] and CMS [29] experiments in their fit [30]. The PDFs determined by MSTW
are shown in Fig. 2 at two different $Q^{2}$ scales. For example, it can be
observed that gluons dominate the low $x$ region and the contributions from
sea quarks become more dominant at higher $Q^{2}$.
Figure 2: Parton distribution functions of the proton as determined for the
MSTW08 PDF set for (left) $Q^{2}=10\,{\mathrm{\ Ge\kern-1.00006ptV}}^{2}$ and
(right) $Q^{2}=10^{4}\,{\mathrm{\ Ge\kern-1.00006ptV}}^{2}$. The bands reflect
the uncertainties at the $68$% confidence level (from Ref. [25]).
### 0.3.2 Jet Production via QCD scattering processes
Two-jet events result when an incoming parton from one hadron scatters off an
incoming parton from the other hadron to produce two high transverse momentum
partons which are observed as jets. The parton processes that contribute at
leading order are shown in Fig. 3. The matrix elements have been calculated at
leading order [31] and next-to-leading order [32, 33]. At the LHC, terms
involving gluons in the initial state are dominant at low $p_{\mathrm{T}}$.
Figure 3: Leading order diagrams for the production of high-$p_{\mathrm{T}}$
jets.
Unlike in lowest order, where a direct correspondence between a jet cross
section and the parton cross section can be made, a prescription is needed to
derive jet cross sections in next-to-leading order. When such prescriptions
are applied, the next-to-leading order cross sections show substantially
smaller sensitivities to variations of the renormalization scale than at
lowest order.
### 0.3.3 $W$ and $Z$ Production
In leading order the production of the vector bosons $W$ and $Z$ is described
by the Drell-Yan process, where a quark and an antiquark from the incoming
hadrons annihilate. This process has been calculated up to next-to-next-to-
leading order in the strong coupling constant $\alpha_{\mathrm{S}}$ [34, 35,
36, 37]. Some of the relevant Feynman diagrams are given in Fig. 4.
Figure 4: Leading order (top) and some next-to-leading order diagrams (bottom)
for the production of $W$ and $Z$ bosons.
When going from LO to NLO the cross sections increase by about 20% and the
factorization and renormalization scale uncertainties decrease. This is nicely
shown in Fig. 5.
Figure 5: Dependence of the production cross section of on-shell $Z$ bosons at
rapidity $y=0$ on the choice of the renormalization and factorization scales.
For each order in perturbation theory (LO, NLO, NNLO), three curves are shown.
The solid curve represents the results obtained under a common variation of
$\mu_{R}=\mu_{F}=\mu$ over the range $M/5<\mu<5M$. The dashed (dotted) curves
represent the results obtained under variations of the factorization
(renormalization) scale alone, holding the other scale fixed (from Ref. [37]).
Including NNLO contributions slightly decreases the cross-sections but the
result is consistent with the NLO prediction within the NLO scale uncertainty,
indicating that the perturbative expansion converges. The impact of higher
order corrections on the predicted rapidity222The rapidity $y$ of a particle
is related to its energy E and the projection of its momentum on the beam axis
$p_{z}$ by
$y=\frac{1}{2}\ln[(E+p_{z})/(E-p_{z})]$. The pseudorapidity $\eta$ is defined
as $\eta=-\ln\tan\frac{\theta}{2}$, where $\theta$ is the polar angle.
distributions of the $W$ and $Z$ bosons is shown in Fig. 6 for proton-proton
collisions with a centre-of-mass energy of $\sqrt{s}=14$ TeV. This figure also
illustrates that larger cross sections for $W^{+}$ production than for $W^{-}$
production are expected at the LHC. This asymmetry results from the dominance
of $u$ over $d$ valence quarks in the incoming protons (see also Fig. 2).
Figure 6: Theory predictions at LO, NLO and NNLO of the rapidity distributions
for $W$ (left) and $Z$ (right) boson production in proton-proton collisions at
$\sqrt{s}=14$ TeV. The bands indicate the factorization and renormalization
scale uncertainties, obtained by scale variations in the range
$M_{W/Z}/2\leq\mu\leq 2M_{W/Z}$ (from Ref. [37]).
Electroweak radiative corrections for the $W$ and $Z$ boson production have
been computed up to next-to-leading order [38, 39]. These corrections change
the production cross sections and affect kinematic properties like the lepton
transverse momenta, lepton rapidities and the transverse and invariant masses
of the lepton pairs. In particular for precision measurements like that of the
$W$ boson mass, it is therefore important to take electroweak radiative
corrections into account.
### 0.3.4 Cross Sections at the LHC
An overview of cross sections of some benchmark processes at proton-proton and
proton-antiproton colliders as a function of the centre-of-mass energy is
shown in Fig 7. The total inelastic proton-proton cross section is dominant
and reaches a huge value of about 70 mb at the LHC. Processes which can
proceed via the strong interaction have a much larger cross section than
electroweak processes. The dominant electroweak process, the production of $W$
and $Z$ bosons is found to be about six and seven orders of magnitude smaller
than the total inelastic proton-proton cross section. However, this process
constitutes the most copious source of prompt high-$p_{\mathrm{T}}$ leptons,
which are important for many physics measurements and searches for new physics
at the LHC. The production processes of the Standard Model Higgs boson and of
other non-coloured heavy new particles have small cross sections and therefore
require a correspondingly high integrated luminosity for their detection. The
Higgs boson production cross section is found to be ten to eleven orders of
magnitude smaller than the inelastic pp cross section. The exact values depend
strongly on the mass of the Higgs boson, as further discussed in Section
0.5.1.
Figure 7: Standard Model cross sections at the Tevatron and LHC colliders as
function of the centre-of-mass energy $\sqrt{s}$. In case of the LHC both the
energy during data taking in 2010/2011 ($\sqrt{s}=7$ TeV) and the nominal
energy ($\sqrt{s}=14$ TeV) are marked (adapted from Ref. [16]).
## 0.4 Measurement of Standard Model processes
### 0.4.1 The production of high-$p_{\mathrm{T}}$ jets
A measurement of the production of high-$p_{\mathrm{T}}$ jets constitutes an
important test of QCD in the new energy regime of the LHC. Events with two
high transverse momentum jets (dijets) arise from parton-parton scattering
where the outgoing scattered partons manifest themselves as hadronic jets. The
measurements of the inclusive jet production cross section or the dijet
production cross section are therefore also sensitive to the structure of the
proton and may lead to further constrains on the PDFs. In addition, the
precise measurement of jet production is important for searches of physics
beyond the Standard Model. New physics may lead to significant deviations from
the expected QCD behaviour. For example, a substructure of quarks may manifest
itself in deviations of the measured inclusive jet-production cross section
from the expected behaviour at high transverse momenta. The measurement of the
dijet cross section as a function of the dijet mass $m_{jj}$ allows for a
sensitive search for physics beyond the Standard Model, such as dijet
resonances or contact interactions of composite quarks.
The production of multijets provides as well an important background in the
search for physics beyond the Standard Model. In many cases, leptons and
missing transverse energy are used as final states signatures. Although they
do not appear at first place in QCD jet production, they might originate from
decays of heavy quarks, from mis-measurements of jet energies or from mis-
identification of jets as leptons. Although the probability for this to happen
is small, the contributions to the background can be sizeable, give the huge
jet production cross sections.
#### Jet reconstruction and calibration
For the reconstruction of jets both the ATLAS and CMS experiments use the
infrared- and collinear-safe anti-$k_{T}$ jet clustering algorithm [40] with
distance parameters $0.4\leq R\leq 0.7$. The inputs are either topological
calorimeter cluster energies [41, 42] in the ATLAS experiment or particle flow
objects [43, 44] in the CMS experiment. For the theoretical comparison the
input can also be four-vectors from stable particles in generator-level
simulations. In all cases residual jet-level corrections are needed to account
for energy losses not detectable on cluster or particle flow level. These jet-
level calibrations are Monte Carlo based correction functions in
pseudorapidity $|\eta|$ and $p_{\mathrm{T}}$. The jet energy scale and the
attached uncertainties are validated with in-situ methods using the balance of
transverse momenta in dijet and $\gamma$-jet events. The systematic jet energy
scale uncertainties are found to be typically in the range of $\pm(3-6)\%$
over a large range of $\eta$ and $p_{\mathrm{T}}$. The larger values are
reached at large $|\eta|$ as well as at very low and very high
$p_{\mathrm{T}}$.
#### Jet cross section measurements
The inclusive jet production cross section has been measured by both the ATLAS
[45] and CMS [46] experiments as a function of the jet transverse momentum
($p_{\mathrm{T}}$) and jet rapidity ($y$). In addition, double differential
cross sections in the maximum jet rapidity $y_{\rm{max}}$ and dijet mass
$m_{\rm{jj}}$ for dijet events are measured [45, 47]. The data are corrected
for migration and resolution effects due to the steeply falling spectra in
$p_{\mathrm{T}}$ and mass. The NLO perturbative parton-level QCD predictions
are corrected for hadronisation and the underlying event activity. Figure 8
(left) shows the inclusive jet cross-section measurement for jets with size
$R=0.4$ as a function of jet transverse momentum from the ATLAS collaboration
[45], based on the total data set collected in 2010 corresponding to an
integrated luminosity of 37 pb-1. The experimental systematic uncertainties
are dominated by the jet energy scale uncertainty. There is an additional
overall uncertainty of $\pm$3.4% due to the luminosity measurement. The
theoretical uncertainties result mainly from the choice of the renormalization
and factorization scales, parton distribution functions, $\alpha_{s}(m_{Z})$
and the modelling of non-perturbative effects.
The cross section measurement as a function of the dijet invariant mass from
the CMS collaboration [47] is shown in Fig. 8 (right). Like for the inclusive
jet cross section measurements, the experimental uncertainties are in the
range 10-20% and are dominated by uncertainties on the jet energy scale and
resolution.
Figure 8: (Left): Inclusive jet double-differential cross section as a
function of jet $p_{\mathrm{T}}$ in different regions of $|y|$ from the ATLAS
collaboration. (Right): Measured double-differential dijet cross sections
(points) as a function of the dijet invariant mass $m_{jj}$ in bins of the
variable $y_{\rm{max}}$ from the CMS collaboration. The data are compared in
both cases to NLO pQCD calculations to which non-perturbative corrections have
been applied. The error bars indicate the statistical uncertainty on the
measurement. The dark-shaded band indicates the quadratic sum of the
experimental systematic uncertainties, excluding the uncertainties from the
luminosity. The theory uncertainty is shown as the light, hatched band (from
Refs. [45, 47]).
Different NLO pQCD predictions, using different PDF sets, are compared to the
data and the corresponding ratios of data to the NLO predictions. Figure 9
shows an example from the ATLAS collaboration [45]. Within the experimental
and theoretical uncertainties the data are well described by the predictions,
although they are found to be systematically higher than the data. The
deviations become larger at large $|y|$ and $p_{\mathrm{T}}$. However, it is
impressive to see that the QCD calculations are able to describe the data over
many orders of magnitude and up to the highest values of $p_{\mathrm{T}}$ and
mass ever observed.
Figure 9: Ratios of inclusive jet double-differential cross sections to the
theoretical predictions. The ratios are shown as a function of jet
$p_{\mathrm{T}}$ in different regions of $|y|$. The theoretical error bands
obtained by using NLOJET++ with different PDF sets (CT10, MSTW 2008, NNPDF
2.1, HERAPDF 1.5) are shown (from Ref. [45]).
The ATLAS and CMS collaborations have performed many further studies on jet
production, including the measurement of dijet angular distributions [48] and
dijet angular decorrelations [45, 49]. At Born level, dijets are produced with
equal transverse momenta $p_{\mathrm{T}}$ and back-to-back in the azimuthal
angle ($\Delta\phi_{\rm{dijet}}=|\phi_{\rm{jet1}}-\phi_{\rm{jet2}}|$). Gluon
emission will decorrelate the two highest $p_{\mathrm{T}}$ jets and cause
smaller angular separations. The measurement of the angular distribution
between the highest $p_{\mathrm{T}}$ jets is therefore also a sensitive test
of perturbative QCD with the advantage that the measurement is not strongly
affected by the dominant systematic uncertainty on the jet energy scale. The
predictions from NLO pQCD are found to be in reasonable agreement with the
measured distributions [45, 49].
Figure 10: Event display of a six-jet event passing the ATLAS multijet
selection requirements. The towers in the bottom right figure represent
transverse energy deposited in the calorimeter projected on a grid of $\eta$
and $\phi$. Jets with transverse momenta ranging from 84 to 203 GeV are
measured in this event (from Ref. [50]).
In addition, the production of multijets was studied [50, 51]. In a data
sample corresponding to an integrated luminosity of 2.4 pb-1 the ATLAS
collaboration has identified 115 events with more than six jets. One such
event is shown in Fig. 10. The transverse energy deposition in the calorimeter
is shown as a function or $\eta$ and $\phi$. For this event the six jets are
well separated spatially. Leading-order Monte Carlo simulations have been
compared to the measured multi-jet inclusive and differential cross sections.
For events containing two or more jets with $p_{\mathrm{T}}>$ 60 GeV, of which
at least one has $p_{\mathrm{T}}>$ 80 GeV, a reasonable agreement is found
between data and leading-order Monte Carlo simulations with parton-shower
tunes that describe adequately the ATLAS $\sqrt{s}=7$ TeV underlying event
data. The agreement is found after the predictions of the Monte Carlo
simulations are normalized to the measured inclusive two-jet cross section.
### 0.4.2 The production of $W$ and $Z$ bosons
$W$ and $Z$ bosons are expected to be produced abundantly at the LHC. The
large dataset and the high LHC energy allow for detailed measurements of their
production properties in a previously unexplored kinematic domain. These
conditions, together with the proton-proton nature of the collisions, provide
new constraints on the parton distribution functions and allow for precise
tests of perturbative QCD. Besides the measurements of the $W$ and $Z$ boson
production cross sections, the measurement of their ratio $R$ and of the
asymmetry between the $W^{+}$ and $W^{-}$ cross sections (see Section 0.3.3)
constitute important tests of the Standard Model. This ratio $R$ can be
measured with a higher relative precision because both experimental and
theoretical uncertainties partially cancel. With larger data sets this ratio
can be used to provide constraints on the $W$-boson width $\Gamma_{W}$ [52].
#### Inclusive cross-section measurements
Measurement of the $W^{+},W^{-}$ and $Z/\gamma^{*}$ boson inclusive production
cross sections are performed using the leptonic decay modes $W\to\ell\nu$ and
$Z\to\ell\ell$. Already in 2010, the two collaborations published first
measurements in the electron and muon decay modes [53, 54]. They were updated
with the full data sample taken in 2010 corresponding to an integrated
luminosity of 36 pb-1 [55, 56]. In this data sample the ATLAS experiment has
observed a total of about 270.000 $W\to\ell\nu$ decays and a total of about
24.000 $\mbox{$Z/\gamma^{*}$}\to\ell\ell$ decays. The measurements in the
electron and muon channels were found to give consistent results and were
combined to obtain a single joint measurement taking into account the
statistical and systematic uncertainties and their correlations. The results
are displayed in Fig. 11 together with previous measurements of the total $W$
and $Z$ production cross sections by the UA1 [57] and UA2 [58] experiments at
$\sqrt{s}=0.63$ TeV at the CERN Sp$\overline{\rm{p}}$S and by the CDF [52] and
DØ [59] experiments at $\sqrt{s}=1.8$ TeV and $\sqrt{s}=1.96$ TeV at the
Fermilab Tevatron collider and by the PHENIX [60] experiment in proton-proton
collisions at $\sqrt{s}=0.5$ TeV at the RHIC collider. These measurements are
compared to the NNLO theoretical predictions for proton-proton and proton-
antiproton collisions. All measurements are in good agreement with the
theoretical predictions and the energy dependence of the total $W$ and $Z$
production cross sections is well described.
Figure 11: The measured values of $\sigma_{W}\cdot\mathrm{BR}$
($W\rightarrow\ell\nu)$ for $W^{+}$, $W^{-}$ and for their sum (left) and of
$\sigma_{Z/\gamma^{*}}\times\mathrm{BR}$ ($Z/\gamma^{*}\rightarrow\ell\ell$)
(right) compared to the theoretical predictions based on NNLO QCD
calculations. Results are shown for the combined measurements of electron and
muon final states. The predictions are shown for both proton-proton ($W^{+}$,
$W^{-}$and their sum) and proton-antiproton colliders ($W$) as a function of
$\sqrt{s}$. In addition, previous measurements at proton-antiproton and
proton-proton colliders are shown. The data points at the various energies are
staggered to improved readability. The data points are shown with their total
uncertainty. The theoretical uncertainties are not shown in this figure (from
Ref. [56]).
The precision of the integrated $W$ and $Z/\gamma^{*}$ cross sections in the
fiducial regions is $\sim\pm 1.2\%$ with an additional uncertainty of
$\pm$3.4% resulting from the knowledge of the luminosity. It should be noted
that the experimental uncertainties are already dominated by systematic
uncertainties. The total integrated cross sections are obtained from an
extrapolation of the measurement in the fiducial regions to the full
acceptance. Due to uncertainties on the acceptance corrections, the
uncertainties on the total cross sections are about twice as large.
A summary of the ratios of the measured total $W^{+},W^{-},W$ and
$Z/\gamma^{*}$ cross sections by the CMS collaboration to the theoretical NNLO
calculations is shown in Fig. 12 (left). Within the experimental and
theoretical uncertainties there is excellent agreement. This figure also
includes a comparison of the measured ratios $R_{W/Z}=\sigma_{W}\cdot
BR(W\rightarrow\ell\nu)/\sigma_{Z}\cdot BR(Z\rightarrow\ell\ell)$ and
$R_{+/-}=\sigma_{W^{+}}\cdot
BR(W^{+}\rightarrow\ell^{+}\nu)/\sigma_{W^{-}}\cdot
BR(W^{-}\rightarrow\ell^{-}\nu)$. Due to the cancellation of uncertainties,
most notably the luminosity uncertainty, the precision of these ratio
measurements is more precise. Also the measured ratios are well described by
the theoretical NNLO calculations.
Figure 12: (Left): Ratio of CMS measurement of $W$ and $Z$ cross sections to
theory expectations. The experimental uncertainty is the sum in quadrature of
the statistical and the systematic uncertainties not including the uncertainty
on the extrapolation to the full acceptance due to parton density functions
(taken from Ref. [55]). (Right): Measurements of the $Z\to\tau\tau$ cross
sections from the ATLAS experiment in various $\tau$ decay modes and
comparison to theoretical predictions and to the measurements in the electron
and muon channels (taken from Ref. [61]).
Meanwhile the cross sections have also been measured in the $W\to\tau\nu$ [62]
and $Z\to\tau\tau$ [63, 61] decay modes, where hadronically decaying $\tau$
leptons are identified and reconstructed. The results obtained by the ATLAS
collaboration in various $\tau$ decay modes are displayed in Fig. 12 (right).
They are found to be in good agreement with theoretical predictions and with
the results obtained in the $Z\to e^{+}e^{-}$ and $Z\to\mu\mu$ final states.
#### Differential cross section measurements
With the complete data set collected in 2010 more differential cross-section
measurements became possible. Both the ATLAS and CMS collaborations have
performed measurements as a function of lepton pseudorapidity $\eta_{\ell}$,
for the $W^{\pm}$ cross sections, and of the boson rapidity, $y_{Z}$, for the
$Z/\gamma^{*}$ cross section [55, 56]. For the $Z/\gamma^{*}$ case, all values
refer to dilepton mass windows from 66 - 116 GeV and 60 - 110 GeV for the
ATLAS and CMS analyses, respectively. The cross sections are measured in well-
defined kinematic regions within the detector acceptance, defined by the
pseudorapidity of the charged lepton and the transverse momentum of the
neutrino.
Figure 13: Differential cross-section measurement $d\sigma/d|\eta_{\ell}|$ for
$W^{+}$ (left) and $W^{-}$ (right) for $W\to\ell\nu$ from the ATLAS
collaboration compared to the NNLO theory predictions using various PDF sets.
The kinematic requirements are $p_{\mathrm{T}}(\ell)$ > 20 GeV,
$p_{\mathrm{T}}(\nu)$ > 25 GeV and $m_{T}$ > 40 GeV. The ratio of theoretical
predictions to data is also shown. Theoretical points are displaced for
clarity within each bin (from Ref. [56]).
The differential $W^{+}$ and $W^{-}$ cross sections as measured by the ATLAS
collaboration are shown in Fig. 13. The measurements for the electron and muon
final states were found to be in good agreement with each other and were
combined. These data are compared with the theoretical NNLO predictions using
various NNLO PDF sets (JR09, ABKM09, HERAPDF1.5 and MSTW08). The differential
$Z/\gamma^{*}$ cross section as a function of the boson rapidity as measured
by the ATLAS and CMS collaborations are shown in Fig. 14. Although the gross
features of these differential $W$ and $Z$ cross-section measurements are well
described by the theoretical calculations, the (pseudo)rapidity dependence
shows some disagreement which carries important information on the underlying
parton density functions. It is expected that these differential measurements
will reduce the uncertainties on the parton density functions. Very recently,
these data have been used together with the $ep$ scattering data from HERA to
extract the ratio or the strange-to-down see quark density at Bjorken $x$
values around 0.01. The ratio is found to be consistent with 1 and supports
the hypothesis that the density of the light sea quarks is flavour independent
[64]. The general agreement between theory and experiment is remarkable and
provides evidence for the universality of the PDFs and the reliability of
perturbative QCD calculations in the kinematic regime of the LHC.
Figure 14: Differential cross-section measurements ${\rm d}\sigma/{\rm
d}|y_{Z}|$ for $Z\to\ell\ell$ from the ATLAS (left) and CMS (right)
collaborations compared to NNLO theory predictions using various PDF sets. The
kinematic requirements are $66<m(\ell\ell)<116$ GeV and
$p_{\mathrm{T}}(\ell)>20$ GeV. The ratio of theoretical predictions to data is
also shown for the ATLAS measurements. Theoretical points are displaced for
clarity within each bin (from Refs. [56, 55]).
#### Measurements of the associated $W$ and $Z$ \+ jet production
The study of the associated production of vector bosons with
high-$p_{\mathrm{T}}$ jets constitutes another important test of the
perturbative QCD. In addition, these final states are a significant background
to studies of other Standard Model processes, such as $t\bar{t}$ or diboson
production, as well as for searches for the Higgs boson and for physics beyond
the Standard Model.
The ATLAS and CMS collaborations have presented detailed measurements of these
processes based on the complete dataset from 2010, corresponding to an
integrated luminosity of 36 pb-1 [65, 66, 67, 68]. Cross sections have been
determined for the associated $W$ and $Z$+jet production as a function of
inclusive jet multiplicity, $N_{\rm{jet}}$, for up to five jets. At each
multiplicity, the cross sections have also been presented as a function of jet
transverse momenta of all jets. The results, corrected for all detector
effects and for all backgrounds such as diboson and top quark pair production,
are compared with particle-level predictions from perturbative QCD. As an
example, the $W$+jets cross-section measurements as a function of jet
multiplicity are shown in Fig. 15 (left) and as a function of the
$p_{\mathrm{T}}$ of the first jet in the event in Fig. 15 (right). Leading-
order multiparton event generators like ALPGEN [69] or SHERPA [70], normalized
to the NNLO total cross section for inclusive $W$-boson production, describe
the data reasonably well for all measured inclusive jet multiplicities. Next-
to-leading-order calculations from MCFM [71], studied for $N_{\rm{jet}}\leq
2$, and BlackHat-Sherpa [72], studied for $N_{\rm{jet}}\leq 4$, are found to
be mostly in good agreement with the data. This also holds for the measurement
of the transverse momentum distributions of the $W$ and $Z$ boson, which are
correlated to the jet activity in the $W$ and $Z$ events.
Figure 15: The measured $W$+jets cross sections as a function of jet
multiplicity (left) and as a function of the $p_{\mathrm{T}}$ of the first jet
in the event (right). The $p_{\mathrm{T}}$ of the first jet is shown
separately for events with $\geq 1$ jet to $\geq 4$ jets. Shown are
predictions from ALPGEN, SHERPA and BlackHat-SHERPA, and the ratio of
theoretical predictions to data (from Ref. [68]).
### 0.4.3 Production of top quarks
#### Measurement of production cross sections
The top quark is the heaviest known elementary particle with a mass of about
173 GeV. Due to its high mass it is believed to play a special role in the
electroweak symmetry breaking and possibly in models of new physics beyond the
Standard Model. In that context it is remarkable that its Yukawa coupling
$\lambda_{t}$ is close to one. We still know little about the properties of
the top quark, like spin, charge, lifetime, decay properties (rare decays) or
the gauge and Yukawa couplings. Another important parameter is the mass of the
top quark, which has, however, been relatively precisely measured at the
Tevatron collider to be $m_{\rm top}=173.2\pm 0.90$ GeV, i.e. with a precision
of 0.5%. A further improvement here is important for a precise test of
electroweak radiative corrections.
Due to the high mass, the top quarks decays mainly via $t\to Wb$ before it
hadronizes. The production of $t\bar{t}$ pairs at the LHC proceeds mainly via
gluon initiated processes and is expected to be a factor of 20 larger at the
LHC with $\sqrt{s}$ = 7 TeV than at the Tevatron. The decays studied are
characterized by the number of charged leptons in the final state. A large
fraction of the branching ratio is devoted to lepton-hadron decays, where one
of the $W$s decays leptonically and the other one into a pair of jets, i.e.
$t\bar{t}\to Wb\ Wb\to\ell\nu b\ qqb$. The final state in this case consists
of a lepton, missing transverse momentum (carried away by the neutrino) and
four jets out of which two are originating from b-quarks. The complementary
dilepton decay mode is also important for top-quark physics at hadron
colliders. The fully hadronic channel is more difficult to trigger on and
shows a worse signal-to-background ratio, but despite this has also been
measured at the LHC.
Both the ATLAS and CMS collaborations measured the production cross section
for the pair production of top quarks in all above-mentioned final states [73,
74, 75, 76, 77]. The results are displayed in Fig. 16. The most precise
measurement comes from the single lepton channel, which shows already a
precision of the order of $\pm$7%. In this channel the cross sections are
measured with and without the requirement of a b-tagged jet. The results
obtained in the dilepton channel are consistent with these results. The
measurements are found to be in good agreement with the approximate NNLO
calculations [78, 79], although the experimental values are found to be about
1$\sigma$ higher in each experiment. The experimental measurement is already
limited by the experimental systematic uncertainties (jet energy scale,
b-tagging, …) and by the uncertainty on the luminosity.
Figure 16: The measured value of $\sigma_{t\bar{t}}$ in the various decay
modes and the combination of these measurements from the ATLAS (left) and CMS
(right) experiments. The approximate NNLO prediction with its uncertainty is
also shown (from Refs. [77, 74]).
Single top quarks can be produced at the LHC via electroweak processes. The
$t$-channel production of single top-quarks has been measured by both the
ATLAS [80] (L = 0.7 fb-1) and CMS [81] (L = 1.5 fb-1) collaborations. The
results are found to be consistent with the Standard Model expectations. The
measured cross section value from the CMS collaboration is shown in Fig. 17 in
comparison to the theoretical expectation and the measurements at the
Tevatron.
#### Measurement of the top-quark mass
Among the various top-quark properties, the ATLAS and CMS collaborations have
already presented first measurements on the top-quark mass $m_{t}$ in several
final states [82, 83, 84, 85]. The most precise result was presented recently
as a preliminary result by the CMS collaboration [84]. The top-quark mass has
been measured using a sample of $t\bar{t}$ candidate events with one muon and
at least four jets in the final state. The full 2011 data set corresponding to
an integrated luminosity of 4.7 fb-1 was used. In this sample 2391 candidate
events were selected and using a likelihood method the top-quark mass was
measured from fits to kinematic distributions, simultaneously with the jet
energy scale (JES). The result of $m_{t}=172.6\pm 0.6\ \rm{(stat+JES)}\ \pm
1.2\ \rm{(syst)}$ GeV is consistent with the Tevatron result (see Fig. 17
(right)). It is impressive that such a precision, in particular on the
systematic uncertainty, can be claimed after only two years of operation of
the LHC. The dominant contribution to this systematic uncertainty results from
uncertainties on the $b$-jet energy scale and from modelling uncertainties
estimated via changes of the factorization scale [84]. However, it remains to
be seen whether the small overall uncertainty can be confirmed by the ATLAS
experiment. The results of the present measurements at the LHC are summarized
in Fig. 17 together with the Tevatron results.
Figure 17: (Left): Measured cross section for single-top quark production via
the $t$-channel process in the CMS experiment in comparison to the theoretical
expectation and the measurements at the Tevatron (from Ref. [81]). (Right):
Compilation of the top-quark mass measurements from the ATLAS, CDF, CMS and DØ
experiments (from Ref. [84]).
### 0.4.4 The production of diboson pairs
The production of pairs of bosons ($W\gamma$, $WW$, $WZ$, $ZZ$) at the LHC is
of great interest since it provides an excellent opportunity to test the
predictions on the structure of the gauge couplings of the electroweak sector
of the Standard Model at the TeV energy scale. In addition, $WW$ and $ZZ$
pairs constitute the irreducible background in important Higgs boson search
channels like $H\to WW$ and $H\to ZZ$.
The dominant Standard Model $W^{+}W^{-}$ production mechanisms are $s$-channel
and $t$-channel quark-antiquark annihilation. The $s$-channel production
occurs only through the triple gauge coupling vertex and accounts for
$\sim$10% of the full $W^{+}W^{-}$ production cross section. The leading-order
Feynman diagrams for the dominant $q\bar{q}^{\prime}\to W^{+}W^{-}$ production
mechanisms at the LHC are shown in the left and middle diagrams of Fig. 18.
The total cross section $\sigma(q\bar{q},q\bar{q}^{\prime})\to W^{+}W^{-}$ are
known at next-to-leading order. The gluon fusion through quark loops, shown in
the right diagram of Fig. 18, contributes an additional 2.9%.
Figure 18: (Left): The Standard Model tree-level Feynman diagram for
$W^{+}W^{-}$ production through the $q\bar{q}$ initial state in the
$t$-channel. (Middle): The corresponding Standard Model tree-level diagram in
the $s$-channel, which contains the $WWZ$ and $WW\gamma$ triple gauge boson
coupling (TGC) vertices. (Right): The gluon fusion process, mediated by quark
loops.
The $ZZ$ production proceeds at leading order via $t$-channel quark-antiquark
interactions. The $ZZZ$ and $ZZ\gamma$ triple gauge boson couplings (nTGCs)
are absent. Hence there is no contribution from $s$-channel $q\bar{q}$
annihilation at tree level. Many models of physics beyond the Standard Model
predict values of nTGCs at the level of 10-4 to 10-3 [86]. The signature of
nonzero nTGCs is an increase of the $ZZ$ cross section at high $ZZ$ invariant
mass and high transverse momentum of the $Z$ bosons [87].
The ATLAS and CMS collaborations have measured the cross sections for all
diboson production processes, $W\gamma$ [88, 89], $WW$ [90, 91, 92], $WZ$ [93,
91], $ZZ$ [94, 91]. Several analyses are already based on the full data set
from 2011. The results are found to be in good agreement with the predictions
from the Standard Model and first constraints on anomalous triple gauge boson
couplings have been set. The agreement between the measurements and the
Standard Model predictions is shown for the $WW$ and $ZZ$ production in Fig.
19. The constraints on the triple gauge boson couplings are still limited by
the number of observed diboson events.
Figure 19: (Left): The distributions of $m_{T}$ of the
dilepton+$E_{\mathrm{T}}^{\mathrm{miss}}$ system for the $W^{+}W^{-}$
candidates in the ATLAS experiment (from Ref. [92]). (Right): The distribution
of the four-lepton invariant mass for the $ZZ$ candidate events in the CMS
experiment (from Ref. [91]).
### 0.4.5 Summary
As discussed in the previous subchapters, the first two years have seen a very
successful operation of the LHC collider and of the experiments. The data
collected have been used to extract precise measurements of many Standard
Model processes. They range from the measurement of $W$ and $Z$ production
with cross sections in the order of picobarns via the measurement of
$t\bar{t}$ production to the measurement of important diboson processes. The
summary of all measured cross sections by the ATLAS collaboration is shown in
Fig. 20 together with the theoretical predictions. Within the uncertainties,
excellent agreement is found for all the processes considered. This is a
remarkable achievement of the Standard Model and the underlying theoretical
concepts, including QCD and factorization. The smallest cross sections
measured, the diboson production cross sections, are extremely relevant for
the Higgs boson search, as discussed in the next section.
Figure 20: Summary of several Standard Model total production cross-section
measurements from the ATLAS collaboration compared to the corresponding
theoretical expectations. The integrated luminosities used for the
measurements are indicated on the figure. The dark error bars represent the
statistical uncertainties. The red error bars represent the full
uncertainties, including systematics and luminosity uncertainties. All
theoretical expectations were calculated at NLO or higher.
## 0.5 Search for the Higgs boson
The Higgs boson is the only Standard Model particle that has not been
discovered so far. Indirectly, high precision electroweak data constrain its
mass via their sensitivity to radiative corrections. Assuming the overall
validity of the Standard Model, a global fit [95] to all electroweak data
leads to $m_{H}=94^{+29}_{-24}$ GeV. On the basis of the present theoretical
knowledge, the Higgs sector in the Standard Model remains largely
unconstrained. While there is no direct prediction for the mass of the Higgs
boson, an upper limit of $\sim$1 TeV can be inferred from unitarity arguments
[96, *higgs-unitarity1, *higgs-unitarity2].
Direct searches at the $e^{+}e^{-}$ collider LEP has led to a lower bound on
its mass of 114.4 GeV [99]. Before the LHC started its operation, the Fermilab
Tevatron $p\overline{p}$ collider with a centre-of-mass energy of 1.96 TeV was
the leading accelerator exploring the energy frontier. Until the end of data
taking in September 2011, the two experiments CDF and DØ have collected data
corresponding to an integrated luminosity of 11.9 fb-1. During the past years,
they were able to exclude Higgs boson masses around 160 GeV, mainly based on
the search for the $H\to WW^{(*)}\rightarrow\ell\nu\ell\nu$ decay mode. In
Summer 2011, the combination of the results from the two experiments, based on
data corresponding to an integrated luminosity of 8.6 fb-1, excluded a mass
range from 156 - 177 GeV [100]. At the same time, the first exclusions from
the ATLAS and CMS experiments, based on a data corresponding to an integrated
luminosity of up to 2.3 fb-1, were presented. The ATLAS experiment excluded
mass ranges from 146 - 230 GeV, 256-282 GeV and 296 - 459 GeV [101]. The CMS
analysis was based on data corresponding to an integrated luminosity of up to
1.7 fb-1 and the Higgs boson was excluded in the mass ranges from 145 - 216
GeV, 226 - 288 GeV and 310 - 400 GeV[102].
Since then the full data set of the LHC taken until the end of 2011 has been
analyzed. Preliminary results were presented in a special seminar at CERN in
December 2011 and updates were presented at the Moriond conference 2012. They
created a lot of attention and excitement in the community since the data show
tantalizing hints of a possible Higgs boson signal in the low mass region
around 125 GeV. However, it must be stressed that the background-only
probability still shows acceptable values, in particular if the look-elsewhere
effect [103] is taken into account.
In the following, these results are summarized since they supersede those
presented at the CERN school in September 2011. Before entering the
discussion, the Higgs boson production at hadron colliders and the Higgs boson
decay properties as well as a few statistical issues are briefly summarized.
### 0.5.1 Higgs boson production at the LHC
At hadron colliders, Higgs bosons can be produced via four different
production mechanisms:
* •
gluon fusion, $gg\to H$, which is mediated at lowest order by a heavy quark
loop;
* •
vector boson fusion (VBF), $qq\to qqH$;
* •
associated production of a Higgs boson with weak gauge bosons,
$qq\to W/Z\ H$ (Higgs strahlung, Drell-Yan like production);
* •
associated Higgs boson production with heavy quarks,
$gg,qq\to t\bar{t}H$, $gg,qq\to bbH$ (and $gb\to bH$).
The dominant production mode is the gluon-fusion process, followed by the
vector boson fusion. In the low mass region it amounts at leading order to
about 20% of the gluon-fusion cross section, whereas it reaches the same level
for masses around 800 GeV. The associated $WH$, $ZH$ and $t\bar{t}H$
production processes are relevant only for the search of a light Standard
Model Higgs boson with a mass close to the LEP limit.
The Higgs boson production cross sections are computed up to next-to-next-to-
leading order (NNLO) [104] in QCD for the gluon-fusion process. In addition,
QCD soft-gluon resummations up to next-to-next-to-leading log (NNLL) improve
the NNLO calculation [105]. The next-to-leading order (NLO) electroweak
corrections [106, *AActis:2008ug] are also applied, assuming factorization
between QCD and electroweak corrections. The cross sections for the VBF
process are calculated with full NLO QCD and electroweak corrections [108],
and approximate NNLO QCD corrections are available [109]. The $W/Z\ H$
processes are calculated at NLO [110] and at NNLO [111], and NLO electroweak
radiative corrections [112] are applied. Also for the associated $t\bar{t}H$
production the full NLO QCD corrections are available [113]. For a more
detailed review of the theoretical aspects of Higgs boson production the
reader is referred to Ref. [114]. The results of the calculations and the
estimated theoretical uncertainties are shown for the different production
processes in Fig. 21 (left) as a function of the Higgs boson mass [114].
Figure 21: (Left): Production cross sections for the different production
processes for a Standard Model Higgs boson as a function of the Higgs boson
mass at the LHC. (Right): Branching ratios of the Standard Model Higgs boson
as a function of Higgs boson mass (from Ref. [114]).
### 0.5.2 Higgs boson decays
The branching ratios and the total decay width of the Standard Model Higgs
boson are shown in Fig. 21 (right) as a function of the Higgs boson mass. They
have been calculated taking into account both electroweak and QCD corrections
[115, 116]. When kinematically accessible, decays of the Standard Model Higgs
boson into vector boson pairs $WW$ or $ZZ$ dominate over all other decay
modes. Above the kinematic threshold, the branching fraction into $t\bar{t}$
can reach up to 20%. All other fermionic decays are only relevant for Higgs
boson masses below $\sim$140 GeV, with $H\rightarrow b\bar{b}$ dominating. The
branching ratios for both $H\rightarrow\tau\tau$ and $H\to gg$ reach up to
about 8% at Higgs boson masses between 100 and 120 GeV. Decays into two
photons, which are of interest due to their relatively clean experimental
signature, can proceed via charged fermion and $W$ loops with a branching
ratio of up to 2 $\cdot$ 10-3 at low Higgs boson masses.
Compared to the mass resolution of hadron collider experiments, the total
decay width of the Standard Model Higgs boson is small at low masses and
becomes significant only above the threshold for decays into $ZZ$. For a Higgs
boson with a mass of $\sim$1 TeV the resonance is broad with a width of about
600 GeV. In this mass regime, the Higgs field is coupling strongly, resulting
in large corrections [114, 117].
### 0.5.3 Search for the Standard Model Higgs boson at the LHC
The Standard Model Higgs boson is searched for at the LHC in various decay
channels, the choice of which is given by the signal rates and the signal-to-
background ratios in the different mass regions. The search strategies and
background rejection methods have been established in many studies over the
past years [118, *physics-tdr-cms, *atlas-csc]. Among the most important
channels are the inclusive $H\to\gamma\gamma$ and $H\rightarrow
ZZ^{(*)}\rightarrow\ell\ell\ell\ell$ decay channels. These channels are
characterized by a very good mass resolution. In the low mass region, the
Higgs boson appears as a sharp resonance, the width of which is dominated by
the detector resolution, on top of flat backgrounds which are dominated by
$\gamma\gamma$ and $ZZ$ continuum production, respectively. In addition, the
$H\to WW$ decay mode contributes in particular in the mass region around 160
GeV, however, due to the neutrinos in the final state, no mass peak can be
reconstructed. Evidence for Higgs boson production is given by a broad peak in
the transverse mass distribution (see below). From the fermionic decays, only
the modes $H\rightarrow\tau\tau$ and $H\rightarrow b\bar{b}$ have a chance to
be detected. For the $H\rightarrow\tau\tau$ decays the selection of the vector
boson fusion production mode is important to improve the signal-to-background
ratio by exploiting forward jet tagging [121]. The $b\bar{b}$ decay mode is
searched for in the associated production of the Higgs boson with a vector
boson or with a $t\bar{t}$ pair [122, 123].
In the following the present status (March 2012) of the Higgs boson search in
the various final states at the LHC is described. Before the individual
channels are discussed some common issues on the statistical treatment are
given. At the end of this section the combination of the results is presented
for both the ATLAS and CMS collaborations.
#### Limit setting, statistical issues
In the following the distributions of reconstructed masses or other
distributions as measured in data are compared to the expectations from
Standard Model background processes. In order to test the compatibility of the
data with the background-only hypothesis a so-called $p_{0}$ probability value
is calculated. It quantifies the probability that a background-only experiment
is more signal-like than that observed. The local $p_{0}$ probability is
assessed for a fixed $m_{H}$ hypothesis and the equivalent formulation in
terms of number of standard deviations is referred to as the local
significance. Since fluctuations of the background could occur at any point in
the mass range the results have to be corrected for the look-elsewhere effect
[103]. The probability for a background-only experiment to produce a local
significance of this size or larger anywhere in a given mass region is
referred to as the global $p_{0}$. The corresponding reduction in the
significance is estimated using the prescription described in Ref. [124].
The data can also be used to set 95% confidence level (C.L.) upper limits
($\sigma_{95}$) on the cross section for Higgs boson production. These cross
sections are usually normalized to the expected Standard Model value
($\sigma_{95}/\sigma_{\rm{SM}}$). All exclusion limits quoted in the following
have been calculated using the $CL_{s}$ method [125]. In addition to the
observed limits based on the observed data, also the expected limits are
calculated. They are calculated as a function of $m_{H}$ and are based on the
central values of the expected background in case no Higgs signal is present.
The 1$\sigma$ and 2$\sigma$ fluctuations around the expected exclusion limits
are calculated as well.
Excluded mass regions are determined from a comparison of the observed cross-
section limit to the Standard Model cross-section value. If the observed value
of $\sigma_{95}/\sigma_{\rm{SM}}$ is smaller than 1 (Standard Model cross-
section expectation) for a particular hypothetical Higgs boson mass, this mass
value can be excluded with a confidence level of 95%. Systematic uncertainties
are incorporated by introducing nuisance parameters with constraints.
Asymptotic formulae [126] are used to derive the limits and $p_{0}$ values.
This procedure has been validated using ensemble tests and a Bayesian
calculation of the exclusion limits with a uniform prior on the signal cross
section. These approaches to the limits typically agree with the asymptotic
median results to within a few percent [127].
#### Search for $H\to\gamma\gamma$ decays
The decay $H\to\gamma\gamma$ is a rare decay mode, which is only detectable in
a limited Higgs boson mass region between 100 and 150 GeV, where both the
production cross section and the decay branching ratio are relatively large.
Excellent energy and angular resolution are required to observe the narrow
mass peak above the irreducible prompt $\gamma\gamma$ continuum. In addition,
there is a reducible background resulting from direct photon production or
from two-jet production via QCD processes. These processes contribute if one
or both jets are misidentified as a photon. The background can be determined
from a fit to the data (sidebands) such that uncertainties from Monte Carlo
predictions or uncertainties due to normalizations in control regions are
avoided. Due to the rather large amount of material in the tracking detectors
of the LHC experiments there is a high probability for a photon to undergo
conversion and therefore both unconverted and converted photons need to be
reconstructed.
Figure 22: (Left): Invariant mass distribution for the selected data sample in
the ATLAS experiment, overlaid with the total background (see text). The
bottom inset displays the residual of the data with respect to the total
background. The Higgs boson expectation for a mass hypothesis of 120 GeV
corresponding to the Standard Model cross section is also shown. (Right):
Observed and expected 95% C.L. limits on the Standard Model Higgs boson
production cross section normalized to the predicted one as a function of
$m_{H}$ (from Ref. [128]).
Both collaborations have presented results on the $H\to\gamma\gamma$ search in
a mass range between 110 and 150 GeV based on the full data set collected
until the end of 2011 [128, 129]. The ATLAS analysis [128] separates events
into nine independent categories. The categorisation is based on the direction
of each photon and whether it was reconstructed as a converted or unconverted
photon, together with the momentum component of the diphoton system transverse
to the thrust axis. The distribution of the invariant mass of the diphoton
events, $m_{\gamma\gamma}$, summed over all categories, is shown in Fig. 22
(left). The fit to the background is performed separately in each category in
the mass range 100 - 160 GeV by using an exponential function with free slope
and normalization parameters. The result for the total sample is superimposed
in Fig. 22. The signal expectation for a Higgs boson with $m_{H}=120$ GeV is
also shown. The mass resolution depends on the classification of the photon
(calorimeter $\eta$ region, conversion status) and is found to be 1.4 GeV in
the best category and 1.7 GeV on average. The residuals of the data with
respect to the total background as a function of $m_{\gamma\gamma}$ is also
shown in Fig. 22. Around a mass of 126 GeV an excess of events above the
background is seen (see discussion below). The 95% C.L. upper limits on the
cross section for Higgs boson production normalized to the Standard Model
value, $\sigma_{95}/\sigma_{SM}$, are shown in Fig. 22 (right). The observed
exclusion limits follow well the expectations over a large mass range, except
in the region around 126 GeV. The ATLAS data allow for a 95% C.L. exclusion of
a Standard Model Higgs boson in the mass ranges between $113{\\--}115$ GeV and
$134.5{\\--}136$ GeV.
Figure 23: (Left): Invariant mass distribution for the selected data sample in
the CMS experiment, overlaid with the total background. (Middle): The
invariant mass distribution for diphotons fulfilling the VBF selection (see
text). The Higgs boson expectation for a mass hypothesis of 120 GeV
corresponding to the Standard Model cross section multiplied by a factor of
two is also shown. (Right): Observed and expected 95% C.L. limits on the
Standard Model Higgs boson production cross section normalized to the
predicted one as a function of $m_{H}$ (from Ref. [129]).
The analysis of the CMS collaboration [129] is done in a similar way. Diphoton
events are split into four categories depending on their $\eta$ value and
shower shape characteristics (to distinguish converted from unconverted
photons). The diphoton mass resolution is best for the class of two central
unconverted photons and reaches a value of 1.2 GeV (full width at half maximum
divided by 2.35) for a Higgs boson mass of 120 GeV. Including the other
classes a weighted average resolution of $\sim$1.8 GeV is found. A further
class of events is introduced to select the vector boson fusion topology (VBF
topology). By requiring two jets with a large separation in pseudorapidity, a
class of events is defined for which the expected signal-to-background ratio
is about an order of magnitude larger than for the events in the four classes
defined by photon properties. The $m_{\gamma\gamma}$ distributions observed in
the data for the sum of the five event classes and for the VBF topology
separately are shown in Fig. 23 (left, middle) together with the background
fits based on polynomial functions. The uncertainty bands shown are computed
from the fit uncertainty on the background yields. The limit set on the cross
section of a Higgs boson decaying to two photons normalized to the Standard
Model value is shown in Fig. 23 (right). The CMS analysis excludes at the 95%
C.L. the Standard Model Higgs boson decaying into two photons in the mass
range 128 to 132 GeV. However, it should be noted that this exclusion, as well
as the ATLAS exclusions in this channel, are lucky since the expected
sensitivities are larger than one and the observed values of
$\sigma_{95}/\sigma_{\rm{SM}}$ are at the edges of the 2$\sigma$ bands. The
fluctuations of the observed limit about the expected limit are consistent
with statistical fluctuations to be expected in scanning the mass range. The
largest deviation in the CMS experiment is seen at $m_{\gamma\gamma}$ =124
GeV.
In order to quantify the fluctuations seen in both experiments, the
probabilities for the background-only hypothesis have been calculated. The
observed and expected local $p_{0}$ values obtained are displayed in Fig. 24
for the ATLAS (left) and CMS (right) data. Before considering the uncertainty
on the signal mass position, the largest excess with respect to the
background-only hypothesis in the mass range $110{\\--}150$ GeV is observed at
126.5 GeV in the ATLAS data with a local significance of 2.9$\sigma$. The
uncertainty on the mass position ($\pm 0.7$ GeV) due to the imperfect
knowledge of the photon energy scale has a small effect on the significance.
When this uncertainty is taken into account, the significance is slightly
reduced to 2.8$\sigma$. The local $p_{0}$ value corresponding to the largest
upwards fluctuation in the CMS data at 124 GeV has a significance of
3.1$\sigma$. The observed significances reduce to 1.5$\sigma$ for ATLAS and
1.8$\sigma$ for CMS, when the look-elsewhere effect is taken into account over
the mass range $110{\\--}150$ GeV.
Figure 24: The observed local $p_{0}$, the probability that the background
fluctuates to the observed number of events or higher, for the ATLAS (left)
and CMS (right) data. In the ATLAS case, the open points indicate the observed
local $p_{0}$ value when energy scale uncertainties are taken into account.
The dotted line shows the expected median local $p_{0}\ $ for the signal
hypothesis when tested at $m_{H}$. In the CMS case, the $p_{0}$ values are
shown for the VBF-tagged class separately (from Refs. [128, 129]).
#### Search for $H\rightarrow ZZ^{(*)}\rightarrow\ell\ell\ell\ell$ decays
The decay channel $H\to ZZ^{(*)}\to\ell\ell\ \ell\ell$ provides a rather clean
signature in the mass range 115 GeV $<m_{H}<2\ m_{Z}$. In addition to the
irreducible backgrounds from $ZZ^{*}$ and $Z\gamma^{*}$ production, there are
large reducible backgrounds from $t\bar{t}$ and $Zb\bar{b}$ production. Due to
the large production cross section, the $t\bar{t}$ background dominates at
production level, whereas the $Zb\bar{b}$ events contain a genuine $Z$ boson
in the final state and are therefore more difficult to reject. In addition,
there is background from $ZZ$ continuum production, where one of the $Z$
bosons decays into a $\tau$ pair, with subsequent leptonic decays of the
$\tau$ leptons, and the other $Z$ decays into an electron or muon pair.
Figure 25: Distributions of the four-lepton invariant mass, $m_{4\ell}$ of the
selected candidates, compared to the background expectation for the 100 - 250
GeV mass range (left) and the full mass range (right) in the ATLAS experiment.
The signal expectations for several $m_{H}$ hypotheses are also shown (from
Ref. [130]).
Figure 26: Distributions of the four-lepton invariant mass, $m_{4\ell}$ of the
selected candidates, compared to the background expectation for the 100 - 160
GeV mass range (left) and the full mass range (right) in the CMS experiment.
The signal expectations for several $m_{H}$ hypotheses are also shown (from
Ref. [131]).
Both collaborations have performed the $H\rightarrow
ZZ^{(*)}\rightarrow\ell\ell\ell\ell$ search for $m_{H}$ hypotheses in the full
110 to 600 GeV mass range using data corresponding to an integrated luminosity
of $\sim$4.8 fb-1 [130, 131]. It has been shown that in both experiments
calorimeter and track isolation requirements together with impact parameter
requirements can be used to suppress the irreducible background well below the
irreducible $ZZ^{*}$ continuum background. The residual irreducible $Z$+jets
and $t\bar{t}$ backgrounds, which have an impact mostly for low invariant
four-lepton masses, are estimated from control regions in the data. The
irreducible $ZZ^{*}$ background is estimated using Monte Carlo simulation. The
events are categorised according to the lepton flavour combinations. Mass
resolutions of approximately 1.5% in the four-muon channel and 2% in the four-
electron channel are achieved at $m_{H}$$\sim$120 GeV [130]. The four-lepton
invariant mass is used as a discriminant variable. The observed and expected
mass distributions for events selected after all cuts are displayed in Figs.
25 and 26 for the ATLAS and CMS experiments, respectively.
The measured mass distributions are again confronted to the background-only
hypotheses. The corresponding $p_{0}$ values are shown in Fig. 27 for the two
experiments. In the ATLAS experiment large upward deviations from the
background-only hypothesis are observed for $m_{H}$ = 125 GeV, 244 GeV and 500
GeV with local significances of 2.1$\sigma$, 2.2$\sigma$ and 2.1$\sigma$,
respectively. After accounting for the look-elsewhere effect none of these
excesses is significant. The CMS collaboration observes excesses of events
around 119 GeV, 126 GeV and 320 GeV. The most significant excess for a mass
value near 119 GeV corresponds to a local (global) significance of about
2.5$\sigma$ (1.0$\sigma$).
Figure 27: The observed local $p_{0}$ values as a function of the Higgs boson
mass in the $H\rightarrow ZZ^{(*)}\rightarrow\ell\ell\ell\ell$ channel in the
ATLAS (left) and CMS (right) experiments. The dashed curve shows the expected
median local $p_{0}$. The horizontal lines indicate values of constant
significance of 1$\sigma$, 2$\sigma$ and 3$\sigma$ (from Refs. [130, 131]).
#### Search for $H\to WW^{(*)}\rightarrow\ell\nu\ell\nu$ decays
The decay mode $H\to WW^{(*)}\rightarrow\ell\nu\ell\nu$ has the highest
sensitivity for Higgs boson masses around 170 GeV. Based on searches in this
channel, mass regions could be excluded by both the Tevatron and the LHC
experiments already in Summer 2011 [100, 101, 102]. However, this channel is
more challenging in the low mass region around 125 GeV since due to the
reduced $H\to WW$ branching ratio the expected signal rates are small. Due to
the presence of neutrinos it is not possible to reconstruct a Higgs boson mass
peak and evidence for a signal must be extracted from an excess of events
above the expected backgrounds. Usually, the $WW$ transverse mass ($m_{T}$),
computed from the leptons and the missing transverse momentum,
$m_{T}=\sqrt{(E_{\rm T}^{\ell\ell}+E_{\mathrm{T}}^{\mathrm{miss}})^{2}-|{\bf
p}_{\rm T}^{\ell\ell}+{\bf p}_{\rm T}^{\rm miss}|^{2}},$
where $E_{\rm T}^{\ell\ell}=\sqrt{|{\bf p}_{\rm
T}^{\ell\ell}|^{2}+m_{\ell\ell}^{2}}$, $|{\bf p}_{\rm T}^{\rm
miss}|=E_{\mathrm{T}}^{\mathrm{miss}}$ and $|{\bf p}_{\rm
T}^{\ell\ell}|=p_{\rm T}^{\ell\ell}$, is used to discriminate between signal
and background. The $WW$, $t\bar{t}$ and single-top production processes
constitute severe backgrounds and the signal significance depends critically
on their absolute knowledge.
The analyses of the ATLAS and CMS collaborations are based on the full data
set ($\sim$4.7 fb-1) [132, 133]. In order to optimize the sensitivity, the
analyses are split into different lepton final states ($ee$, $e\mu$ and
$\mu\mu$) and different jet multiplicities. In addition, they have been
optimized for different mass regions (low and high mass). Typical selection
cuts require the presence of two isolated high $p_{\mathrm{T}}$ leptons with a
significant missing transverse energy and a small azimuthal angular
separation. The latter requirement is motivated by the decay characteristics
of a spin-0 Higgs boson decaying into two $W$ bosons with their subsequent
$W\to\ell\nu$ decay [134]. The various jet categories are sensitive to
different Higgs boson production mechanisms and have very different background
compositions. The 0-jet category is mainly sensitive to the gluon-fusion
process and has the non-resonant $WW$ production as major background. The
2-jet category is more sensitive to the vector-boson fusion process, with
$t\bar{t}$ as dominant background. As a final discriminant the $WW$ transverse
mass distribution is used. This distribution is shown in Fig. 28 (left) for
events passing the 0-jet selection in ATLAS. The observed data are well
described by the expected background contributions which are dominated by the
$WW$ production.
Figure 28: (Left): The distribution of the transverse mass $m_{T}$ in the H+0
jet channel of the ATLAS analysis. The expected signal for a Standard Model
Higgs boson with $m_{H}$ = 125 GeV is superimposed. (Right): Expected (dashed)
and observed (solid) 95% C.L. upper limits on the cross section, normalized to
the Standard Model cross section, as a function of $m_{H}$. The results at
neighbouring mass points are highly correlated due to the limited mass
resolution in this final state (from Ref. [132]).
Figure 29: (Left): The distribution of the azimuthal angle separation
$\Delta\phi_{\ell\ell}$ in the H+0 jet channel of the CMS analysis. The
expected signal for a Standard Model Higgs boson with $m_{H}$ = 130 GeV is
superimposed. (Right): Expected (dashed) and observed (solid) 95% C.L. upper
limits on the cross section, normalized to the Standard Model cross section,
as a function of $m_{H}$. The results at neighbouring mass points are highly
correlated due to the limited mass resolution in this final state (from Ref.
[133]).
As another example, Fig. 29 (left) shows the distribution of the azimuthal
angle difference ($\Delta\phi_{\ell\ell}$) between the two selected leptons
for events in the 0-jet category in the CMS experiment. Also this distribution
is well described by the expected background processes. To enhance the
sensitivity, the CMS experiment exploits two different analysis strategies for
the 0-jet and 1-jet categories, the first one using a cut-based approach and
the second one using a multivariate technique [133].
Since no significant excesses of events are found in any of the event
categories in both the ATLAS and CMS experiments, upper limits on the
production cross section are set, as shown in Figs. 28 and 29. The ATLAS
experiment excludes the existence of a Standard Model Higgs boson over a mass
range from 130 - 260 GeV, while the expected exclusion, in case no Higgs boson
is present, is 127 $\leq m_{H}\leq$ 234 GeV. The CMS experiments excludes a
mass range from 129 - 270 GeV, with an expected range from 127 - 270 GeV.
#### Search for $H\rightarrow\tau\tau$ and $H\rightarrow b\bar{b}$ decays
In addition to the searches described above, the search for the Higgs boson
has also been performed in the $H\rightarrow\tau\tau$ [135, 136] and
$H\rightarrow b\bar{b}$ final states [137, 138]. These searches do not yet
reach the sensitivity of the others described above. They are, however,
included in the overall combination of the results of the two collaborations
[139, 140]. The observed and expected cross section limits are included in
Fig. 31.
#### Search for the Higgs boson in the high mass region
For higher Higgs boson masses ($m_{H}>2m_{Z}$) the decays $H\to WW$ and $H\to
ZZ$ dominate. Due to the higher mass and improved signal-to-background
conditions, also the decays $H\to ZZ\to\ell\ell\ \nu\nu$ [141, 142], $H\to
ZZ\to\ell\ell\ qq$ [143, 144], $H\to ZZ\to\ell\ell\ \tau\tau$ [145] and $H\to
WW\to\ell\nu\ qq$ [146] provide additional sensitivity.
The $H\to ZZ\to\ell\ell\nu\nu$ is the most sensitive channel. The selection of
two leptons and large missing transverse energy gives rather good signal-to-
background conditions. The dominant backgrounds are from diboson and
$t\bar{t}$ production. Also in this case the transverse mass $m_{T}$ of the
$\ell\ell-E_{\mathrm{T}}^{\mathrm{miss}}$ system is used as discriminating
variable. The distributions are shown in Fig. 30 for the ATLAS (left) and CMS
(right) experiments together with expected signals at 400 GeV. No indications
for excesses are seen and upper limits on the Higgs boson production cross
sections are set. They are included as well in Fig. 31.
Figure 30: The distributions of the transverse mass of $H\to
ZZ\to\ell\ell\nu\nu$ candidates in the ATLAS (left) and CMS (right)
experiments. Expected signals for a Higgs boson with a mass of 400 GeV are
superimposed (from Refs. [142, 141]).
### 0.5.4 Combination results of searches for the Standard Model Higgs boson
#### Excluded mass ranges
The ATLAS and CMS experiments have combined their respective search results on
the Standard Model Higgs boson [139, 140]. The combination procedure is based
on the profile likelihood ratio test statistic $\lambda({\mu})$ [126], which
extracts the information on the signal strength $\mu=\sigma/\sigma_{\rm{SM}}$
from a full likelihood including all the parameters describing the systematic
uncertainties and their correlations. More details on the statistical
procedure used are described in Ref. [124].
Figure 31: The observed (solid) and expected (dashed) 95% C.L. cross section
upper limits for the individual search channels in the ATLAS (left) and CMS
(right) experiments, normalized to the Standard Model Higgs boson production
cross section, as a function of the Higgs boson mass. The expected limits are
those for the background-only hypothesis, i.e. in the absence of a Higgs boson
signal (from Refs. [139, 140]).
In Figure 31 the expected and observed 95% C.L. limits are shown from the
individual channels entering this combination, separately for the ATLAS and
CMS experiments. The combined 95% C.L. exclusion limits are shown in Fig. 32
as a function of $m_{H}$ for the full mass range and for the low mass range.
The combined expected 95% C.L. exclusion regions for the two experiments are
very similar and cover the $m_{H}$ range from 120 to 555 GeV for the ATLAS and
from 118 to 543 GeV for the CMS experiment. Based on the observed limit, the
ATLAS experiment excludes at the 95% C.L. the Standard Model Higgs boson in
three mass ranges: from 110.0 - 117.5 GeV, from 118.5 to 122.5 GeV and from
129 to 539 GeV. The 95% C.L. CMS exclusion covers the range 127 - 600 GeV. The
observed exclusion covers a large part of the expected exclusion range, with
the exception of the low mass region where an excess of events above the
expected background is observed. It is striking that both experiments are not
able to cover the mass window from about 118 to 129 GeV, despite their
sensitivity in this range.
Figure 32: The observed (full line) and expected (dashed line) 95% C.L.
combined upper limits on the Standard Model Higgs boson production cross
section divided by the Standard Model expectation as a function of $m_{H}$ in
the full mass range considered in the analyses (top) and in the low mass range
(bottom) for the ATLAS (left) and CMS (right) experiments. The dotted curves
show the median expected limit in the absence of a signal and the green and
yellow bands indicate the corresponding 68% and 95% intervals (from Refs.
[139, 140]).
#### Compatibility with the background-only hypothesis
Figure 33: The local probability $p_{0}$ for a background-only experiment to
be more signal-like than the observation in the low mass range as a function
of $m_{H}$ for the ATLAS (left) and CMS (right) experiments. The $p_{0}$
values are shown for individual channels as well as for the combination. The
dashed curves show the median expected local $p_{0}$ under the hypothesis of a
Standard Model Higgs boson production signal at that mass. The horizontal
dashed lines indicate the $p$ values corresponding to significances of
1$\sigma$ to 5$\sigma$ (from Refs. [139, 140]).
Excesses of events are observed in the ATLAS experiment near 126 GeV in the
$H\to\gamma\gamma$ and $H\rightarrow ZZ^{(*)}\rightarrow\ell\ell\ell\ell$
channels, both of which provide fully reconstructed candidates with high-
resolution in invariant mass. The CMS experiment observes two localized
excesses, one at 119.5 GeV associated with three $Z\to 4\ell$ events and the
other one at 124 GeV, arising mainly from the $\gamma\gamma$ channel. In
addition, a broad offset of about one standard deviation is seen for the low
resolution channels $H\to WW,H\to\tau\tau$ and $H\to b\bar{b}$. The observed
local $p_{0}$ values, calculated using the asymptotic approximation, as a
function of $m_{H}$ and the expected value in the presence of a Standard Model
Higgs boson signal are shown in Fig. 33 in the low mass region for the two
experiments.
In the ATLAS data the local significance for the combined result reaches
2.6$\sigma$ for $m_{H}$=126 GeV with an expected value in the presence of a
signal at that mass of 2.9$\sigma$. The local significance for the combination
of the CMS channels at $m_{H}$ = 124 GeV amounts to 3.1$\sigma$
The significance of the excesses is mildly sensitive to energy scale
systematic (ESS) uncertainties and the resolution for photons and electrons.
The observed effect of the ESS uncertainty is small and reduces the maximum
local significance in the ATLAS experiment from 2.6$\sigma$ to 2.5$\sigma$.
The global $p_{0}$ for local excesses depends on the range of $m_{H}$ and the
channels considered. The global probability for an excess as large as the one
observed in the ATLAS combination at 126 GeV to occur anywhere in the mass
range 110–600 GeV is estimated to be approximately 30%, decreasing to 10% in
the range 110–146 GeV, which is not excluded at the 99% confidence level by
the LHC combined Standard Model Higgs boson search [147]. The global
significance for the CMS excess is estimated to be 1.5$\sigma$ for the full
search range from 110–600 GeV and 2.1$\sigma$ for the restricted search range
from 110–145 GeV.
The best-fit value of $\mu$, denoted $\hat{\mu}$, is displayed for the
combination of all channels for the two experiments in Fig. 34. The bands
around $\hat{\mu}$ illustrate the $\mu$ interval corresponding to
$-2\ln\lambda(\mu)<1$ and represent an approximate $\pm 1\sigma$ variation.
The excess observed for $m_{H}=126\leavevmode\nobreak\ {\mathrm{\
Ge\kern-1.00006ptV}}$ in the ATLAS experiment corresponds to $\hat{\mu}$ of
approximately $0.9^{+0.4}_{-0.3}$, which is compatible with the signal
strength expected from a Standard Model Higgs boson at that mass ($\mu=1$).
Also for the CMS experiment the $\hat{\mu}$ values are within one sigma of
unity in the mass range from 117–126 GeV.
Figure 34: The best-fit signal strength $\hat{\mu}$ as a function of the Higgs
boson mass hypothesis in the full mass range for the combination of the ATLAS
(left) and CMS (right) analyses. The $\mu$ value indicates by what factor the
Standard Model Higgs boson cross section would have to be scaled to best match
the observed data. The band shows the interval around $\hat{\mu}$
corresponding to a variation of $-2\ln\lambda(\mu)<1$ (from Refs. [139, 140]).
## 0.6 Search for Supersymmetric Particles
Due to the high centre-of-mass energy of 7 TeV, the LHC has a large discovery
potential for new heavy particles beyond the Tevatron limits. This holds in
particular for particles with colour charge, such as squarks and gluinos in
supersymmetry (SUSY) [148, *SUSY-refs2]. However, due to the excellent
luminosity performance of the LHC in 2011, sensitivity also exists for
electroweak production of charginos and neutralinos, the supersymmetric
partners of the electroweak gauge bosons and the Higgs boson. In the following
a few results of the searches by the ATLAS and CMS collaborations for
supersymmetry with up to 2 fb-1 of LHC $pp$ data at $\sqrt{s}$ = 7 TeV are
summarized. Since none of the analyses have observed any excess above the
Standard Model expectations, limits on SUSY parameters or masses of SUSY
particles are set. The discussion presented here follows largely the review of
Ref.[150] on the results from the ATLAS collaboration.
### 0.6.1 Searches with jets and missing momentum
Assuming conservation of R-parity, the lightest supersymmetric particle (LSP)
is stable and weakly interacting, and will typically escape detection. If the
primary produced particles are squarks or gluinos (and assuming a negligible
lifetime of these particles), they will decay to final states with energetic
jets and significant missing transverse momentum. This final state can be
produced in a large number of R-parity conserving models [151, *susy:Fayet1],
in which squarks, $\tilde{q}$, and gluinos, $\tilde{g}$, can be produced in
pairs as $\tilde{g}\tilde{g}$, $\tilde{g}\tilde{q}$, or $\tilde{q}\tilde{q}$.
They can decay via $\tilde{q}\to q\mathchoice{\displaystyle\raise
1.72218pt\hbox{$\displaystyle\tilde{\chi}^{0}_{1}$}}{\textstyle\raise
1.72218pt\hbox{$\textstyle\tilde{\chi}^{0}_{1}$}}{\scriptstyle\raise
0.90417pt\hbox{$\scriptstyle\tilde{\chi}^{0}_{1}$}}{\scriptscriptstyle\raise
0.64583pt\hbox{$\scriptscriptstyle\tilde{\chi}^{0}_{1}$}}$ and $\tilde{g}\to
q\bar{q}\mathchoice{\displaystyle\raise
1.72218pt\hbox{$\displaystyle\tilde{\chi}^{0}_{1}$}}{\textstyle\raise
1.72218pt\hbox{$\textstyle\tilde{\chi}^{0}_{1}$}}{\scriptstyle\raise
0.90417pt\hbox{$\scriptstyle\tilde{\chi}^{0}_{1}$}}{\scriptscriptstyle\raise
0.64583pt\hbox{$\scriptscriptstyle\tilde{\chi}^{0}_{1}$}}$ to weakly
interacting neutralinos, $\textstyle\tilde{\chi}^{0}_{1}$. However, also
charginos or heavier neutralinos might appear in the decay cascade and these
particles may produce high transverse momentum leptons in their decays into
the LSP.
The ATLAS and CMS collaborations have carried out analyses with a lepton veto
[153, 154], requiring one isolated lepton [155, 156], or requiring two or more
leptons [157, 158]. In addition, a dedicated search was performed for events
with high jet multiplicity with six or more jets [159]. Data samples
corresponding to integrated luminosities between 1.0 and 1.3 fb-1 were used.
Events are triggered either on the presence of a jet plus large missing
momentum, or on the presence of at least one high-$p_{T}$ lepton. Backgrounds
to the searches arise from Standard Model processes such as vector boson
production plus jets ($W$ \+ jets, $Z$ \+ jets), top quark pair production and
single top production, QCD multijet production, and diboson production. They
are estimated in a semi-data-driven way, using control regions in combination
with a transfer factor obtained from simulation. The results are interpreted
in the MSUGRA/CMSSM model [160], and in particular as limits in the plane
spanned by the common scalar mass parameter at the GUT scale $m_{0}$ and the
common gaugino mass parameter at the GUT scale $m_{1/2}$, for values of the
common trilinear coupling parameter $A_{0}$ = 0, Higgs mixing parameter $\mu$
> 0, and ratio of the vacuum expectation values of the two Higgs doublets
$\tan\beta$ = 10. Figure 35 (left) shows the results for the analyses of the
ATLAS collaboration with $\geq 2,\geq 3$ or $\geq 4$ jets plus missing
transverse momentum, and the multijets plus missing momentum analysis. For a
choice of parameters leading to equal squark and gluino masses, squark and
gluino masses below approximately 1 TeV are excluded. The 1-lepton and
2-lepton results are less constraining in MSUGRA/CMSSM for this choice of
parameters, but these analyses are complementary, and therefore no less
important. The exclusion contours obtained by the CMS collaboration in
different final states, including the lepton channels, are shown in Fig. 36.
Figure 35: (Left): Exclusion contours in the MSUGRA/CMSSM
($m_{0}-m_{1/2}$)-plane for $A_{0}$ = 0, $\tan\beta$ = 10 and $\mu$ > 0,
arising from the analysis of the ATLAS collaboration with $\geq 2,\geq 3$ or
$\geq 4$ jets plus missing transverse momentum, and the multijets plus missing
momentum analysis (from Ref. [159]). (Right): Exclusion contours from the
ATLAS analyses in the squark-gluino mass plane for three values of the LSP
mass using the simplified model description (see text) (from Ref. [161]).
### 0.6.2 Simplified model interpretation
The various analyses have also been interpreted in simplified models assuming
specific production and decay modes. The constraints implied by the
MSUGRA/CMSSM models [160] are relaxed, leaving more freedom for the variation
of particle masses and decay modes. Interpretations in simplified models thus
show better the limitations of the analyses as a function of the relevant
kinematic variables.
Inclusive search results with jets and missing momentum are interpreted using
simplified models with either pair production of squarks or of gluinos, or
production of squark-gluino pairs. Direct squark decays ($\tilde{q}\rightarrow
q\tilde{\chi}^{0}_{1}$) or direct gluino decays ($\tilde{g}\rightarrow
q\tilde{q}\tilde{\chi}^{0}_{1}$) are dominant if all other particle masses
have multi-TeV values, so that those do not play a role. Using these
assumptions, the excluded mass regions are sensitive to the mass of the LSP
($\tilde{\chi}^{0}_{1}$). Figure 35 (right) shows the ATLAS results
interpreted in terms of limits on (first and second generation) squark and
gluino masses, for three values of the LSP ($\tilde{\chi}^{0}_{1}$) mass, and
assuming that all other SUSY particles are very massive [161]. Further
interpretations are also done in terms of limits on gluino mass versus LSP
mass assuming high squark masses, or in terms of limits on squark mass vs LSP
mass assuming large gluino masses [155, 161].
Figure 36: Summary of exclusion contours in the MSUGRA/CMSSM
($m_{0}-m_{1/2}$)-plane for the parameters $A_{0}$ = 0, $\tan\beta$ = 10 and
$\mu$ > 0 for various analyses and different final states from the CMS
collaboration (from Ref. [156]).
The results of the inclusive jets plus missing momentum searches, interpreted
in these simplified models, indicate that masses of first and second
generation squarks and of gluinos must be above approximately 750 GeV. An
important caveat in this interpretation is the fact that this is only true for
neutralino LSP masses below approximately 250 GeV (as in MSUGRA/CMSSM [160]
for values of $m_{1/2}$ below $\sim$600 GeV). For higher LSP masses, the
squark and gluino mass limits are significantly less restricting. It will be a
challenge for further analyses to extend the sensitivity of inclusive squark
and gluino searches to the case of heavy neutralinos. If the LSP is heavy,
events are characterized by less energetic jets and less missing transverse
momentum. This will be more difficult to trigger on, and lead to higher
Standard Model backgrounds in the analysis.
### 0.6.3 Search for stop and sbottom production
Important motivations for electroweak-scale supersymmetry are the facts that
SUSY might provide a natural solution to the hierarchy problem by preventing
‘unnatural’ fine-tuning of the Higgs sector, and that the lightest stable SUSY
particle is an excellent dark matter candidate. It is instructive to consider
what such a motivation really requires from SUSY: a relatively light top quark
partner (the stop, $\tilde{t}$ and an associated sbottom-left,
$\tilde{b}_{\mathrm{L}}$), a gluino not much heavier than about 1.5 TeV to
keep the stop light, given that it receives radiative corrections from loops
like $\tilde{t}\to\tilde{g}t\to\tilde{t}$, and electroweak gauginos below the
TeV scale [162]. There are no strong constraints on first and second
generation squarks and sleptons; in fact heavy squarks and sleptons make it
easier for SUSY to satisfy the strong constraints from flavour physics.
Motivated by these considerations, the ATLAS and CMS collaborations have also
carried out a number of searches for supersymmetry with $b$-tagged jets, which
are sensitive to sbottom and stop production, either in direct production or
in production via gluino decays. Jets are tagged as originating from
$b$-quarks by an algorithm that exploits both track impact parameter and
secondary vertex information.
Direct sbottom pair production is searched for in a data sample corresponding
to an integrated luminosity of 2 fb-1 by requiring two $b$-tagged jets with
$p_{T}$ > 130, 50 GeV and significant missing transverse momentum of more than
130 GeV [163]. The final discriminant in the ATLAS analysis is the boost-
corrected contransverse mass $m_{CT}$ [164], and signal regions with $m_{CT}$
> 100, 150, 200 GeV are considered. No excesses are observed above the
expected backgrounds from top, $W$+heavy flavour and $Z$+heavy flavour
production. Figure 37 (left) shows the resulting limits in the sbottom-
neutralino mass plane, assuming sbottom pair production and sbottom decays
into a $b$-quark plus a neutralino (LSP) with a 100% branching fraction. Under
these assumptions, sbottom masses up to 390 GeV are excluded for neutralino
masses below 60 GeV.
The ATLAS collaboration has searched for stop quark production in gluino
decays [165] using an analysis requiring at least four high-$p_{T}$ jets of
which at least one should be $b$-tagged, one isolated lepton, and significant
missing transverse momentum. Since the number of observed events agrees with
the expectations from Standard Model processes, limits are set in the gluino-
stop mass plane, assuming the gluino to decay as
$\tilde{g}\rightarrow\tilde{t}t$, and the stop quark to decay as
$\tilde{t}\rightarrow b\tilde{\chi}^{\pm}_{1}$. The obtained mass limits are
shown in Fig. 37 (right).
Figure 37: (Left): Exclusion contours from the ATLAS analyses in the sbottom-
neutralino mass plane resulting from the analysis searching for sbottom pair
production assuming $\tilde{b}_{1}\to b\tilde{\chi}^{0}_{1}$ decays (from Ref.
[163]). (Right): Exclusion contours from the ATLAS analyses in the gluino-stop
mass plane resulting from the analysis searching for stop production via
gluino decays. The assumptions made to derive the limits are given in the
figure (from Ref. [165]).
Further searches for direct stop pair production are in progress. These
searches are challenging due to the similarity with the top-quark pair-
production final state for stop masses similar to the top mass, and due to the
low cross section for the production of stops with high mass. The ATLAS
collaboration has searched for signs of new phenomena in events passing a top-
quark pair selection with large missing transverse momentum [166]. Such an
analysis is sensitive to pair production of massive partners of the top quark,
decaying to a top quark and a long-lived undetected neutral particle. No
excess above background was observed, and limits on the cross section for pair
production of top quark partners are set. These limits constrain fermionic
exotic fourth generation quarks, but not yet scalar partners of the top quark,
such as the stop quark [166].
### 0.6.4 Search for supersymmetry in multilepton final states
The search for final states with several leptons and missing transverse
momentum are sensitive to the production of charginos and/or heavier
neutralinos (other than the LSP), decaying leptonically into the LSP. These
analyses comprise the golden search modes at the Tevatron, but are also
rapidly gaining relevance at the LHC and both the ATLAS and CMS collaborations
have performed corresponding analyses [157, 158]. The ATLAS collaboration has
published results of various analyses searching for dilepton events plus
missing momentum in data corresponding to an integrated luminosity of 1.0 fb-1
[157]. Three searches are performed for new phenomena in final states with
opposite-sign and same-sign dileptons and missing transverse momentum. These
searches also include signal regions that place requirements on the number and
$p_{\mathrm{T}}$ of energetic jets in the events. For all signal regions good
agreement is found between the numbers of observed events and the predictions
of expected events from Standard Model processes. Additionally, in opposite-
sign events, a search is made for an excess of same-flavour over different-
flavour lepton pairs. Effective production cross sections in excess of 9.9 fb
for opposite-sign events with missing transverse momentum greater than 250 GeV
are excluded at 95% C.L. For same-sign events with missing transverse momentum
greater than 100 GeV, effective production cross sections in excess of 14.8 fb
are excluded at 95% C.L. The latter limit is interpreted in a simplified
electroweak gaugino production model excluding chargino masses up to 200 GeV,
under the assumption that slepton decays are dominant [157].
The CMS collaboration has presented preliminary results, based on data
corresponding to an integrated luminosity of 2.1 fb-1, on the search for
supersymmetric particles in three- and four-lepton final states, including
hadronic decays of $\tau$ leptons [158]. The backgrounds from Standard Model
processes are suppressed by requiring missing transverse energy, Z-mass vetos
of the invariant dilepton mass or high jet activity. Control samples in data
are used to obtain reliable background estimates. Within the statistical and
systematic uncertainties the numbers of observed events are consistent with
the expectations from Standard Model processes. These results are used to
exclude previously unexplored regions of the supersymmetric parameter space
assuming R-parity conservation with the lightest supersymmetric particle being
a neutralino. The corresponding exclusion contours in the MSUGRA/CMSSM [160]
interpretation are shown in Fig. 38 in the ($m_{0}-m_{1/2}$) plane for $A_{0}$
= 0, $\mu$ > 0, and for $\tan\beta$ values of 3 and 10. They extend
significantly the regions excluded by the CDF [167] and DØ[168] experiments
and those excluded with previous searches at the LHC [169].
Figure 38: Exclusion contours in the MSUGRA/CMSSM ($m_{0}-m_{1/2}$)-plane for
the parameters $A_{0}$ = 0, $\mu$ > 0 and $\tan\beta$ = 3 (left) and
$\tan\beta$ = 10 (right) obtained from searches for SUSY production in final
states with multileptons by the CMS collaboration (from Ref. [158]).
### 0.6.5 Summary and outlook on SUSY searches
Many different searches for the production of supersymmetric particles have
been performed in a large variety of final states by the ATLAS and CMS
collaborations at the LHC. Data corresponding to integrated luminosites in the
range between 1.0 and 4.7 fb-1 taken during the year 2011 have been analyzed.
In all channels, the number of observed events is in agreement with the
expectations from Standard Model processes and no evidence for the production
of supersymmetric particles has been found so far. The data have been used to
set already rather strong limits on the masses of possible supersymmetric
particles. A summary of the most important mass limits is given in Fig. 39.
In addition to the analyses summarized here, many other analyses have been
performed and many different final states have been explored. There are
investigations of SUSY searches in gauge mediated supersymmetry breaking
models, by using final states with photons or multileptons. In addition, in
many models (split SUSY, R-hadrons, anomaly-mediated SUSY breaking and in
certain parts of the phase space of gauge-mediated SUSY breaking scenarios)
SUSY particles may be long-lived either because their decay is kinematically
suppressed or due to very small couplings, e.g. in R-parity violating models.
Many of these scenarii have already been explored and the reader is referred
to the corresponding publications of the ATLAS and CMS collaborations. Also in
all these searches for more exotic SUSY scenarios the number of observed
events in in agreement with the expectations from background from Standard
Model processes.
Figure 39: Summary of excluded mass ranges from a variety of searches for the
production of supersymmetric particles from the ATLAS (left) and CMS (right)
collaborations. Only a representative selection of available results is shown.
The CMS results indicate the change of the limits under variation of the
neutralino mass from 0 to 200 GeV.
Although no signs of supersymmetry have been found so far, it is important to
realize that actual tests of ‘natural’ supersymmetry are only just beginning.
In this respect, the LHC run of 2012, with an expected luminosity of more than
10 fb-1, possibly at $\sqrt{s}$ = 8 TeV, will be very important. However,
experimentally there will be considerable challenges in triggering and in
dealing with high pile-up conditions. In the longer term, increasing the LHC
beam energy to > 6 TeV will again enable the crossing of kinematical barriers
and open the way for multi-TeV SUSY searches.
## 0.7 Search for other Physics Scenarios Beyond the Standard Model
As already mentioned in Section 2, the Standard Model is an extremely
successful effective theory which has been extensively tested over the past
forty years. However, a number of fundamental questions are left unanswered.
Many models for physics Beyond the Standard Model (BSM) have been proposed and
the ATLAS and CMS experiments have used the data collected in 2010 and 2011 to
search for indications of new physics. An impressive list of analyses has been
performed. So far, no indications for deviations from the Standard Model have
been found. The event numbers and kinematical distributions in all final
states considered agree with the expectations from Standard Model processes.
Therefore, these analyses have been used to constrain the parameter space of
many BSM models.
Since it is impossible to present and discuss all analyses in such a summary
paper, a few benchmark processes are selected and the search results are
presented in the following. This concerns the search for new vector bosons, or
more general the search for heavy dilepton resonances, the search for
compositeness and the search for dijet resonances. Finally the results from
other searches are briefly summarized.
### 0.7.1 Search for heavy dilepton resonances
The ATLAS and CMS collaborations have performed searches for narrow high-mass
neutral and charged resonances decaying into $e^{+}e^{-}$ or $\mu^{+}\mu^{-}$
pairs or $e\nu$ or $\mu\nu$, respectively. In several extensions of the
Standard Model new heavy spin-1 neutral gauge bosons such as $Z^{\prime}$
[170, 171, 172], technimesons [173, 174, 175], as well as spin-2 Randall-
Sundrum gravitons, $G^{*}$, [176] are predicted. Additional heavy charged
gauge bosons appear e.g. in left-right-symmetric models [177].
The benchmark models considered in the analyses for the $Z^{\prime}$ are the
Sequential Standard Model [170], with the same couplings to fermions as the
$Z$ boson, and the $E_{6}$ grand unified symmetry group [172], broken into
$SU(5)$ and two additional $U(1)$ groups, leading to new neutral gauge fields
$\psi$ and $\chi$. The particles associated with the additional fields can mix
in a linear combination to form the $Z^{\prime}$ candidate:
$Z^{\prime}(\theta_{E_{6}})=Z_{\psi}^{\prime}\cos\theta_{E_{6}}+Z_{\chi}^{{}^{\prime}}\sin\theta_{E_{6}}$
, where $\theta_{E_{6}}$ is the mixing angle between the two gauge bosons. The
pattern of spontaneous symmetry breaking and the value of $\theta_{E_{6}}$
determine the $Z^{\prime}$ couplings to fermions.
Other models predict additional spatial dimensions as a possible explanation
for the gap between the electroweak symmetry breaking scale and the
gravitational energy scale. The Randall-Sundrum (RS) model [176] predicts
excited Kaluza-Klein modes of the graviton, which appear as spin-2 resonances.
These modes have a narrow intrinsic width when $k/\bar{M}_{\rm{Pl}}$ < 0.1,
where $k$ is the spacetime curvature in the extra dimension, and
$\bar{M}_{\rm{Pl}}=M_{\rm{Pl}}/\sqrt{8\pi}$ is the reduced Planck scale.
The search performed by the ATLAS experiment [178] is based on a dataset
corresponding to an integrated luminosity of up to 1.2 fb-1. The observed
invariant mass spectrum is shown in Fig. 40 (left) for the $e^{+}e^{-}$ final
state after final selections. The backgrounds from Drell-Yan, $t\bar{t}$,
diboson and $W$+jets production are determined from Monte Carlo simulation
after normalization to the respective (N)NLO cross sections. The background
from QCD multijet production is estimated using data-driven methods with the
inversion of lepton identification criteria. The simulated backgrounds are
rescaled so that the total sum of the backgrounds matches the observed number
of events observed in data in the 70-110 GeV mass interval. The scaling factor
is within 1% of unity. The advantage of this approach is that the uncertainty
on the luminosity and any mass independent uncertainties on efficiencies,
cancel between the $Z^{\prime}$ ($G^{*}$) and the $Z$ boson. The dilepton
invariant mass distributions are well described by the prediction from
Standard Model processes. Figure 40 (left) also displays the expected
$Z^{\prime}$ signals in the Sequential Standard Model for three mass
hypotheses.
Figure 40: (Left): Distribution of the dielectron invariant mass after final
selections in the ATLAS experiment, compared to the stacked sum of all
expected backgrounds, with three example $Z_{\rm{SSM}}^{{}^{\prime}}$ signals
overlaid. The bin width is constant in $\log m_{\ell\ell}$ (from Ref. [178]).
(Right): Distribution of the $\mu-E_{\mathrm{T}}^{\mathrm{miss}}$ transverse
mass after final selections in the CMS experiment. The expected signal from a
hypothetical $W^{\prime}$ boson with a mass of 1.5 TeV is superimposed (from
Ref. [179]).
Given the good agreement between the data and the Standard Model expectations,
limits are set on the cross section times branching ratio for the different
$Z^{\prime}$ models. The resulting mass limits are 1.83 TeV for the Sequential
Standard Model $Z^{\prime}$ boson, 1.49-1.64 TeV for various $E_{6}$-motivated
$Z^{\prime}$ bosons, and 0.71-1.63 TeV for a Randall-Sundrum graviton with
couplings ($k/\bar{M}_{\rm{Pl}}$) in the range 0.01-0.1. Similar analyses have
been performed by the CMS collaboration [180] and comparable limits have been
extracted. They are included in the summary of results from different
experiments for various physics models in Table 1.
The benchmark model considered in the search for the $W^{\prime}$ is the
Sequential Standard Model [170], with the same couplings to fermions as the
$W$ boson. In this case the transverse mass of the lepton and
$E_{\mathrm{T}}^{\mathrm{miss}}$ system is used as discriminating variable. As
an example, the measured transverse mass distribution in the muon final state
in the CMS experiment [179] is shown in Fig. 40 (right). The expectation for a
$W^{\prime}$ signal with a mass of 1.5 TeV is superimposed. Also the
transverse mass distributions measured by the LHC experiments are well
described by the prediction from Standard Model processes and the data allow
to exclude heavy $W^{\prime}$ bosons with masses below 2.15 TeV (ATLAS) [181]
and 2.25 TeV (CMS) [179] at the 95% C.L.
The mass limits obtained at the LHC are the most stringent to date, including
indirect limits set by LEP2. It is striking to see how fast the LHC
experiments have superseded the limits obtained with much higher luminosity at
the Tevatron. The analyses based on the data from 2010 ($L_{\rm{int}}$ = 36
pb-1) resulted in comparable limits to those obtained at the Tevatron based on
an integrated luminosity of 5.5 fb-1 (see Table 1).
Table 1: Observed 95% C.L. mass lower limits on $Z^{\prime}$, $G^{*}$ gravitons and $W^{\prime}$ resonances obtained for various models in the ATLAS and CMS experiments. The results from searches at the Tevatron are included for comparison. Model | Experiment | $L_{\rm{int}}$ | 95% C.L. limits | Ref.
---|---|---|---|---
| | | $e^{+}e^{-}$ | $\mu^{+}\mu^{-}$ | $\ell^{+}\ell^{-}$ |
| | (fb-1) | (TeV) | (TeV) | (TeV) |
$Z^{{}^{\prime}}_{SSM}$ | CDF/DØ | 5.5 | | | 1.07 | [182, *cdftab1]
| ATLAS/CMS | 0.036 | 0.96 | 0.83 | 1.05/1.14 | [184, *CMStab1]
| ATLAS | 1.1 / 1.2 | 1.70 | 1.61 | 1.83 | [178]
| CMS | 1.1 | | | 1.94 | [180]
$Z^{{}^{\prime}}$ $E_{6}$ models | ATLAS | 1.1 / 1.2 | | | 1.49 - 1.64 | [178]
| CMS | 1.1 | | | 1.62 | [186]
$G^{*}$ $k/\bar{M}_{\rm{Pl}}$ = 0.01 | ATLAS | 1.1 / 1.2 | | | 0.71 | [178]
$G^{*}$ $k/\bar{M}_{\rm{Pl}}$ = 0.03 | | 1.1 / 1.2 | | | 1.03 |
$G^{*}$ $k/\bar{M}_{\rm{Pl}}$ = 0.05 | | 1.1 / 1.2 | | | 1.33 |
$G^{*}$ $k/\bar{M}_{\rm{Pl}}$ = 0.10 | | 1.1 / 1.2 | | | 1.63 |
$G^{*}$ $k/\bar{M}_{\rm{Pl}}$ = 0.05 | CMS | 1.1 | | | 1.45 | [180]
$G^{*}$ $k/\bar{M}_{\rm{Pl}}$ = 0.10 | | 1.1 | | | 1.78 |
$W^{{}^{\prime}}_{SSM}$ | ATLAS | 1.04 | 2.08 | 1.98 | 2.15 | [181]
| CMS | 1.1 | | | 2.27 | [179]
### 0.7.2 Limits on new physics from jet production
The measurements on inclusive and dijet production, as discussed in Section
0.4.1, can also be used to constrain contributions from new physics that would
modify the expected QCD behaviour in the jet production cross sections. Two
examples are discussed in the following.
#### Substructure of quarks
Both collaborations have searched for quark compositeness by investigating the
angular distribution of jet events [187, 188]. At small scattering angles in
the centre-of-mass system of the two partons, the angular distribution is
expected to be proportional to the Rutherford cross section,
$d\hat{\sigma}/d\cos\theta^{*}\sim 1/(1-\cos\theta^{*})^{2}$. For the
scattering of massless partons, which are assumed to be collinear with the
beam protons, the longitudinal boost of the parton-parton centre-of-mass frame
with respect to the proton-proton centre-of-mass frame, $y_{\rm{boost}}$, and
$\theta^{*}$ are obtained from the rapidities $y_{1}$ and $y_{2}$ of the jets
from the two scattered partons by $y_{\rm{boost}}=\frac{1}{2}(y_{1}+y_{2})$
and $|\cos\theta^{*}|=\tanh y^{*}$, where $y^{*}=\frac{1}{2}|y_{1}-y_{2}|$ and
where $\pm y^{*}$ are the rapidities of the two jets in the parton-parton
centre-of-mass frame. The variable $\chi_{\rm{dijet}}=e^{2y^{*}}$ is used to
measure the dijet angular distribution, which for collinear massless-parton
scattering takes the form
$\chi_{\rm{dijet}}=(1+|\cos\theta^{*}|)/(1-|\cos\theta^{*}|)$. This choice of
$\chi_{\rm{dijet}}$, rather than $\theta^{*}$, is motivated by the fact that
$d{\sigma_{\rm{dijet}}}/d\chi_{\rm{dijet}}$ is flat for Rutherford scattering.
Figure 41: (Left): Normalized dijet angular distributions in several ranges of
the dijet mass as measured by the CMS collaboration. The data points include
statistical and systematic uncertainties. The results are compared with the
predictions of pQCD at NLO (shaded bands) and with the predictions including a
contact interaction term of compositeness scale $\Lambda^{+}$ = 5 TeV (dashed
histogram) and $\Lambda^{-}$ = 5 TeV (dotted histogram). The shaded bands show
the effect on the NLO pQCD predictions due to $\mu_{R}$ and $\mu_{F}$ scale
variations and PDF uncertainties, as well as the uncertainties from the non-
perturbative corrections added in quadrature (from Ref. [188]). (Right): The
reconstructed dijet mass distribution (filled points) measured by the ATLAS
collaboration fitted with a smooth functional form describing the QCD
background. The bin-by-bin significance of the data-background difference is
shown in the lower panel. Vertical lines show the most significant excess
found (from Ref. [189]).
The differential dijet angular distributions for different $m_{\rm{jj}}$
ranges, and corrected for detector effects as measured by the CMS experiment
using the 2010 data ($L_{\rm{int}}$ = 36 pb-1) are shown in Fig. 41 (left).
The data are found to be in good agreement with pQCD predictions at NLO
calculated with NLOJET++ [32, 33], which are superimposed on the figure. The
measured dijet angular distributions can be used to set limits on quark
compositeness parametrized by a four-fermion contact interaction term in
addition to the QCD Lagrangian. The value of the mass scale $\Lambda$
characterizes the strengths of the quark substructure binding interactions and
the physical size of the composite states. A color- and isospin-singlet
contact interaction (CI) of left-handed quarks gives rise to an effective
Lagrangian term [190, 191]
$L_{qq}=\eta_{0}\frac{2\pi}{\Lambda^{2}}(\bar{q}_{L}\gamma^{\mu}q_{L})(\bar{q}_{L}\gamma_{\mu}q_{L}),$
(4)
where $\eta_{0}$ = +1 corresponds to destructive interference between the QCD
and the new physics term, and $\eta_{0}$ = -1 to constructive interference.
From the measured $\chi_{\rm{dijet}}$ distribution, lower limits on the
contact interaction scale of $\Lambda^{+}$ = 5.6 TeV and $\Lambda^{-}$ = 6.7
TeV for destructive and constructive interference, respectively, have been set
by the CMS collaboration at the 95% C.L. [188]. The expected limits in case of
no substructure are 5.0 TeV and 5.8 TeV, respectively. The ATLAS collaboration
has performed a similar analysis and excludes at the 95% C.L. quark contact
interactions with a scale $\Lambda<$ 9.5 ${\mathrm{\ Te\kern-1.00006ptV}}$
[187]. However, it should be noted that this observed limit is significantly
above the expected limit of 5.7 TeV for the data sample corresponding to an
integrated luminosity of 36 pb-1. Very recently, the CMS collaboration has
published the results of an updated analysis based on data corresponding to an
integrated luminosity of 2.2 fb-1 [192] and taking NLO calculations for the
QCD predictions into account. Also this larger data set has been found to be
in good agreement with the QCD expectations. For the contact interaction model
described above, 95% C.L. limits of $\Lambda^{+}$ = 7.5 TeV and $\Lambda^{-}$
= 10.5 TeV have been set. The expected limits are 7.0 TeV and 9.7 TeV,
respectively.
#### Dijet resonances
The ATLAS and CMS collaborations have also examined the dijet mass spectrum
for resonances due to new phenomena localised near a given mass, employing
data-driven background estimates that do not rely on detailed QCD calculations
[189, 193]. The searches are based on data corresponding to an integrated
luminosity of 1.0 fb-1. As an example, the observed dijet mass distribution
measured in the ATLAS experiment, which extend up to masses of $\sim 4$ TeV is
displayed in Fig. 41 (right). It is found to be in good agreement with a
smooth function representing the Standard Model expectation. Since no evidence
for the production of new resonances is found, 95% C.L. mass limits have been
set in the context of several models of new physics: excited quarks (q∗) [194,
195], axigluons [196, 197, 198], scalar colour octet states [199] and scalar
diquarks predicted in Grand Unified Theories based on the $E_{6}$ gauge group
[200]. The results are summarized in Table 2. Also these limits are the most
stringent ones to date.
Table 2: The 95% C.L. mass lower limits on dijet resonance models. Model | Experiment | $L_{\rm{int}}$ | 95% C.L. limits | Ref.
---|---|---|---|---
| | (fb-1) | Expected (TeV) | Observed (TeV) |
Excited quark $q*$ | ATLAS | 1.0 | 2.81 | 2.99 | [189]
| CMS | 1.0 | 2.68 | 2.49 | [193]
Axigluon | ATLAS | 1.0 | 3.07 | 3.32 | [189]
| CMS | 1.0 | 2.66 | 2.47 | [193]
Colour Octet Scalar | ATLAS | 1.0 | 1.77 | 1.92 | [189]
$E_{6}$ diquarks | CMS | 1.0 | 3.28 | 3.52 | [193]
### 0.7.3 Summary of results on other searches
Many different searches for the Beyond the Standard Model processes have been
performed by the ATLAS and CMS collaborations at the LHC. Data corresponding
to integrated luminosities in the range between 1.0 and 4.7 fb-1 taken during
the year 2011 have been analyzed and many different final states have been
investigated. So far, no indications for deviations from the Standard Model
have been found. The event numbers and kinematical distributions in all final
states considered agree with the expectations from Standard Model processes.
Therefore, these analyses have been used to constrain the parameter space of
many BSM models. A summary of the most important limits from the ATLAS
collaboration is given in Fig. 42. Comparable limits have been set by the CMS
collaboration.
Figure 42: Summary of excluded mass ranges from a variety of searches from the
ATLAS experiment for Beyond the Standard Model physics processes. Only a
representative selection of available results is shown.
## 0.8 Conclusions
With the start-up of the operation of the LHC at high energies particle
physics has entered a new era. Both the accelerator and the detectors have
worked magnificently. Until the end of 2011 data corresponding to an
integrated luminosity of 5.5 fb-1 have been recorded with high efficiency by
the LHC experiments. Based on these data, many tests of the predictions of the
Standard Model and searches for physics Beyond the Standard Model have been
performed in the new energy regime. So far, all measurements have been found
to be in good agreement with the predictions from the Standard Model. Towards
the end of 2011, the experiments have reached sensitivity for the Standard
Model Higgs boson. A large fraction of the possible Higgs boson mass range has
already been excluded by the ATLAS and CMS experiments with a confidence level
of 95%. However, it is striking that both experiments are not able to exclude
the existence of the Higgs boson in the mass range from 118 - 129 GeV, despite
their sensitivity in this range. In addition, tantalizing hints for a Higgs
boson signal have been seen by both experiments in the two high resolution
channels $H\to\gamma\gamma$ and $H\rightarrow
ZZ^{(*)}\rightarrow\ell\ell\ell\ell$. However, the statistical significance is
not sufficient to claim evidence. More data are needed to clarify the
situation. With a successful run of the LHC in 2012 a final conclusion on the
existence of the Standard Model Higgs boson might be reached and the year 2012
might enter as the “Year of the Higgs Boson” into the history of Physics.
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|
arxiv-papers
| 2012-06-29T13:42:36 |
2024-09-04T02:49:32.385606
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Karl Jakobs (U. Freiburg)",
"submitter": "Annette Holtkamp",
"url": "https://arxiv.org/abs/1206.7024"
}
|
1206.7067
|
# Faster Geometric Algorithms via Dynamic Determinant Computation
Vissarion Fisikopoulos National and Kapodistrian University of Athens,
Department of Informatics & Telecommunications, Athens, Greece.
{vfisikop,lpenaranda}@di.uoa.gr Luis Peñaranda11footnotemark: 1
###### Abstract
Determinant computation is the core procedure in many important geometric
algorithms, such as convex hull computations and point locations. As the
dimension of the computation space grows, a higher percentage of the
computation time is consumed by these predicates. In this paper we study the
sequences of determinants that appear in geometric algorithms. We use dynamic
determinant algorithms to speed-up the computation of each predicate by using
information from previously computed predicates.
We propose two dynamic determinant algorithms with quadratic complexity when
employed in convex hull computations, and with linear complexity when used in
point location problems. Moreover, we implement them and perform an
experimental analysis. Our implementations outperform the state-of-the-art
determinant and convex hull implementations in most of the tested scenarios,
as well as giving a speed-up of $78$ times in point location problems.
Keywords: computational geometry, determinant algorithms, orientation
predicate, convex hull, point location, experimental analysis
## 1 Introduction
Determinantal predicates are in the core of many important geometric
algorithms. Convex hull and regular triangulation algorithms use _orientation_
predicates, the Delaunay triangulation algorithms also involve the _in-sphere_
predicate. Moreover, algorithms for exact volume computation of a convex
polytope rely on determinantal volume formulas. In general dimension $d$, the
orientation predicate of $d+1$ points is the sign of the determinant of a
matrix containing the homogeneous coordinates of the points as columns. In a
similar way, the volume determinant formula and in-sphere predicate of $d+1$
and $d+2$ points respectively can be defined. In practice, as the dimension
grows, a higher percentage of the computation time is consumed by these core
procedures. For this reason, we focus on algorithms and implementations for
the exact computation of the determinant. We give particular emphasis to
division-free algorithms. Avoiding divisions is crucial when working on a ring
that is not a field, _e.g._ , integers or polynomials. Determinants of
matrices whose elements are in a ring arise in combinatorial problems [Kra05],
in algorithms for lattice polyhedra [BP99] and secondary polytopes [Ram02] or
in computational algebraic geometry problems [CLO05].
Our main observation is that, in a sequence of computations of determinants
that appear in geometric algorithms, the computation of one predicate can be
accelerated by using information from the computation of previously computed
predicates. In this paper, we study orientation predicates that appear in
convex hull computations. The convex hull problem is probably the most
fundamental problem in discrete computational geometry. In fact, the problems
of regular and Delaunay triangulations reduce to it.
Our main contribution is twofold. First, we propose an algorithm with
quadratic complexity for the determinants involved in a convex hull
computation and linear complexity for those involved in point location
problems. Moreover, we nominate a variant of this algorithm that can perform
computations over the integers. Second, we implement our proposed algorithms
along with division-free determinant algorithms from the literature. We
perform an experimental analysis of the current state-of-the-art packages for
exact determinant computations along with our implementations. Without taking
the dynamic algorithms into account, the experiments serve as a case study of
the best implementation of determinant algorithms, which is of independent
interest. However, dynamic algorithms outperform the other determinant
implementations in almost all the cases. Moreover, we implement our method on
top of the convex hull package triangulation [BDH09] and experimentally show
that it attains a speed-up up to $3.5$ times, results in a faster than state-
of-the-art convex hull package and a competitive implementation for exact
volume computation, as well as giving a speed-up of $78$ times in point
location problems.
Let us review previous work. There is a variety of algorithms and
implementations for computing the determinant of a $d\times d$ matrix. By
denoting $O(d^{\omega})$ their complexity, the best current $\omega$ is
$2.697263$ [KV05]. However, good asymptotic complexity does not imply good
behavior in practice for small and medium dimensions. For instance, LinBox
[DGG+02] which implements algorithms with state-of-the-art asymptotic
complexity, introduces a significant overhead in medium dimensions, and seems
most suitable in very high dimensions (typically $>100$). Eigen [GJ+10] and
CGAL [CGA] implement decomposition methods of complexity $O(n^{3})$ and seem
to be suitable for low to medium dimensions. There exist algorithms that avoid
divisions such as [Rot01] with complexity $O(n^{4})$ and [Bir11] with
complexity $O(nM(n))$ where $M(n)$ is the complexity of matrix multiplication.
In addition, there exists a variety of algorithms for determinant sign
computation [BEPP99, ABM99]. The problem of computation of several
determinants has also been studied. TOPCOM [Ram02], the reference software for
computing triangulations of a set of points, efficiently precomputes all
orientation determinants that will be needed in the computation and stores
their signs. In [EFKP12], a similar problem is studied in the context of
computational algebraic geometry. The computation of orientation predicates is
accelerated by maintaining a hash table of computed minors of the
determinants. These minors appear many times in the computation. Although,
applying that method to the convex hull computation does not lead to a more
efficient algorithm.
Our main tools are the Sherman-Morrison formulas [SM50, Bar51]. They relate
the inverse of a matrix after a small-rank perturbation to the inverse of the
original matrix. In [San04] these formulas are used in a similar way to solve
the dynamic transitive closure problem in graphs.
The paper is organized as follows. Sect. 2 introduces the dynamic determinant
algorithms and the following section presents their application to the convex
hull problem. Sect. 4 discusses the implementation, experiments, and
comparison with other software. We conclude with future work.
## 2 Dynamic Determinant Computations
In the dynamic determinant problem, a $d\times d$ matrix $A$ is given.
Allowing some preprocessing, we should be able to handle updates of elements
of $A$ and return the current value of the determinant. We consider here only
non-singular updates, _i.e._ , updates that do not make $A$ singular. Let
$(A)_{i}$ denote the $i$-th column of $A$, and $e_{i}$ the vector with $1$ in
its $i$-th place and $0$ everywhere else.
Consider the matrix $A^{\prime}$, resulting from replacing the $i$-th column
of $A$ by a vector $u$. The Sherman-Morrison formula [SM50, Bar51] states that
$(A+wv^{T})^{-1}=A^{-1}-\frac{(A^{-1}w)(v^{T}A^{-1})}{1+v^{T}A^{-1}w}$. An
$i$-th column update of $A$ is performed by substituting $v=e_{i}$ and
$w=u-(A)_{i}$ in the above formula. Then, we can write $A^{\prime-1}$ as
follows.
$\displaystyle
A^{\prime-1}=(A+(u-(A)_{i})e_{i}^{T})^{-1}=A^{-1}-\frac{(A^{-1}(u-(A)_{i}))\
(e_{i}^{T}A^{-1})}{1+e_{i}^{T}A^{-1}(u-(A)_{i})}$ (1)
If $A^{-1}$ is computed, we compute $A^{\prime-1}$ using Eq. 1 in
$3d^{2}+2d+O(1)$ arithmetic operations. Similarly, the matrix determinant
lemma [Har97] gives Eq. 2 below to compute $det(A^{\prime})$ in $2d+O(1)$
arithmetic operations, if $det(A)$ is computed.
$\displaystyle
det(A^{\prime})=det(A+(u-(A)_{i})e_{i}^{T})=(1+e_{i}^{T}A^{-1}(u-(A)_{i})det(A)$
(2)
Eqs. 1 and 2 lead to the following result.
###### Proposition 1
[SM50] The dynamic determinant problem can be solved using $O(d^{\omega})$
arithmetic operations for preprocessing and $O(d^{2})$ for non-singular one
column updates.
Indeed, this computation can also be performed over a ring. To this end, we
use the adjoint of $A$, denoted by $A^{adj}$, rather than the inverse. It
holds that $A^{adj}=det(A)A^{-1}$, thus we obtain the following two equations.
$\displaystyle A^{\prime adj}$
$\displaystyle=\frac{1}{\det(A)}(A^{adj}\det(A^{\prime})-(A^{adj}(u-(A)_{i}))\
(e_{i}^{T}A^{adj}))$ (3) $\displaystyle det(A^{\prime})$
$\displaystyle=det(A)+e_{i}^{T}A^{adj}(u-(A)_{i})$ (4)
The only division, in Eq. 3, is known to be exact, _i.e._ , its remainder is
zero. The above computations can be performed in $5d^{2}+d+O(1)$ arithmetic
operations for Eq. 3 and in $2d+O(1)$ for Eq. 4. In the sequel, we will call
dyn_inv the dynamic determinant algorithm which uses Eqs. 1 and 2, and dyn_adj
the one which uses Eqs. 3 and 4.
## 3 Geometric Algorithms
We introduce in this section our methods for optimizing the computation of
sequences of determinants that appear in geometric algorithms. First, we use
dynamic determinant computations in incremental convex hull algorithms. Then,
we show how this solution can be extended to point location in triangulations.
Let us start with some basic definitions from discrete and computational
geometry. Let ${\mathcal{A}}\subset{\mathbb{R}}^{d}$ be a pointset. We define
the convex hull of a pointset ${\mathcal{A}}$, denoted by
conv(${\mathcal{A}}$), as the smallest convex set containing ${\mathcal{A}}$.
A hyperplane supports conv(${\mathcal{A}}$) if conv(${\mathcal{A}}$) is
entirely contained in one of the two closed half-spaces determined by the
hyperplane and has at least one point on the hyperplane. A face of
conv(${\mathcal{A}}$) is the intersection of conv(${\mathcal{A}}$) with a
supporting hyperplane which does not contain conv(${\mathcal{A}}$). Faces of
dimension $0,1,d-1$ are called vertices, edges and facets respectively. We
call a face $f$ of conv(${\mathcal{A}}$) visible from $a\in{\mathbb{R}}^{d}$
if there is a supporting hyperplane of $f$ such that conv(${\mathcal{A}}$) is
contained in one of the two closed half-spaces determined by the hyperplane
and $a$ in the other. A $k$-simplex of ${\mathcal{A}}$ is an affinely
independent subset $S$ of ${\mathcal{A}}$, where $dim(\text{conv}(S))=k$. A
triangulation of ${\mathcal{A}}$ is a collection of subsets of
${\mathcal{A}}$, the _cells_ of the triangulation, such that the union of the
cells’ convex hulls equals conv(${\mathcal{A}}$), every pair of convex hulls
of cells intersect at a common face and every cell is a simplex.
Denote $\vec{a}$ the vector $(a,1)$ for $a\in{\mathbb{R}}^{d}$. For any
sequence $C$ of points $a_{i}\in{\mathcal{A}}\text{, }i=1\ldots d+1$, we
denote $A_{C}$ its orientation $(d+1)\times(d+1)$ matrix. For every $a_{i}$,
the column $i$ of $A_{C}$ contains $\vec{a}_{i}$’s coordinates as entries. For
simplicity, we assume general position of ${\mathcal{A}}$ and focus on the
Beneath-and-Beyond (BB) algorithm [Sei81]. However, our method can be extended
to handle degenerate inputs as in [Ede87, Sect. 8.4], as well as to be applied
to any incremental convex hull algorithm by utilizing the dynamic determinant
computations to answer the predicates appearing in point location (see Cor.
2). In what follows, we use the dynamic determinant algorithm dyn_adj, which
can be replaced by dyn_inv yielding a variant of the presented convex hull
algorithm.
The BB algorithm is initialized by computing a $d$-simplex of ${\mathcal{A}}$.
At every subsequent step, a new point from ${\mathcal{A}}$ is inserted, while
keeping a triangulated convex hull of the inserted points. Let $t$ be the
number of cells of this triangulation. Assume that, at some step, a new point
$a\in{\mathcal{A}}$ is inserted and $T$ is the triangulation of the convex
hull of the points of ${\mathcal{A}}$ inserted up to now. To determine if a
facet $F$ is visible from $a$, an orientation predicate involving $a$ and the
points of $F$ has to be computed. This can be done by using Eq. 4 if we know
the adjoint matrix of points of the cell that contains $F$. But, if $F$ is
visible, this cell is unique and we can map it to the adjoint matrix
corresponding to its points.
Our method (Alg. 1), as initialization, computes from scratch the adjoint
matrix that corresponds to the initial $d$-simplex. At every incremental step,
it computes the orientation predicates using the adjoint matrices computed in
previous steps and Eq. 4. It also computes the adjoint matrices corresponding
to the new cells using Eq. 3. By Prop. 1, this method leads to the following
result.
Input : pointset ${\mathcal{A}}\subset{\mathbb{R}}^{d}$
Output : convex hull of ${\mathcal{A}}$
sort ${\mathcal{A}}$ by increasing lexicographic order of coordinates, _i.e._
, ${\mathcal{A}}=\\{a_{1},\dots,a_{n}\\}$;
$T\leftarrow\\{\text{\,$d$-face of }conv(a_{1},\dots,a_{d+1})\\}$; //
$conv(a_{1},\dots,a_{d+1})$ is a $d$-simplex
$Q\leftarrow\\{$ facets of $conv(a_{1},\dots,a_{d+1})\\}$;
compute
$A_{\\{a_{1},\dots,a_{d+1}\\}}^{adj}\text{,}\det(A_{\\{a_{1},\dots,a_{d+1}\\}})$;
foreach _$a\in\\{a_{d+2},\dots,a_{n}\\}$ _ do
$Q^{\prime}\leftarrow Q$;
foreach _$F\in Q$_ do
$C\leftarrow$ the unique $d$-face s.t. $C\in T$ and $F\in C$;
$u\leftarrow$ the unique vertex s.t. $u\in C$ and $u\notin F$;
$C^{\prime}\leftarrow F\cup\\{a\\}$;
$i\leftarrow$ the index of $u$ in $A_{C}$;
// both $det(A_{C})$ and $A_{C}^{adj}$ were computed in a previous step
$det(A_{C^{\prime}})\leftarrow det(A_{C})+(A_{C}^{adj})^{i}(\vec{u}-\vec{a})$;
if _$\det(A_{C^{\prime}})\det(A_{C}) <0$ and $\det(A_{C^{\prime}})\neq 0$_
then
$A_{C^{\prime}}^{adj}\leftarrow\frac{1}{\det(A_{C})}(A_{C}^{adj}\det(A_{C^{\prime}})-A_{C}^{adj}(\vec{u}-\vec{a})(e_{i}^{T}A_{C}^{adj}))$;
$T\leftarrow T\cup\\{\text{$d$-face of }conv(C^{\prime})\\}$;
$Q^{\prime}\leftarrow Q^{\prime}\,\ominus\,\\{(d-1)\text{-faces of
}C^{\prime}\\}$; // symmetric difference
$Q\leftarrow Q^{\prime}$;
return $Q$;
Algorithm 1 Incremental Convex Hull (${\mathcal{A}}$)
###### Theorem 1
Given a $d$-dimensional pointset all, except the first, orientation predicates
of incremental convex hull algorithms can be computed in $O(d^{2})$ time and
$O(d^{2}t)$ space, where $t$ is the number of cells of the constructed
triangulation.
Essentially, this result improves the computational complexity of the
determinants involved in incremental convex hull algorithms from
$O(d^{\omega})$ to $O(d^{2})$. To analyze the complexity of Alg. 1, we bound
the number of facets of $Q$ in every step of the outer loop of Alg. 1 with the
number of $(d-1)$-faces of the constructed triangulation of
conv(${\mathcal{A}}$), which is bounded by $(d+1)t$. Thus, using Thm. 1, we
have the following complexity bound for Alg. 1.
###### Corollary 1
Given $n$ $d$-dimensional points, the complexity of BB algorithm is $O(n\log
n+d^{3}nt)$, where $n\gg d$ and $t$ is the number of cells of the constructed
triangulation.
Note that the complexity of BB, without using the method of dynamic
determinants, is bounded by $O(n\log n+d^{{\omega}+1}nt)$. Recall that $t$ is
bounded by $O(n^{\lfloor d/2\rfloor})$ [Zie95, Sect.8.4], which shows that
Alg. 1, and convex hull algorithms in general, do not have polynomial
complexity. The schematic description of Alg. 1 and its coarse analysis is
good enough for our purpose: to illustrate the application of dynamic
determinant computation to incremental convex hulls and to quantify the
improvement of our method. See Sect. 4 for a practical approach to incremental
convex hull algorithms using dynamic determinant computations.
The above results can be extended to improve the complexity of geometric
algorithms that are based on convex hulls computations, such as algorithms for
regular or Delaunay triangulations and Voronoi diagrams. It is straightforward
to apply the above method in orientation predicates appearing in point
location algorithms. By using Alg. 1, we compute a triangulation and a map of
adjoint matrices to its cells. Then, the point location predicates can be
computed using Eq. 4, avoiding the computation of new adjoint matrices.
###### Corollary 2
Given a triangulation of a $d$-dimensional pointset computed by Alg. 1, the
orientation predicates involved in any point location algorithm can be
computed in $O(d)$ time and $O(d^{2}t)$ space, where $t$ is the number of
cells of the triangulation.
## 4 Implementation and Experimental Analysis
We propose the hashed dynamic determinants scheme and implement it in C++. The
design of our implementation is modular, that is, it can be used on top of
either geometric software providing geometric predicates (such as orientation)
or algebraic software providing dynamic determinant algorithm implementations.
The code is publicly available from http://hdch.sourceforge.net.
The hashed dynamic determinants scheme consists of efficient implementations
of algorithms dyn_inv and dyn_adj (Sect. 2) and a hash table, which stores
intermediate results (matrices and determinants) based on the methods
presented in Sect. 3. Every $(d-1)$-face of a triangulation, _i.e._ , a common
facet of two neighbor cells (computed by any incremental convex hull package
which constructs a triangulation of the computed convex hull), is mapped to
the indices of its vertices, which are used as keys. These are mapped to the
adjoint (or inverse) matrix and the determinant of one of the two adjacent
cells. Let us illustrate this approach with an example, on which we use the
dyn_adj algorithm.
###### Example 1
Let
$A=\\{a_{1}=(0,1),\,a_{2}=(1,2),\,a_{3}=(2,1),\,a_{4}=(1,0),\,a_{5}=(2,2)\\}$
where every point $a_{i}$ has an index $i$ from $1$ to $5$. Assume we are in
some step of an incremental convex hull or point location algorithm and let
$T=\\{\\{1,2,4\\},\,\\{2,3,4\\}\\}$ be the $2$-dimensional triangulation of
${\mathcal{A}}$ computed so far. The cells of $T$ are written using the
indices of the points in ${\mathcal{A}}$. The hash table will store as keys
the set of indices of the $2$-faces of $T$, _i.e._ ,
$\\{\\{1,2\\},\\{2,4\\},\\{1,4\\}\\}$ mapping to the adjoint and the
determinant of the matrix constructed by the points $a_{1},a_{2},a_{4}$.
Similarly, $\\{\\{2,3\\},\\{3,4\\},\\{2,4\\}\\}$ are mapped to the adjoint
matrix and determinant of $a_{2},a_{3},a_{4}$. To insert $a_{5}$, we compute
the determinant of $a_{2},a_{3},a_{5}$, by querying the hash table for
$\\{2,3\\}$. Adjoint and determinant of the matrix of $a_{2},a_{3},a_{4}$ are
returned, and we perform an update of the column corresponding to point
$a_{4}$, replacing it by $a_{5}$ by using Eqs. 3 and 4.
To implement the hash table, we used the Boost libraries [boo]. To reduce
memory consumption and speed-up look-up time, we sort the lists of indices
that form the hash keys. We also use the _GNU Multiple Precision arithmetic
library_ (GMP), the current standard for multiple-precision arithmetic, which
provides integer and rational types mpz_t and mpq_t, respectively.
We perform an experimental analysis of the proposed methods. All experiments
ran on an Intel Core i5-2400 $3.1$GHz, with $6$MB L2 cache and $8$GB RAM,
running 64-bit Debian GNU/Linux. We divide our tests in four scenarios,
according to the number type involved in computations: (a) rationals where the
bit-size of both numerator and denominator is $10000$, (b) rationals converted
from doubles, that is, numbers of the form $m\times 2^{p}$, where $m$ and $p$
are integers of bit-size $53$ and $11$ respectively, (c) integers with bit-
size $10000$, and (d) integers with bit-size $32$. However, it is rare to find
in practice input coefficients of scenarios (a) and (c). Inputs are usually
given as $32$ or $64$-bit numbers. These inputs correspond to the coefficients
of scenario (b). Scenario (d) is also very important, since points with
integer coefficients are encountered in many combinatorial applications (see
Sect. 1).
We compare state-of-the-art software for exact computation of the determinant
of a matrix. We consider LU decomposition in CGAL [CGA], optimized LU
decomposition in Eigen [GJ+10], LinBox asymptotically optimal algorithms
[DGG+02] (tested only on integers) and Maple 14 LinearAlgebra [Determinant]
(the default determinant algorithm). We also implemented two division-free
algorithms: Bird’s [Bir11] and Laplace expansion [Poo06, Sect.4.2]. Finally,
we consider our implementations of dyn_inv and dyn_adj.
We test the above implementations in the four coefficient scenarios described
above. When coefficients are integer, we can use integer exact division
algorithms, which are faster than quotient-remainder division algorithms. In
this case, Bird, Laplace and dyn_adj enjoy the advantage of using the number
type mpz_t while the rest are using mpq_t. The input matrices are constructed
starting from a random $d\times d$ matrix, replacing a randomly selected
column with a random $d$ vector. We present experimental results of the four
input scenarios in Tables 1, 2, 3, 4. We tested a fifth coefficient scenario
(rationals of bit-size $32$), but do not show results here because timings are
quite proportional to those shown in Table 1. We stop testing an
implementation when it is slow and far from being the fastest (denoted with
’-’ in the Tables).
On one hand, the experiments show the most efficient determinant algorithm
implementation in the different scenarios described, without considering the
dynamic algorithms. This is a result of independent interest, and shows the
efficiency of division-free algorithms in some settings. The simplest
determinant algorithm, Laplace expansion, proved to be the best in all
scenarios, until dimension $4$ to $6$, depending on the scenario. It has
exponential complexity, thus it is slow in dimensions higher than $6$ but it
behaves very well in low dimensions because of the small constant of its
complexity and the fact that it performs no divisions. Bird is the fastest in
scenario (c), starting from dimension $7$, and in scenario (d), in dimensions
$7$ and $8$. It has also a small complexity constant, and performing no
divisions makes it competitive with decomposition methods (which have better
complexity) when working with integers. CGAL and Eigen implement LU
decomposition, but the latter is always around two times faster. Eigen is the
fastest implementation in scenarios (a) and (b), starting from dimension $5$
and $6$ respectively, as well as in scenario (d) in dimensions between $9$ and
$12$. It should be stressed that decomposition methods are the current
standard to implement determinant computation. Maple is the fastest only in
scenario (d), starting from dimension $13$. In our tests, Linbox is never the
best, due to the fact that it focuses on higher dimensions.
On the other hand, experiments show that dyn_adj outperforms all the other
determinant algorithms in scenarios (b), (c), and (d). On each of these
scenarios, there is a threshold dimension, starting from which dyn_adj is the
most efficient, which happens because of its better asymptotic complexity. In
scenarios (c) and (d), with integer coefficients, division-free performs much
better, as expected, because integer arithmetic is faster than rational. In
general, the sizes of the coefficients of the adjoint matrix are bounded. That
is, the sizes of the operands of the arithmetic operations are bounded. This
explains the better performance of dyn_adj over the dyn_inv, despite its worse
arithmetic complexity.
$d$ | Bird | CGAL | Eigen | Laplace | Maple | dyn_inv | dyn_adj
---|---|---|---|---|---|---|---
3 | 16.61 | 17.05 | 15.02 | 11.31 | 16.234 | 195.38 | 191.95
4 | 143.11 | 98.15 | 71.35 | 63.22 | 115.782 | 746.32 | 896.58
5 | 801.26 | 371.85 | 239.97 | 273.27 | 570.582 | 2065.08 | 2795.53
6 | 3199.79 | 1086.80 | 644.62 | 1060.10 | 1576.592 | 4845.38 | 7171.81
7 | 10331.30 | 2959.80 | 1448.60 | 7682.24 | 4222.563 | – | –
Table 1: Determinant tests, inputs of scenario (a): rationals of bit-size $10000$. Times in milliseconds, averaged over 1000 tests. Light blue highlights the best non-dynamic algorithm while yellow highlights the dynamic algorithm if it is the fastest over all. $d$ | Bird | CGAL | Eigen | Laplace | Maple | dyn_inv | dyn_adj
---|---|---|---|---|---|---|---
3 | .013 | .021 | .014 | .008 | .058 | .046 | .023
4 | .046 | .050 | .033 | .020 | .105 | .108 | .042
5 | .122 | .110 | .072 | .056 | .288 | .213 | .067
6 | .268 | .225 | .137 | .141 | .597 | .376 | .102
7 | .522 | .412 | .243 | .993 | .824 | .613 | .148
8 | .930 | .710 | .390 | – | 1.176 | .920 | .210
9 | 1.520 | 1.140 | .630 | – | 1.732 | 1.330 | .310
10 | 2.380 | 1.740 | .940 | – | 2.380 | 1.830 | .430
11 | – | 2.510 | 1.370 | – | 3.172 | 2.480 | .570
12 | – | 3.570 | 2.000 | – | 4.298 | 3.260 | .760
13 | – | 4.960 | 2.690 | – | 5.673 | 4.190 | 1.020
14 | – | 6.870 | 3.660 | – | 7.424 | 5.290 | 1.360
15 | – | 9.060 | 4.790 | – | 9.312 | 6.740 | 1.830
Table 2: Determinant tests, inputs of scenario (b): rationals converted from double. Each timing (in milliseconds) corresponds to the average of computing 10000 (for $d<7$) or 1000 (for $d\geq 7$) determinants. Highlighting as in Table 1. $d$ | Bird | CGAL | Eigen | Laplace | Linbox | Maple | dyn_inv | dyn_adj
---|---|---|---|---|---|---|---|---
3 | .23 | 3.24 | 2.58 | .16 | 132.64 | .28 | 27.37 | 2.17
4 | 1.04 | 14.51 | 10.08 | .61 | 164.80 | 1.36 | 76.76 | 6.59
5 | 3.40 | 45.52 | 28.77 | 2.02 | 367.58 | 4.52 | 176.60 | 14.70
6 | 8.91 | 114.05 | 67.85 | 6.16 | – | 423.08 | 325.65 | 27.97
7 | 20.05 | 243.54 | 138.80 | 42.97 | – | – | 569.74 | 48.49
8 | 40.27 | 476.74 | 257.24 | – | – | – | 904.21 | 81.44
9 | 73.90 | 815.70 | 440.30 | – | – | – | 1359.80 | 155.70
10 | 129.95 | 1358.50 | 714.40 | – | – | – | 1965.30 | 224.10
11 | 208.80 | – | – | – | – | – | – | 328.50
12 | 327.80 | – | – | – | – | – | – | 465.00
13 | 493.90 | – | – | – | – | – | – | 623.80
14 | 721.70 | – | – | – | – | – | – | 830.80
15 | 1025.10 | – | – | – | – | – | – | 1092.30
16 | 1422.80 | – | – | – | – | – | – | 1407.20
17 | 1938.40 | – | – | – | – | – | – | 1795.60
18 | 2618.30 | – | – | – | – | – | – | 2225.80
19 | 3425.70 | – | – | – | – | – | – | 2738.00
20 | 4465.60 | – | – | – | – | – | – | 3413.40
Table 3: Determinant tests, inputs of scenario (c): integers of bit-size $10000$. Times in milliseconds, averaged over 1000 tests for $d<9$ and 100 tests for $d\geq 9$. Highlighting as in Table 1. $d$ | Bird | CGAL | Eigen | Laplace | Linbox | Maple | dyn_inv | dyn_adj
---|---|---|---|---|---|---|---|---
3 | .002 | .021 | .013 | .002 | .872 | .045 | .030 | .008
4 | .012 | .041 | .028 | .005 | 1.010 | .094 | .058 | .015
5 | .032 | .080 | .048 | .016 | 1.103 | .214 | .119 | .023
6 | .072 | .155 | .092 | .040 | 1.232 | .602 | .197 | .033
7 | .138 | .253 | .149 | .277 | 1.435 | .716 | .322 | .046
8 | .244 | .439 | .247 | – | 1.626 | .791 | .486 | .068
9 | .408 | .689 | .376 | – | 1.862 | .906 | .700 | .085
10 | .646 | 1.031 | .568 | – | 2.160 | 1.014 | .982 | .107
11 | .956 | 1.485 | .800 | – | 10.127 | 1.113 | 1.291 | .133
12 | 1.379 | 2.091 | 1.139 | – | 13.101 | 1.280 | 1.731 | .160
13 | 1.957 | 2.779 | 1.485 | – | – | 1.399 | 2.078 | .184
14 | 2.603 | 3.722 | 1.968 | – | – | 1.536 | 2.676 | .222
15 | 3.485 | 4.989 | 2.565 | – | – | 1.717 | 3.318 | .269
16 | 4.682 | 6.517 | 3.391 | – | – | 1.850 | 4.136 | .333
Table 4: Determinant tests, inputs of scenario (d): integers of bit-size $32$.
Times in milliseconds, averaged over 10000 tests. Highlighting as in Table 1.
For the experimental analysis of the behaviour of dynamic determinants used in
convex hull algorithms (Alg. 1, Sect. 3), we experiment with four state-of-
the-art exact convex hull packages. Two of them implement incremental convex
hull algorithms: triangulation [BDH09] implements [CMS93] and beneath-and-
beyond (bb) in polymake [GJ00]. The package cdd [Fuk08] implements the double
description method, and lrs implements the gift-wrapping algorithm using
reverse search [Avi00]. We propose and implement a variant of triangulation,
which we will call hdch, implementing the hashed dynamic determinants scheme
for dimensions higher than $6$ (using Eigen for initial determinant and
adjoint or inverse matrix computation) and using Laplace determinant algorithm
for lower dimensions. The main difference between this implementation and Alg.
1 of Sect. 3 is that it does not sort the points and, before inserting a
point, it performs a point location. Thus, we can take advantage of our scheme
in two places: in the orientation predicates appearing in the point location
procedure and in the ones that appear in construction of the convex hull. We
design the input of our experiments parametrized on the number type of the
coefficients and on the distribution of the points. The number type is either
rational or integer. From now on, when we refer to rational and integer we
mean scenario (b) and (d), respectively. We test three uniform point
distributions: (i) in the $d$-cube $[-100,100]^{d}$, (ii) in the origin-
centered $d$-ball of radius $100$, and (iii) on the surface of that ball.
We perform an experimental comparison of the four above packages and hdch,
with input points from distributions (i)-(iii) with either rational or integer
coefficients. In the case of integer coefficients, we test hdch using mpq_t
(hdch_q) or mpz_t (hdch_z). In this case hdch_z is the most efficient with
input from distribution (ii) (Fig. 1; distribution (i) is similar to this)
while in distribution (iii) both hdch_z and hdch_q perform better than all the
other packages (see Fig. 1). In the rational coefficients case, hdch_q is
competitive to the fastest package (Fig. 2). Note that the rest of the
packages cannot perform arithmetic computations using mpz_t because they are
lacking division-free determinant algorithms. Moreover, we perform experiments
to test the improvements of hashed dynamic determinants scheme on
triangulation and their memory consumption. For input points from distribution
(iii) with integer coefficients, when dimension ranges from $3$ to $8$, hdch_q
is up to $1.7$ times faster than triangulation and hdch_z up to $3.5$ times
faster (Table 5). It should be noted that hdch is always faster than
triangulation. The sole modification of the determinant algorithm made it
faster than all other implementations in the tested scenarios. The other
implementations would also benefit from applying the same determinant
technique. The main disadvantage of hdch is the amount of memory consumed,
which allows us to compute up to dimension $8$ (Table 5). This drawback can be
seen as the price to pay for the obtained speed-up.
A large class of algorithms that compute the exact volume of a polytope is
based on triangulation methods [BEF98]. All the above packages compute the
volume of the polytope, defined by the input points, as part of the convex
hull computation. The volume computation takes place at the construction of
the triangulation during a convex hull computation. The sum of the volumes of
the cells of the triangulation equals the volume of the polytope. However, the
volume of the cell is the absolute value of the orientation determinant of the
points of the cell and these values are computed in the course of the convex
hull computation. Thus, the computation of the volume consumes no extra time
besides the convex hull computation time. Therefore, hdch yields a competitive
implementation for the exact computation of the volume of a polytope given by
its vertices (Fig. 1).
$|{\mathcal{A}}|$ | $d$ | hdch_q | hdch_z | triangulation
---|---|---|---|---
time | memory | time | memory | time | memory
(sec) | (MB) | (sec) | (MB) | (sec) | (MB)
260 | 2 | 0.02 | 35.02 | 0.01 | 33.48 | 0.05 | 35.04
500 | 2 | 0.04 | 35.07 | 0.02 | 33.53 | 0.12 | 35.08
260 | 3 | 0.07 | 35.20 | 0.04 | 33.64 | 0.20 | 35.23
500 | 3 | 0.19 | 35.54 | 0.11 | 33.96 | 0.50 | 35.54
260 | 4 | 0.39 | 35.87 | 0.21 | 34.33 | 0.82 | 35.46
500 | 4 | 0.90 | 37.07 | 0.47 | 35.48 | 1.92 | 37.17
260 | 5 | 2.22 | 39.68 | 1.08 | 38.13 | 3.74 | 39.56
500 | 5 | 5.10 | 45.21 | 2.51 | 43.51 | 8.43 | 45.34
260 | 6 | 14.77 | 1531.76 | 8.42 | 1132.72 | 20.01 | 55.15
500 | 6 | 37.77 | 3834.19 | 21.49 | 2826.77 | 51.13 | 83.98
220 | 7 | 56.19 | 6007.08 | 32.25 | 4494.04 | 90.06 | 102.34
320 | 7 | swap | swap | 62.01 | 8175.21 | 164.83 | 185.87
120 | 8 | 86.59 | 8487.80 | 45.12 | 6318.14 | 151.81 | 132.70
140 | 8 | swap | swap | 72.81 | 8749.04 | 213.59 | 186.19
Table 5: Comparison of hdch_z, hdch_q and triangulation using points from
distribution (iii) with integer coefficients; swap means that the machine used
swap memory.
Figure 1: Comparison of convex hull packages for $6$-dimensional inputs with
integer coefficients. Points are uniformly distributed (a) inside a $6$-ball
and (b) on its surface.
Figure 2: Comparison of convex hull packages for $6$-dimensional inputs with
rational coefficients. Points are uniformly distributed (a) inside a $6$-ball
and (b) on its surface.
Finally, we test the efficiency of hashed dynamic determinants scheme on the
point location problem. Given a pointset, triangulation constructs a data
structure that can perform point locations of new points. In addition to that,
hdch constructs a hash table for faster orientation computations. We perform
tests with triangulation and hdch using input points uniformly distributed on
the surface of a ball (distribution (iii)) as a preprocessing to build the
data structures. Then, we perform point locations using points uniformly
distributed inside a cube (distribution (i)). Experiments show that our method
yields a speed-up in query time of a factor of $35$ and $78$ in dimension $8$
to $11$, respectively, using points with integer coefficients (scenario (d))
(see Table 6).
| $d$ | $|{\mathcal{A}}|$ | preproc. | data | # of | query time
---|---|---|---|---|---|---
| time | structs. | cells in | (sec)
| (sec) | (MB) | triangul. | 1K | 1000K
hdch_z | 8 | 120 | 45.20 | 6913 | 319438 | 0.41 | 392.55
triang | 8 | 120 | 156.55 | 134 | 319438 | 14.42 | 14012.60
hdch_z | 9 | 70 | 45.69 | 6826 | 265874 | 0.28 | 276.90
triang | 9 | 70 | 176.62 | 143 | 265874 | 13.80 | 13520.43
hdch_z | 10 | 50 | 43.45 | 6355 | 207190 | 0.27 | 217.45
triang | 10 | 50 | 188.68 | 127 | 207190 | 14.40 | 14453.46
hdch_z | 11 | 39 | 38.82 | 5964 | 148846 | 0.18 | 189.56
triang | 11 | 39 | 181.35 | 122 | 148846 | 14.41 | 14828.67
Table 6: Point location time of 1K and 1000K (1K=1000) query points for hdch_z
and triangulation (triang), using distribution (iii) for preprocessing and
distribution (i) for queries and integer coefficients.
## 5 Future Work
It would be interesting to adapt our scheme for gift-wrapping convex hull
algorithms and implement it on top of packages such as [Avi00]. In this
direction, our scheme should also be adapted to other important geometric
algorithms, such as Delaunay triangulations.
In order to overcome the large memory consumption of our method, we shall
exploit hybrid techniques. That is, to use the dynamic determinant hashing
scheme as long as there is enough memory and subsequently use the best
available determinant algorithm (Sect. 4), or to clean periodically the hash
table.
Another important experimental result would be to investigate the behavior of
our scheme using filtered computations.
### Acknowledgments.
The authors are partially supported from project “Computational Geometric
Learning”, which acknowledges the financial support of the Future and Emerging
Technologies (FET) programme within the Seventh Framework Programme for
Research of the European Commission, under FET-Open grant number: 255827. We
would like to thank I.Z. Emiris for his advice and support, E. Tsigaridas for
bibliographic suggestions and M. Karavelas for discussions on efficient
dynamic determinant updates.
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|
arxiv-papers
| 2012-06-29T16:24:28 |
2024-09-04T02:49:32.406490
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Vissarion Fisikopoulos, Luis Pe\\~naranda",
"submitter": "Luis Pe\\~naranda",
"url": "https://arxiv.org/abs/1206.7067"
}
|
1207.0047
|
# Can we distinguish between black holes and wormholes by their Einstein-ring
systems?
Naoki Tsukamoto,111Electronic address:11ra001t@rikkyo.ac.jp Tomohiro Harada
and Kohji Yajima Department of Physics, Rikkyo University, Tokyo 171-8501,
Japan
###### Abstract
For the last decade, gravitational lensing in the strong gravitational field
has been studied eagerly. It is well known that, for the lensing by a black
hole, an infinite number of Einstein rings are formed by the light rays which
wind around the black hole nearly on the photon sphere, which are called
relativistic Einstein rings. This is also the case for the lensing by a
wormhole. In this paper, we study the Einstein ring and relativistic Einstein
rings for the Schwarzschild black hole and the Ellis wormhole, the latter of
which is an example of traversable wormholes of the Morris-Thorne class. Given
the configuration of the gravitational lensing and the radii of the Einstein
ring and relativistic Einstein rings, we can distinguish between a black hole
and a wormhole in principle. We conclude that we can detect the relativistic
Einstein rings by wormholes which have the radii of the throat $a\simeq 0.5$pc
at a galactic center with the distance $10$Mpc and which have $a\simeq 10$AU
in our galaxy using by the most powerful modern instruments which have the
resolution of $10^{-2}$arcsecond such as a $10$-meter optical-infrared
telescope. The black holes which make the Einstein rings of the same size as
the ones by the wormholes are galactic supermassive black holes and the
relativistic Einstein rings by the black holes are too small to measure with
the current technology. We may test the hypotheses of astrophysical wormholes
by using the Einstein ring and relativistic Einstein rings in the future.
###### pacs:
04.20.-q, 04.70.-s
††preprint: RUP -11-5
## I I. Introduction
Gravitational lensing is a very useful tool for astrophysics and cosmology. At
first the gravitational lensing mainly was investigated on a theoretical basis
in the weak gravitational field. Using the gravitational lensing, we determine
the cosmological constant, the distribution of dark matter and the Hubble
constant, the existence of extrasolar planets and so on (see Schneider et al.
Gravitational_lenses and Perlick Perlick_2004_Living_Rev ; Perlick_2010 for
the detail of the gravitational lens, and references therein).
For the last decade, gravitational lensing in the strong gravitational field
has been studied eagerly (see Virbhadra and Keeton Virbhadra_Keeton_2008 ,
Virbhadra Virbhadra_2009 , Bozza Bozza_2010 , Bozza and Mancini
Bozza_Mancini_2012 and references therein). Frittelli et al.
Frittelli_Kling_Newman_2000 , Virbhadra and Ellis Virbhadra_Ellis_2000 ;
Virbhadra_Ellis_2002 and Bozza et al. Bozza_Capozziello_Iovane_Scarpetta_2001
studied the gravitational lensing in the strong field with the Schwarzschild
spacetime and found the infinite Einstein rings which are too close to each
other to separately resolve. In this paper, we call these rings relativistic
Einstein rings. The gravitational lensing in the strong field on the
spherically symmetric static spacetime was investigated by Bozza Bozza_2002 ,
Hasse and Perlick Hasse_Perlick_2001 and Perlick Perlick_2004_Phys_Rev_D .
They showed that the relativistic Einstein rings are formed not only in the
Schwarzschild spacetime but also in the other spherically symmetric static
spacetime.
General relativity permits nontrivial topology of the spacetime such as
wormhole spacetimes (see Visser Lorentzuan_Wormholes for the details of
wormholes). Some hypotheses of astrophysical wormholes have been investigated
Harko_Kovacs_Lobo_2009 ; Abdujabbarov_Ahmedov_2009 ; Pozanenko_Shatskiy_2010 .
For example, Kardashev et al. suggest that some active galactic nuclei and
other compact astrophysical objects may be explained as wormholes
Kardashev_Novikov_Shatskiy_2007 . We may test these hypotheses by using the
gravitational lensing in the future.
Kim and Cho Kim_Cho_1994 and Cramer et al.
Cramer_Forward_Morris_Visser_Benford_Landis_1995 pioneered gravitational
lensing effects by wormholes. Since then, the gravitational lensing effects by
various wormholes have been investigated Rahaman_Kalam_Chakraborty_2007 ;
Safonova_Torres_2002 ; Safonova_Torres_Romero_2002 ;
Safonova_Torres_Romero_2001 ; Eiroa_Romero_Torres_2001 ;
Nandi_Zhang_Zakharov_2006 ; Cramer_Forward_Morris_Visser_Benford_Landis_1995 .
The Ellis spacetime which was investigated by Ellis Ellis_1973 is an example
of traversable wormholes of the Morris-Thorne class Morris_Thorne_1988 ;
Morris_Thorne_Yurtsever_1988 . The deflection angle of light in the Ellis
wormhole geometry was studied by Chetouani and Clement Chetouani_Clement_1984
and recently Nakajima and Asada Nakajima_Asada_2012 . The gravitational
lensing on the Ellis geometry was studied by Dey and Sen Dey_Sen_2008 , Abe
Abe_2010 and Toki et al. Toki_Kitamura_Asada_Abe_2011 in the weak
gravitational field and Perlick Perlick_2004_Phys_Rev_D , Nandi et al.
Nandi_Zhang_Zakharov_2006 and Tejeiro and Larranaga Tejeiro_Larranaga_2005
in the strong gravitational field.
Perlick Perlick_2004_Phys_Rev_D , Nandi et al. Nandi_Zhang_Zakharov_2006 and
Tejeiro and Larranaga Tejeiro_Larranaga_2005 pointed out that the qualitative
features of the gravitational lensing in the Ellis spacetime are very similar
to the ones in the Schwarzschild spacetime for their photon spheres and their
asymptotic flatness.
In this paper, we will consider the Einstein ring and relativistic Einstein
rings in the Ellis spacetime and the Schwarzschild spacetime, both of which
are static and spherically symmetric ones. We ask whether we can distinguish
the Einstein-ring systems on the Schwarzschild spacetime and on the Ellis
spacetime. To answer this question, we focus on the relations between the
Einstein ring and the relativistic Einstein rings.
This paper is organized as follows. In Sec. II we will review the deflection
angle on the Ellis wormhole spacetime. In Sec. III we give the radii of the
Einstein ring and relativistic Einstein rings in the Ellis spacetime. In Sec.
IV we will compare the Einstein ring and the relativistic Einstein rings in
the Ellis spacetime to the ones in the Schwarzschild spacetime. In Sec. V we
summarize and discuss our result. In this paper we use the units in which
$c=1$.
## II II. Ellis wormhole spacetime and deflection angle
In this section, we review the deflection angle on the Ellis wormhole
spacetime Chetouani_Clement_1984 ; Nakajima_Asada_2012 . The line element in
the Ellis wormhole solution is written in the following form:
$ds^{2}=-dt^{2}+dr^{2}+(r^{2}+a^{2})d\Omega^{2},$ (1)
where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$ and $a$ is a positive
constant. Introducing $\rho^{2}=r^{2}+a^{2}$, we can rewrite this into
$ds^{2}=-dt^{2}+\left(1-\frac{a^{2}}{\rho^{2}}\right)^{-1}d\rho^{2}+\rho^{2}d\Omega^{2},$
(2)
where $\rho=\pm a$ corresponds to the wormhole throat. The spacetime has the
Killing vectors $t^{\mu}\partial_{\mu}=\partial_{t}$ and
$\phi^{\mu}\partial_{\mu}=\partial_{\phi}$ for stationarity and axial
symmetry.
We can concentrate ourselves on the equatorial plane because of spherical
symmetry. Using the conservation of the energy $E\equiv-
g_{\mu\nu}k^{\mu}t^{\nu}$ and angular momentum $L\equiv
g_{\mu\nu}k^{\mu}\phi^{\nu}$ and $k^{\mu}k_{\mu}=0$, where $k^{\mu}$ is the
photon wave number, the photon trajectory is then given by
$\frac{1}{\rho^{4}}\left(\frac{d\rho}{d\phi}\right)^{2}=\frac{1}{b^{2}}\left(1-\frac{a^{2}}{\rho^{2}}\right)\left(1-\frac{b^{2}}{\rho^{2}}\right),$
(3)
where $b\equiv L/E$ is the impact parameter of the photon.
We can see that the photon is scattered if $|b|>a$, while it reaches the
throat if $|b|<a$. Since we are interested in the scattering problem, we
assume $|b|>a$. Using $u=1/\rho$, we find
$\left(\frac{du}{d\phi}\right)^{2}=\frac{1}{b^{2}}(1-a^{2}u^{2})(1-b^{2}u^{2}).$
(4)
Putting
$G(u)=a^{2}(a^{-2}-u^{2})(b^{-2}-u^{2}),$ (5)
the azimuthal angle $\phi$ can be given as a function of $u$ by
$\phi=\pm\int^{b^{-1}}_{u}\frac{du}{\sqrt{G(u)}}.$ (6)
Here we have set $\phi(b^{-1})=0$. The deflection angle $\alpha$ is then
calculated to give
$\alpha=2\int^{b^{-1}}_{0}\frac{du}{\sqrt{G(u)}}-\pi.$ (7)
In the present case, we find
$\int^{b^{-1}}_{0}\frac{du}{\sqrt{G(u)}}=\int^{\pi/2}_{0}\frac{d\theta}{\sqrt{1-\left(\frac{a}{b}\right)^{2}\sin^{2}\theta}}=K\left(\frac{a}{b}\right),$
(8)
where we have transformed $u=b^{-1}\sin\theta$ and $K(k)$ denotes the complete
elliptic integral of the first kind (for example, see
Handbook_of_Elliptic_Integrals_for_Engineers_and_Scientists ), which is
defined as
$K(k)=\int^{\pi/2}_{0}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}.$ (9)
Hence, the deflection angle is given by
$\alpha=2K\left(\frac{a}{b}\right)-\pi.$ (10)
Since $K(k)$ admits a power series
$K(k)=\frac{\pi}{2}\sum_{n=0}^{\infty}\left[\frac{(2n-1)!!}{(2n)!!}\right]^{2}k^{2n},$
(11)
where $n!!$ denotes the double factorial of $n$ and $(-1)!!=1$, we get the
deflection angle
$\alpha=\pi\sum_{n=1}^{\infty}\left[\frac{(2n-1)!!}{(2n)!!}\right]^{2}\left(\frac{a}{b}\right)^{2n}.$
(12)
Thus, the deflection angle is approximately given in the weak-field regime
$|b|\gg a$ by
$\alpha\simeq\frac{\pi}{4}\left(\frac{a}{b}\right)^{2}.$ (13)
In general, the deflection angle is always greater than its weak-field
approximation and is diverging as $|b|\to a$.
## III III. Einstein ring and relativistic Einstein rings of Ellis wormhole
In this section, we examine the diameter angles of the Einstein ring and the
relativistic Einstein rings on the Ellis spacetime. Now we will consider the
case that both the observer and the source object are far from the lensing
object, or $D_{l}\gg b$ and $D_{ls}\gg b$, where $D_{l}$ and $D_{ls}$ are the
separations between the observer and lens and between the lens and source,
respectively. The configuration of the gravitational lensing is given in Fig.
1.
Figure 1: The configuration of the gravitational lensing. The light rays
emitted by the source $S$ are deflected by the lens $L$ (a wormhole or a black
hole) and reach the observer $O$ with the angle of the lensed image $\theta$,
instead of the real angle $\phi$. $b$ and $\bar{\alpha}$ are the impact
parameter and the effective deflection angle, respectively. $D_{l}$ and
$D_{ls}$ are the separations between the observer and the lens and between the
lens and the source, respectively.
Then, the lens equation is given by
$D_{ls}\bar{\alpha}=D_{s}(\theta-\phi),$ (14)
where $\bar{\alpha}=(\alpha~{}\mbox{mod}~{}2\pi)$ is the effective deflection
angle, $\theta$ and $\phi$ are the angles of the lensed image and the real
image from the observer, respectively, and $D_{s}=D_{l}+D_{ls}$ is the
separation between the observer and source. Note that we have assumed
$|\bar{\alpha}|\ll 1$, $|\theta|\ll 1$ and $|\phi|\ll 1$. The deflection angle
can be expressed $\alpha=\bar{\alpha}+2\pi n$, where $n$ is a non-negative
integer, denoting the winding number of the light ray.
The ring image corresponds to the image angle $\theta$ for vanishing real
angle $\phi=0$. By the symmetry, the image is necessarily a ring with the
diameter angle $\theta$.
Since $b=D_{l}\theta$, we find that the ring image is given by
$\theta_{n}=\frac{a}{D_{l}}\frac{1}{k_{n}},$ (15)
where $k_{n}\in(0,1)$ is a unique root of the transcendental equation
$\displaystyle 2K(k)-\frac{\eta}{k}$ $\displaystyle=$
$\displaystyle\left(2n+1\right)\pi,$ (16) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{D_{s}}{D_{l}D_{ls}}a.$ (17)
We should note that $2K(k)-\eta/k$ is monotonically increasing with respect to
$k$ and changes from $-\infty$ to $\infty$ as $k$ increases from $0$ to $1$.
The uniqueness of the root follows from the monotonicity. Moreover, we can
conclude that $k_{n}$ monotonically increases and approaches 1 as $n\to\infty$
and hence the image angle $\theta_{n}$ monotonically decreases and approaches
$a/D_{l}$.
In the weak-field regime $|b|\gg a$, the winding number $n$ should be $n=0$.
Using the deflection angle (13), we can solve the equation (16) approximately
and get the diameter angle of the Einstein ring
$\displaystyle\theta_{0}$
$\displaystyle\simeq\left(\frac{\pi}{4}\frac{D_{ls}}{D_{s}D_{l}^{2}}a^{2}\right)^{\frac{1}{3}}$
(18) $\displaystyle\simeq
2.0\,\textrm{arcsecond}\left(\frac{D_{ls}}{10\textrm{Mpc}}\right)^{\frac{1}{3}}\left(\frac{20\textrm{Mpc}}{D_{s}}\right)^{\frac{1}{3}}\left(\frac{10\textrm{Mpc}}{D_{l}}\right)^{\frac{2}{3}}\left(\frac{a}{0.5\textrm{pc}}\right)^{\frac{2}{3}}.\qquad$
This approximation is good for $D_{l}\gg a$ and $D_{ls}\gg a$. The relative
error is $\sim 10^{-2}$ for $a=0.5$pc and $D_{l}=D_{ls}=10$ Mpc.
In the especially strong-field regime, where the winding number $n$ becomes
$n\geq 1$, we can easily check that $a\simeq b$ or $k_{n}\simeq 1$ satisfies
the transcendental equation (16) in numerical calculations. Physically this
means that the light rays which wind around the wormhole nearly on the photon
sphere make the relativistic Einstein rings Bozza_2002 ;
Perlick_2004_Phys_Rev_D . Then the diameter angles of the relativistic
Einstein rings are approximately given by
$\displaystyle\theta_{n\geq 1}$ $\displaystyle\simeq$
$\displaystyle\frac{a}{D_{l}}$ (19) $\displaystyle\simeq$ $\displaystyle
1.0\times
10^{-2}\,\textrm{arcsecond}\left(\frac{10\textrm{Mpc}}{D_{l}}\right)\left(\frac{a}{0.5\textrm{pc}}\right).\qquad$
Regardless of the values of $D_{ls}$, $D_{l}$ and $a$, the relative error of
the above approximation to the direct numerical solution of the outermost
relativistic Einstein ring ($n=1$) is $\sim 10^{-3}$ and those of the other
relativistic Einstein rings ($n\geq 2$) are smaller than $10^{-5}$. This
implies that it is difficult to resolve each relativistic Einstein ring
separately.
Thus, we conclude that there is one Einstein-ring image and countably infinite
relativistic Einstein-ring images, the latter of which accumulate to form the
apparently single ring image of the throat with the diameter $a/D_{l}$. This
conclusion does not depend on the value of $\eta$.
If we are given the distance $D_{s}$ to the source from the observer, the
distance $D_{l}$ to the lens from the observer and the radius $\theta_{0}$ of
the Einstein ring, we can determine the radius of the throat $a$ from Eq.
(18). Then, we can use $\theta_{n\geq 1}$ (19) to test the assumption that the
lens object is a wormhole.
From Eqs. (18) and (19) we obtain the relation between $\theta_{0}$ and
$\theta_{n\geq 1}$ by
$\displaystyle\theta_{n\geq
1}\simeq\left(\frac{4}{\pi}\frac{D_{s}}{D_{ls}}\right)^{\frac{1}{2}}\theta_{0}^{\frac{3}{2}}.$
(20)
This relation generally holds in astrophysical situations, as long as $a\ll
D_{l}$ and $a\ll D_{ls}$ are satisfied. If the lens is identified with a
wormhole, we can even estimate the radius of the wormhole throat in terms of
$D_{s}$, $\theta_{0}$ and $\theta_{n}$ through
$a\simeq\theta_{n\geq 1}\left(1-\frac{4}{\pi}\theta_{n\geq
1}^{-2}\theta_{0}^{3}\right)D_{s}.$ (21)
This follows from Eqs. (19) and (20).
## IV IV. Comparison between wormholes and black holes
In this section we compare the Einstein ring and the relativistic Einstein
rings in the Ellis spacetime to the ones in the Schwarzschild spacetime and
show that we can distinguish between black holes and wormholes.
Now, we will briefly review the deflection angle, Einstein ring and
relativistic Einstein rings for the Schwarzschild spacetime
Bisnovatyi_Tsupko_2008 ; Bozza_Capozziello_Iovane_Scarpetta_2001 ; Muller_2008
; Virbhadra_Ellis_2000 ; Virbhadra_Ellis_2002 ; Hasse_Perlick_2001 and
present the relation between $\theta_{0}$ and $\theta_{n\geq 1}$.
In the weak-field regime $b\gg r_{g}$, where $r_{g}=2GM$ is the Schwarzschild
radius of the black hole of mass $M$, the deflection angle is approximately
given by
$\alpha\simeq\frac{2r_{g}}{b}.$ (22)
The diameter angle of the Einstein ring is given by
$\displaystyle\theta_{0}$
$\displaystyle\simeq\sqrt{\frac{2D_{ls}}{D_{l}D_{s}}r_{g}}$ (23)
$\displaystyle\simeq
2.0\,\textrm{arcsecond}\left(\frac{D_{ls}}{10\textrm{Mpc}}\right)^{\frac{1}{2}}\left(\frac{M}{10^{10}M_{\odot}}\right)^{\frac{1}{2}}\left(\frac{10\textrm{Mpc}}{D_{l}}\right)^{\frac{1}{2}}\left(\frac{20\textrm{Mpc}}{D_{s}}\right)^{\frac{1}{2}}.\qquad$
We can determine $r_{g}$ in the same way as the radius of the throat $a$.
In the especially strong-field regime, where the winding number $n$ becomes
$n\geq 1$, the impact parameter $b$ that satisfies the lens equation should be
nearly the critical impact parameter $b\simeq(3\sqrt{3}/2)r_{g}$ (see
Bozza_Capozziello_Iovane_Scarpetta_2001 ; Bozza_2002 ; Virbhadra_Ellis_2000 ).
Then the diameter angles of the inseparable relativistic Einstein rings
$\theta_{n\geq 1}$ are given by
$\displaystyle\theta_{n\geq 1}$ $\displaystyle\simeq$
$\displaystyle\frac{3\sqrt{3}}{2}\frac{r_{g}}{D_{l}}$ (24)
$\displaystyle\simeq$ $\displaystyle 5.1\times
10^{-5}\,\textrm{arcsecond}\left(\frac{M}{10^{10}M_{\odot}}\right)\left(\frac{10\textrm{Mpc}}{D_{l}}\right).\qquad$
It is useful to remember that the leading term of the deflection angle in the
weak-field regime is the second order of the small amount $a/b$ on the Ellis
geometry (13), while it is the first order of the small amount $r_{g}/b$ on
the Schwarzschild geometry (22). So the relation between $\theta_{0}$ and
$\theta_{n\geq 1}$ for the Schwarzschild spacetime
$\displaystyle\theta_{n\geq
1}\simeq\frac{3\sqrt{3}}{4}\frac{D_{s}}{D_{ls}}\theta_{0}^{2}$ (25)
is different from that on the Ellis spacetime (20). Figure. 2 shows the angle
of the relativistic Einstein ring $\theta_{n\geq 1}$ versus the angle of the
Einstein ring $\theta_{0}$ for $D_{l}=D_{ls}=10$Mpc. Thus, we can distinguish
between black holes and wormholes in principle if we are given $D_{s}$,
$D_{l}$, $\theta_{0}$ and $\theta_{n\geq 1}$.
Figure 2: The angle of the relativistic Einstein ring $\theta_{n\geq 1}$
versus the angle of the Einstein ring $\theta_{0}$ for $D_{l}=D_{ls}=10$Mpc.
The broken (green) and solid (red) lines plot the cases where the lens objects
are a wormhole and a black hole, respectively.
If the lens is identified with a black hole, we can estimate the Schwarzschild
radius and,hence, the black hole mass by
$r_{g}\simeq\frac{2}{3\sqrt{3}}\theta_{n\geq
1}\left(1-\frac{3\sqrt{3}}{4}\theta_{0}^{2}\theta_{n\geq 1}^{-1}\right)D_{s}$
(26)
in terms of $\theta_{0}$, $\theta_{n\geq 1}$ and $D_{s}$. This follows from
Eqs. (24) and (25).
## V V. Discussion and conclusion
It is well known that the qualitative features of the gravitational lensing on
the Ellis spacetime are very similar to the ones on the Schwarzschild
spacetime for their photon spheres and their asymptotic flatness
Perlick_2004_Phys_Rev_D ; Nandi_Zhang_Zakharov_2006 ; Tejeiro_Larranaga_2005 .
However, we realize that their quantitative features are very different due to
their different weak-field behaviors.
We consider the experimental situation where we know the separation $D_{s}$
between the observer and the source and the separation $D_{l}$ between the
observer and the lens. We assume that we do not know whether the lens object
is a black bole or a wormhole and do not know its parameter, i.e., the mass
$M$ or the radius $a$ of the throat in advance.
We need at least two observable quantities to determine whether the lens
object is a black hole or wormhole since the lens system has one parameter in
this situation. First, we observe an Einstein ring and determine the parameter
for both possibilities. Second, we observe relativistic Einstein rings and
tell the wormhole from the black hole. If the predicted relativistic ring
angles by the black hole and by the wormhole were of similar size, we could
not discern the difference. However, Eqs. (20), (25) and Fig. 2 show that we
do not confuse them.
We conclude that we can detect the relativistic Einstein rings by wormholes
which have $a\simeq 0.5$pc at a galactic center with the distance
$D_{l}=D_{ls}=10$Mpc and which have $a\simeq 10$AU in our galaxy with the
distance $D_{l}=D_{ls}=10$kpc using the most powerful modern instruments which
have the resolution of $10^{-2}$arcsecond such as a $10$-meter optical-
infrared telescope. Note that the corresponding black holes which have the
Einstein rings of the same size are galactic supermassive black holes with
$10^{10}M_{\odot}$ and $10^{7}M_{\odot}$, respectively, and that the
relativistic Einstein rings by these black holes are too small to measure with
the current technology.
In fact, our results imply that we can distinguish between slowly rotating
Kerr-Newmann black holes and the Ellis wormholes with their Einstein-ring
systems. This is because the leading term of the deflection angle for the
lensing by the Kerr-Newmann black holes in the weak-field regime is equal to
the one for the lensing by the Schwarzschild black holes, while the black hole
charge and small spin only slightly change the radii of the relativistic
Einstein rings Bozza_2002 ; Eiroa_Romero_Torres_2002 ; Sereno_2004 ;
Bozza_2003 ; Sereno_Luca_2006 . Moreover, this also suggests that it is much
more challenging to determine the charge and/or small spin of black holes than
to distinguish between black holes and the Ellis wormholes.
We assumed that the observer, the lensing object and the source object are
directly-aligned, though such a configuration is fairly rare. In general the
strong gravitational lensing effect is observed as broken-ring images which
are called relativistic images Virbhadra_Ellis_2000 . Therefore, more
realistic problem is to size the relativistic images. Our result suggests that
we can distinguish black holes and wormholes by using the relativistic images.
To observe the relativistic images is one of the challenging works with many
difficulties. Bozza et al. pointed out that relativistic images are always
very faint with respect to the weak field images
Bozza_Capozziello_Iovane_Scarpetta_2001 . The Very Large Telescope
Interferometer (VLTI) has high resolution Delplancke_Gorski_Richichi_2001 ;
Eckart_Bertram_et_al_2003 but it will not work because of this demagnifying
effect.
We also assumed point-like sources, although astrophysical sources have their
own size. If the source object is a galaxy, it may conceal the relativistic
Einstein rings, especially, in the case that the lens object is a black hole.
Testing some hypotheses of astrophysical wormholes by using the relativistic
Einstein rings and the Einstein ring is left as future work.
Tejeiro and Larranaga Tejeiro_Larranaga_2005 investigated the gravitational
lensing effect of the wormhole solution obtained by connecting the Ellis
solution as an interior region and the Schwarzschild solution as an exterior
region Lemos_Lobo_Quinet_2003 . They concluded that we cannot distinguish the
Schwarzschild black hole and the wormhole unless the discontinuity of the
magnification curve at the boundary is observed. This does not contradict our
results because their wormhole solution behaves as the Schwarzschild solution
in the weak-field regime.
Acknowledgements:
The authors would like to thank U. Miyamoto, H. Asada and F. Abe for valuable
comments and discussion. The work of N. T. and K. Y. was supported in part by
Rikkyo University Special Fund for Research. T. H. would also thank the
Astronomy Unit, Queen Mary, University of London for its hospitality. T. H.
was supported by the Grant-in-Aid for Young Scientists (B) (No. 21740190) and
the Grant-in-Aid for Challenging Exploratory Research (No. 23654082) for
Scientific Research Fund of the Ministry of Education, Culture, Sports,
Science and Technology, Japan. N. T. thanks the Yukawa Institute for
Theoretical Physics at Kyoto University, where this work progressed during the
YITP workshop YITP-W-12-08 on ”Summer School on Astronomy & Astrophysics
2012”.
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|
arxiv-papers
| 2012-06-30T04:55:30 |
2024-09-04T02:49:32.420636
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Naoki Tsukamoto, Tomohiro Harada and Kohji Yajima",
"submitter": "Naoki Tsukamoto",
"url": "https://arxiv.org/abs/1207.0047"
}
|
1207.0052
|
# The Complexity of Learning Principles and Parameters Grammars
Jacob Andreas
###### Abstract
We investigate models for learning the class of context-free and context-
sensitive languages (CFLs and CSLs). We begin with a brief discussion of some
early hardness results which show that unrestricted language learning is
impossible, and unrestricted CFL learning is computationally infeasible; we
then briefly survey the literature on algorithms for learning restricted
subclasses of the CFLs. Finally, we introduce a new family of subclasses, the
principled parametric context-free grammars (and a corresponding family of
principled parametric context-sensitive grammars), which roughly model the
“Principles and Parameters” framework in psycholinguistics. We present three
hardness results: first, that the PPCFGs are not efficiently learnable given
equivalence and membership oracles, second, that the PPCFGs are not
efficiently learnable from positive presentations unless
$\mathsf{P}=\mathsf{NP}$, and third, that the PPCSGs are not efficiently
learnable from positive presentations unless integer factorization is in P.
## 1 Introduction
A great deal modern psycholinguistics has concerned itself with resolving the
problem of the so-called “poverty of the stimulus”—the claim that natural
languages are unlearnable given only the data available to infants, and
consequently that some part of syntax must be “native” (i.e. prespecified)
rather than learned. Gold’s theorem (described below), which states that there
exists a superfinite class of languages which is not learnable in the limit
from positive presentations, is often offered as proof of this fact (though
the extent to which the theorem is psycholinguistically informative remains a
contentious issue). [gor90]
But how is innate linguistic knowledge represented? One mechanism usually
offered is the Chomskian “Principles and Parameters” framework [cho93], which
suggests that there is a set of universal principles of grammar which inhere
in the structure of the brain. In this framework, the process of language
learning simply consists of determining appropriate settings for a finite
number of parameters which determine how those principles are applied.
While this problem is generally supposed to be easier than unrestricted
language learning, we are not aware of any previous work specifically aimed at
studying the Principles and Parameters model in a computational setting. In
this report, we introduce a family of subclasses of the context-free languages
which we believe roughly captures the intuition behind the Principles and
Parameters model, and explore the difficulty of learning that model in various
learning environments.
We begin by presenting an extremely brief survey of the existing literature on
the hardness of language learning; we then introduce three hardness results,
one unconditional, one complexity-theoretic and one cryptographic, which
suggest that the existence of a generalized algorithm for learning in the
principles and parameters framework is highly unlikely. While we obviously
cannot produce any psychologically definitive results in this setting, we at
least hope to challenge the notion that the Principles and Parameters
framework is somehow a computationally satisfying explanation of the language
learning process.
## 2 Background
### 2.1 Definitions
#### 2.1.1 Learnability in the limit
Gold defines the language learning problem as follows: [gol67]
###### Definition 1.
Given a class of languages $\mathcal{L}$ and an algorithm $A$, we say $A$
identifies $\mathcal{L}$ in the limit from positive presentations if $\forall
L$, $\forall i_{1},i_{2},\cdots\in L$, there is a time $t$ such that for all
$u>t$, $h_{u}=h_{t}=A(i_{1},i_{2},\cdots,i_{t})$.
#### 2.1.2 Exact identification using queries
Modeling the language learning process as being entirely dependent on positive
examples seems rather extreme; it’s useful to consider enviornments in which
the learner has access to a richer representation of the language. Angluin
[ang90] describes a model of language learnability from oracle queries, as
follows:
###### Definition 2.
An equivalence oracle for a language $L$ takes as input the representation of
a language $r(L)$ and outputs “true” if $L=L^{*}$, or some $w\in L\Delta
L^{*}$ (the symmetric difference of the languages) otherwise. There is an
obvious equivalence, first pointed out by Littlestone [lit88], between the
equivalence query model and the online mistake bound model.
###### Definition 3.
A membership oracle for a language $L$ with start symbol $S$ takes a string
$w$, and outputs true if $S\Rightarrow^{*}w$ and false otherwise.
###### Definition 4.
A nonterminal membership oracle for a language $L$ takes a string $w$ (not
necessarily in $L$) and a nonterminal $A$, and outputs whether
$A\Rightarrow^{*}w$ (i.e. whether the set of possible derivations with $A$ as
a start symbol includes $w$).
###### Definition 5.
A class of languages $\mathcal{L}$ is learnable from an equivalence oracle (or
analogously from an equivalence oracle and a membership oracle, sometimes
referred to as a “minimal adequate teacher”) if there exists a learning
algorithm with runtime polynomial in the size of the representation of the
class and length of the longest counterexample.
### 2.2 Hardness of language learning
###### Theorem (Gold).
There exists a class of languages not learnable in the limit from positive
presentations.
###### Proof sketch.
Construct an infinite sequence of languages $L_{1}\subset L_{2}\subset\cdots$,
all finite, and let $L_{\infty}=\bigcup_{i}L_{i}$. Suppose there existed some
algorithm $A$ that could identify each $L_{i}$ from positive presentations.
Then there is a positive presentation of $L_{\infty}$ that causes $A$ to make
an infinite number of mistakes. First present a set of examples, all in
$L_{1}$, that force $A$ to identify $L_{1}$. Then present a set of examples
forcing it to identify $L_{2}$, then $L_{3}$, and so on. An infinite number of
mistakes can be forced in this way, so $L_{\infty}$ is not learnable in the
limit.
While space does not permit us to discuss the proof here, we also note the
following important result for CFL learning:
###### Theorem (Angluin [ang80]).
There exists a class of context-free languages with “natural” representations
which are not learnable from equivalence queries in time polynomial in the
size of the representation.
### 2.3 Learnable subclasses of the CFLs
While this last result rules out the possibility of a general algorithm for
learning CFLs, subsets of the CFLs have been shown to be learnable when given
slightly more powerful oracles. These include simple deterministic languages
[ish90], one-counter languages [ber87] and so-called very simple languages
[yok91]. Particularly heartening is Angluin’s result that $k$-bounded CFGs can
be learned in polynomial time if nonterminal membership queries are permitted
[ang87].
## 3 Principled Parametric Grammars
We now introduce a formal model of the “principles and parameters” framework
described in the introduction.
### 3.1 Motivation
Before moving on to the details of the construction, it’s useful to consider a
few example “principles” and “parameters” suggested by proponents of the
model.
* •
The pro-drop parameter: does this language allow pronoun dropping? If PNP is a
non-terminal symbol designating a pronoun, this parameter determines whether
or not a rule of the form $\mathtt{PNP}\rightarrow\varepsilon$ exists in the
language.
* •
The ergative/nominative parameter: ergative languages distinguish between
transitive and intransitive senses of verb by marking the subject, while
nominative languages (like English) mark the object. Let $\mathtt{NP}$ and
$\mathtt{VP}$ be non-terminal symbols for noun and verb phrases respectively,
and let $\mathtt{NP_{trans}}$ and $\mathtt{VP_{trans}}$ be distinguished
versions of those symbols for ergative/nominative marking. Now, any language
with Verb-Subject-Object order, there will be a rule
$\mathtt{S}\rightarrow\mathtt{NP}\ \mathtt{VP}$. In an ergative language,
there is additionally a rule of the form
$\mathtt{S}\rightarrow\mathtt{NP_{trans}}\ \mathtt{VP}$, and in a nominative
language a rule of the form $\mathtt{S}\rightarrow\mathtt{NP}\
\mathtt{VP_{trans}}$.
In each of these cases, a pattern holds: for every possible possible parameter
setting, there is some finite set of context-free productions in the native
grammar, from which only one must be selected as the element of the learned
grammar. This leads very naturally to the following development of principled
parametric context-free grammars as a model of the principles and parameters
model.
### 3.2 Construction
###### Definition 6.
An $\mathbf{n}$-principled, $\mathbf{k}$-parametric context-free grammar
($(n,k)$-PPCFG) $\Gamma$ is a 4-tuple $(V,\Sigma,\Pi,S)$, where:
1. 1.
$V$ is a finite alphabet of nonterminal symbols
2. 2.
$\Sigma$ is a finite alphabet of terminal symbols
3. 3.
$\Pi$ is a set of $n$ production groups of the form
$(A_{i,1}\rightarrow\alpha_{i,1}),\cdots,(A_{i,j}\rightarrow\alpha_{i,j}),\cdots,(A_{i,k}\rightarrow\alpha_{i,k})$
where each $\alpha\in(V\cup\Sigma)^{*}$, i.e. is a finite sequence of
terminals and nonterminals. Let $\Pi_{i,j}$ denote the production
$(A_{i,j}\rightarrow\alpha_{i,j})$.
4. 4.
$S\in V$ is the start symbol.
###### Definition 7.
A parameter setting $p=(p_{1},p_{2},\cdots,p_{n})$ is a sequence of length
$n$, with each $p_{i}\in 1..k$. Then define $\Gamma_{p}$ to be the ordinary
context-free grammar ($V$, $\Sigma$, $R$, $S$) with $R=\\{\Pi_{i,p_{i}}:i\in
1..n\\}$.
As usual, let $L(G)$ denote the context free language represented by the CFG
$G$. Then let $\Lambda(\Gamma)=\\{L(G):\exists p:G=\Gamma_{p}\\}$.
###### Definition 8.
An algorithm $A$ learns the PPCFGs from an equivalence oracle if $\forall$
PPCFGs $\Gamma$ and languages $l\in\Gamma$, after a finite number of oracle
queries, $A$ outputs some $p$ such that $L(\Gamma_{p})=l$, or determines that
no such $p$ exists.
###### Definition 9.
$A$ efficiently learns the PPCFGs from an equivalence oracle if the number of
oracle queries it makes is bounded by some polynomial function $poly(n,k)$.
###### Definition 10.
Finally, a principled parametric context-sensitive grammar is defined exactly
as above, with corresponding learning definitions, but with context-sensitive
productions in each production group.
### 3.3 Equivalence
Some useful facts about the PPCFGs:
###### Observation.
A “heterogeneous PPCFG” with a variable number of right hand sides can be
transformed into a “homogeneous PPCFG” of the kind described above by
“padding” out the shorter principles with duplicate rules (i.e. to insert an
unambiguous production $A\rightarrow\alpha$ into an $(n,2)$-PPCFG, add to
$\Pi$ the production group $(A\rightarrow\alpha),(A\rightarrow\alpha)$).
###### Observation.
A $(n,k)$-PPCFG can be converted into an $(n(k-1),2)$-PPCFG as follows:
replace each principle
$A\rightarrow(\alpha_{1},\alpha_{2},\dots,\alpha_{k})$
with a set of principles
$\displaystyle A_{1}$ $\displaystyle\rightarrow(\alpha_{1},A_{2})$
$\displaystyle A_{2}$ $\displaystyle\rightarrow(\alpha_{2},A_{3})$
$\displaystyle\vdots$ $\displaystyle A_{k-1}$
$\displaystyle\rightarrow(\alpha_{k-1},\alpha_{k})$
Thus without loss of generality we may treat every PPCFG as a $(n,2)$-PPCFG.
The conversion above results in only a polynomial increase in the number of
principles, so any algorithm which is polynomial in $n$, and which assumes
$k=2$, can be used to solve $k>2$ with only a polynomial increase in running
time. This also means that we may specify an individual language in a PPCFG by
a bit string of length $n$.
Finally, note that a $k$-PPCFG with $n$ rules contains at most $k^{n}$
languages.
## 4 Generic hardness results for PPCFGs
We will construct a minimal adequate teacher $T$ consisting of two oracles
$EQ$ (an equivalence oracle) and $M$ (a membership oracle), such that any
algorithm $A$ requires an exponential number of queries to identify the
correct parameter setting $p$ from a PPCFG $\Gamma$.
###### Theorem 1.
Without condition, there exists no algorithm $A$ capable of learning the
PPCFGs from equivalence queries and membership queries in polynomial time.
###### Proof.
Fix some number $N$. Construct the PPCFG $\Gamma$ with
$\displaystyle V$ $\displaystyle=X_{i}:i\in 1..N$ $\displaystyle S$
$\displaystyle=\mathtt{START}$ $\displaystyle\Sigma$ $\displaystyle=\\{0,1\\}$
and $\Pi$ as defined as follows:
$\displaystyle(\mathtt{START}\rightarrow X_{1}X_{2}\cdots X_{N})$
$\displaystyle(X_{k}\rightarrow 0,X_{k}\rightarrow 1)$ $\displaystyle\forall
k\in 1..N$
Every parameter setting $p$ in this grammar allows it to derive precisely 1
string: every production is deterministic. Consequently, the $N$ possible
settings of the grammar derive $2^{N}$ unique strings. Given some algorithm
$A$ for learning PPCFGs, the procedure specified below describes an
adversarial distinguisher for this PPCFG which forces the learner to make a
total of $2^{N}-1$ queries.
$i\leftarrow 0$
while $i<2^{N}-1$ do
on query $EQ(\Gamma^{\prime})$
if $\Gamma^{\prime}$ has not been previously queried then
$i\leftarrow i+1$
end if
return FALSE, $L(\Gamma^{\prime})$ $\triangleright$ $L(\Gamma^{\prime})$
contains only one string
end query
on query $M(w)$
if $w$ has not been previously queried then
$i\leftarrow i+1$
end if
return FALSE
end query
end while
on query
return TRUE $\triangleright$ Only one language is consistent with the evidence
end query
After each query, the number of grammars still possible given the evidence
provided so far decreases by precisely 1 (because each grammar is capable of
producing only string), so after $2^{N}-1$ queries of either kind, the oracle
must output true.
Thus, only after $2^{N}-1$ queries (superpolynomial in $|\Gamma|$ and the
length of the longest production) can the learner halt, so the grammar is not
efficiently learnable from membership and equivalence queries.
## 5 Complexity-theoretic hardness results for
PPCFGs
We will construct a reduction from 3SAT to PPCFG learning. Let $X=\\{x_{i}\\}$
be a set of variables and $C=\\{c_{i}\\}$ be a set of clauses. Let us write
$x_{j}\in c_{i}$ if the $j$th $i$th clause is satisfied by the $j$th variable,
and $\bar{x}_{j}\in c_{i}$ if the $i$th clause is satisfied by the negation of
the $j$th variable.
Then construct the PPCFG $\Gamma$ with
$V=X\cup\\{\mathtt{START}\\},\Sigma=C,S=\mathtt{START}$, and $\Pi$ with the
following production groups:
$\displaystyle(\mathtt{START}\rightarrow x_{1}x_{2}\cdots x_{n})$
$\displaystyle(x_{i}\rightarrow x_{i,T}),(x_{1}\rightarrow x_{i,F})$
$\displaystyle\forall x_{i}\in X$ $\displaystyle(x_{i}\rightarrow\varepsilon)$
$\displaystyle\forall x_{i}\in X$ $\displaystyle(x_{j,T}\rightarrow c_{i})$
$\displaystyle\forall c_{i}\in C,\forall x_{j}\in c_{i}$
$\displaystyle(x_{j,F}\rightarrow c_{i})$ $\displaystyle\forall c_{i}\in
C,\forall\bar{x}_{j}\in c_{i}$
Note that only for production groups of the form $(x_{i}\rightarrow
x_{i,T}),(x_{1}\rightarrow x_{i,F})$ does the parameter setting change the
resulting language. These groups may be thought of as assigning truth values
to the variables.
###### Proposition 1.
If there exists some $l\in\Gamma$ such that $\forall c_{i}\in C:c_{i}\in l$,
then the 3SAT instance is satisfiable.
###### Proof.
Set $x_{i}$ true if the rule $x_{i}\rightarrow x_{i,T}$ is chosen, and false
otherwise. For any $c_{i}$ in the language, there is a derivation from
$\mathtt{START}\Rightarrow^{*}c_{i}$ of the following form:
$\displaystyle\mathtt{START}$ $\displaystyle\Rightarrow x_{1}x_{2}\cdots
x_{n}$ $\displaystyle\Rightarrow x_{j}$ $\displaystyle\Rightarrow x_{j,a}$
$\displaystyle\Rightarrow c_{i}$
Then $x_{j}$ satisfies $c_{i}$.
###### Proposition 2.
If the 3SAT instance is satisfiable, there exists some $l\in\Gamma$ such that
$\forall c_{i}\in C:c_{i}\in l$.
###### Proof.
Choose the rule $(x_{i}\rightarrow x_{i,T})$ if $x_{i}$ is set true in the
satisfying assignment, and $(x_{i}\rightarrow x_{i,F})$ if $x_{i}$ is set
false. These settings determine $l$. Then, consider any string $c_{i}$. There
is some variable $x_{j}$ with truth value $a$ which satisfies the
corresponding clause; then by assignment $l$ contains a production of the form
$x_{j}\rightarrow x_{j,a}$, and by definition contains a production of the
form $x_{j,a}\rightarrow c_{i}$, so derivation identical to the one in the
previous proposition must exist.
###### Theorem 2.
If $\mathsf{P}\neq\mathsf{NP}$, no efficient algorithm exists for learning
PPCFGs from positive presentations.
###### Proof.
Assume that there exists some algorithm $A$ which efficiently learns the
PPCFGs from positive presentations. We will use $A$ to construct a SAT solver
$S$ by simulating the oracle. Construct $\Gamma$ from the SAT instance as
described above. Then $S$’s interaction with $A$ takes the following form:
By assumption, after observing polynomially many positive presentations, and
performing polynomially many computations, $A$ outputs a parameter setting $p$
which produces every $c_{i}\in C$, or a signal indicating no such assignment
exists. From Propositions 1 and 2, such a $p$ exists if and only if the SAT
instance is satisfiable. Thus $S$ determines in a polynomial number of steps
whether the SAT instance is satisfiable, and the existence of $A$ implies
$\mathsf{P}=\mathsf{NP}$.
## 6 Cryptographic hardness results for PPCSGs
We will construct another reduction, this time from integer factorization to
PPCSG learning. Let $N$ be a product of two $(n-1)$-digit primes.
Let $A$ be a set of non-terminal symbols $A_{0}..A_{\lceil\lg\sqrt{N}\rceil}$,
and $B,C,Z$ be similar sets of nonterminals of cardinality $\lceil\lg
N\rceil+1$. Then construct the PPCSG $\Gamma$ with
$\displaystyle V$ $\displaystyle=A\cup B\cup C\cup Z\cup\\{\mathtt{S}\\}$
$\displaystyle\Sigma$ $\displaystyle=\\{c_{k}\\}$ $\displaystyle\forall k\in
0..\lceil\lg N\rceil$ $\displaystyle S$ $\displaystyle=\mathtt{S}$
and $\Pi$ with the following production groups:
$\displaystyle(\mathtt{S}\rightarrow A_{\lceil\lg\sqrt{N}\rceil}\mathtt{S})$
$\displaystyle(\mathtt{S}\rightarrow\varepsilon)$
$\displaystyle(A_{0}\rightarrow B_{0}),(A_{0}\rightarrow\varepsilon)$
$\displaystyle(A_{j}\rightarrow A_{j-1}),(A_{j}\rightarrow B_{j}A_{j-1})$
$\displaystyle\forall j\in 1..\lceil\lg\sqrt{N}\rceil$
$\displaystyle(B_{j}\rightarrow B_{j-1}B_{j-1})$ $\displaystyle\forall j\in
1..\lceil\lg\sqrt{N}\rceil$ $\displaystyle(B_{0}\rightarrow C_{0})$
$\displaystyle(C_{k}C_{k}\rightarrow C_{k}Z_{k})$
$\displaystyle(C_{k}Z_{k}\rightarrow C_{k+1}Z_{k})$ $\displaystyle\forall k\in
0..\lceil\lg{N}\rceil$ $\displaystyle(C_{k+1}Z_{k}\rightarrow C_{k+1})$
$\displaystyle(C_{k}\rightarrow c_{k})$
Intuitively, the parameter settings in this grammar $(A_{j}\rightarrow
A_{j-1}),(A_{j}\rightarrow B_{j}A_{j-1})$ fix some number $m$ between $1$ and
$\sqrt{N}$. Each $A_{\lceil\lg\sqrt{N}\rceil}\Rightarrow^{*}C^{m}$, so
$S\Rightarrow^{*}C^{mk}$ for all $k$, i.e. the unary representation of all
multiples of $m$. This unary string may then be collapsed into a $\lceil\lg
N\rceil$-ary representation as a string of terminal $c_{i}$s.
Let $l$ be the language consisting of the single string $w$, where $w$ is the
concatenation of every $a_{i}$ such that the $i$th digit of the binary
representation of $N$ is 1.
Given a parameter setting $s$ for $\Gamma$, for each production group
$(A_{j}\rightarrow A_{j-1}),$ $(A_{j}\rightarrow B_{j}A_{j-1})$ in $s$, let
$p_{i}=0$ if the first setting is chosen and $1$ if the second setting is
chosen. Let $P_{s}$ be the number whose binary representation is given by the
$p_{i}$s. Alternatively, given a binary number $P$ let $s_{P}$ be the
parameter setting induced by $P$’s bits.
Finally, some notation: given a sequence of strings $S$, let
$\bigparallel_{s\in S}s_{i}$ denote the concatenation of all $s_{i}$s.
###### Proposition 3.
Given numbers $P$ and $Q$, $P\leq Q$, if $PQ=N$ then $w\in L(\Gamma_{s_{P}})$.
###### Proof.
In $\Gamma_{s_{T}}$,
$\displaystyle\mathtt{S}$ $\displaystyle\Rightarrow^{*}S^{Q}$
$\displaystyle\Rightarrow^{*}(A_{\lceil\lg\sqrt{N}\rceil})^{Q}$
$\displaystyle\Rightarrow^{*}\left(\bigparallel_{\begin{subarray}{c}0\leq
i\leq\lceil\lg\sqrt{N}\rceil\\\ T_{i}=1\end{subarray}}B_{i}\right)^{Q}$
$\displaystyle\Rightarrow^{*}B_{0}^{PQ}=B_{0}^{N}$
$\displaystyle\Rightarrow^{*}C_{0}^{N}$ $\displaystyle\Rightarrow^{*}w$
###### Proposition 4.
If $w\in L(\Gamma_{s})$, then there exists $Q$ such that $P_{s}Q=N$.
###### Proof.
Certainly if $w\in L(\Gamma_{s})$, $C_{0}^{N}\Rightarrow^{*}w$. But
$\mathtt{S}\Rightarrow^{*}C_{0}^{P_{s}k}$ for all $k$ (using the derivation in
Proposition 3); then there exists some $Q$ such that $PQ=N$.
###### Theorem 3.
If integer factorization is hard, no efficient algorithm exists for learning
random PPCSGs with non-negligible probability from an equivalence oracle.
###### Proof.
Assume that there exists some algorithm $A$ which, given $\Gamma$ and the
positive presentation of the single string $w$ as specified above, outputs a
parameter setting $P$ for $\Gamma$ such that $w\in L(\Gamma_{P})$ with non-
negligible probability a polynomial number of computations. Then we will
construct a factorizer $F$ that decomposes $N$ into $P$ and $Q$.
From the preceding conjectures, if an acceptable $P$ is found then $PQ=N$, for
some $Q$, so if $A$ can find a parameter setting in polynomial time then this
algorithm finds a factorization in polynomial time.
This final proof is neither particularly interesting or satisfying: even the
task of finding a derivation in a CSG is known to be PSPACE-complete (though
it’s easy to see that a polynomial-time parsing algorithm for this particular
family of grammars exists). Note that the only context-sensitive production
groups employed in this production are used to guarantee a compact encoding of
$w$; we suspect that there is an alternative way of constructing this “grammar
arithmetic” that requires only weaker rules, perhaps mildly context-sensitive
or even context-free. We thus close with the following:
###### Open Problem.
If integer factorization is hard, does there exist a polynomial-time algorithm
for learning random PPCFGs with non-negligible probability from positive
presentations?
## 7 Conclusion
We have introduced a new model, the princpled parametric context-free (also
context-sensitive) grammars as a model of the “Principles and Parameters”
model in psycholinguistics, and presented three hardness-of-learning results
for the class of PPCFGs and PPCSGs. While these results certainly do not
demonstrate definitively that learning under the Principles and Parameters
framework is completely impossible (all that is required for human language
learning to be possible is that one PPCFG be efficiently learnable), we have
shown that there is likely no generic algorithm for learning a class of PPCFGs
given either oracle and membership queries or a positive presentation. In
general, these results prove that even radically restricting the class of
candidate grammars does not guarantee a successful outcome when attempting to
learn CFGs and CSGs.
## References
* [ang80] Angluin, Dana. “Inductive inference of formal languages from positive data.” Info. Contr. 1980, Vol. 45, No. 2.
* [ang87] Angluin, Dana. “Learning $k$-bounded context-free grammars.” Yale Technical Report RR-557, 1987.
* [ang90] Angluin, Dana. “Negative Results for Equivalence Queries.” In Machine Learning 1990, Vol. 2, No. 2.
* [ber87] Berman, Piotr, and Roos, Robert. “Learning one-counter languages in polynomial time.” In Foundations of Computer Science, 1987\.
* [cho93] Chomsky, Noam and Lasnik, Howard. “Principles and Parameters Theory.” In Syntax: An International Handbook of Contemporary Research, Berlin: De Gruyter.
* [gol67] Gold, E. Mark. “Language Identification in the Limit.” In Information and Control 1967, Vol. 10, No. 5.
* [gor90] Gordon, Peter. “Learnability and Feedback.” In Developmental Psychology 1990, Vol. 26, No. 2.
* [ish90] Ishizawa, Hiroki. “Polynomial Time Learnability of Simple Deterministic Languages.” In Machine Learning 1990, Vol. 5, No. 2.
* [lit88] Littlestone, Nick. “Learning Quickly When Irrelevant Attributes Abound.” In Machine Learning 1988, Vol. 2, No. 1.
* [yok91] Yokomori, Takashi. “Polynomial Time Learning of Very Simple Grammars from Positive Data.” In proceedings of COLT, 1991.
|
arxiv-papers
| 2012-06-30T06:36:04 |
2024-09-04T02:49:32.428190
|
{
"license": "Public Domain",
"authors": "Jacob Andreas",
"submitter": "Jacob Andreas",
"url": "https://arxiv.org/abs/1207.0052"
}
|
1207.0107
|
www.math.univ-toulouse.fr/ sauloy/ Le $q$-analogue du groupe fondamental
sauvage et le problème inverse de la théorie de Galois aux $q$-différences
# The $q$-analogue of the wild fundamental group and the inverse problem of
the Galois theory of $q$-difference equations
Jean-Pierre Ramis Institut de France (Académie des Sciences) and Institut de
Mathématiques, CNRS UMR 5219, Équipe Émile Picard, U.F.R. M.I.G., Université
Paul Sabatier (Toulouse 3), 31062 Toulouse CEDEX 9 ramis.jean-
pierre@wanadoo.fr Jacques Sauloy Institut de Mathématiques, CNRS UMR 5219,
Équipe Émile Picard, U.F.R. M.I.G., Université Paul Sabatier (Toulouse 3),
31062 Toulouse CEDEX 9 sauloy@math.univ-toulouse.fr
###### Abstract
In [RS1, RS2], we defined $q$-analogues of alien derivations for linear
analytic $q$-difference equations with integral slopes and proved a density
theorem (in the Galois group) and a freeness theorem. In this paper, we
completely describe the wild fundamental group and apply this result to the
inverse problem in $q$-difference Galois theory.
###### Key words and phrases:
q-difference equations, Stokes phenomenon, alien derivations, Galois theory,
inverse problem
###### 1991 Mathematics Subject Classification:
39A13. Secondary: 34M50
|
arxiv-papers
| 2012-06-30T15:14:45 |
2024-09-04T02:49:32.435413
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jean-Pierre Ramis and Jacques Sauloy",
"submitter": "Jacques Sauloy",
"url": "https://arxiv.org/abs/1207.0107"
}
|
1207.0174
|
eurm10 msam10
# Aspect ratio dependence of heat transport by turbulent Rayleigh-Bénard
convection in rectangular cells
Quan ZHOU Email address for correspondence: qzhou@shu.edu.cn Bo-Fang LIU
Chun-Mei LI and Bao-Chang ZHONG Shanghai Institute of Applied Mathematics and
Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering,
Shanghai University, Shanghai 200072, China
(?? and in revised form ??)
###### Abstract
We report high-precision measurements of the Nusselt number $Nu$ as a function
of the Rayleigh number $Ra$ in water-filled rectangular Rayleigh-Bénard
convection cells. The horizontal length $L$ and width $W$ of the cells are
50.0 cm and 15.0 cm, respectively, and the heights $H=49.9$, 25.0, 12.5, 6.9,
3.5, and 2.4 cm, corresponding to the aspect ratios $(\Gamma_{x}\equiv
L/H,\Gamma_{y}\equiv W/H)=(1,0.3)$, $(2,0.6)$, $(4,1.2)$, $(7.3,2.2)$,
$(14.3,4.3)$, and $(20.8,6.3)$. The measurements were carried out over the
Rayleigh number range $6\times 10^{5}\lesssim Ra\lesssim 10^{11}$ and the
Prandtl number range $5.2\lesssim Pr\lesssim 7$. Our results show that for
rectangular geometry turbulent heat transport is independent of the cells’
aspect ratios and hence is insensitive to the nature and structures of the
large-scale mean flows of the system. This is slightly different from the
observations in cylindrical cells where $Nu$ is found to be in general a
decreasing function of $\Gamma$, at least for $\Gamma=1$ and larger. Such a
difference is probably a manifestation of the finite plate conductivity
effect. Corrections for the influence of the finite conductivity of the top
and bottom plates are made to obtain the estimates of $Nu_{\infty}$ for plates
with perfect conductivity. The local scaling exponents $\beta_{l}$ of
$Nu_{\infty}\sim Ra^{\beta_{l}}$ are calculated and found to increase from
0.243 at $Ra\simeq 9\times 10^{5}$ to 0.327 at $Ra\simeq 4\times 10^{10}$.
## 1 Introduction
Convection is ubiquitous in nature and in our everyday life. It can be found
in the stars and planets, in the Earth’s mantle and outer core, in the oceans
and atmosphere, as well as in heat transport and mass mixing in many
engineering applications. The paradigmatic example for natural convection is
the Rayleigh-Bénard (RB) convection of an enclosed fluid layer between the
colder top and the warmer bottom plates (Ahlers, Grossmann $\&$ Lohse 2009 _b_
; Lohse $\&$ Xia 2010). A key issue in the study of turbulent RB convection is
to understand how heat is transported upwards across the fluid layer by
convective flows. The global heat transport by convection is usually expressed
in terms of the Nusselt number, namely,
$Nu=\frac{QH}{\lambda_{f}\Delta},$ (1)
where $Q$ is the heat current density across a fluid layer of thermal
conductivity $\lambda_{f}$ with height $H$ and with an applied temperature
difference $\Delta$. The dynamics of the system is determined by the
geometrical configuration of the convection cell and by two dimensionless
control parameters, i.e. the Rayleigh number and the Prandtl number,
$Ra=\frac{\alpha g\Delta H^{3}}{\nu\kappa}\mbox{\ \ and\ \
}Pr=\frac{\nu}{\kappa}.$ (2)
Here, $g$ is the acceleration due to gravity and $\alpha$, $\nu$, and $\kappa$
are the isobaric thermal expansion coefficient, the kinematic viscosity, and
the thermal diffusivity of the working fluid, respectively. The cell geometry
is usually described in terms of one or more aspect ratios, such as
$\Gamma\equiv D/H$ for a cylindrical cell of inner diameter $D$ and
$(\Gamma_{x}\equiv L/H,\Gamma_{y}\equiv W/H)$ for a rectangular cell of
horizontal length $L$ and width $W$. The $Ra$\- and $Pr$-dependence of $Nu$
for various working fluids and cell geometries have been studied, both
experimentally and numerically, in great detail for many years (Castaing _et
al._ 1989; Kerr 1996; Chavanne _et al._ 1997, 2001; Du $\&$ Tong 2000; Kerr
$\&$ Herring 2000; Niemela _et al._ 2000; Ahlers $\&$ Xu 2001; Roche _et al._
2002, 2005; Xia, Lam $\&$ Zhou 2002; Verzicco $\&$ Camussi 2003; Niemela $\&$
Sreenivasan 2003; Shishkina $\&$ Wagner 2007; Ahlers, Funfschilling $\&$
Bodenschatz 2009 _a_ ; Funfschilling, Bodenschatz $\&$ Ahlers 2009; Song $\&$
Tong 2010; Stevens, Verzicco $\&$ Lohse 2010; Stevens, Lohse $\&$ Verzicco
2011 _a_ ; Silano, Sreenivasan $\&$ Verzicco 2010; He _et al._ 2012). In
addition, various theoretical models have been advanced to predict the
behavior of convective heat transport Castaing et al. (1989); Shraiman &
Siggia (1990); Grossmann & Lohse (2000, 2001, 2003, 2004, 2011); Dubrulle
(2001, 2002). For more detailed elucidation of the problem, we refer
interested readers to the recent review paper by Ahlers et al. (2009b). On the
other hand, there are fewer measurements focusing on the aspect-ratio-
dependence (Xu, Bajaj $\&$ Ahlers 2000; Fleischer $\&$ Goldstein 2002; Cheung
2004; Nikolaenko _et al._ 2005; Funfschilling _et al._ 2005; Sun _et al._ 2005
_a_ ; Niemela $\&$ Sreenivasan 2006; Roche _et al._ 2010), and those
measurements were all performed in containers with a cylindrical geometry. The
objective of the present experimental investigation is to fill this gap by
making high-precision measurements of $Nu$ over a wide range of the aspect
ratio in convection cells with a rectangular geometry.
A lateral sidewall is indispensable for any convection experiment in the
laboratory. The interaction between the sidewall and fluids would change the
velocity and temperature distributions in the cell, and in turn change the
flow structures of the system. Indeed, previous experimental studies for both
cylindrical (du Puits, Resagk $\&$ Thess 2007) and rectangular (Xia, Sun $\&$
Cheung 2008) cells have shown that with increasing the aspect ratio the large-
scale circulation (LSC) departs from a single-roll structure and becomes a
multi-roll pattern. Thus, the $\Gamma$-dependence of $Nu$ may reflect the
influence of flow structures on heat transport characteristics via the
influence on the boundary layers. Measurements in cylindrical samples using
water as working fluid ($Pr\approx 4.3$) revealed that for $\Gamma\lesssim 6$
$Nu$ decreases, albeit only a few percent, with increasing $\Gamma$
Funfschilling et al. (2005); Sun et al. (2005a). This suggests that the global
heat transport properties of the RB system are not very sensitive to the flow
structures for such parameter ranges. However, the situation is very different
for lower $Pr$ and for the two-dimensional (2D) case. Using 3D direct
numerical simulation (DNS), Bailon-Cuba, Emran $\&$ Schumacher (2010) found
for $Pr=0.7$ that the minimum of $Nu$ occurs at the aspect ratio where the LSC
undergoes a transition from a single-roll to a double-roll structure and the
variations in $Nu$ for different $\Gamma$ are significant and can yield
$11.3\%$ for $Ra=10^{8}$. For 2D steady-state calculations, Ching & Tam (2006)
obtained a power-law of $Nu\sim\Gamma^{-1}$ for $\Gamma\leqslant 3$. In 2D
numerical RB flow, van der Poel, Stevens $\&$ Lohse (2011) identified
different turbulent states for both $Pr=0.7$ and 4.3, corresponding to
different roll structures and associated with different overall heat
transfers. Transitions among these states thus lead to jumps and sharp
transitions in $Nu(\Gamma)$. By connecting the structures of $Nu(\Gamma)$ to
the way the flow organizes itself in the sample, van der Poel et al. (2011)
explained why the aspect-ratio dependence of $Nu$ is more pronounced for small
$Pr$. Compared with the 3D cases, we note that the 2D simulations show larger
variations in $Nu(\Gamma)$ for both $Pr=0.7$ and 4.3, which may be explained
by the different flow structures formed in the 2D and 3D cases.
Different geometrical shapes represent different symmetries, and hence may
lead to different features of the flow and heat transfer. This prompts us to
study the $\Gamma$-dependence of $Nu$ in a non-cylindrical system. In the
present study, we choose rectangle as the shape of the cells, which has been
widely used in the past (Xia, Sun $\&$ Zhou 2003; Gasteuil _et al._ 2007;
Maystrenko, Resagk $\&$ Thess 2007; Zhou $\&$ Xia 2008, 2010 _a_ ,_b_ ; Zhou
_et al._ 2007, 2010). It was found that the convective flows in cells with
such geometry share some similar dynamics as those in 2D RB system, such as
the reversals of the LSC Sugiyama et al. (2010).
The remainder of this paper is organized as follows. We give detailed
descriptions of the experimental apparatus and conditions in $\S$2\.
Experimental results are presented and analyzed in $\S$3, which is divided
into two parts. In $\S$3.1, we compare the measured $Nu$ for different aspect
ratios and with those obtained in cylindrical cells. In $\S$3.2 we consider
the finite conductivity corrections of the top and bottom plates and estimate
$Nu_{\infty}$ for perfectly conducting plates. The scaling behaviors of
$Nu_{\infty}$ are also discussed in $\S$3.2. We summarize our findings and
conclude in $\S$4.
## 2 Experimental apparatus and methods
Figure 1: Schematic diagram of front view of a rectangular cell. A: the
Plexiglas cover; B: the copper top plate; Ci and Ci’ ($i=1,2,3,4$): nozzles
connecting the channels to the refrigerated circulators; D: thermistors; E:
the Plexiglas sidewall; F: the copper bottom plate; G: the copper cover for
the bottom plate, H1: nozzle for transferring fluid into the cell; H2: nozzle
for letting air out of the cell.
Figure 1 is a schematic drawing of the front view of our apparatus and the
drawing corresponds to $(\Gamma_{x},\Gamma_{y})=(1,0.3)$. The sidewall of the
cell, indicated as E in the figure, is composed by four transparent Plexiglas
plates of 1.2 cm in thickness. The inner length $L$ and inner width $W$ of the
cell are 50 cm and 15 cm, respectively. Six sidewalls of heights $H=49.9$,
25.0, 12.5, 6.9, 3.5, and 2.4 cm were used in the experiment. The
corresponding aspect ratios are $(\Gamma_{x}\equiv L/H,\Gamma_{y}\equiv
W/H)=(1,0.3)$, $(2,0.6)$, $(4,1.2)$, $(7.3,2.2)$, $(14.3,4.3)$, and
$(20.8,6.3)$, respectively. As $\Gamma_{y}$ is proportional to $\Gamma_{x}$,
in the remainder of the paper we use only $\Gamma_{x}$ to indicate the cell’s
aspect ratio for ease of presentation. The top (B) and bottom (F) plates are
made of pure copper of 56 cm in length and 21 cm in width and their fluid-
contact surfaces are electroplated with a thin layer of nickel to prevent the
oxidation by water. The thickness of the top plate is 3.5 cm and that of the
bottom plate is 1.5 cm. Silicon O-rings are placed between the copper plates
and the sidewall plates to avoid fluid leakage. Eight stainless steel posts
(not shown) hold the top and bottom plates together. They are insulated from
the plates by Teflon sleeves and washers. Four parallel channels (not shown)
of 1.5 cm in width and 2 cm in depth are machined into the top plate and the
separation between adjacent channels is 1.5 cm. The channels start and end,
respectively, at the two diagonal ends of the long edges. A silicon rubber
sheet (not shown) and a Plexiglas plate (A) of 1.4 cm in thickness are fixed
on the top to form the cover and also to prevent interflow between the
adjacent channels. At two ends of the $i^{th}$ channel ($i=1,2,3,4$), there
are two nozzles (Ci and Ci’) located, through which the channel is connected
to a separate refrigerated circulator (Polyscience 9712) that has a
temperature stability of 0.01 ∘C. The channels and the circulators are
connected such that the incoming cooler fluid and the outgoing warmer fluid in
adjacent channels always flow in opposite directions. To provide constant and
uniform heating, two rectangular Kapton film heaters of 25 cm in length and 15
cm in width are sandwiched between two copper plates (F and G) and are
connected in parallel to a d.c. power supply (SGI 330X15D) with $99.99\%$
long-term stability. Therefore, the experiments were conducted under constant
heating of the bottom plate while maintaining a constant temperature at the
top plate. Note that recent high-resolution 2D Johnston & Doering (2009) and
3D Stevens et al. (2011a) simulations have revealed that turbulent thermal
convection with boundary conditions of constant temperature and constant heat
flux display identical heat transport at sufficient high Rayleigh numbers.
Figure 2: The normalized horizontal temperature differences,
$(T_{i}-T_{m})/\Delta$, for the top (_a_ , _c_) and bottom (_b_ , _d_) plates
and for $\Gamma_{x}=1$ (_a_ , _b_) and 7.3 (_c_ , _d_). The index $i$
($=1,2,3,4,5$) of the thermistor is listed in figure 1 (see the top plate B
and the bottom plate F).
Degassed water was used as the convecting fluid and the cell was leveled to
better than $0.1^{\circ}$. During the measurements the entire cell was wrapped
with several layers of Styrofoam. The temperature of each conducting plate was
measured by five thermistors (D), which are embedded uniformly beneath the
fluid-contact surface of the respective plate. When calculating the
temperature difference $\Delta$ between the bottom and top plates, a
correction was made for the temperature change between the thermistor position
and the fluid-contact surface. In each measurement after $Ra$ was changed it
took about 4 to 8 hours for the system to reach the steady state and we waited
for at least 12 hours to start the measurements. A typical measurement lasted
over 12 hours and more than 24 hours for low-$\Delta$ experiments ($\Delta<4$
∘C). No long-term drift of the mean temperature in the plates was observed
over the duration of the measurement and the standard deviations were less
than $0.5\%$ of $\Delta$ for all measurements.
To see the temperature uniformity of each conducting plate, we plot in figure
2 the normalized temperature variation, $(T_{i}-T_{m})/\Delta$, for both the
top and bottom plates. Here, $T_{i}$ ($i=1,2,3,4,5$) is the time-averaged
temperature measured by the $i^{th}$ thermistor in a given plate and $T_{m}$
is the mean value of all the five thermistors in the same plate. It is found
that the $\Gamma_{x}=1$ cell has the largest variation of plate temperature
and we plot the $\Gamma_{x}=1$ results in figures 2(_a_) and (_b_). The
temperature variation for the other five $\Gamma_{x}$ are similar and we
choose $\Gamma_{x}=7.3$ as an example and plot the results in figures 2(_c_)
and (_d_). One sees that $(T_{i}-T_{m})/\Delta$ is at most $4\%$ for
$\Gamma_{x}=1$ and is less than $2\%$ for all $\Gamma_{x}=7.3$ measurements.
We thus estimate that a systematic error of the order of $1\%$ could be
introduced into the measured $Nu$ by this horizontal temperature inhomogeneity
of the conducting plates. Several thermistors were placed around the cell
sidewall and around the bottom plate to monitor the environment temperature,
based on which heat leakages to the environment and conduction by the posts
and Plexiglas sidewall were calculated. The relative leakages to the total
heat current were found to become more significant with decreasing $\Delta$
and thus we kept heat leakages from all sources to be less than $7\%$ of the
total applied heat current by working with sufficiently large $\Delta$. We
found that the largest source of leakage, especially for the small aspect-
ratio cells, is through the cell sidewall which is insulated by multi-layers
of Styrofoam. The errors in calculating the leaks come mainly from
uncertainties in the thermal conductivities of the material involved, which
are estimated to be less than $20\%$. This translates into an uncertainty of
less than $1.5\%$ in the results of $Nu$.
## 3 Results and discussion
### 3.1 $Nu$ vs. $Ra$ for different $\Gamma_{x}$
Figure 3: (color online). Compensated $Nu/Ra^{1/3}$ on a linear scale vs. $Ra$ on a logarithmic scale. $\bigtriangledown$, $\Gamma_{x}=20.8$; $\bigcirc$, $\Gamma_{x}=14.3$; $\bigtriangleup$, $\Gamma_{x}=7.3$; $\square$, $\Gamma_{x}=4$; $\vartriangleleft$, $\Gamma_{x}=2$; $\vartriangleright$, $\Gamma_{x}=1$ and $Pr\approx 7.01$; $\lozenge$, $\Gamma_{x}=1$ and $Pr\approx 5.45$. Note that the data are as measured, without the correction for the finite plate conductivity. Table 1: Experimental parameters and results for the $\Gamma_{x}=1$ cell. Here, $Nu_{\infty}$ is calculated using Eq. (3) and the parameters $a=0.275$ and $b=0.39$ obtained by Brown et al. (2005) in cylindrical cells of two sets of plates of different thermal conductivities (Cu and Al). It is also seen that $\alpha\Delta\lesssim 0.01$ for all cases, which is considered to be sufficiently small. Note that two points are listed per line and the data are listed in chronological order. $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$ | $|$ | $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$
---|---|---|---|---|---|---|---|---|---|---|---|---
| | | | | ($10^{-3}$) | $|$ | | | | | | ($10^{-3}$)
$\Gamma_{x}=1$ | | | | | | $|$ | | | | | |
$3.44$ | $1.06\times 10^{10}$ | $5.45$ | $128.8$ | $130.3$ | $1.04$ | $|$ | $5.24$ | $1.63\times 10^{10}$ | $5.41$ | $147.4$ | $149.5$ | $1.59$
$15.16$ | $4.73\times 10^{10}$ | $5.40$ | $205.5$ | $210.6$ | $4.62$ | $|$ | $18.33$ | $5.57\times 10^{10}$ | $5.48$ | $216.6$ | $222.4$ | $5.49$
$24.36$ | $7.71\times 10^{10}$ | $5.40$ | $239.8$ | $247.2$ | $7.42$ | $|$ | $9.97$ | $3.03\times 10^{10}$ | $5.47$ | $178.3$ | $181.9$ | $2.99$
$12.65$ | $3.77\times 10^{10}$ | $5.52$ | $191.9$ | $196.2$ | $3.74$ | $|$ | $7.52$ | $2.30\times 10^{10}$ | $5.46$ | $163.5$ | $166.4$ | $2.27$
$6.41$ | $1.96\times 10^{10}$ | $5.46$ | $155.4$ | $157.9$ | $1.93$ | $|$ | $8.74$ | $2.66\times 10^{10}$ | $5.47$ | $170.4$ | $173.6$ | $2.63$
$30.93$ | $9.27\times 10^{10}$ | $5.51$ | $255.3$ | $264.0$ | $9.19$ | $|$ | $5.30$ | $9.25\times 10^{9}$ | $7.01$ | $121.9$ | $123.1$ | $1.09$
$7.02$ | $1.22\times 10^{10}$ | $7.02$ | $133.1$ | $134.6$ | $1.44$ | $|$ | $3.20$ | $5.56\times 10^{9}$ | $7.02$ | $105.3$ | $106.1$ | $0.66$
$11.98$ | $2.11\times 10^{10}$ | $6.98$ | $160.2$ | $162.8$ | $2.49$ | $|$ | $4.23$ | $7.34\times 10^{9}$ | $7.03$ | $114.6$ | $115.6$ | $0.87$
$15.10$ | $2.64\times 10^{10}$ | $7.01$ | $171.6$ | $174.7$ | $3.12$ | $|$ | $2.63$ | $4.60\times 10^{9}$ | $7.00$ | $99.3$ | $100.0$ | $0.54$
$21.96$ | $3.87\times 10^{10}$ | $6.99$ | $193.1$ | $197.3$ | $4.56$ | $|$ | $3.02$ | $5.26\times 10^{9}$ | $7.02$ | $102.5$ | $103.2$ | $0.62$
$31.40$ | $5.55\times 10^{10}$ | $6.98$ | $216.9$ | $222.5$ | $6.54$ | $|$ | $18.52$ | $3.23\times 10^{10}$ | $7.02$ | $182.0$ | $185.6$ | $3.82$
Table 2: Experimental results for the $\Gamma_{x}=2$, 4, 7.3, 14.3, and 20.8 cells. Note that two points are listed per line and the data are listed in chronological order. $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$ | $|$ | $\Delta$ (K) | $Ra$ | $Pr$ | $Nu$ | $Nu_{\infty}$ | $\alpha\Delta$
---|---|---|---|---|---|---|---|---|---|---|---|---
| | | | | ($10^{-3}$) | $|$ | | | | | | ($10^{-3}$)
$\Gamma_{x}=2$ | | | | | | $|$ | | | | | |
$17.18$ | $6.67\times 10^{9}$ | $5.26$ | $111.0$ | $114.1$ | $5.41$ | $|$ | $9.39$ | $3.64\times 10^{9}$ | $5.26$ | $90.8$ | $92.6$ | $2.95$
$21.63$ | $8.46\times 10^{9}$ | $5.24$ | $119.5$ | $123.2$ | $6.84$ | $|$ | $7.12$ | $2.79\times 10^{9}$ | $5.23$ | $82.5$ | $84.0$ | $2.25$
$33.33$ | $1.28\times 10^{10}$ | $5.28$ | $135.6$ | $140.5$ | $11.0$ | $|$ | $6.02$ | $2.40\times 10^{9}$ | $5.19$ | $78.5$ | $79.7$ | $1.92$
$29.86$ | $1.17\times 10^{10}$ | $5.23$ | $130.7$ | $135.3$ | $9.45$ | $|$ | $14.62$ | $5.69\times 10^{9}$ | $5.25$ | $105.3$ | $107.9$ | $4.61$
$5.17$ | $2.01\times 10^{9}$ | $5.25$ | $74.7$ | $75.8$ | $1.63$ | $|$ | $27.15$ | $1.08\times 10^{10}$ | $5.18$ | $129.3$ | $133.8$ | $8.37$
$14.94$ | $5.85\times 10^{9}$ | $5.25$ | $106.1$ | $108.9$ | $4.57$ | $|$ | $3.82$ | $1.56\times 10^{9}$ | $5.23$ | $69.2$ | $70.1$ | $1.21$
$\Gamma_{x}=4$ | | | | | | $|$ | | | | | |
$17.02$ | $8.90\times 10^{8}$ | $5.25$ | $57.4$ | $59.1$ | $5.36$ | $|$ | $1.60$ | $8.33\times 10^{7}$ | $5.25$ | $29.1$ | $29.4$ | $0.50$
$35.83$ | $1.88\times 10^{9}$ | $5.25$ | $73.8$ | $76.8$ | $11.3$ | $|$ | $2.26$ | $1.18\times 10^{8}$ | $5.25$ | $31.9$ | $32.3$ | $0.71$
$32.61$ | $1.68\times 10^{9}$ | $5.29$ | $70.7$ | $73.4$ | $10.2$ | $|$ | $3.03$ | $1.59\times 10^{8}$ | $5.24$ | $34.9$ | $35.3$ | $0.96$
$29.21$ | $1.52\times 10^{9}$ | $5.26$ | $68.0$ | $70.5$ | $9.18$ | $|$ | $5.10$ | $2.67\times 10^{8}$ | $5.26$ | $40.2$ | $40.9$ | $1.61$
$25.98$ | $1.36\times 10^{9}$ | $5.25$ | $66.0$ | $68.4$ | $8.19$ | $|$ | $6.93$ | $3.62\times 10^{8}$ | $5.25$ | $43.9$ | $44.7$ | $2.18$
$22.96$ | $1.19\times 10^{9}$ | $5.27$ | $63.2$ | $65.3$ | $7.20$ | $|$ | $9.16$ | $4.83\times 10^{8}$ | $5.23$ | $47.8$ | $48.8$ | $2.90$
$19.87$ | $1.04\times 10^{9}$ | $5.24$ | $60.4$ | $62.3$ | $6.28$ | $|$ | $11.66$ | $6.06\times 10^{8}$ | $5.27$ | $51.1$ | $52.3$ | $3.66$
$14.25$ | $7.44\times 10^{8}$ | $5.26$ | $54.5$ | $56.0$ | $4.49$ | $|$ | $3.96$ | $2.07\times 10^{8}$ | $5.25$ | $37.3$ | $37.9$ | $1.25$
$1.87$ | $9.78\times 10^{7}$ | $5.26$ | $30.4$ | $30.7$ | $0.59$ | $|$ | $5.88$ | $3.08\times 10^{8}$ | $5.25$ | $41.8$ | $42.5$ | $1.85$
$3.44$ | $1.79\times 10^{8}$ | $5.26$ | $35.9$ | $36.4$ | $1.08$ | | | | | | |
$\Gamma_{x}=7.3$ | | | | | | $|$ | | | | | |
$16.16$ | $1.42\times 10^{8}$ | $5.24$ | $34.0$ | $35.1$ | $5.10$ | $|$ | $2.91$ | $2.57\times 10^{7}$ | $5.25$ | $21.1$ | $21.4$ | $0.92$
$13.53$ | $1.20\times 10^{8}$ | $5.24$ | $32.1$ | $33.1$ | $4.27$ | $|$ | $4.63$ | $4.08\times 10^{7}$ | $5.25$ | $23.6$ | $24.1$ | $1.46$
$22.21$ | $1.96\times 10^{8}$ | $5.25$ | $36.6$ | $37.9$ | $7.01$ | $|$ | $2.18$ | $1.92\times 10^{7}$ | $5.25$ | $19.4$ | $19.6$ | $0.69$
$9.87$ | $8.69\times 10^{7}$ | $5.25$ | $29.2$ | $30.0$ | $3.11$ | $|$ | $1.75$ | $1.54\times 10^{7}$ | $5.25$ | $18.6$ | $18.9$ | $0.55$
$6.55$ | $5.74\times 10^{7}$ | $5.26$ | $26.1$ | $26.7$ | $2.06$ | $|$ | $3.74$ | $3.31\times 10^{7}$ | $5.24$ | $22.3$ | $22.7$ | $1.18$
$1.51$ | $1.32\times 10^{7}$ | $5.26$ | $17.8$ | $18.0$ | $0.47$ | $|$ | $25.14$ | $2.21\times 10^{8}$ | $5.26$ | $38.4$ | $39.9$ | $7.91$
$8.36$ | $7.37\times 10^{7}$ | $5.25$ | $27.5$ | $28.2$ | $2.64$ | $|$ | $28.43$ | $2.49\times 10^{8}$ | $5.27$ | $39.5$ | $41.1$ | $8.93$
$2.04$ | $1.80\times 10^{7}$ | $5.25$ | $19.1$ | $19.3$ | $0.64$ | $|$ | $19.31$ | $1.69\times 10^{8}$ | $5.26$ | $35.0$ | $36.1$ | $6.07$
$\Gamma_{x}=14.3$ | | | | | | | | | | | |
$12.35$ | $1.42\times 10^{7}$ | $5.25$ | $17.8$ | $18.4$ | $3.89$ | $|$ | $2.65$ | $3.03\times 10^{6}$ | $5.26$ | $11.6$ | $11.8$ | $0.83$
$8.96$ | $1.03\times 10^{7}$ | $5.26$ | $16.3$ | $16.8$ | $2.82$ | $|$ | $1.61$ | $1.84\times 10^{6}$ | $5.26$ | $10.6$ | $10.7$ | $0.51$
$5.98$ | $6.84\times 10^{6}$ | $5.27$ | $14.5$ | $14.9$ | $1.88$ | $|$ | $14.95$ | $1.71\times 10^{7}$ | $5.26$ | $18.6$ | $19.3$ | $4.70$
$2.00$ | $2.29\times 10^{6}$ | $5.26$ | $10.9$ | $11.1$ | $0.63$ | $|$ | $18.96$ | $2.19\times 10^{7}$ | $5.24$ | $19.8$ | $20.6$ | $5.99$
$3.44$ | $3.94\times 10^{6}$ | $5.26$ | $12.4$ | $12.6$ | $1.08$ | $|$ | $23.45$ | $2.66\times 10^{7}$ | $5.28$ | $20.8$ | $21.7$ | $7.34$
$4.28$ | $4.90\times 10^{6}$ | $5.26$ | $13.0$ | $13.3$ | $1.35$ | $|$ | $26.52$ | $3.02\times 10^{7}$ | $5.27$ | $21.5$ | $22.4$ | $8.32$
$1.39$ | $1.59\times 10^{6}$ | $5.26$ | $10.1$ | $10.3$ | $0.44$ | | | | | | |
$\Gamma_{x}=20.8$ | | | | | | | | | | | |
$11.75$ | $4.34\times 10^{6}$ | $5.26$ | $12.9$ | $13.4$ | $3.70$ | $|$ | $3.19$ | $1.18\times 10^{6}$ | $5.26$ | $9.1$ | $9.3$ | $1.01$
$16.55$ | $6.12\times 10^{6}$ | $5.26$ | $14.2$ | $14.8$ | $5.21$ | $|$ | $5.59$ | $2.07\times 10^{6}$ | $5.26$ | $10.6$ | $10.9$ | $1.76$
$19.15$ | $7.11\times 10^{6}$ | $5.24$ | $14.8$ | $15.4$ | $6.05$ | $|$ | $1.85$ | $6.82\times 10^{5}$ | $5.27$ | $8.1$ | $8.3$ | $0.58$
$22.00$ | $8.18\times 10^{6}$ | $5.24$ | $15.3$ | $16.0$ | $6.95$ | $|$ | $3.97$ | $1.47\times 10^{6}$ | $5.26$ | $9.6$ | $9.8$ | $1.25$
$14.18$ | $5.24\times 10^{6}$ | $5.26$ | $13.4$ | $13.9$ | $4.46$ | $|$ | $2.48$ | $9.16\times 10^{5}$ | $5.26$ | $8.7$ | $8.8$ | $0.78$
$7.51$ | $2.81\times 10^{6}$ | $5.23$ | $11.3$ | $11.6$ | $2.38$ | $|$ | $1.71$ | $6.30\times 10^{5}$ | $5.26$ | $8.1$ | $8.2$ | $0.54$
$9.59$ | $3.55\times 10^{6}$ | $5.25$ | $12.0$ | $12.3$ | $3.02$ | $|$ | $25.00$ | $9.30\times 10^{6}$ | $5.24$ | $15.7$ | $16.4$ | $7.90$
$1.62$ | $5.96\times 10^{5}$ | $5.26$ | $7.9$ | $8.1$ | $0.51$ | | | | | | |
The measured $Nu$ with corresponding values of $\Delta$, $Ra$, and $Pr$ are
given in tables 1 and 2. The $\Gamma_{x}=1$ measurements were made at mean
temperatures of 20∘C and 30∘C, corresponding to $Pr=7.01$ and 5.45,
respectively, and the measurements for the other five values of $\Gamma_{x}$
were conducted at 31.3∘C, corresponding to $Pr=5.25$. Previous studies have
revealed that with increasing Prandtl number $Nu$ first increases, reaches its
maximum value at around $Pr\approx 4$ (depends on $Ra$), and then decreases
slightly or remains independent of $Pr$ Ahlers & Xu (2001); Grossmann & Lohse
(2001); Xia et al. (2002); Silano et al. (2010); Stevens et al. (2011a).
Therefore, within the present $Pr$ range, $Nu$ is expected to depend very
weakly on $Pr$. Indeed, as shown in figures 3 and 5, no significant
differences are observed between the $Pr=5.45$ (circles) and 7.01 (right-
triangles) results. Another source of uncertainty in the measured $Nu$ could
be the non-Boussinesq effects, as some of our measurements were made at larger
$\Delta$. Funfschilling et al. (2005) argued that the applied temperature
difference $\Delta$ should be limited to $\lesssim 15^{\circ}$C to strictly
conform to the Boussinesq conditions. However, it can be seen from figure 2
and tables 1 and 2 that some of our data have $\Delta$ much larger than
$15^{\circ}$C. As we shall see below, the measured large-$\Delta$ data show
the same trend as those of small $\Delta$. This suggests that some of our data
being not strictly Boussinesq will not change the main conclusions of the
present work. Indeed, Ahlers et al. (2006) have shown that for water as the
working fluid the non-Boussinesq effects could only slightly reduce the
measured $Nu$ by at most $1.4\%$ for $\Delta=40$ K and thus $Nu$ is rather
insensitive against even significant deviations from the Boussinesq
conditions. Hence, for completeness all data are listed in the tables and are
plotted in the figures.
Figure 3 shows $Nu/Ra^{1/3}$ as a function of $Ra$ for all six $\Gamma_{x}$.
It is seen that data points for $\Gamma_{x}=20.8$ (down-triangles) collapse
well on top of those for $\Gamma_{x}=14.3$ (diamonds) (within their overlap
$Ra-$range), which in turn collapse well on top of those for $\Gamma_{x}=7.3$
(up-triangles). This implies that all sets of data can be described by a
single curve over such a wide range of $\Gamma_{x}$, i.e., no significant
$\Gamma_{x}$-dependence is observed. As discussed in $\S$1,
$\Gamma$-dependence of $Nu$ essentially reflects the influence of flow
structures on heat transport characteristics. Indeed, using particle image
velocimetry (PIV), Xia et al. (2008) have shown in rectangular cells that the
number of the convection rolls depends systematically on the aspect ratio of
the system: only one convection roll is observed in the $\Gamma_{x}=1$ and 2
cells and the LSC breaks into (horizontally arranging) multi-roll structure
for $\Gamma_{x}$ larger than or equal to 4. Therefore, our present results
suggest that for rectangular geometry turbulent heat transport is very
insensitive to the nature and structures of the large-scale mean flows of the
system. We note that our present results are different from those obtained in
2D numerical case van der Poel et al. (2011), which were made for both
$Pr=0.7$ and 4.3. For the 2D simulation, the stable states with $n$ rolls are
found to enable larger heat transfer than those with $n+1$ rolls for
vertically arranging LSC rolls. One possible reason for this difference may be
attributed to different alignments of the LSC rolls, i.e. the aspect ratios of
the 2D simulation vary between 0.4 and 1.25 and the LSC rolls are stacked
vertically, whereas for the present geometry we have horizontally stacked
rolls Xia et al. (2008).
In figure 4, we compare the present results with those obtained in cylindrical
cells and at $Pr\approx 4.4$. We mainly consider two recent data sets: one is
from Funfschilling et al. (2005) (referred to as FBNA) and the other is from
Sun et al. (2005a) (referred to as SRSX). As our measured $Nu$ is independent
of the cells’ aspect ratio, here we do not distinguish our data for different
$\Gamma_{x}$. Note that all data in figure 4 are as measured, without the
correction for the finite plate conductivity. For the FBNA data, one sees that
the $\Gamma=6$ data of FBNA are in excellent agreement with ours, but their
small-$\Gamma$ data are a few percent larger. Nevertheless, we note that the
$\Gamma=1$ data of FBNA (down-triangles) show similar trends to ours for
$Ra\lesssim 10^{9}$ and $Ra\gtrsim 10^{10}$ and thus the small difference
between the two sets of data may be attributed to different system errors. The
SRSX data and ours are much closer. One sees that parts of the $\Gamma=5$ and
10 data of SRSX agree well with ours and the others lie slightly below our
measured $Nu$.
Figure 4: (color online). Comparison among $Nu/Ra^{-1/3}$ from the present
work ($\bigcirc$), from Funfschilling et al. (2005) ($\bigtriangledown$,
$\Gamma=1$; $\times$, $\Gamma=2$; $*$, $\Gamma=3$; $\square$, $\Gamma=6$; for
clarity their $\Gamma=1.5$ data are not shown) and from Sun et al. (2005a)
($\bigtriangleup$, $\Gamma=2$; $\vartriangleright$, $\Gamma=5$;
$\vartriangleleft$, $\Gamma=10$; $\bigcirc$, $\Gamma=20$). As our results show
that for rectangular geometry $Nu/Ra^{-1/3}$ is independent of the cells’
aspect ratio, here we do not distinguish our data for different $\Gamma_{x}$.
Note that all data are as measured, without the correction for the finite
plate conductivity.
One noticeable difference between the present results and those of FBNA and
SRSX is worthy of note. For both FBNA and SRSX the larger-$\Gamma$ results lie
consistently below those of smaller ones, i.e. $Nu$ is generally smaller for
larger $\Gamma$. This is true even for the largest-$\Gamma$ (i.e. $\Gamma=10$
and 20) data of SRSX (see figure 4). Whereas, our measured $Nu$ for all six
values of $\Gamma_{x}$ fall into a single curve, i.e. $Nu$ is essentially
independent of $\Gamma_{x}$ for our results. To understand such a difference,
we note that all data plotted in figure 4 have not been corrected for the
influence of the finite conductivity of the top and bottom plates Verzicco
(2004). In a cylindrical cell of $\Gamma=0.5$, Sun, Xi $\&$ Xia (2005 _b_)
argued that because of the finite conductivity and finite heat capacity of the
plates the azimuthal sweeping of the circulation plane of the LSC would make
heat transfer more efficient than the case when the LSC is locked in a
particular orientation. Moreover, Xi & Xia (2008) studied the azimuthal motion
of the LSC in cylindrical cells of $\Gamma=0.5$, 1.0, and 2.3 and their
results showed that the LSC’s azimuthal motion is more confined in
larger-$\Gamma$ cells. If we generalize the above two findings to large
$\Gamma$ and taken them together, a possible scenario can be achieved: the
larger-$\Gamma$ cell confines the azimuthal sweeping motion of the LSC which
in turn reduces the measured $Nu$. This scenario could be valid for
cylindrical cells due to their azimuthal symmetry. But for rectangular cells,
this is not expected to work because the rectangular geometry has already
locked the orientation of the LSC Xia et al. (2003). Therefore, our present
$\Gamma_{x}$-independent results are consistent with the lack of the azimuthal
sweeping motion of the LSC in rectangular cells. What we should stress is that
the large-$\Gamma$ cell contains multi-roll structures of the LSC du Puits et
al. (2007); Xia et al. (2008); Bailon-cuba et al. (2010) and thus the
azimuthal motion of the LSC and its influence on heat transfer should be more
complicated for large $\Gamma$. However, as we shall see in figure 6, when
corrections for the finite conductivity of the plates are made,
$\Gamma$-dependence of $Nu_{\infty}$ becomes weaker for both the FBNA and SRSX
data, which suggests that the finite plate conductivity effect is indeed a
major factor for the observed difference of the behaviors of heat transfer in
rectangular and cylindrical cells.
Figure 5: (color online). (_a_) Compensated $Nu_{\infty}/Ra^{1/3}$ on a linear
scale vs. $Ra$ on a logarithmic scale: symbols as figure 3. Here,
$Nu_{\infty}$ is calculated using Eq. (3) and the parameters $a=0.275$ and
$b=0.39$ obtained by Brown et al. (2005) in cylindrical cells. For comparison,
the uncorrected data $Nu/Ra^{1/3}$ of figure 3 are replotted in (_b_).
### 3.2 $Nu_{\infty}$ vs. $Ra$ for different $\Gamma_{x}$
Brown et al. (2005) suggested an empirical correction factor
$f(X)=1-exp[-(aX)^{b}]$, namely
$Nu=f(X)Nu_{\infty}=\\{1-exp[-(aX)^{b}]\\}Nu_{\infty}(Ra,Pr),$ (3)
to obtain estimates of the ideal Nusselt number $Nu_{\infty}$ for plates with
perfect conductivity from the measured $Nu$. Here, $X=R_{f}/R_{p}$ is the
ratio of the effective thermal resistance of the working fluid,
$R_{f}=H/(\lambda_{f}Nu)$, to the thermal resistance of the plates,
$R_{p}=e/\lambda_{p}$, $\lambda_{p}$ ($=401$ W/m K) is the conductivity of
plates (Cu), $\lambda_{f}$ ($\lambda_{f}=0.614$ W/m K for $Pr=5.25$ and
$\lambda_{f}=0.589$ W/m K for $Pr=7$) is the conductivity of water, and $e$
($=1.5$ cm) is the mean thickness of the top (the part of the top plate below
the cooling channels is used here) and bottom plates Verzicco (2004). To apply
the relation (3) to our measured $Nu$, one needs to determine the values of
$a$ and $b$. The best way to do this is to use plates of different thermal
conductivities, as was done by Brown et al. (2005), who used two sets of
plates made of Cu and Al. However, the lack of aluminum-plate measurements in
the present study prevents us to determine the values of $a$ and $b$.
Alternatively, to estimate $Nu_{\infty}$, we use the parameters $a=0.275$ and
$b=0.39$ obtained by Brown et al. (2005) in cylindrical samples of 50 cm in
diameter.
Figure 5(_a_) shows the calculated $Nu_{\infty}/Ra^{1/3}$ as a function of
$Ra$. For comparison, the uncorrected data $Nu/Ra^{1/3}$ of figure 3 are
replotted in figure 5(_b_). The corrected data in figure 5(_a_) seem to
display a small aspect ratio dependence, e.g., near $Ra=2\times 10^{8}$ the
$\Gamma_{x}=7.3$ data lie slightly above the $\Gamma_{x}=4$ data and near
$Ra=2\times 10^{9}$ the $\Gamma_{x}=4$ data lie slightly above the
$\Gamma_{x}=2$ data. We note that this dependence is consistent with the FBNA
data (see figure 6, where data points for $\Gamma=6$ lie slightly above those
for $\Gamma=3$, which in turn lie slightly above those for $\Gamma=2$).
Nevertheless, the differences of $Nu_{\infty}$ for the two
adjacent-$\Gamma_{x}$ sets of data are only a few percent, which is smaller
than or comparable with the experimental errors. Hence, we may also conclude
that no significant $\Gamma_{x}$-dependence of heat transfer is observed.
We want to emphasize that here we adopted the parameters of $a=0.275$ and
$b=0.39$ determined by Brown et al. (2005) for cylindrical samples with a
diameter of 50 cm. Brown et al. (2005) have shown experimentally that $a$ and
$b$ are independent of the aspect ratio, but vary with the diameter of the
sample. In the present study, we chose the rectangle as the geometry of the
convection cells, which can possibly have an influence on the finite plate
correction, as we have discussed at the end of $\S$ 3.1. However, without the
measurement performed with aluminum plates it is difficult to assess whether
the geometry would (significantly) influence the values of $a$ and $b$. By
using different sets of $a$ and $b$ to perform the finite conductivity
correction, we estimate that the uncertainties of $a$ and $b$ may yield a few
percent uncertainty on the obtained $Nu_{\infty}$, which is the same order of
the difference between $Nu$ and $Nu_{\infty}$. We should also stress that the
best way to estimate $Nu_{\infty}$ is to use plates with different thermal
conductivities, as was done by Brown et al. (2005) for cylindrical samples.
Therefore, new measurements with Al plates are essential for concretely
settling the problem and this will be the objective of future studies.
Figure 6: (color online). Comparison among $Nu_{\infty}/Ra^{-1/3}$ from the
present work, from Funfschilling et al. (2005), and from Sun et al. (2005a).
Note that the data have been corrected for the finite plate conductivity.
In figure 6 we directly compare our $Nu_{\infty}$ and those of FBNA and SRSX.
It is seen that our data display the similar $Ra$-dependence as the $\Gamma=1$
data of FBNA, and an excellent agreement between our data and those of SRSX
can be found for nearly all the overlap $Ra$ range. As the three sets of data
were taken independently from the samples with very different geometries, this
agreement is just remarkable.
Table 3: Fitted parameters from equations (4). $\Gamma_{x}$ | | 1 | | 2 | | 4 | | 7.3 | | 14.3 | | 20.8
---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{(}\Gamma_{x})$ | | 0.074 | | 0.059 | | 0.108 | | 0.173 | | 0.212 | | 0.242
$\beta(\Gamma_{x})$ | | 0.324 | | 0.335 | | 0.307 | | 0.282 | | 0.271 | | 0.262
Figure 7: (color online). Local scaling exponent $\beta_{l}$ of
$Nu_{\infty}(Ra)$, determined from a power-law fit over a sliding window of
half a decade, as a function of $Ra$ for the present data (solid circles) and
for the $\Gamma=1$ data of Funfschilling et al. (2005) (open triangles).
Dashed line marks $\beta_{l}=1/3$ for reference. Note that both sets of
$\beta_{l}$ were obtained from the corrected data.
Finally, we studied the power-law relation between $Nu_{\infty}$ and $Ra$,
namely,
$Nu_{\infty}=A(\Gamma_{x})Ra^{\beta(\Gamma_{x})}.$ (4)
Table 3 displays the fitted results for the power-law relation (4) at fixed
aspect ratios. One sees that as $\Gamma_{x}$ increases in general the
prefactor $A$ increases and the scaling exponent $\beta$ decreases. Here,
$\Gamma_{x}$-dependences of $A$ and $\beta$ essentially reflect their
$Ra$-dependence as our results do not reveal significant aspect-ratio
dependence of $Nu_{\infty}$. We thus do not distinguish the data for different
$\Gamma_{x}$ and take them together to study the scaling behaviors of
$Nu_{\infty}(Ra)$. The local scaling exponent $\beta_{l}$ is obtained by a
power-law fit, $Nu_{\infty}\sim Ra^{\beta_{l}}$, to the data for
$Nu_{\infty}(Ra)$ within a sliding window that covers half a decade of $Ra$.
Figure 7 shows the results for $\beta_{l}$ as a function of $Ra$. It is seen
that $\beta_{l}$ roughly increases linearly with $\log{Ra}$ from
$\beta_{l}=0.243$ at $Ra\approx 9\times 10^{5}$ to $\beta_{l}=0.342$ at
$Ra\approx 3.6\times 10^{9}$. At small $Ra$, our results differ very much from
the DNS results for cylindrical samples of unit aspect ratio by Wagner,
Shishkina $\&$ Wagner (2012), who found that $\beta_{l}$ increases again as
$Ra$ decreases below $2\times 10^{7}$ and grows to about 0.30 at $Ra=10^{6}$
[see figure 2 of Wagner _et al._ (2012)]. The apparent differences in
$\beta_{l}$ may be explained by the different Prandtl number in the two
studies, i.e., Wagner _et al._ (2012) carried out their simulations at
$Pr=0.786$, while our measurements were made at $Pr\approx 5.45$. For higher
$Ra$, $\beta_{l}$ drops slightly and fluctuates around 0.32. The FBNA results
for $\beta_{l}$ are also displayed together with ours in figure 7 for
comparison. For $Ra\geq 10^{8}$ the two data sets both increase with $Ra$ with
ours being a little scatter. A source of uncertainty for our data could be the
non-Boussinesq effect, i.e. the FBNA data were obtained in the strictly
Boussinesq range, while some of our data are beyond the Boussinesq range (i.e.
$\Delta>15^{\circ}$C). However, as have discussed in $\S$3.1, some of our data
being not strictly Boussinesq will not change our main conclusions. What is
worthy of note is that around $Ra\approx 10^{10}$ our measured $\beta_{l}$ has
a value that is close to the value of 1/3. The exponent $\beta\approx 1/3$ was
obtained before in cylindrical cells of $\Gamma=1$ by FBNA at $Ra\approx
7\times 10^{10}$ (see open triangles in figure 7) and of $\Gamma=4$ by Niemela
& Sreenivasan (2006) for $Ra>10^{10}$. Here, our results in rectangular cells
seem to be qualitatively consistent with these findings
## 4 Conclusion
In conclusion, our high-precision measurements of $Nu$ in rectangular cells
with $(\Gamma_{x},\Gamma_{y})$ varying from $(1,0.3)$ to $(20.8,6.3)$ show
that $Nu$ is independent of the aspect ratio. This is slightly different from
the observations by both FBNA and SRSX in cylindrical cells where $Nu$ is
found to be in general a decreasing function of $\Gamma$, at least for
$\Gamma\sim 1$ and larger. Such a difference may be attributed to different
azimuthal dynamics of the large-scale circulation (LSC) and is probably a
manifestation of the finite plate conductivity effect. To make finite
conductivity corrections, an empirical correction factor
$f(X)=1-exp[-(aX)^{b}]$, together with the parameters $a=0.275$ and $b=0.39$
obtained by Brown et al. (2005) in cylindrical samples, were adopted to
estimate $Nu_{\infty}$ for plates with perfect conductivity from the measured
$Nu$. The obtaiend $Nu_{\infty}$ were found to be consistent with the FBNA and
SRSX data measured in cylindrical samples to only a few percent. The scaling
behaviors between $Nu_{\infty}$ and $Ra$ were studied for all six aspect
ratios. The local scaling exponents $\beta_{l}$ of $Nu_{\infty}\sim
Ra^{\beta_{l}}$ were calculated and found to increase with increasing $Ra$.
Around $Ra\approx 10^{10}$ our measured $\beta_{l}$ has a value that is close
to the value of $1/3$.
###### Acknowledgements.
This work was supported by the Natural Science Foundation of China (Nos.
11161160554, 11002085, and 11032007) and Shanghai Program for Innovative
Research Team in Universities.
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|
arxiv-papers
| 2012-07-01T02:26:58 |
2024-09-04T02:49:32.444504
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Quan Zhou, Bo-Fang Liu, Chun-Mei Li and Bao-Chang Zhong",
"submitter": "Quan Zhou",
"url": "https://arxiv.org/abs/1207.0174"
}
|
1207.0194
|
# BD +36 3317: An Algol Type Eclipsing Binary in Delta Lyrae Cluster
O. Özdarcan orkun.ozdarcan@ege.edu.tr E. Sipahi H. A. Dal Ege University,
Science Faculty, Department of Astronomy and Space Sciences, 35100 Bornova,
İzmir, Turkey
###### Abstract
In this paper, we present standard Johnson $UBV$ photometry of the eclipsing
binary BD +36 3317 which is known as a member of Delta Lyrae (Stephenson 1)
cluster. We determined colors and brightness of the system, calculated
$E(B-V)$ color excess. We discovered that the system shows total eclipse in
secondary minimum. Using this advantage, we found that the primary component
of the system has $B9-A0$ spectral type. Although there is no published
orbital solution, we tried to estimate the physical properties of the system
from simultaneous analysis of $UBV$ light curves with 2003 version of Wilson -
Devinney code. Then we considered photometric solution results together with
evolutionary models and estimated the masses of the components as $M_{1}$ =
2.5 $M_{\odot}$ and $M_{2}$ = 1.6 $M_{\odot}$. Those estimations gave the
distance of the system as 353 pc. Considering the uncertainties in distance
estimation, resulting distance is in agreement with the distance of Delta
Lyrae cluster.
###### keywords:
techniques: photometric - (stars:) binaries: eclipsing - stars: individual:
(BD +36 3317)
††journal: New Astronomy
## 1 Introduction
BD +36 3317 (SAO 67556, $\alpha_{2000}$ = 18h 54m 22s, $\delta_{2000}$ = 36∘
51′ 07′′, $V$ = 8m.77) is an Algol type eclipsing binary which is in the same
area with Delta Lyrae cluster ($\alpha_{2000}$ = 18h 54m, $\delta_{2000}$ =
36∘ 49′, Kharchenko et al. (2005)) in the sky. The system is considered as a
member of the cluster (Eggen, 1972, 1983; Anthony-Twarog, 1984). The existence
of the cluster was suggested by Stephenson (1959) for the first time. Further
photometric evidence for the existence of the cluster was provided by Eggen
(1968). However, possible members of the cluster, included BD +36 3317, have
considerably different distance modulus values (Anthony-Twarog, 1984) and this
makes harder to confirm real members, even the existence of cluster. After
all, mean distance modulus for the cluster is given as 7m.29 with $E(B-V)$ =
0m.04 by Anthony-Twarog (1984). More recent distance modulus value was given
by Kharchenko et al. (2005) as 7m.98 (their $E(B-V)$ value is 0m.04, too) with
the distance of 373 pc. Kharchenko et al. (2004) calculated the Delta Lyrae
membership probability of BD +36 3317 as 0.67, 0.91 and 1.0, in terms of
proper motion, photometry and spatial position, respectively. This calculation
seems to support the membership of BD +36 3317 to the cluster.
Spectroscopic binary nature of the star has been first noticed by Eggen (1968)
via its large radial velocity variation from $-$90 km s-1 to 17 km s-1.
However, eclipses of the system has been discovered by Violat-Bordonau and
Arranz Heras (2008), many years after the discovery of its spectroscopic
binary nature. No orbital solution via radial velocity measurements has been
published up to know. At that point, BD +36 3317 has the advantage of to be an
eclipsing binary in terms of estimating its absolute physical properties and
distance. Furthermore, one can calculate its distance and compare it with the
distance of the cluster and check the membership of the system to the cluster.
For those purposes, we obtained standard Johnson $UBV$ observations of the
system in 2008 and 2009. By using standard photometric data, we calculated
$E(B-V)$ value of the system and estimated their spectral type. Although the
lack of spectroscopic mass ratio and orbital solution is a disadvantage in
terms of determination of the absolute dimension of the system, we made a fair
estimation for physical properties of BD +36 3317, including the distance of
the system, by using the photometric solution. In the next section, we give
summary of our observations. In section 3 we refine the light elements of the
system via $O-C$ analysis. In section 4, we give basic photometric properties
of the system and investigate the effect of interstellar extinction. In
section 5 we present the simultaneous analysis of $UBV$ light curves and our
estimation for the absolute dimension of the system. In the last section we
discuss the results.
## 2 Observations
We carried out Johnson $UBV$ observations of the star at Ege University
Observatory ($EUO$). We observed the star on 41 separate nights in 2008 and
2009. Our instrumental setup was 0.3 m Schmidth-Cassegrain telescope equipped
with uncooled SSP5 photometer. BD +36 3314 was comparison star in our
observations, while BD +36 3313 was check star. In many observing run, when no
primary or secondary minimum occurred, we only performed a short observing
sequence as $S-C2-C1-VVVV-S-C2-C1-VVVV-C1-C2-S$, where $S$ is sky, $C2$ is
check star, $C1$ is comparison star and $V$ is variable star. For those kind
of nights, we used average atmospheric extinction coefficients of $EUO$ in
order to correct all differential magnitudes in terms of $V-C1$, $V-C2$ and
$C2-C1$. For other nights, when a primary or a secondary minimum occurred, we
performed an all night observing run. For those nights, we calculate
atmospheric extinction coefficients via measurements of $C1$ and made all
corrections for atmospheric extinction on differential magnitudes according to
those coefficients. We estimate average standard deviations of observations
from $C2-C1$ measurements and resulting average standard deviations are
0m.053, 0m.019 and 0m.017 for $U$, $B$ and $V$ filters, respectively. We
observed 11 stars from IC 4665 cluster (Menzies and Marang, 1996) on 29th July
2008 and 9 stars from the list of Andruk et al. (1995) on 17th August 2009, in
order to calculate coefficients of the transformation of the instrumental
system to the standard one. By those coefficients, we applied color
corrections for all differential measurements. Then, we directly calculated
average standard magnitudes and colors of the comparison star from those two
nights. Finally, by using standard magnitude and colors of the comparison
star, we calculate standard magnitude and colors of the variable.
## 3 $O-C$ Analysis
It is not possible to make a comprehensive $O-C$ analysis for BD +36 3317
since there is not enough minimum time observations. Only the first ephemeris
of the system is available in literature (Violat-Bordonau and Arranz Heras,
2008) as
$(HJD)_{MinI}=2,454,437.25921+4^{d}.30216\ \times\ E\ .$ (1)
where $(HJD)_{MinI}$ is epoch, which corresponds to a time of a primary
minimum of BD +36 3317, and $E$ is integer eclipse cycle number. We have
already had three primary minima (Type I) and one secondary minimum (Type II)
in our observations (Sipahi et al., 2009). In Table 1, we list those minima
with corresponding $O-C$ values.
Table 1: The times of light minima of BD +36 3317. In the first column, the errors are given for the last digit of the measurements. HJD | E | $O-C$ | Filter | Type
---|---|---|---|---
(24 00000 +) | | (day) | |
54652.3522(4) | 50.0 | $-$0.0025 | $UBV$ | I
54667.4148(5) | 53.5 | 0.0026 | $UBV$ | II
55052.4561(2) | 143.0 | 0.0004 | $UBV$ | I
55078.2683(4) | 149.0 | $-$0.0004 | $UBV$ | I
Application of linear least squares method to primary and secondary minima
data gives very small amount of corrections in the ephemeris. The new light
elements and their errors are as follows:
$(HJD)_{MinI}=2,454,437.2466(30)+4^{d}.302162(27)\ \times\ E\ .$ (2)
For further analysis of light curves, we use those final light elements.
## 4 Basic Photometric Parameters and Interstellar Extinction
We show phased light and color curves of the system in Figure 1.
Figure 1: Phase folded and standardized light and color curves of BD +36 3317.
One can easily notice clear variations in $B-V$ in primary and secondary
minima, which indicate quite large temperature difference among the components
of the system. We determined brightnesses and colors of the system in maximum
light, primary and secondary minima, which would give us some hints about
components. We list those values in Table 2.
Table 2: Magnitude and colors of BD +36 3317. | Max | Min I | Min II
---|---|---|---
| (mag) | (mag) | (mag)
V | 8.798 | 9.715 | 9.053
U-B | -0.108 | -0.090 | -0.119
B-V | 0.066 | 0.167 | 0.009
We can inspect effect of interstellar extinction via $UBV$ color - color
diagram, by considering the colors at maximum light. We used $UBV$ standard
star data from Drilling and Landolt (2000) for this purpose. First, we used
magnitude and colors of maximum light (see $Max$ column in Table 2) by
assuming $E(U-B)/E(B-V)=0.72$ as the slope of the reddening vector in $UBV$
color - color diagram. Resulting $E(B-V)$ and $A_{v}$ values are 0m.139 and
0m.43, respectively, under the assumption of $R$ = 3.1. Then, de-reddened
total color for the system is $(B-V)_{0}$ = -0m.07. Now we can make an
estimation for the temperature of the primary component, by assuming that the
maximum contribution to the total light comes from the primary component.
However, during the light curve analysis, we noticed that orbital inclination
value of the system shows very small variations around 89.4 degrees which
makes the secondary minimum total eclipse. Then, the secondary component is
completely hidden behind the primary component at the middle of the secondary
minimum, hence, the magnitude and colors of that phase corresponds to the
direct measurements of the primary component. Here, we refer the reader to the
next section for justification of
${}^{\prime\prime}total~{}eclipse~{}at~{}secondary~{}minimum^{\prime\prime}$
case. In this case, we have direct measurements of the primary component
(third column in Table 2) and total colors and magnitude of the system (first
column in Table 2), therefore we can easily calculate magnitude and colors of
the secondary component as $V$ = 10m.497, $U-B$ = $-$0m.053 and $B-V$ =
0m.316. This case is also an advantage in terms of more accurate determination
of the interstellar reddening and intrinsic colors of the components
separately. Therefore, we repeated the calculation of the interstellar
reddening as described above, but this time by using the direct measurements
of the primary component at the middle of the secondary minimum. We list
intrinsic colors of the components together with the more accurate
interstellar reddening and extinction estimation in Table 3.
Table 3: Intrinsic colors and magnitude of the components of BD +36 3317 together with the amount of interstellar reddening and extinction. | Primary | Secondary
---|---|---
| (mag) | (mag)
$V_{0}$ | 8.833 | 10.277
$(U-B)_{0}$ | -0.170 | -0.104
$(B-V)_{0}$ | -0.062 | 0.245
$E(B-V)$ | 0.07
$A_{v}$ | 0.22
According to Table 3, colors of the primary component indicates $B9$ spectral
type while secondary components corresponds about $A8$ spectral type.
## 5 Analysis of Light Curves
Under the assumption of total eclipse in secondary minimum, we can estimate
the effective temperature of the primary component directly, which is very
critical for light curve analysis. We adopted calibration of Gray (2005) for
effective temperature estimation of the primary component and resulting
temperature for $(B-V)_{0}$ = $-$0m.062 is $T_{1}$ = 10750 $K$ with the error
of $\sigma_{T_{1}}$ = 470 $K$. The error of the temperature is calculated from
the standard deviation of $B-V$ color in our observations.
Before starting analysis, we chose 0.25 phase as normalisation phase and
converted all magnitude measurements into normalised flux according to the
light level at that phase. For light curve analysis, we used 2003 version of
the Wilson - Devinney code (Wilson and Devinney, 1971; Wilson, 1979, 1990).
Photometric properties of the components gives us hints for some parameters to
reduce the number of free parameters in photometric solution. We adopt the
gravity brightening coefficients $g_{1}$ = $g_{2}$ = 1 and albedos $A_{1}$ =
$A_{2}$ = 1 for stars which have radiative envelopes. We assume synchronised
rotation for both components, so $F_{1}$ = $F_{2}$ = 1. We took the band-pass
dependent ($x_{1,2}$, $y_{1,2}$) and bolometric ($x_{1,2}(bol)$,
$y_{1,2}(bol)$) limb darkening coefficients from van Hamme (1993) by assuming
square root law (Díaz-Cordovés and Giménez, 1992) which is more appropriate
for stars hotter than 8500 $K$. Since there is no radial velocity study in
literature, we do not have any information about mass ratio, which is another
critical parameter for light curve analysis. Although it is not an efficient
way to determine $q$ from photometry in detached systems, we searched for the
best solution for different $q$ values, starting from $q$ = 0.30 until $q$ =
0.90, by using $UBV$ data simultaneously. Orbital inclination ($i$), effective
temperature of the secondary component ($T_{2}$), $\Omega$ potentials of
primary and secondary ($\Omega_{1}$, $\Omega_{2}$) and luminosity of the
primary component ($L_{1}$) are free parameters in the solution. The errors of
the best solutions for individual $q$ values are very close to each other
between $q$ = 0.45 and $q$ = 0.70. At that point, we considered mass -
luminosity relation as $L\propto M^{4}$ by using the resulting absolute
luminosities at the end of the solution (see later results in this section).
We repeated solutions for many $q$ values between $q$ = 0.45 and $q$ = 0.70.
In most cases, $L\propto M^{4}$ relation indicates the $q$ value close to
0.65, hence, we finally accepted $q$ = 0.65 and applied a final light curve
solution. In Table 4, we give final light curve analysis results. We note that
the error of $T_{2}$ is internal to the Wilson - Devinney code and its error
should be similar to the error of $T_{2}$. In the table, $\langle
r_{1,2}\rangle$ denotes average of three fractional radius values ($pole$,
$side$ and $back$ values in solution output) relative to the semi-major axis
of the orbit, for corresponding component. We give the normalised fluxes for
$U$, $B$, and $V$ filters and corresponding theoretical light curves for our
final solution in Figure 2.
Table 4: Light curve analysis results of BD +36 3317. Errors are given in parenthesis. $q(=M{{}_{2}}/M{{}_{1}})$ | 0.65 (fixed)
---|---
$T_{1}(K)$ | 10750 (fixed)
$g_{1}$ = $g_{2}$ | 1.0
$A_{1}$ = $A_{2}$ | 1.0
$F_{1}$ = $F_{2}$ | 1.0
$i~{}(^{\circ})$ | 89.61(11)
$T_{2}(K)$ | 7711 (10)
$\Omega_{1}$ | 10.571 (25)
$\Omega_{2}$ | 8.997 (23)
$L_{1}$/($L_{1}$+$L_{2})_{U}$ | 0.851(17)
$L_{1}$/($L_{1}$+$L_{2})_{B}$ | 0.831(15)
$L_{1}$/($L_{1}$+$L_{2})_{V}$ | 0.789(15)
$x_{1}(bol),x_{2}(bol)$ | 0.558, 0.215
$y_{1}(bol),y_{2}(bol)$ | 0.172, 0.525
$x{{}_{1}},y{{}_{1}}(U)$ | 0.082 , 0.590
$x{{}_{1}},y{{}_{1}}(B)$ | $-$0.058 , 0.846
$x{{}_{1}},y{{}_{1}}(V)$ | $-$0.047 , 0.723
$x{{}_{2}},y{{}_{2}}(U)$ | 0.184 , 0.646
$x{{}_{2}},y{{}_{2}}(B)$ | 0.105 , 0.822
$x{{}_{2}},y{{}_{2}}(V)$ | 0.096 , 0.711
$\langle r_{1}\rangle,\langle r_{2}\rangle$ | 0.1009(3), 0.0832(2)
rms | 0.011
Figure 2: Representation of observational data (points) in terms of normalised
flux and theoretical solution (continuous line).
In Figure 3, we zoom to the primary (left panels) and secondary (right panels)
minima to show the shapes of the eclipses. One can notice that the secondary
minimum is certainly total eclipse which lasts for a short phase range. At the
primary minimum, we observe non-flat bottomed light variation which shows the
effects of annular eclipse and limb darkening together.
Figure 3: Close look to the primary and secondary minima in each filter. In
secondary minimum, one can notice the short total eclipse.
Comparison among photometric solution results, mass - luminosity relation and
evolutionary models of Girardi et al. (2000) indicates that the mass of the
primary $M_{1}$ is close to the 2.5 $M_{\odot}$, therefore we assume $M_{1}$ =
2.5 $M_{\odot}$. This assumption makes the mass of the secondary component
$M_{2}$ = 1.6 $M_{\odot}$ according to the $q$ value.
Since we estimated the masses of the components, we can go one step further
and calculate the absolute parameters and the distance of the system. By the
aid of Kepler’s third law, we can calculate semi-major axis of the orbit $a$.
After that point, one can easily calculate absolute radii of the components by
using average fractional radii in Table 4. Now we have effective temperatures
and absolute radii of the components and we can calculate their luminosities
in solar unit via $Stefan-Boltzmann$ law, by using $T_{\odot}$ = 5770 K. We
list absolute parameters of the system in Table 5. Finally, we can calculate
the distance of the system by using primary and secondary component
separately, via their photometric and absolute properties. In distance
calculation, we adopted bolometric corrections from Gray (2005). Photometric
properties of the primary component leads to a distance of 353 pc. We only use
primary component in order to estimate the distance since its signal is very
strong relative to the secondary component.
Table 5: Absolute Physical Properties of BD +36 3317. Solar $M_{bol}$ value of 4.m74 is used to calculate $M_{bol}$ values of the components. Parameter | Primary | Secondary
---|---|---
Mass $(M_{\odot})$ | 2.5 | 1.6
Radius $(R_{\odot})$ | 1.8 | 1.5
Luminosity $(L_{\odot})$ | 39 | 7
$M_{bol}$ (mag) | 0.76 | 2.62
$log(g)$ (cgs) | 4.32 | 4.31
## 6 Summary and conclusions
We presented standard Johnson $UBV$ photometry of the Algol type eclipsing
binary BD +36 3317 with a fairly good phase coverage and reasonably accurate
observational data. During observations, we obtained three primary and one
secondary minima. We refined the epoch and the period of the system by
applying linear least squares method to the timings of those light minima. We
determined standard colors and magnitude of the system in maximum light,
primary minimum and secondary minimum. During the analysis, we noticed that
the secondary minimum is total eclipse, which is an advantage in analysis and
means that the measurement at that phase corresponds to direct measurements of
the primary component. Using this advantage, we first determined interstellar
reddening and extinction via direct measurements of the primary component.
Then, we were able to calculate de-reddened colors and magnitudes of the
components, separately. This case was an another advantage in order to make a
more accurate estimation of $T_{1}$, hence enabled us to reduce the number of
free parameters more reliably in simultaneous $UBV$ light curve solution.
Photometric solution justified that the secondary minimum was total eclipse.
Lack of an orbital solution based on radial velocity measurements prevented us
from determining the accurate absolute parameters of the system and their
uncertainties. Hence, we can only make a rough estimation for the
uncertainties and check how those uncertainties affect our results.
If we assume that the $q$ is between 0.6 and 0.7 and $M_{1}$ is between 2.4
$M_{\odot}$ and 2.6 $M_{\odot}$, then we can calculate a lower and upper limit
for $a$ via Kepler’s third law. Those lower and upper limits of $a$ put
$R_{1}$ between 1.76 $R_{\odot}$ \- 1.84 $R_{\odot}$ when we consider the
average fractional radius of the primary component from photometric solution.
Same calculation puts $R_{2}$ between 1.45 $R_{\odot}$ \- 1.52 $R_{\odot}$. A
similar method can be used for the luminosities of the components by using
$Stefan-Boltzmann$ law and $T_{\odot}$ = 5770 K. Assuming a lower and upper
limits for $T_{1}$ and $T_{2}$ via $\sigma_{T_{1}}$ = 470 $K$, we can
calculate the ranges of $L_{1}$ and $L_{2}$ as 31 $L_{\odot}$ \- 49
$L_{\odot}$ and 5 $L_{\odot}$ \- 9 $L_{\odot}$, respectively.
We show preliminary plots of the components in log $T_{eff}$ \- log $L$ plane
in Figure 4. Evolutionary tracks for solar abundance ($Z$ = 0.019) comes from
Girardi et al. (2000). The components of the system seem close to the ZAMS and
in good agreement with solar metal abundance. However, spectroscopic analysis
is necessary to revise or refine it.
Figure 4: Positions of the components (filled stars) in log $T_{eff}$ \- log
$L$ plane. All tracks for $Z$ = 0.019 are from Girardi et al. (2000).
We can make an estimation for the uncertainty of the distance in a similar
way, as described above. If we assume the same ranges for $q$, $M_{1}$ and
$T_{1}$ as in previous uncertainty estimations, we can calculate the range of
absolute bolometric magnitude of the primary component as 1m.01 - 0m.52. Here,
we adopt bolometric corrections from Gray (2005) for corresponding temperature
limits in order to calculate limit visual absolute magnitudes ($M_{V}$) of the
primary component. Using de-reddened $V$ magnitude and $M_{V}$ limits of the
primary component in distance modulus, we can calculate the range of the
distance as 329 - 377 pc and this leads a mean value of 353 pc which is our
estimation in previous section. The uncertainty in the distance might be
slightly exaggerated since we make a rough uncertainty estimation for related
parameters. However, our distance estimation differs only by 20 pc from
distance value of the cluster (Kharchenko et al., 2005). This confirms the
membership of BD +36 3317 to Delta Lyrae cluster. Nevertheless, comprehensive
spectroscopic study of the system, in terms of radial velocity measurements,
would help to refine or revise the physical properties and distance of the
system. Further spectroscopy would also give a chance to check the metal
abundance of the system which also contains some hints about the nature of the
cluster.
## Acknowledgments
The authors acknowledge generous allotments of observing time at the Ege
University Observatory.
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|
arxiv-papers
| 2012-07-01T09:14:46 |
2024-09-04T02:49:32.454291
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "O. \\\"Ozdarcan, E. Sipahi, H. A. Dal",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1207.0194"
}
|
1207.0204
|
# Discovering bright quasars at intermediate redshifts based on optical/near-
IR colors
Xue-Bing Wu, Wenwen Zuo, Jinyi Yang, Qian Yang, Feige Wang
Department of Astronomy, Peking University, Beijing 100871, China
email: wuxb@pku.edu.cn
###### Abstract
Identifications of quasars at intermediate redshifts ($2.2<z<3.5$) are
inefficient in most previous quasar surveys as their optical colors are
similar to those of stars. The near-IR K-band excess technique has been
suggested to overcome this difficulty. Our recent study also proposed to use
optical/near-IR colors for selecting $z<4$ quasars. To verify the
effectiveness of this method, we selected a list of 105 unidentified bright
targets with $i\leq 18.5$ from the quasar candidates of SDSS DR6 with both
SDSS ugriz optical and UKIDSS YJHK near-IR photometric data, which satisfy our
proposed Y-K/g-z criterion and have photometric redshifts between 2.2 and 3.5
estimated from the 9-band SDSS-UKIDSS data. 43 of them were observed with the
BFOSC instrument on the 2.16m optical telescope at Xinglong station of NAOC in
the spring of 2012. 36 of them were spectroscopically identified as quasars
with redshifts between 2.1 and 3.4. High success rate of discovering these
quasars in the SDSS spectroscopic surveyed area further demonstrates the
robustness of both the Y-K/g-z selection criterion and the photometric
redshift estimation technique. We also used the above criterion to investigate
the possible star contamination rate to the quasar candidates of SDSS DR6, and
found that it is much higher in selecting $3<z<3.5$ quasar candidates than
selecting lower redshift ones ($z<2.2$). The significant improvement in the
photometric redshift estimation by using the 9-band SDSS-UKIDSS data than
using the 5-band SDSS data is demonstrated and a catalog of 7,727 unidentified
quasar candidates in SDSS DR6 selected with the optical/near-IR colors and
with photometric redshifts between 2.2 and 3.5 is provided. We also tested the
Y-K/g-z selection criterion with the recently released SDSS-III/DR9 quasar
catalog, and found 96.2% of 17,999 DR9 quasars with UKIDSS Y and K-band data
satisfy our criterion. With some available samples of red quasars and type II
quasars, we find that 88% and 96.5% of them can be selected by the Y-K/g-z
criterion respectively, which supports that using the Y-K/g-z criterion we can
efficiently select both unobscured and obscured quasars. We discuss the
implications of our results to the ongoing and upcoming large optical and
near-IR sky surveys.
galaxies: active — galaxies:high-redshift — quasars: general — quasars:
emission lines
## 1 Introduction
Quasars are important extragalactic objects in astrophysics due to their
unusual properties. They not only can be used to probe the physics of
supermassive black holes and accretion/jet process, but also are closely
related to the studies of galaxy evolution, intergalactic medium, large scale
structure and cosmology. The number of quasars has increased steadily since
their discovery in 1963 (Schmidt 1963; Hazard, Mackey & Shimmins 1963; Oke
1963; Greenstein & Matthews 1963). Especially, a large number of quasars have
been discovered in the last two decades in two large spectroscopic surveys,
namely, the Two-Degree Fields (2DF) quasar survey (2QZ; Boyle et al. 2000) and
the Sloan Digital Sky Survey (SDSS; York et al. 2000). 2DF mainly selected low
redshift ($z<2.2$) quasar candidates with UV-excess (Smith et al. 2005) and
has discovered more than 20,000 blue quasars (Croom et al. 2004), while SDSS
adopted a multi-band optical color selection method (Richards et al. 2002) and
has identified more than 120,000 quasars (Schneider et al. 2010). 90% of SDSS
quasars have low redshifts ($z<2.2$), though some dedicated methods were also
proposed for finding high redshift quasars ($z>3.5$) (Fan et al. 2001a,b;
Richards et al. 2002).
However, in the redshift range $2.2<z<3.5$, the selection of SDSS quasars is
inefficient. Richards et al. (2006) have demonstrated this problem by checking
the efficiency of SDSS quasar selection with the FIRST radio quasars and found
that the efficiency drops substantially in the redshift range $2.2<z<3.5$.
There are two remarkable dips, one around $z=2.8$ and another around $z=3.4$.
The reason for this problem is well understood. At redshift $2.2<z<3.5$, the
spectral energy distributions of quasars show similar optical colors to that
of normal stars, and quasar selections using the optical color-color diagrams
become very inefficient due to the serious contaminations of stars (Fan 1999;
Richards et al. 2002; 2006; Schneider et al. 2007; Hennawi et al. 2010). In
addition, SDSS preferentially selects $3<z<3.5$ quasars with intervening H I
Lyman limit systems, which can also result in a lower efficiency in
identifying quasars with redshift around $z=3.4$ (Worseck & Prochaska 2011).
Because of the importance of using Ly$\alpha$ forest of $z>2.2$ quasars to
study cosmic baryon acoustic oscillation (BAO) (White 2003; McDonald &
Eisenstein 2007) and using these quasars to construct the accurate luminosity
function to study quasar evolution in the mid-redshift universe (Wolf et al.
2003; Jiang et al. 2006), we need to explore other more efficient ways to
identify the $2.2<z<3.5$ quasars than using optical colors alone. We notice
that significant efforts have been made recently for the quasar target
selections in the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS;
Eisenstein et al. 2011, Ross et al. 2012), and much more quasars at
intermediate redshifts have been found in SDSS-III/DR 9 (Paris et al. 2012) .
One possible way to identify the $2.2<z<3.5$ quasars is to use optical
variability, as this is one of the well known quasar properties (Hook et al.
1994; Cristiani et al. 1996; Giveon et al. 1999). Selecting quasars based on
variability is usually thought to be less biased to the optical colors, though
more variation in the shorter wavelength have been found for SDSS quasars
(Vanden Berk et al. 2004; Zuo et al. 2012). Recently, Schmidt et al. (2010),
MacLeod et al. (2011) and Butler & Bloom (2011) have proposed to select quasar
candidates by constructing various intrinsic variability parameters from the
light-curves of known quasars in SDSS Stripe 82 (hereafter S82; see also Sesar
et al. 2007). They claimed that with their methods they can efficiently
separate quasars from stars and substantially increase the number of quasars
at $\rm 2.5<z<3.0$. Moreover, recent results from SDSS-III/BOSS (Eisenstein et
al. 2011) also confirmed the high success rate of spectroscopically
identifying variability selected quasars, which leads to a significant
increase of $z>2.2$ quasar density in S82 than that based on optical color
selected quasars (Palanque-Delabrouille et al. 2011; Ross et al. 2012).
However, only for very limited sky areas the multi-epoch observational data
have been publicly available, so at present the variability methods can not be
broadly used for selecting quasars over a large sky area.
Another possible way for separating $z>2.2$ quasars from stars is to utilize
their near-IR colors. Due to the different radiative mechanisms of stars and
quasars, the continuum emission from stars usually has a blackbody-like
spectrum and decreases more rapidly from optical to near-IR wavelengths than
that of quasars, which usually display a power-law spectra over a broad range
of wavelength plus thermal emissions from accretion disks and dusts. This
leads to obvious color differences in near-IR band between stars and quasars,
even though their optical spectra are similar. Because of this difference, a
K-band excess technique has been proposed for identifying quasars at $z>2.2$
(e.g. Warren, Hewett & Foltz 2000; Croom, Warren & Glazebrook 2001; Sharp et
al. 2002; Hewett et al. 2006; Chiu et al. 2007; Maddox et al. 2008,2012; Smail
et al. 2008; Wu & Jia 2010). Based on a sample of 8498 quasars and a sample of
8996 stars complied from the photometric data in the ugriz bands of SDSS and
YJHK bands of UKIRT InfraRed Deep Sky Surveys (UKIDSS111 The UKIDSS project is
defined in Lawrence et al. (2007). UKIDSS uses the UKIRT Wide Field Camera
(WFCAM; Casali et al. 2007) and a photometric system described in Hewett et
al. (2006). The pipeline processing and science archive are described in
Hambly et al. (2008).), Wu & Jia (2010) proposed an efficient empirical
criterion, (i.e. $\rm Y-K>0.46(g-z)+0.82$, where YK magnitudes are Vega
magnitudes and $\rm gz$ magnitudes are AB magnitudes) for selecting $z<4$
quasars. A check with the VLA-FIRST (Becker, White & Helfand 1995) radio-
detected SDSS quasars, which are thought to be free of color selection bias
(see McGreer, Helfand & White (2009), however), also proved that with this
Y-K/g-z criterion they can achieve the completeness higher than 95% for these
radio-detected quasars at $z<3.5$, which seems to be difficult when using the
SDSS optical color selection criteria alone where two dips around $z\sim 2.7$
and $z\sim 3.4$ obviously exist (Richards et al. 2002,2006; Schneider et al.
2007,2010). Recently, Peth, Ross & Schneider (2011) extended the study of Wu &
Jia (2010) to a larger sample of 130,000 SDSS-UKIDSS selected quasar
candidates and re-examined the near-IR/optical colors of them.
Although by combining the variability and optical/near-IR color we may achieve
the maximum efficiency in identifying $2.2<z<3.5$ quasars (Wu et al. 2011),
for most sky areas we still lack publicly available variability data. One may
also think of using radio and X-ray data, but they can only be helpful for
selecting specific quasar samples(White et al. 2000; Green et al. 1995), which
represent only a small fraction of the whole quasar population. Therefore,
using optical/near-IR colors is probably still the most important way for
selecting $2.2<z<3.5$ quasars. Although we have done some observations to
identify a few $z>2.2$ quasars (Wu et al. 2010a,b; Wu et al. 2011), more
efforts are still needed to check whether using our Y-K/g-z criterion can help
us to discover more quasars at $2.2<z<3.5$, especially in the SDSS
spectroscopically surveyed area.
The paper is organized as follows. In Section 2 we will describe the target
selections and spectroscopic observations, and report our discovery of 36 new
$2.2<z<3.5$ quasars. In section 3 we will use our Y-K/g-z criterion to check
the possible contaminations of stars in the quasar candidate catalog of SDSS
DR6 and investigate the improvement in the photometric redshift estimation by
using the 9-band SDSS-UKIDSS data. In section 4 we will check the
effectiveness of our Y-K/g-z criterion in selecting quasars with the recently
released SDSS-III/Dr 9 quasar catalog. In section 5 we will check whether we
can use the Y-K/g-z criterion to select red quasars and type II quasars. In
section 6 we will estimate how many $2.2<z<3.5$ quasars we can select in
additional to the SDSS-III/BOSS quasars. Finally we will discuss our results
and their implications in Section 7.
## 2 Target selections and spectroscopic observations
### 2.1 Target selections
The main purpose of our study is to use our proposed Y-K/g-z criterion to
discover more quasars at $2.2<z<3.5$ in the SDSS spectroscopically surveyed
area. We started from the 1 million quasar candidates in SDSS DR6 given by
Richards et al. (2009) (hereafter R09) from the Bayesian classification, and
selected 8,845 unidentified quasar candidates brighter than $i=18.5$. From
them we further selected 2,149 candidates with photometric redshifts given in
R09 greater than 2.2 and redshift probability higher than 0.5. Then we cross-
matched these quasar candidates with the UKIDSS/Large Area Survey(LAS) DR7
using a position offset within 3′′ and got 401 candidates with both SDSS ugriz
and UKIDSS YJHK photometric data. 126 of them satisfy our $\rm
Y-K>0.46(g-z)+0.82$ criterion for selecting $z<4$ quasars (Wu & Jia 2010). 17
among these 126 candidates have been spectroscopically identified as quasars
after the SDSS DR6, and 15 of them have redshifts greater than 2.2. Another
one candidate was identified as a white dwarf. After excluding these 18 known
objects, we got a list of 108 quasar candidates.
Since these quasar candidates have 9-band SDSS and UKIDSS photometric data,
the photometric redshifts obtained from these 9-band data are more accurate
than those given by the 5-band SDSS photometric data (Wu & Jia 2010). We use
our developed program to estimate the photometric redshifts based on the
9-band data and found that 15 objects among 108 quasar candidates have
photometric redshifts smaller than 2, while for other quasar candidates our
results are consistent with those obtained in R09 from the SDSS photometric
data. After excluding these 15 objects, we get a target list of 93 quasar
candidates with reliable photometric redshifts between 2.2 and 3.6. After the
public release of UKIDSS/LAS DR8 in April, 2012, we also added other 12 quasar
candidates, which have UKIDSS/LAS DR8 data and satisfy our Y-K/g-z selection
criterion but were not included in the UKIDSS/LAS DR7, making our final list
containing 105 quasar candidates. The procedures of our target selections are
also listed in Table 1.
### 2.2 Spectroscopic observations
Among these 105 quasar candidates, 92 have Right Ascension (RA) between 7hr
and 17hr. In March to May of 2012, we made 8-night spectroscopic observations
on 43 bright quasar candidates (sampling rate of 46.7%) with photometric
redshifts at $2.2<z_{ph}<3.5$ using the BAO Faint Object Spectrograph and
Camera (BFOSC) of the 2.16m optical telescope at the Xinglong station of
National Astronomical Observatories, Chinese Academy of Sciences (NAOC) . A
low resolution grism with the dispersion of 198Å/mm, the wavelength coverage
from 3850 to 7000Å, and the spectral resolution of 2.97$\AA$ was used. During
our observations, the typical seeing varied from 1.5′′ to 2.5′′, so we adopted
the long slit width of 1.8′′ and 3.6′′ accordingly (no choice in between). In
Table 2 we summarize the parameters and details of the observations for these
43 quasar candidates, including the observation date, exposure time, SDSS and
UKIDSS magnitudes, photometric redshift, identification result and spectral
redshift.
Among the 43 quasar candidates, 36 were spectroscopically identified as
quasars with redshifts from 2.1 to 3.4, 2 were identified as stars and 5
remain unidentified due to the lower signal-to-noise ratios of their spectra.
The spectra of 36 new quasars, after the standard flat-field corrections, flux
and wavelength calibrations, were plotted in Fig. 1.
The spectra of these quasars were analyzed following the method described
detailedly in Shen et al. (2008) and Shen & Liu (2012). First the spectra were
redshift corrected to the rest-frame and were corrected for the Galactic
extinction using the extinction map of Schlegel et al. (1998). They are then
fitted based on an IDL code MPFIT (Markwardt 2009). We fitted the spectra with
the pseudo-continuum model consisting of the featureless non-stellar continuum
and Fe II emissions. The featureless non-stellar continuum is assumed to be a
power-law, so two free parameters (amplitude and slope) are required.
Templates for Fe II emissions have been constructed from the spectrum of the
narrow-line Seyfert 1 galaxy, I Zw I (Boroson & Green 1992), by convolving
with a velocity dispersion and shifting with a velocity. We used the UV
template generated by Vestergaard & Wilkes (2001) and Tsuzuki et al. (2006) in
the wavelength range of 1000-3500Å.
After constructing the pseudo-continuum, the broad CIV component is fitted
with two Gaussians and the narrow component is fitted with one Gaussian.
However, as the spectra of 13 quasars have low signal-to-noise ratio ($\sim$
6.5), we used only one Gaussian to fit the whole CIV emission line profile for
them. We measured the Full Width at Half Maximum of CIV line (FWHM(CIV)),
luminosity at 1350Å($L_{1350}$) from the spectra. We note that the line widths
for 13 quasars with lower signal-to-noise ratio are rough estimates. The black
hole mass is calculated based on FWHM(CIV) and $L_{1350}$ with Eq.(7) in
Vestergarrd & Peterson (2006)(see also Kong et al. 2006). Using a scaling
relation between $L_{1350}$ and bolometric luminosity $L_{bol}$,
$L_{bol}=4.62L_{1350}$, we estimate the bolometric luminosity for these
quasars. Based on the obtained black hole mass and bolometric luminosity, we
also calculate their Eddington ratios($L_{\rm bol}/L_{Edd}$, where $L_{Edd}$
is the Eddington luminosity). The results are summarized in Table 3. Although
we noticed that the uncertainties of these values are probably quite large due
to the low spectral quality and the unusual properties of CIV, the overall
properties of these quasars, including the line width, continuum luminosity,
black hole mass and Eddington ratio, are consistent with those of typical SDSS
quasars with redshift greater than 2.2 (Shen et al. 2011). The continuum slope
parameter, $\alpha_{\lambda}$, is given for each quasar in Table 3. The median
value is -1.315. If we convert it to $\alpha_{\nu}$, the median value of
$\alpha_{\nu}$ is then -0.685, which is not too different from -0.517 and
-0.862 obtained from SDSS DR9 and DR7 quasars, respectively (Paris et al.
2012).
We also investigate the Broad Absorption Line (BAL) quasars in our sample. The
Balincity Index is calculated for each quasar using the traditional method
(Weymann et al. 1991). 14 quasars have positive BI values, indicating that
they are probably BAL quasars. However, by visually inspecting the quasar
spectra, we find that this traditional method risks identifying false troughs
from noisy and poor continuum fitting. To avoid these false identifications,
we calculate the Balnicity Index (BI) and Absorption Index (AI), adding the
same extra minimum depth and width requirement in the emission line region (
for more details see section 4.4 of Trump et al. (2006)). With these procedure
we found 4 BALs. However, SDSS J124605.36+071128.2 is likely not a real BAL
due to the lack of the spectrum shortward to the C IV line center. The
remained 3 BALs are SDSS J115531.45-014611.9, SDSS J1359420+022426.0 and SDSS
J142405.57+044105.5. Their BI values are 3379.27, 381.18 and 178.70 km/s,
respectively. The high velocities (3837 - 21293 km/s) derived from their broad
absorption features of the CIV lines are consistent with those of quasars with
higher UV luminosity (Gibson et al. 2009). This is also expected if the BAL
outflow is produced by the strong radiation pressure (Murray et al. 1995).
### 2.3 Success rate of finding $2.2<z<3.5$ quasars
Our spectroscopic observations identified 36 quasars at $2.1<z<3.4$ and 2
stars from 43 candidates, which indicated a success rate of at least 83.7% in
identifying the bright quasars at intermediate redshifts because 5 candidates
still remain unidentified. This high success rate is largely due to the quasar
candidate selection procedures we adopted, especially by using the Y-K/g-z
criterion and 9-band SDSS-UKIDSS photometric redshifts to select $2.2<z<3.5$
quasar candidates. As we stated before, using photometric redshifts given in
R09 greater than 2.2 and redshift probability higher than 0.5 enables us to
reduce the number of quasar candidates brighter than $i=18.5$ from 8845 to
2149. In addition, using the Y-K/g-z criterion we can reduce the number of
quasar candidates with SDSS-UKIDSS data from 401 to 126. Therefore, our high
success rate of identifying $2.2<z<3.5$ quasar is not a surprise because we
can efficiently exclude the star contaminations by using the Y-K/g-z criterion
and select most reliable $2.2<z<3.5$ quasar candidates by using the
photometric redshifts obtained from the SDSS or SDSS-UKIDSS photometric data.
## 3 Accuracy of photometric redshifts and star contaminations in the quasar
candidate catalog of SDSS DR6
We selected the quasar targets from the SDSS DR6 1 million quasar candidate
catalog of R09, and used both the SDSS and UKIDSS photometric data for further
selections and photometric redshift estimations to achieve the high success
rate of identifying $2.2<z<3.5$ quasars. With the SDSS-UKIDSS optical/near-IR
data and our proposed quasar selection criterion, we may also investigate the
accuracy of photometric redshifts and the possible star contaminations in the
quasar candidate catalog of SDSS DR6 (R09), which will be helpful for the
future spectroscopic observations.
We cross-matched the SDSS DR6 1 million quasar candidate catalog of R09 with
the UKIDSS/LAS DR8 data by using the positional offset of 3′′ for finding only
the closest counterpart, and obtained 97,923 sources with full detections in
SDSS and UKIDSS 9 photometric bands. This SDSS-UKIDSS quasar candidate sample
is much bigger than the previous one with 42,133 sources from the UKIDSS/LAS
DR3 (Peth, Ross & Schneider 2011). Among these 97,923 sources, there are
24,878 known quasars and 73,011 unidentified quasar candidates in SDSS DR6.
First we checked the improvement of photometric redshift estimations using the
9-band SDSS-UKIDSS photometric data than using the SDSS data alone. We used
our photometric redshift estimation program (Wu & Jia 2010; Wu, Zhang & Zhou
2004) to obtain the photometric redshifts of all unidentified quasar
candidates and known quasars in R90 based on the SDSS-UKIDSS data, and
compared them with the photometric redshifts given in R90 and the spectral
redshifts for known quasars in SDSS DR6. In two upper panels of Fig. 2, we
compare the photometric redshifts in R90 and ours for 73,011 unidentified
quasar candidates in R09, and show the histogram distribution of their
differences. For 59.8% of these unidentified quasar candidates, the
differences between two kinds of photometric redshifts are less than 0.2.
However, there are still obvious differences, especially when the photometric
redshifts are smaller than 3. Comparing with our results, the photometric
redshifts given in R09 are systematically larger for some low-redshift quasar
candidates. In two middle panels and two lower panels of Fig. 2, we compare
the photometric redshifts given in R90 and by us for 24,878 known quasars with
spectral redshifts, respectively (two middle panels are similar to Figure 7 in
Peth et al. (2011) but with more known quasars because we used the data in
UKIDSS/LAS DR8). For 76.1% of the known quasars, R90 gave the photometric
redshifts within the difference smaller than 0.2 from their spectral
redshifts. By using the SDSS-UKIDSS 9-band photometric data to estimate the
photometric redshifts, such a fraction increases to 85.2%. This significant
improvement can be clearly observed from Fig. 2, and demonstrates again that
by adding the near-IR photometric data to the SDSS optical data we can achieve
substantially higher accuracy in photometric redshift estimations (Wu & Jia
2010; Wu et al. 2012).
Next we checked the possible star contaminations in the quasar candidate
catalog of SDSS DR6 (R09), using the Y-K/g-z quasar selection criterion. In
Fig. 3 we show the distributions of 24,878 known quasars and 73,011
unidentified quasar candidates in R09 in the Y-K/g-z color-color diagram, as
well as our Y-K/g-z quasar selection criterion (Wu & Jia 2010). For 24,648
known $z<4$ quasars, using the Y-K/g-z criterion can select 24,295 of them
(98.6%). For 61,489 unidentified quasar candidates in R09 with the photometric
redshifts of $z_{ph}<2.2$ (we adopted the photometric redshifts estimated with
the SDSS-UKIDSS 9-band photometric data), using the Y-K/g-z criterion can
select 60,412 of them (98.3%). For 10,687 unidentified quasar candidates in
R09 with photometric redshift of $2.2<z_{ph}<4$, using the Y-K/g-z criterion
we can select 8,934 of them (83.6%). Therefore, the quasar candidates
selection in R09 are well consistent with our Y-K/g-z selection for
$z_{ph}<2.2$ quasar candidates, but there are substantial contaminations from
stars for selecting $2.2<z_{ph}<4$ quasar candidates. This can be also seen
from the lower panel of Fig. 3, where the green dots below the line most
probably represent the star contaminations.
To better understand the quasar selection efficiency and the star
contaminations at different redshift, in Fig. 4 we plot the photometric
redshift dependences of the fraction of 24,648 known $z<4$ quasars selected by
the Y-K/g-z criterion and the fraction of 72,176 unidentified quasar
candidates in R09 with photometric redshifts of $z_{ph}<4$ selected by the
Y-K/g-z criterion. For known $z<4$ quasars, using the Y-K/g-z criterion can
reach the efficiency higher than 90% at almost all redshift, except for
$z>3.5$. For unidentified quasar candidates, R09 selection has the similar
efficiency (higher than 90%) as using the Y-K/g-z criterion for selecting
$z_{ph}<2.6$ quasars but has higher star contaminations for selecting
$z_{ph}>2.6$ quasars than using the Y-K/g-z criterion. On may think that the
decrease of quasar selection fraction at $z_{ph}>2.6$ (denoted by the blue
dotted line in Fig. 4) is due to both the mis-identifications of quasars as
stars by the Y-K/g-z criterion and the true star contaminations. After the
deduction of the mis-identification rate of quasars as stars (which can be
estimated from the known quasar selection fraction denoted as the black solid
line in Fig.4) by the Y-K/g-z criterion at different redshift, we can obtain
the possible star contamination rate in R09 at different redshift (denoted by
the red dashed line in Fig. 4). It is clear that the star contamination rate
becomes substantially higher for selecting $z_{ph}>2.6$ quasars than the lower
redshift ones, even up to 30% to 40% for selecting quasars at redshift
$3<z_{ph}<3.5$. We must notice that the real star contaminations are probably
much higher than those we estimated with the Y-K/g-z criterion. Therefore, the
star contamination rate in R09 we obtained for selecting $2.6<z_{ph}<4$
quasars could be underestimated. In fact, based on the clustering study of
Myers et al. (2006), the star contamination in the ’mid-z’ range of R09 was
estimated to be higher than 50% (Richards et al. 2009b). Nevertheless, we
believe that using the Y-K/g-z criterion can help us to exclude the star
contaminations significantly and obtain higher efficiency in selecting
$2.6<z_{ph}<4$ quasars.
From R09, we can obtain a list of SDSS DR6 unidentified quasar candidates with
UKIDSS/LAS DR8 full detections in the YJHK bands and with the photometric
redshifts (given in R09) at $2.2\leq z_{ph}(R09)\leq 3.5$, which consists of
17,719 objects. However, if we adopt our photometric redshifts obtained from
the 9-band SDSS-UKIDSS photometric data and use our Y-K/g-z criterion to do
further selection, such a list consists of only 7,727 quasar candidates at
$2.2\leq z_{ph}\leq 3.5$. The substantial decrease of the size is mainly due
to the increase of photometric redshift reliability and the deduction of star
contaminations by using the Y-K/g-z criterion. In Table 4 we list the name,
photometric redshift, SDSS and UKIDSS magnitudes for these 7727 quasar
candidates with our estimated photometric redshift at $2.2\leq z_{ph}\leq
3.5$. We noticed that some of them have been identified after SDSS DR6,
including this work. Future spectroscopy on these unidentified quasar
candidates will provide further checks to the robustness of both the quasar
selection criterion and the photometric redshift estimation method.
## 4 Comparisons with SDSS-III DR9 quasars
Very recently, SDSS-III/BOSS has released the DR 9 quasar catalog, which
consists of 87,822 quasars (78,086 are new and 61,931 have redshifts higher
than 2.15) detected over a sky area of 3,275 deg2 (Paris et al. 2012). This
provides us a chance to check the effectiveness of our proposed Y-K/g-z
criterion with the largest sample of $z>2.1$ quasars currently available.
After cross-matching the SDSS-III DR9 quasar catalog with the UKIDSS/LAS DR8
catalog, 17,999 among 87,822 quasars have available Y and K-band data, with a
sampling rate of 20.5%. 17,308 of these 17,999 quasars satisfy the Y-K/g-z
selection criterion for $z<4$ quasars, with a completeness of 96.2%. In the
upper panel of Fig. 5, we show the distributions of 17,999 SDSS-
III/DR9-UKIDSS/LAS/DR8 (hereafter DR9-UKIDSS) quasars in the Y-K/g-z color-
color diagram and compare them with our proposed Y-K/g-z selection criterion
for $z<4$ quasars. Similar to the upper panel of Fig. 3, this comparison also
clearly demonstrates the effectiveness in using the Y-K/g-z selection
criterion to select $z<4$ quasars.
We also check whether the Y-K/g-z selection depends on the magnitude and
redshift. In the middle panel of Fig. 5 we show the magnitude dependence of
the selection fraction of 17,999 DR9-UKIDSS quasars by our Y-K/g-z criterion,
and the normalized magnitude distributions (fraction between the number of
quasars in each magnitude bin and the total number) for 17,308 quasars
selected by the Y-K/g-z criterion and for 87,822 SDSS-III/DR9 quasars. The
comparison shows that using UKIDSS data we do select optically brighter
quasars (especially those brighter than $i=20.5$) due to the limited
sensitivity of UKIDSS. However, the selection efficiency of using the Y-K/g-z
criterion does not significantly depend on the magnitudes for quasars with
UKIDSS data at $i<20.5$. In the lower panel of Fig. 5 we show the redshift
dependence of the selection fraction of 17,999 DR9-UKIDSS quasars by our
Y-K/g-z criterion, which clearly states that our criterion is robust for
selecting $z<4$ quasars. Comparing the normalized redshift distributions
(fraction between the number of quasars in each redshift bin and the total
number) for 17,308 quasars selected by the Y-K/g-z criterion and for 87,822
SDSS-III/DR9 quasars also demonstrates the similar redshift distribution of
quasar sample selected by the Y-K/g-z criterion and the SDSS-III/DR9 quasar
sample at $z<4$.
In addition, we need to check whether using the Y-K/g-z criterion we actually
select quasars with specific colors. In Fig. 6 we show the distributions of
$\Delta(g-i)$ versus redshift for 17,308 quasars selected by the Y-K/g-z
criterion and for 87,822 SDSS-III/DR9 quasars. Obviously they significantly
overlap at $z<4$. From the distribution of median $\Delta(g-i)$ values in each
redshift bin the quasars selected by the Y-K/g-z criterion have slightly
redder $g-i$ color than the SDSS-III/DR9 quasars at $z<4$. The median
$\Delta(g-i)$ value for 17,308 quasars selected by the Y-K/g-z criterion is
0.088, which is slightly larger that the median value 0.035 for 87,822 SDSS-
III/DR9 quasars. This is mainly because using the UKIDSS near-IR data we
mostly select quasars with $i<20.5$ (see the middle panel of Fig. 5). Most
SDSS-III/DR9 quasar with $i>20.5$ have smaller or negative $\Delta(g-i)$
values. We also noticed that the SDSS-III/DR9 quasars with $i<20.5$ have
median $\Delta(g-i)$ value of 0.080, which is close to median value 0.088 of
the Y-K/g-z selected quasars.
Finally we check whether using the Y-K/g-z criterion can help us to select
more BAL quasars. In SDSS-III/DR9 quasar catalog, 7,533 quasars are found to
be BAL quasars after the visual inspection. 2,173 of them have UKIDSS/LAS DR8
data and 1,974 quasars satisfy the Y-K/g-z criterion (With a percentage of
90.8%). This fraction is only slightly smaller than 96.2% for the Y-K/g-z
selection of all DR9-UKIDSS quasars, implying that using the Y-K/g-z criterion
we can also efficiently select BAL quasars. We also noticed that 3 quasars in
our 36 newly-discovered quasars are BAL quasars (see section 2), which is
consistent with the fraction of 7,533 BAL quasars in 87,822 SDSS-III/DR9
quasars, though with small number statistics.
## 5 Selecting red quasars and type II quasars with the Y-K/g-z criterion
In this section we check whether we can also efficiently select red quasars
and type II quasars by using the Y-K/g-z criterion.
Glikman et al. (2007,2012) presented a sample of 128 red quasars, with
redshifts up to 3.05, selected from the FIRST-2MASS radio and near-IR surveys.
These quasars with red color, J-K$>$1.7, are thought to be transient objects
between heavily obscured quasars and normal blue quasars. Urrutia et al.
(2009) also identified 57 red quasars with J-K$>$1.3 from 122 candidates
selected from FIRST, 2MASS and SDSS, with a high fraction of BAL quasars. By
cross-matching with the UKIDSS LAS/DR8 catalog, we get 26 and 16 red quasars
with Y and K-band detections from these two samples, respectively. We plot
them in the Y-K/g-z color-color diagram (see the upper panel of Fig. 7) and
find that 23 among 26 red quasars, and 14 among 16 red quasars in these two
samples satisfy the Y-K/g-z criterion. The selection efficiency is 88.5% and
87.5% for these two samples, respectively. This comparison clearly
demonstrates the high efficiency of selecting red quasars with the Y-K/g-z
criterion.
Because the SDSS-III/DR9 quasar catalog does not provide the information about
how many type II quasars are included in it, we use the existed type II quasar
catalog in SDSS to check the efficiency of selecting type II quasars with the
Y-K/g-z criterion. Using the catalog provided by Reyes et al. (2010), which
includes 887 optically selected $0.3<z<0.83$ type II SDSS quasars, we find
that 282 among 887 type II quasars have available Y and K-band data from
UKIDSS LAS/DR8. We plot them in the Y-K/g-z color-color diagram, in comparison
with the Y-K/g-z selection criterion (see the lower panel of Fig. 7). 272 of
these 282 type II quasars (with a percentage of 96.5%) satisfy our Y-K/g-z
criterion. This selection fraction is similar as 96.2% for the Y-K/g-z
selection of all DR9-UKIDSS quasars, which indicates that using the Y-K/g-z
criterion we can also efficiently discover type II quasars.
In Fig. 7, we also plot the predicted color tracks for type I quasars and type
II quasars at different redshifts (up to 4.3, to the right side in Fig. 7),
using the related spectral templates from Polletta et al. (2007). We can
clearly see that although type II quasars have redder Y-K and g-z colors than
type I quasars, using the Y-K/g-z criterion we can efficiently select both
type I and type II quasars with redshifts up to 4.
## 6 Can we use the Y-K/g-z criterion to select more $2.2<z<3.5$ quasars?
Although our above test with the recently released SDSS-III/DR9 quasar catalog
does indicate the effectiveness of using the Y-K/g-z critirion in selecting
$z<4$ quasars, one question still needs to be addressed. Can we use the
Y-K/g-z criterion to select additional $2.2<z<3.5$ quasars which SDSS-III/BOSS
does not select?
To answer this question, we should find a sky area which covered both by SDSS-
III/BOSS and UKIDSS, and compare the quasar candidates selected by the SDSS-
III/BOSS and Y-K/g-z selection criterion. Thanks to Adam Myers in the SDSS-
III/BOSS team, who kindly provides us the photometric catalog of a 15 square-
degree region in SDSS Stripe 82 (with $36^{o}<RA<42^{o}$ and
$-1.25^{o}<Dec<1.25^{o}$), which has relatively complete spectroscopic
observations on SDSS-III/BOSS quasar targets. By cross-matching the 90,922
SDSS sources in this area with UKIDSS and selecting sources with SDSS
photometric parameters type=6, r$<$21.85 and g$<$22 (the same as in SDSS-
III/BOSS, r and g magnitudes are Galactic extinction corrected), we get 24,627
point sources with UKIDSS/LAS Y and K-band detections. Among them, 21,715 were
unidentified while 2,912 were identified (including 743 quasars and 2,169
stars). 135 of 743 identified quasars are in the redshift range between 2.2
and 3.5. 130 of them were included in the SDSS-III/DR9 quasar catalog and 5 of
them were identified by other BOSS ancillary programs. We also found that 712
of 743 identified quasars, including 126 of 135 $2.2<z<3.5$ quasars, satisfy
the Y-K/g-z selection criterion.
Among 21,715 unidentified sources, 340 of them satisfy the Y-K/g-z selection
criterion. The photometric redshifts of these 340 quasar candidates were
estimated based on their SDSS and UKIDSS data, and 140 sources are found to
have photometric redshifts between 2.2 and 3.5. If we further require their
$\chi^{2}$ values of photometric redshift estimations (Wu & Jia 2010; Wu et
al. 2004; Weinstein et al. 2004) smaller than 10, which is satisfied by 94% of
743 identified quasars in this sky area, the number of $2.2<z<3.5$ quasar
candidates selected by the Y-K/g-z criterion becomes 86. Even if we require
the $\chi^{2}$ values smaller than 6, which is satisfied by 80% of 743 known
quasars in this area, the number of Y-K/g-z selected $2.2<z<3.5$ quasar
candidates becomes 52. Because there are 470 known $2.2<z<3.5$ quasars (421
are SDSS-III/DR9 quasars) in this sky area and 135 of them (mostly brighter
ones) have UKIDSS Y and K-band detections, the Y-K/g-z selected additional
$2.2<z<3.5$ quasar candidates may add at least 10% to the total number of
SDSS-III/BOSS $2.2<z<3.5$ quasars. However, whether these candidates are real
$2.2<z<3.5$ quasars still needs to be confirmed by the future spectroscopic
observations. Therefore, with this check we believe that SDSS-III/BOSS has
selected most $2.2<z<3.5$ quasars, and the Y-K/g-z selection may add about 10%
additional $2.2<z<3.5$ quasars in the UKIDSS surveyed area.
## 7 Discussion
We have presented the spectroscopic observations on 43 bright quasar
candidates selected from R09, which have photometric redshifts at
$2.2<z_{ph}<3.5$ estimated from the 9-band SDSS and UKIDSS photometric data
and satisfy our Y-K/g-z criterion, and successfully identified 36 of them to
be real quasars with redshifts between 2.1 and 3.4. The high efficiency of
spectroscopic identifications provides further support for discovering more
quasars at intermediate redshifts based on the optical and near-IR color
selections. We also found substantial improvement of photometric redshift
estimation from using the 9-band SDSS-UKIDSS data than using the SDSS data
alone. We investigated the star contamination rate of quasar candidates in
R09, which could be much higher for selecting quasars at photometric redshift
of $3<z_{ph}<3.5$ than the lower redshift ones ($z<2.2$). By using our
photometric redshifts estimated from the SDSS and UKIDSS photometric data and
the Y-K/g-z criterion to exclude the star contaminations, we obtained a
catalog of 7727 SDSS-UKIDSS unidentified quasar candidates with photometric
redshifts at $2.2<z_{ph}<3.5$. The ongoing and future spectroscopic
observations, such as SDSS-III/BOSS(Eisenstein et al. 2011), will provide
further check to the robustness of this catalog, though the UKIDSS near-IR
data were not used for selecting the majority of quasar candidates in BOSS
(Ross et al. 2012).
Using the recently released SDSS-III/DR9 quasar catalog and UKIDSS/LAS DR8
data, we find that 96.2% of UKIDSS detected DR9 quasars, including 90.8% of
BAL quasars, satisfy the Y-K/g-z criterion. This provides further support to
this criterion for selecting $z<4$ quasars, including BAL quasars. We also
check the efficiency of using the Y-K/g-z criterion to select red quasars and
type II quasars with some available samples, and find that about 88% of red
quasars with $z<3.05$ and 96.5% of type II quasars with $z<0.83$ satisfy the
Y-K/g-z criterion. These results, together with the predicted color tracks by
using different spectral templates of quasars, support the robustness of using
the Y-K/g-z criterion to discover both unobscured and obscured quasars. Our
test in a small sky area of SDSS Stripe 82 also proves that with the Y-K/g-z
selection criterion we may add about 10% additional $2.2<z<3.5$ quasars to the
SDSS-III/BOSS quasars in the UKIDSS surveyed area.
Since UKIDSS only covers a very limited sky area, we still need much deeper
optical/near-IR photometry in a larger sky area for taking the full advantages
of the optical/near-IR color for selecting quasars, especially for $z>2.2$
quasars. The recently released Wide-field Infrared Survey Explorer (WISE) all-
sky data (Wright et al. 2010) also provided abundant photometric data in the
near(middle)-IR bands, which will be very helpful for quasar selections (Wu et
al. 2012; Stern et al. 2012; Edelson & Malkan 2012; Yan et al. 2013)
Fortunately, several ongoing optical and near-IR photometric sky surveys will
also provide us further oppotunities to apply our optical/near-IR color
selections of quasars to larger and deeper fields. In addition to SDSS III
(Eisenstein et al. 2011), which has taken 2,500 deg2 further imaging in the
south galactic cap, the SkyMapper (Keller et al. 2007) and Dark Energy Survey
(DES; The Dark Energy Survey Collaboration 2005) will also present the multi-
band optical photometry in 20,000/5,000 deg2 of the southern sky, with the
magnitude limit of 22/24 mag in $i$-band, respectively. The Visible and
Infrared Survey Telescope for Astronomy (VISTA; Arnaboldi et al. 2007) is
carrying out the VISTA Hemisphere Survey (VHS) in the near-IR YJHK bands for
20,000 deg2 of the southern sky with a magnitude limit at K=20.0, which is
about five and two magnitude deeper than the Two Micron ALL Sky Survey (2MASS;
Skrutskie et al. 2006) and UKIDSS/LAS limits (Lawrence et al. 2007),
respectively. Therefore, the optical and near-IR photometric data obtained
with these ongoing surveys will provide us a large database for quasar
selections. Needless to say, the ongoing Panoramic Survey Telescope & Rapid
Response System (Pan-STARRS; Kaiser et al. 2002) and the future Large Synoptic
Survey Telescope (LSST; Ivezic et al. 2008) will also provide us with multi-
epoch photometry in multi-bands covering a large area of the sky, which will
undoubtedly help us to construct a much larger sample of quasars based on both
optical/near-IR colors and variability features.
On the other hand, the spectroscopic observations are still crucial to
determine the quasar nature and redshifts for the quasar candidates selected
from the optical/near-IR colors. The ongoing SDSS-III/BOSS is expected to
obtain the spectra of 150,000 quasars at $2.2<z<4$ (Eisenstein et al. 2011;
Ross et al. 2011). We believe that many $2.2<z<3.0$ quasars, including the
candidates we listed in this paper, should be spectroscopically identified by
BOSS. In addition, the Chinese GuoShouJing telescope (LAMOST; Su et al. 1998;
Cui et al. 2012; Zhao et al. 2012), a spectroscopic telescope with 4000
fibers, which is currently in the final stage of commissioning and will start
the regular spectroscopic survey in the fall of 2012, is also aiming at
discovering 0.3 million quasars from 1 million candidates with magnitudes
bright than $i=20.5$ in the next 5 years (Wu et al. 2010a,b; Wu 2011). By
using the optical/near-IR colors, we hope the larger input catalogs of
reliable quasar candidates will be provided to these quasar surveys for future
spectroscopic observations. We expect that a much larger and more complete
quasar sample covering a wider range of redshift will be constructed in the
near future. In addition, the results we presented in this work may be also
helpful to the future spectroscopic surveys of quasars, like those in eBOSS
and Big BOSS (Schlegel et al. 2011).
We thank the referee, Nicholas Ross, for very constructive suggestions to
improve the paper, and Adam Myers for providing us a photometric catalog of an
eBOSS test region in SDSS Stripe 82. This work was supported by the National
Natural Science Foundation of China (grant No. 11033001) and by the Open
Project Program of the Key Laboratory of Optical Astronomy, NAOC, CAS. Funding
for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation,
the Participating Institutions, the National Science Foundation, the US
Department of Energy, the National Aeronautics and Space Administration, the
Japanese Monbukagakusho, the Max Planck Society and the Higher Education
Funding Council for England. The SDSS web site is http://www.sdss.org/. The
SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions. The Participating Institutions are the American Museum of
Natural History, Astrophysical Institute Potsdam, University of Basel,
University of Cambridge, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences
(LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for
Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico
State University, Ohio State University, University of Pittsburgh, University
of Portsmouth, Princeton University, the United States Naval Observatory and
the University of Washington. Facilities: Sloan (SDSS), UKIDSS,2.16m/NAOC
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Table 1: Procedures of target selection Step | Number of candidates | Description
---|---|---
1 | 1,015,082 | All quasar candidates in R09
2 | 925,899 | Excluding known quasars in SDSS DR6
3 | 8,845 | Brighter than i=18.5
4 | 2,149 | Photometric redshift $z_{phot}>2.2$ and probability $z_{prob}>0.5$ in R09
5 | 401 | With UKIDSS LAS DR7 data
6 | 126 | Satisfying Y-K$>$0.46(g-z)+0.82 selection criterion
7 | 108 | Excluding 17 quasars and 1 white dwarf identified after SDSS DR6
8 | 93 | Excluding 15 candidates with SDSS-UKIDSS based photometric redshifts smaller than 2
9 | 105 | Adding 12 candidates not included in UKIDSS/LAS DR 7 but included in UKIDSS/LAS DR8
Table 2: Parameters and observation details of 43 quasar candidates
Name | Date | Exposure | u | g | r | i | z | Y | J | H | K | $z_{ph}(R09)$ | $z_{ph}$ | Result | $z_{sp}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(SDSS J) | | (s) | | | | | | | | | | | | |
075746.08+232054.2 | 2012-03-13 | 3600 | 19.06 | 18.46 | 18.48 | 18.47 | 18.27 | 17.69 | 17.39 | 16.82 | 16.02 | 2.365 | 2.375 | quasar | 2.532
081545.72+264847.1 | 2012-04-14 | 1800 | 18.27 | 17.14 | 17.06 | 17.04 | 16.87 | 16.24 | 15.86 | 15.5 | 15.29 | 2.755 | 2.825 | F star |
081617.55+225604.5 | 2012-05-15 | 2400 | 19.95 | 18.63 | 18.43 | 18.35 | 18.35 | 17.74 | 17.37 | 16.92 | 16.57 | 2.905 | 2.875 | quasar | 2.931
083255.70+004710.1 | 2012-03-13 | 3600 | 19.80 | 18.40 | 18.29 | 18.30 | 18.28 | 17.89 | 17.36 | 16.92 | 16.54 | 2.905 | 2.875 | quasar | 2.919
084659.42+253940.9 | 2012-04-16 | 3600 | 19.75 | 18.44 | 18.46 | 18.43 | 18.25 | 17.46 | 17.03 | 16.53 | 15.99 | 2.795 | 2.875 | quasar | 2.892
085152.98+091808.5 | 2012-05-16 | 3600 | 18.65 | 18.13 | 17.95 | 17.94 | 17.79 | 17.15 | 16.95 | 16.54 | 15.85 | 2.505 | 2.575 | low S/N |
085825.51+283258.5 | 2012-03-13 | 3600 | 20.54 | 18.73 | 18.44 | 18.50 | 18.49 | 17.68 | 17.20 | 16.74 | 16.15 | 3.535 | 2.975 | quasar | 3.226
090233.19+034131.2 | 2012-05-17 | 3600 | 18.49 | 17.92 | 17.79 | 17.77 | 17.6 | 17.06 | 16.88 | 16.39 | 15.63 | 2.465 | 2.575 | quasar | 2.532
090827.71+011322.5 | 2012-04-15 | 2400 | 18.33 | 17.64 | 17.54 | 17.47 | 17.29 | 16.83 | 16.51 | 16.07 | 15.26 | 2.565 | 2.325 | quasar | 2.083
091756.58+100836.7 | 2012-04-16 | 2400 | 18.54 | 17.72 | 17.68 | 17.67 | 17.58 | 16.9 | 16.55 | 16.1 | 15.67 | 2.695 | 2.675 | quasar | 2.760
091857.36+025205.4 | 2012-04-15 | 3000 | 19.39 | 17.9 | 17.79 | 17.72 | 17.61 | 17.06 | 16.75 | 16.39 | 15.9 | 2.905 | 2.875 | quasar | 2.789
092021.02-020113.7 | 2012-03-14 | 3600 | 19.96 | 18.58 | 18.51 | 18.46 | 18.44 | 17.76 | 17.45 | 17.15 | 16.71 | 2.905 | 2.875 | quasar | 2.826
093655.11+305855.3 | 2012-05-17 | 2700 | 20.37 | 18.55 | 18.45 | 18.49 | 18.37 | 17.50 | 17.31 | 16.8 | 16.37 | 2.945 | 2.875 | quasar | 3.005
094137.09+102650.9 | 2012-04-16 | 2700 | 18.67 | 18.04 | 17.90 | 17.9 | 17.75 | 17.01 | 16.77 | 16.38 | 15.84 | 2.535 | 2.625 | quasar | 2.584
095118.43+083959.9 | 2012-03-13 | 3600 | 18.90 | 18.25 | 18.21 | 18.19 | 18.02 | 17.50 | 17.18 | 16.64 | 15.85 | 2.365 | 2.375 | quasar | 2.523
100834.73-022302.5 | 2012-04-14 | 3600 | 19.93 | 18.13 | 17.91 | 17.88 | 17.85 | 17.39 | 16.95 | 16.54 | 16.01 | 3.005 | 2.975 | quasar | 3.086
103301.49+065106.5 | 2012-05-16 | 2700 | 18.73 | 18.03 | 17.91 | 17.84 | 17.69 | 17.15 | 16.88 | 16.45 | 16.05 | 2.565 | 2.625 | quasar | 2.637
114608.05+094216.5 | 2012-05-15 | 5400 | 19.54 | 18.49 | 18.49 | 18.38 | 17.96 | 17.05 | 16.83 | 16.22 | 15.35 | 2.615 | 2.475 | quasar | 2.580
115531.45-014611.9 | 2012-03-13 | 6000 | 23.25 | 19.36 | 18.91 | 18.50 | 18.50 | 17.94 | 17.39 | 17.00 | 16.36 | 3.495 | 3.350 | quasar(BAL) | 3.196
121510.62+142834.5 | 2012-04-15 | 3600 | 19.20 | 18.45 | 18.34 | 18.33 | 18.15 | 17.47 | 17.20 | 16.75 | 16.11 | 2.595 | 2.675 | quasar | 2.480
122043.86+011122.1 | 2012-05-17 | 3600 | 19.21 | 18.39 | 18.38 | 18.27 | 18.00 | 17.20 | 16.86 | 16.31 | 15.61 | 2.395 | 2.725 | quasar | 2.565
122619.73+104953.5 | 2012-04-15 | 3600 | 19.06 | 18.35 | 18.27 | 18.27 | 18.10 | 17.49 | 17.23 | 16.78 | 16.12 | 2.565 | 2.625 | quasar | 2.375
124605.36+071128.2 | 2012-03-14 | 3600 | 19.20 | 18.56 | 17.82 | 17.52 | 17.33 | 16.88 | 16.55 | 15.97 | 15.25 | 3.435 | 3.550 | quasar | 2.044
125934.29+075200.7 | 2012-05-16 | 2700 | 18.29 | 17.78 | 17.67 | 17.68 | 17.63 | 17.2 | 16.93 | 16.45 | 15.93 | 2.475 | 2.625 | quasar | 2.370
130318.32+030809.4 | 2012-04-14 | 2700 | 18.39 | 17.65 | 17.58 | 17.47 | 17.35 | 16.6 | 16.25 | 15.87 | 15.37 | 2.595 | 2.675 | quasar | 2.664
131008.67+084405.0 | 2012-04-16 | 2700 | 18.48 | 17.85 | 17.73 | 17.72 | 17.63 | 17.15 | 16.92 | 16.43 | 15.85 | 2.535 | 2.625 | quasar | 2.232
135942.50+022426.0 | 2012-03-14 | 3600 | 24.10 | 18.82 | 18.33 | 18.29 | 18.18 | 17.69 | 17.26 | 16.85 | 16.33 | 3.475 | 3.350 | quasar(BAL) | 3.265
142405.57+044105.5 | 2012-05-16 | 3600 | 18.72 | 18.17 | 18.01 | 18.01 | 17.83 | 17.36 | 17.09 | 16.50 | 15.89 | 2.465 | 2.625 | quasar(BAL) | 2.232
142543.33+024759.8 | 2012-04-15 | 2700 | 18.48 | 17.79 | 17.77 | 17.75 | 17.67 | 16.94 | 16.65 | 16.16 | 15.70 | 2.605 | 2.675 | quasar | 2.689
142854.09+132259.0 | 2012-05-17 | 5400 | 20.89 | 19.46 | 18.94 | 18.37 | 18.10 | 17.26 | 16.86 | 16.38 | 15.83 | 2.735 | 2.925 | quasar | 3.093
144526.15+023906.8 | 2012-05-15 | 3600 | 19.05 | 18.00 | 18.00 | 17.95 | 17.78 | 17.24 | 16.91 | 16.46 | 15.98 | 2.695 | 2.675 | quasar | 2.706
145230.38+130227.3 | 2012-05-17 | 2700 | 18.26 | 17.77 | 17.65 | 17.68 | 17.51 | 16.81 | 16.58 | 16.03 | 15.33 | 2.465 | 2.525 | quasar | 2.468
151321.18+012502.2 | 2012-04-16 | 3600 | 19.45 | 18.18 | 18.28 | 18.17 | 18.05 | 17.34 | 17.01 | 16.36 | 15.81 | 2.865 | 2.825 | quasar | 2.753
152808.87+005211.8 | 2012-04-16 | 3600 | 18.97 | 18.11 | 18.09 | 18.07 | 17.95 | 17.20 | 16.89 | 16.54 | 16.12 | 2.675 | 2.675 | quasar | 2.610
153303.54+064032.9 | 2012-04-15 | 3600 | 22.74 | 18.99 | 18.43 | 18.35 | 18.26 | 17.84 | 17.26 | 16.81 | 16.15 | 3.405 | 3.350 | quasar | 3.422
153319.44+043257.3 | 2012-04-14 | 5400 | 18.81 | 18.10 | 17.99 | 17.96 | 17.74 | 17.28 | 17.08 | 16.74 | 15.95 | 2.535 | 2.575 | low S/N |
153515.55+291038.5 | 2012-04-14 | 2700 | 19.08 | 17.71 | 17.50 | 17.36 | 17.30 | 16.79 | 17.29 | 16.17 | 16.10 | 2.905 | 2.775 | F star |
153550.13+063352.8 | 2012-05-17 | 3600 | 19.08 | 18.43 | 18.32 | 18.32 | 18.12 | 17.90 | 17.57 | 17.15 | 16.37 | 2.535 | 2.275 | low S/N |
153551.88+044416.4 | 2012-05-16 | 3600 | 18.73 | 18.19 | 18.03 | 18.01 | 17.92 | 17.79 | 17.50 | 17.02 | 16.13 | 2.535 | 2.275 | quasar | 2.377
153951.05+020133.8 | 2012-05-16 | 3600 | 18.90 | 18.31 | 18.24 | 18.21 | 18.08 | 17.32 | 17.11 | 16.78 | 16.13 | 2.365 | 2.575 | quasar | 2.569
154503.23+015614.7 | 2012-05-16 | 2700 | 18.36 | 17.90 | 17.64 | 17.65 | 17.51 | 17.03 | 16.71 | 16.21 | 15.42 | 2.505 | 2.225 | low S/N |
162352.69+230119.6 | 2012-05-15 | 5400 | 20.15 | 19.21 | 18.79 | 18.45 | 17.96 | 17.20 | 16.57 | 16.08 | 15.76 | 2.715 | 2.925 | low S/N |
162620.89+282924.7 | 2012-04-15 | 3600 | 19.20 | 18.42 | 18.32 | 18.33 | 18.17 | 17.39 | 17.04 | 16.81 | 16.30 | 2.605 | 2.675 | quasar | 2.534
Note: The SDSS ugriz magnitudes are given in AB system and the UKIDSS YJHK
magnitudes are given in Vega system. $z_{ph1}$, $z_{ph2}$ and $z_{sp}$ are the
photometric redshifts obtained from the 5-band SDSS data by Richards et al.
(2009) and obtained from the 9-band SDSS-UKIDSS data by us, and the spectral
redshifts from our observations, respectively.
Table 3: Spectral parameters and black hole masses of 36 new quasars
Name | redshift | slopea | $\log(L_{1350})$ | FWHM(CIV) | $\log(M_{BH})$ | $\log(L_{\rm bol})$ | $\log(R_{\rm EDD})$
---|---|---|---|---|---|---|---
(SDSS J) | | | (erg/s) | (km/s) | ($M_{\odot}$) | (erg/s) |
J075746.08+232054.2 | 2.532$\pm$0.007 | -1.59 | 46.39 | 6000 | 9.48 | 47.06 | -0.53
081617.55+225604.5 | 2.931$\pm$0.007 | -0.23 | 46.45 | 11538b | 10.08 | 47.11 | -1.07
083255.70+004710.1 | 2.919$\pm$0.007 | -0.95 | 46.43 | 4778 | 9.31 | 47.10 | -0.31
084659.42+253940.9 | 2.892$\pm$0.029 | -1.80 | 46.54 | 4073 | 9.22 | 47.20 | -0.12
085825.51+283258.5 | 3.226$\pm$0.008 | -1.32 | 46.74 | 6038 | 9.68 | 47.41 | -0.37
090233.19+034131.2 | 2.532$\pm$0.028 | -0.39 | 46.30 | 7817b | 9.66 | 46.96 | -0.80
090827.71+011322.5 | 2.083$\pm$0.081 | -1.58 | 46.82 | 8314 | 9.99 | 47.48 | -0.61
091756.58+100836.7 | 2.760$\pm$0.026 | -2.44 | 46.86 | 6797 | 9.84 | 47.52 | -0.42
091857.36+025205.4 | 2.789$\pm$0.038 | -0.71 | 46.52 | 6306 | 9.59 | 47.18 | -0.51
092021.02-020113.7 | 2.826$\pm$0.025 | -3.27 | 46.61 | 6278 | 9.64 | 47.27 | -0.47
093655.11+305855.3 | 3.005$\pm$0.012 | 0.01 | 46.44 | 4699 | 9.30 | 47.10 | -0.29
094137.09+102650.9 | 2.584$\pm$0.024 | -0.99 | 46.54 | 4779b | 9.36 | 47.20 | -0.26
095118.43+083959.9 | 2.523$\pm$0.006 | -0.82 | 46.27 | 6448 | 9.48 | 46.94 | -0.65
100834.73-022302.5 | 3.086$\pm$0.012 | -2.04 | 46.39 | 5209 | 9.36 | 47.05 | -0.41
103301.49+065106.5 | 2.637$\pm$0.027 | -1.60 | 46.46 | 6817b | 9.63 | 47.13 | -0.61
114608.05+094216.5 | 2.580$\pm$0.020 | -0.90 | 46.11 | 6631 | 9.42 | 46.77 | -0.75
115531.45-014611.9 | 3.196$\pm$0.047 | -1.87 | 46.29 | 8483b | 9.73 | 46.96 | -0.88
121510.62+142834.5 | 2.480$\pm$0.073 | -2.20 | 46.62 | 7208 | 9.76 | 47.29 | -0.58
122043.86+011122.1 | 2.565$\pm$0.042 | 1.00 | 45.95 | 3379b | 8.75 | 46.62 | -0.24
122619.73+104953.5 | 2.375$\pm$0.046 | -2.06 | 46.54 | 7050 | 9.70 | 47.20 | -0.60
124605.36+071128.2 | 2.044$\pm$0.019 | -0.10 | 46.53 | 10662b | 10.06 | 47.19 | -0.96
125934.29+075200.7 | 2.370$\pm$0.030 | -2.02 | 46.91 | 11521b | 10.33 | 47.57 | -0.85
130318.32+030809.4 | 2.664$\pm$0.037 | -1.91 | 46.84 | 5772 | 9.69 | 47.50 | -0.29
131008.67+084405.0 | 2.232$\pm$0.050 | -1.57 | 46.55 | 7262 | 9.73 | 47.21 | -0.62
135942.50+022426.0 | 3.265$\pm$0.014 | -0.42 | 46.77 | 8215b | 9.96 | 47.44 | -0.62
142405.57+044105.5 | 2.232$\pm$0.050 | -1.05 | 46.56 | 11934b | 10.17 | 47.22 | -1.05
142543.33+024759.8 | 2.689$\pm$0.035 | -2.51 | 46.88 | 5984 | 9.74 | 47.55 | -0.30
142854.09+132259.0 | 3.093$\pm$0.015 | 0.02 | 46.10 | 5404b | 9.24 | 46.77 | -0.57
144526.15+023906.8 | 2.706$\pm$0.017 | -1.31 | 46.37 | 6040 | 9.48 | 47.03 | -0.55
145230.38+130227.3 | 2.468$\pm$0.015 | -0.79 | 46.31 | 7693 | 9.66 | 46.98 | -0.78
151321.18+012502.2 | 2.753$\pm$0.035 | -1.31 | 46.10 | 7759b | 9.55 | 46.77 | -0.89
152808.87+005211.8 | 2.610$\pm$0.014 | -0.77 | 46.17 | 12187b | 9.98 | 46.83 | -1.25
153303.54+064032.9 | 3.422$\pm$0.021 | -3.41 | 46.89 | 12183 | 10.36 | 47.56 | -0.91
153551.88+044416.4 | 2.377$\pm$0.025 | -1.43 | 46.35 | 9027b | 9.82 | 47.01 | -0.90
153951.05+020133.8 | 2.569$\pm$0.028 | -1.88 | 46.29 | 8925 | 9.78 | 46.96 | -0.92
162620.89+282924.7 | 2.534$\pm$0.034 | -0.85 | 46.48 | 7456 | 9.72 | 47.15 | -0.67
Note: a The slope of the fitted power-law continuum. b Only 1 Gaussian is
fitted to the whole CIV line profile.
Table 4: A catalog of 7727 SDSS-UKIDSS quasar candidates with $2.2\leq
z_{ph}\leq 3.5$ selected from R09
Name | $z_{ph}(R09)$ | $z_{ph}$ | u | g | r | i | z | Y | J | H | K
---|---|---|---|---|---|---|---|---|---|---|---
(SDSS J) | | | | | | | | | | |
000005.95+145310.1 | 2.255 | 2.725 | 21.31 | 20.64 | 20.45 | 20.19 | 19.93 | 19.41 | 18.83 | 18.51 | 17.76
000035.59-003146.1 | 2.255 | 2.725 | 21.63 | 21.04 | 20.83 | 20.4 | 20.36 | 19.19 | 18.92 | 18.31 | 17.51
000041.87-001207.3 | 2.905 | 2.925 | 21.02 | 19.62 | 19.44 | 19.32 | 19.19 | 18.45 | 18.14 | 17.56 | 16.86
000050.59+010959.1 | 2.605 | 2.575 | 19.85 | 19.08 | 19.02 | 19.09 | 18.89 | 18.33 | 18.17 | 17.69 | 16.9
000201.15+001707.4 | 2.465 | 2.475 | 21.48 | 20.77 | 20.69 | 20.65 | 20.18 | 19.71 | 19.36 | 18.77 | 17.68
Note: The SDSS ugriz magnitudes are given in AB system and the UKIDSS YJHK
magnitudes are given in Vega system. $z_{ph}(R09)$ and $z_{ph}$ are the
photometric redshifts obtained from the 5-band SDSS data by Richards et al.
(2009) and obtained from the 9-band SDSS-UKIDSS data by us, respectively. Only
a portion of the table is shown here. The whole table is available in the
electronic version.
Figure 1: The spectra of the 36 new quasars at $2.2<z<3.5$ identified with
the BFOSC of the Xinglong 2.16m telescope, NAOC. The strongest emission line
in each spectrum is Ly$\alpha$+$\rm N\,{\small V}$.
Fig. 1: (Continued)
Figure 2: Upper panels: Comparison of the photometric redshifts in R90 and
ours based on SDSS-UKIDSS 9-band data for 73,011 unidentified quasar
candidates in R09 and the histogram distribution of their differences. Middle
panels: Comparison of the photometric redshifts in R90 with the spectral
redshifts for 24,878 known quasars and the histogram distribution of their
differences. Lower panels: Comparison of the photometric redshifts given by us
with the spectral redshifts for 24878 known quasars and the histogram
distribution of their differences. Figure 3: Upper panel: The distribution of
24,878 known quasars in R09 in the Y-K/g-z color-color diagram. Black dots
represent $z<4$ quasars and red crosses represent $z>4$ quasars. Middle panel:
The distribution of 61,489 unidentified quasar candidates in R09 with
photometric redshift $z_{ph}<2.2$ in the Y-K/g-z color-color diagram. Lower
panel: The distribution of 10687 unidentified quasar candidates in R09 with
photometric redshift $2.2<z_{ph}<4$ (green dots) and and 835 unidentified
quasar candidates in R09 with photometric redshift $z_{ph}>4$ (red crosses) in
the Y-K/g-z color-color diagram. The error bars in the lower-right part of
each panel denote the typical color uncertanty of quasars at different
redshifts. Figure 4: The black solid line denotes the redshift dependence of
the fraction of 24648 known $z<4$ quasars in R09 selected by the Y-K/g-z
criterion. The blue dotted line denotes the fraction of 72,176 unidentified
quasar candidates in R09 with photometric redshift $z_{ph}<4$ selected by the
Y-K/g-z criterion as a function of photometric redshifts. The red dotted line
denotes the possible star contamination rates of these unidentified quasar
candidates in R09 at different photometric redshifts. Figure 5: Upper panel:
The distributions of 17,999 DR9-UKIDSS quasars in the Y-K/g-z color-color
diagram. Quasars with redshift smaller (higher) than 4 are denoted as black
(red) points. The black (red) error bars in the lower-right part denote the
typical color uncertanty of quasars with redshifts smaller (higher) than 4\.
Blue line represents our proposed Y-K/g-z selection criterion for $z<4$
quasars. Middle panel: Black line represents the magnitude dependence of the
selection fraction of 17,999 DR9-UKIDSS quasars by the Y-K/g-z criterion. The
blue and red histograms show the normalized magnitude distribution (fraction
between the number of quasars in each magnitude bin and the total number) for
17,308 quasars selected by the Y-K/g-z criterion and for 87,822 DR9 quasars,
respectively. Lower panel: Black line represents the redshift dependence of
the selection fraction of 17,999 DR9-UKIDSS quasars by our Y-K/g-z criterion.
The blue and red histograms show the normalized redshift distribution
(fraction between the number of quasars in each redshift bin and the total
number) for 17,308 quasars selected by the Y-K/g-z criterion and for 87,822
DR9 quasars, respectively. For clarity all histograms in the middle and lower
panels are magnified by a factor of 5. Figure 6: The distributions of
$\Delta(g-i)$ versus redshift for 17,308 quasars selected by the Y-K/g-z
criterion (blue points) and for 87,822 SDSS-III/DR9 quasars (black points).
The red solid and dashed lines show the distribution of median $\Delta(g-i)$
values in each redshift bin for these two quasar samples. The blue solid and
black dashed histograms shows the distributions of normalized $\Delta(g-i)$
(fraction between the number of quasars in each $\Delta(g-i)$ bin and the
total number of quasars) in the whole range of redshift of these two samples.
The quasar sample selected by the Y-K/g-z criterion is slightly redder than
the SDSS-III/DR9 quasar sample at $z<4$. Figure 7: Upper panel: Red quasars
with UKIDSS data in the Y-K/g-z diagram. Red crosses and squares denote red
quasars from Glikman et al. (2012) and Urrutia e al. (2009) respectively.
There are 11 common quasars in these two samples. Lower panel: Optically
selected type II quasars with UKIDSS data in the Y-K/g-z diagram. Filled
circles denote type II quasars from Reyes et al. (2010). In both panel, black
lines denote the Y-K/g-z selection criterion. Blue and cyan dashed lines
represent the predicted colors of type I quasars and type II quasars,
respectively, at different redshift (up to $z=4.3$ to the right) using the
templates from Polletta et al. (2007).
|
arxiv-papers
| 2012-07-01T13:36:12 |
2024-09-04T02:49:32.461331
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xue-Bing Wu, Wenwen Zuo, Jinyi Yang, Qian Yang, Feige Wang (Peking\n University)",
"submitter": "Xue-Bing Wu",
"url": "https://arxiv.org/abs/1207.0204"
}
|
1207.0277
|
# Various Correlations in Anisotropic Heisenberg XYZ Model with
Dzyaloshinskii-Moriya Interaction
Mamtimin Tursun School of Physics and Electronic Engineering, Xinjiang Normal
University, Urumchi 830054, China Ahmad Abliz aahmad@126.com School of
Physics and Electronic Engineering, Xinjiang Normal University, Urumchi
830054, China Rabigul Mamtimin School of Physics and Electronic Engineering,
Xinjiang Normal University, Urumchi 830054, China Ablimit Abliz School of
Physics and Electronic Engineering, Xinjiang Normal University, Urumchi
830054, China Qiao Pan-Pan School of Physics and Electronic Engineering,
Xinjiang Normal University, Urumchi 830054, China
###### Abstract
$\bf Abstract$ Various thermal correlations as well as the effect of intrinsic
decoherence on the correlations are studied in a two-qubit Heisenberg XYZ spin
chain with the Dzyaloshinski-Moriya ($DM$) interaction along the $z$ direction
($D_{z}$). It is found that tunable parameter $D_{z}$ may play a constructive
role to the concurrence ($C$), classical correlation ($CC$) and quantum
discord ($QD$) in thermal equilibrium while it plays a destructive role to the
$C$, $CC$ and $QD$ in the intrinsic decoherence case.
thermal quantum discord; classical correlation; Intrinsic decoherence
Heisenberg XYZ model; Dzyaloshinski-Moriya interaction
###### pacs:
03.65.Ud, 03.67.Mn, 75.10.Pq
## I introduction
Entanglement is a kind of quantum nonlocal correlation and has been deeply
studied in the past years J1 ; J2 ; J3 ; J4 . The quantum discord ($QD$) which
measures a more general type of quantum correlation, is found to have nonzero
values even for separable mixed states J5 . $QD$ is built on the fact that two
classical equivalent ways of defining the mutual information turn out to be
inequivalent in the quantum domain. In addition, $QD$ is responsible for the
quantum computational efficiency of deterministic quantum computation with one
pure qubit J6 ; J7 ; J8 albeit in the absence of entanglement.
In recent years, the $QD$ has been intensively investigated in the literature
both theoretically J9 ; J10 ; J11 ; J12 ; J13 ; J14 ; J15 ; J16 ; J17 ; J18 ;
J19 ; J20 ; J21 ; J22 ; J23 ; J24 ; J25 ; J26 ; J27 ; J28 ; J29 ; J30 and
experimentally J7 ; J31 . Generally, it is somewhat difficult to calculate
$QD$ and the analytical solutions can hardly be obtained except for some
particular cases, such as the so-called $X$ states J10 . Some researches show
that $QD$, concurrence ($C$) and classical correlation ($CC$) are respectively
independent measures of correlations with no simple relative ordering and $QD$
is more practical than entanglement J7 . B.Dakic et al J24 have introduced an
easily analytically computable quantity, geometric measure of discord ($GMD$),
and given a necessary and sufficient condition for the existence of nonzero
$QD$ for any dimensional bipartite states. Moreover, the dynamical behavior of
$QD$ in terms of decoherence J27 ; J32 ; J33 in both Markovian J11 and Non-
Markovian J12 ; J34 ; J35 cases is also discussed.
In the previous studies, the QD of a two-qubit one-dimensional XYZ Heisenberg
chain with an external magnetic field in thermal equilibrium has been
studied,J36 where many unexpected ways different from the thermal
entanglement have been shown. In Ref.J37 the authors investigated the effect
of Dzyaloshinski–Moriya (DM) interaction,J38 which arises from spin-orbit
coupling, on QD in an anisotropic XXZ model and shown that with the increase
of the DM interaction the QD gradually reduces at finite temperature. The
effect of DM interaction on QD in Heisenberg XY model has also been discussed
in Ref.J16 , in which the authors showed that QD can describe more information
about quantum correlation than quantum entanglement. There are interesting
papers discussing the QD qualitatively and quantitatively in Heisenberg spin
chain models with various factors such as temperature, anisotropies and
magnetic field.J17 In this Letter, we study the QD, CC and $C$ in an
anisotropic Heisenberg XYZ model with the DM interaction both in the thermal
equilibrium case and the intrinsic decoherence case, and discuss how the DM
interaction influence the correlations in such a system. The present study of
the correlations in Heisenberg spin chain model will help us to understand the
effect of DM interaction on the correlations and the phase decoherence
resistance of the correlations more comprehensively.
## II The Model and Definitions of The Various Correlations
We consider the anisotropic $XYZ$ Heisenberg model with the anisotropic,
antisymmetric $DM$ interaction along the $z$ direction
$D_{z}(\sigma^{x}_{1}\sigma^{y}_{2}\times\sigma^{y}_{1}\sigma^{x}_{2}).$ Then
the Hamiltonian of such a model can be expressed as
$\displaystyle\hskip
25.60747ptH=\frac{1}{2}[J_{x}\sigma^{x}_{1}\sigma^{x}_{2}+J_{y}\sigma_{1}^{y}\sigma^{y}_{2}+J_{z}\sigma_{1}^{z}\sigma_{2}^{z}\qquad$
$\displaystyle\qquad\qquad+D_{z}\left(\sigma^{x}_{1}\sigma^{y}_{2}-\sigma^{y}_{1}\sigma^{x}_{2}\right)],$
(1)
where $J_{x}$, $J_{y}$ and $J_{z}$ are the coupling constants;
$\sigma^{x}_{i}$, $\sigma^{y}_{i}$ and $\sigma^{z}_{i}$ are the Pauli
operators acting on qubit $i(i=1,2).$ In the standard basis
${|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle},$
the Hamiltonian can be expressed in the following matrix form
$\displaystyle
H=\frac{1}{2}\left(\begin{array}[]{cccc}J_{z}&0&0&J_{x}-J_{y}\\\
0&-J_{z}&\beta&0\\\ 0&\beta^{\dagger}&-J_{z}&0\\\
J_{x}-J_{y}&0&0&J_{z}\end{array}\right),$ (6)
where $\beta=J_{x}+J_{y}+2iD_{z}.$ We give a brief overview of various
correlation measures. Given a bipartite quantum state $\rho_{AB}$ in a
composite Hilbert space $\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$,
the concurrence J2 as an indicator for entanglement between the two-qubits is
$\displaystyle
C(\rho_{AB})=\max\\{\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4},0\\},$ (7)
where $\lambda_{i}(i=1,2,3,4)$ are the square roots of the eigenvalues of the
“spin-flipped” density operator
$R=\rho\widetilde{\rho}=\rho(\sigma^{y}_{1}\otimes\sigma^{y}_{2})\rho^{*}(\sigma^{y}_{1}\otimes\sigma^{y}_{2})$
in descending order. $\sigma_{y}$ is the Pauli matrix and $\rho^{\ast}$
denotes the complex conjugation of the matrix $\rho$ in the standard basis
${|\uparrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\downarrow\downarrow\rangle}$.
Let us now recalling the original definition of $QD$. In the classical
information theory, the total correlations in a bipartite quantum system $(A)$
and $(B)$ are measured by the quantum mutual information defined as
$\displaystyle{I}(\rho_{A};\rho_{B})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$
(8)
where $\rho_{A(B)}=Tr_{B(A)}(\rho_{AB})$ is the reduced density matrix of the
subsystem $A(B)$ by tracing out the subsystem $B(A)$. The quantum
generalization of the conditional entropy is not the simply replacement of
Shannon entropy with Von Neumann entropy, but through the process of
projective measurement on the subsystem $B$ by a set of complete projectors
$B_{k}$, with the outcomes labeled by $k$, then the conditional density matrix
$\rho_{k}$ becomes
$\displaystyle\rho_{k}=\frac{1}{p_{k}}(\mathbbm{l}_{A}\otimes{B_{k}})\rho(\mathbbm{l}_{A}\otimes{B_{k}}),$
(9)
which is the locally post-measurement state of the subsystem $B$ after
obtaining the outcome $k$ on the subsystem $A$ with the probability
$\displaystyle
p_{k}=\mathrm{Tr}[(\mathbbm{l}_{A}\otimes{B_{k}})\rho(\mathbbm{l}_{A}\otimes{B_{k}})],$
(10)
where $\mathbbm{l}_{A}$ is the identity operator on the subsystem $A$. The
projectors ${B_{k}}$ can be parameterized as ${B_{k}}=V|k\rangle\langle
k|V^{\dagger},k=0,1$ and the transform matrix $V\in U(2)$ J9 is
$V=\left(\begin{array}[]{cccc}\cos\theta&e^{-i\phi}\sin\theta\\\
e^{i\phi}\sin\theta&-\cos\theta\\\ \end{array}\right).$ (11)
Then the conditional Von Neumann entropy (quantum conditional entropy) and
quantum extension of the mutual information can be defined as J5
$\displaystyle S(\rho|\\{B_{k}\\})=\sum_{k}p_{k}S(\rho_{k}),$ (12)
following the definition of the $CC$ in Ref.J5
$\displaystyle
CC(\rho_{AB})=\sup_{\\{B_{k}\\}}\\{S(\rho_{A}(t))-S(\rho_{AB}(t)|\\{B_{k}\\})\\},$
(13)
then $QD$ defined by the difference between the quantum mutual information
$I(\rho_{AB})$ and the $CC(\rho_{AB})$ is given by
$QD(\rho_{AB})=I(\rho_{AB})-\ CC(\rho_{AB}).$ If we denote
$\mathrm{S_{\min}(\rho_{AB})=\min_{\\{B_{k}\\}}S(\rho_{AB}|\\{B_{k}\\}),}$
then a variant expression of $CC$ and $QD$ in Ref.J5 ; J12
$\displaystyle
CC(\rho_{AB})=S(\rho_{A})-min_{\\{B_{k}\\}}S(\rho_{AB}|\\{B_{k}\\}),$ (14)
$\displaystyle QD(\rho_{AB})=S(\rho_{B})-S(\rho_{AB})+S_{\min}(\rho_{AB}).$
(15)
## III Effects of DM interaction on Various thermal correlations
A typical solid-state system at thermal equilibrium in temperature $T$
(canonical ensemble) is $\rho(T)=e^{-\frac{H}{KT}}/Z$, with
$Z=Tr[e^{-\frac{H}{KT}}]$ the partition function and $K$ is the Boltzmann
constant. Usually we work with natural unit system $\hbar=K=1$ for simplicity
and henceforth. This density matrix can be worked out as
$\rho(T)=\frac{1}{Z}\left(\begin{array}[]{cccc}\rho_{11}&0&0&\rho_{41}\\\
0&\rho_{22}&\rho_{23}^{\dagger}&0\\\ 0&\rho_{23}&\rho_{22}&0\\\
\rho_{41}&0&0&\rho_{11}\end{array}\right),$ (16)
where the elements of the matrix have been defined as
$\displaystyle\rho_{11}=\frac{1}{2}[e^{-\frac{J_{x}+J_{y}+J_{z}}{2T}}(e^{\frac{J_{x}}{T}}+e^{\frac{J_{y}}{T}})],$
$\displaystyle\rho_{41}=\frac{1}{2}[e^{-\frac{J_{x}+J_{y}+J_{z}}{2T}}(-e^{\frac{J_{x}}{T}}+e^{\frac{J_{y}}{T}})],$
$\displaystyle\rho_{22}=\frac{e^{{\frac{J_{z}}{2T}}}\cosh[\frac{\sqrt{(J_{x}+J_{y})^{2}+4D^{2}_{z}}}{2T}]}{Z},$
$\displaystyle\rho_{23}=\frac{(J_{x}+J_{y}-2iD_{z})e^{{\frac{J_{z}}{2T}}}\sinh[\frac{\sqrt{(J_{x}+J_{y})^{2}+4D^{2}_{z}}}{2T}]}{\sqrt{(J_{x}+J_{y})^{2}+4D^{2}_{z}}Z},$
and
$\displaystyle
Z=e^{-\frac{J_{x}+J_{y}+J_{z}}{2T}}(e^{\frac{J_{x}}{T}}+e^{\frac{J_{y}}{T}})$
$\displaystyle+2e^{{\frac{J_{z}}{2T}}}\cosh[\frac{\sqrt{(J_{x}+J_{y})^{2}+4D^{2}_{z}}}{2T}].$
Figure 1: (Color online) The concurrence (a), classical correlation (b) and
quantum discord (c) versus $T$ and $D_{z}$. Here $J_{x}$=0.2, $J_{y}$=0.4,
$J_{z}$=0.8.
According to the above definitions of $C$, $CC$ and $QD$, we will now discuss
them with the corresponding plots. Fig. 1(a) shows that in the case of the
temperature $T$ with finite value, $C(T)$ increases monotonously with the
increasing of $D_{z}$, by which one can also achieve maximum entanglement even
at finite low temperatures. Both $QD(T)$ and $CC(T)$ are zero when the
temperature is zero, which is totally different from the case of $C(T)$ (it
takes the maximum in this case). But there is an apparent increase, which is
sharper for larger $D_{z}$, followed by a gradual decrease when the
temperature is increased gradually starting from zero. More interestingly, the
$QD(T)$ and $CC(T)$ show the same characteristics in their behavior following
the increasing of the absolute value of $D_{z}$, which is different from the
entanglement. The saddle-like structure of $QD(T)$ and $CC(T)$ in this case
reveals the constructive role of $D_{z}$ for the two correlations, one
quantum, one classical, which is one of the interesting results of this work.
All of the above three correlations do not undergo sudden death, instead, they
tend asymptotically towards zero as the temperature is increased.
In the overall, we conclude that $D_{z}$ is an efficient parameter in
increasing various correlations such as $C$, $CC$ and $QD$ at finite
temperature. This is partly contrary to the result for the case of XXZ model,
in which the increase of the $DM$ interaction suppresses the $QD$ J38 .
Moreover, the $QD$ shows a different behavior from the $C$ in the response to
the variation of $DM$ interaction.
## IV Intrinsic Decoherence of Various Correlations
Now, we take the influence of intrinsic decoherence on the various
correlations into account. According to the Milburn’s equation J39 followed
by the assumption that a system does not evolve continuously under unitary
transformation for sufficiently short time steps, the master equation for pure
phase decoherence is given by
$\displaystyle\frac{d\rho(t)}{dt}=-i[H,\rho]-\frac{1}{2\gamma}[H,[H,\rho(t)]],$
(17)
where $\gamma$ is the phase decoherence rate. In the limit $\gamma\rightarrow
0$ the Schodinger,s equation is recovered. The formal solution of the master
equation above can be given by J40
$\displaystyle\rho(t)=\sum_{k=0}^{\infty}\frac{l^{k}}{k!}M^{k}(t)\rho(0)M^{\dagger
k}(t),$ (18)
where $\rho(0)$ is the density operator of the initial system and $M^{k}(t)$
is defined by
$\displaystyle M^{k}(t)=H^{k}e^{-iHt}e^{-\frac{t}{2\gamma}H^{2}}.$ (19)
By inserting the completeness relation
$\sum_{n}|\psi_{n}\rangle\langle\psi_{n}|=1$ of the energy eigenstate into
master equation J39 , we can write the explicit expression of the density
matrix of the states as
$\displaystyle\hskip
17.07164pt\rho(t)=\sum_{mn}\mathrm{exp}\bigg{[}-\frac{\gamma
t}{2}(E_{m}-E_{n})^{2}-i(E_{m}-E_{n})t\bigg{]}\qquad$
$\displaystyle\qquad\times\langle\psi_{m}|\rho(0)|\psi_{n}\rangle|\psi_{m}\rangle\langle\psi_{n}|.$
(20)
We assume that the system is initially prepared in the Bell state
$|\Psi(0)\rangle=\frac{1}{\sqrt{2}}\big{(}|01\rangle+|10\rangle\big{)}$. From
the Eq. (16), the time evolution for this initial state can be obtained as
$\rho(t)=\left(\begin{array}[]{cccc}0&0&0&0\\\ 0&\rho_{22}&\rho_{23}&0\\\
0&\rho_{32}&\rho_{33}&0\\\ 0&0&0&0\end{array}\right),$ (21)
where the elements of the matrix can be defined as
$\displaystyle\rho_{22}=\frac{1}{2}+\frac{D_{z}e^{-\frac{1}{2}\gamma\mu^{2}t}\sin[\mu
t]}{\mu^{2}},$
$\displaystyle\rho_{23}=\frac{J_{x}+J_{y}-2iD_{z}e^{-\frac{1}{2}t\gamma\mu^{2}}\cos[t\mu]}{2(J_{x}+J_{y}-2iD_{z})},$
$\displaystyle\rho_{32}=\frac{J_{x}+J_{y}+2iD_{z}e^{-\frac{1}{2}\gamma\mu^{2}t}\cos[\mu
t]}{2(J_{x}+J_{y}+2iD_{z})},$
$\displaystyle\rho_{33}=\frac{1}{2}-\frac{D_{z}e^{-\frac{1}{2}\gamma\mu^{2}t}\sin[\mu
t]}{\mu^{2}}$
and where
$\displaystyle\mu=\sqrt{J^{2}_{x}+2J_{x}J_{y}+J^{2}_{y}+4D^{2}_{z}}.$
Figure 2: (Color online) The lower part of the figure is the concurrence
(black line), quantum discord (blue line) versus time $t$ with the system
parameters fixed as $J_{x}$=0.03, $J_{y}$=0.06, $\gamma=0.01$ and $D_{z}$=6.
The upper part of the figure is the concurrence (black line), quantum discord
(blue line), classical correlation (red line) versus time $t$ with the system
parameters fixed as $J_{x}$=3, $J_{y}$=0.6, $\gamma=0.1$ and $D_{z}$=0.1
(dotted line), $D_{z}$=0.3 (thin line).
In order to highlight the effect of the pase decoherence $\gamma$ on the
various correlations, we plot the time evolutions of correlations with
different values of $D_{z}$ in Fig. 2. It can be seen from the lower part of
the figure that the time evolution of the entanglement and quantum discord
exhibit the interesting phenomena of ”sudden death” and ”sudden revival”,J4
which occur when the spin-orbit coupling is large and spin-spin coupling is
small. Secondly, with the aim to clarify the joint influence of the system
parameters with the phase decoherence on the time evolution, the combination
of the system parameters for the upper part of the figure is chosen as the
optimum one based on the numerical analysis. All of the three correlations
exhibit oscillatory behavior, which ultimately ends with a steady state value.
Oscillations are suppressed obviously with the increase of $\gamma$. The CC
ends with the maximum value, while the other two end with a smaller steady
state value with respect to the starting maximum value. Importantly, the final
steady state values of the entanglement and quantum discord are all still high
despite the phase decoherence, implying that the optimum combination of the
system parameters can keep the correlations highly immune to the pure phase
decoherence. Moreover, one can see that the quantum discord is more fragile
under the phase decoherence than the entanglement, which is different from the
result that it is more resistant against the environment than entanglement.
Last but not least, the larger the DM interaction, the severer the collapse of
the correlations, which is the opposite of the thermal case.
## V Conclusion
In conclusion, we have studied the various correlations, particularly the
quantum discord in an anisotropic two-qubit Heisenberg XYZ model with the
presence of DM interaction. Results are presented both for the case at thermal
equilibrium and under phase decoherence and they show that the roles of the DM
interactions in controlling the thermal quantum discord are opposite to the
case of the XXZ. It is found constructive in the case of XYZ model under our
consideration. However, this is not the same story for the case of phase
decoherence, where the DM becomes destructive. The time evolution of the
entanglement and quantum discord shows the famous phenomena of collapse and
revival. Though the quantum discord is shown to be more sensitive to the phase
decoherence than the entanglement, optimum combination of the system
parameters can protect the correlations effectively against the influence of
the phase decoherence on the whole.
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|
arxiv-papers
| 2012-07-02T04:56:41 |
2024-09-04T02:49:32.481976
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mamtimin Tursun, Ahmad Abliz, Rabigul Mamtimin, Ablimt Abliz, Pan\n Pan-Qiao",
"submitter": "Mamtimin Tursun",
"url": "https://arxiv.org/abs/1207.0277"
}
|
1207.0330
|
# Fonctions continues sur un compact totalement ordonné
Daher Mohammad et Khalil SAADI m.daher@orange.fr
kh_saadi@yahoo.fr
###### Abstract.
Dans ce travail, nous construisons un compact $K$ (non métrisable) totalement
ordonné à partir d’un ensemble $E$ totalement ordonné (muni de la topologie de
l’ordre) et d’une mesure positive sur $E$.
Si $(E,\tau_{0})$ est analytique, nous montrons que $K$ est un compact de
Rosenthal.
Dans la suite, nous prouvons que $K$ est isomorphe à une partie de
$E\times\left\\{0,1\right\\},$ muni de la topologie de l’ordre lexicographique
lorsque $(E,\tau_{0})$ est connexe et $-\infty,+\infty\in E$.
Finalement nous étudions le cas où $E$ est un groupe satisfaisant quelques
propriétés.
Abstract:In this work, we construct a totally compact ordered space $K$ from a
totally ordered space $E$ and a positive measure on E.
If $(E,\tau_{0})$ is an analytic space, we show that $K$ is a Rosenthal
compact set.
In the case where $(E,\tau_{0})$ is connected and $-\infty,+\infty\in E$, we
show that $K$ is isomorphic to subset of $E\times\left\\{0,1\right\\}$ (we
equip this set the order topology given by the lexicographic order).
Finally we study a particular case, when $E$ is a group satisfies some
properties.
Introduction
Soit $(E,\leq)$ un ensemble totalement ordonné; la topologie de l’ordre
$\tau_{0}$ sur $E$ est la topologie engendrée par les intervalles de la forme,
$\left\\{x\in E;\text{ }x<a\right\\},$ ou de la forme, $\left\\{x\in E;\text{
}x>b\right\\},$ $a,b\in E.$ Cette topologie est séparée,
En effet,
Soit $x,y\in E$ tel que $x<y;$ supposons qu’il existe $a\in E$ verifiant que
$x<a<b.$ Choisissons $V_{x}=\left\\{t\in E;\text{ }t<a\right\\}$ et
$V_{y}=\left\\{t\in E;\text{ }t>a\right\\}$; il est clair que $V_{x}$ est un
ouvert contenant $x$, $V_{y}$ est un ouvert contenant $y$ et $V_{x}\cap
V_{y}=\emptyset.$
S’il n’existe aucun élément entre $x$ et $y,$ on considère $V_{x}=\left\\{t\in
E;\text{ }t<y\text{ }\right\\}$ et $V_{y}=\left\\{t\in E;\text{
}t>x\right\\},$ on a encore $V_{x}\cap\cap V_{y}=\emptyset.$
Désignons par $\tau_{1}$ (resp.$\tau_{2})$ la topologie engendrée par les
intervalles de la forme, $\left\\{x\in E;\text{ }x>a\right\\},$ ou de la
forme, $\left\\{x\in E;\text{ }x\leq b\right\\},$ $a,b\in E$ $\ (resp.$ de la
forme $\left\\{x\in E;\text{ }x<a\right\\}$ ou de la forme, $\left\\{x\in
E;\text{ }x\geq b\right\\},$ $a,b\in E).$
Rappelons que l’ordre lexicographique sur $E\times\left\\{0,1\right\\}$ est
défini comme suivant:
$(r,i)\leq(s,j)$ si et seulement si $r<s$ ou $r=s$ et $i\leq j$
$\forall(r,i),$ $(s,j)\in E\times\left\\{0,1\right\\}.$
Le but de ce travail est de construire un compact $K,$ totalement ordonné, à
partir d’un ensemble de référence $E$ totalement ordonné et d’une mesure
(naturelle) positive $\mu$ sur $E$. Si $(E,\tau_{0})$ est analytique nous
obtenons un compact de Rosenthal.
Dans la suite, nous supposons que $(E,\tau_{0})$ est connexe et
$-\infty,+\infty\in E$. Nous montrons que $K$ est isomorphe à une partie de
$E\times\left\\{0,1\right\\}$ muni de la topologie de l’ordre lexicographique.
Nous signalons que si $K$ est un compact totalement ordonné séparable, $K$ est
isomorphe à un sous-ensemble de $\left[0,1\right]\times\left\\{0,1\right\\},$
muni de la topologie de l’ordre lexicographique [Osta].
Finalement nous montrons qu’il existe une topologie séparée $\tau^{\prime}$
sur $C(K)$ moins fine que la topologie de la convergence simple $\tau_{p}$ et
deux sous-espaces $Y_{1},Y_{2},$ $\tau_{p}$ fermés de $C(K)$ tels que
$(Y_{j},\tau^{\prime})=(Y_{j},\tau_{p}),$ $j=1,2$,
($Y_{1},\tau_{p})\oplus(Y_{2},\tau_{p})=(C(K),\tau_{p})$ et
$(C(K),\tau^{\prime})$ n’est pas universellement mesurable, lorsque $E$ est un
groupe localement compact séparable totalement ordonné satisfaisant les deux
conditions suivantes:
$\forall\text{ }a,b\in E\text{ tel que }a<b\text{, }\exists\text{ }c\in
E\text{ v\'{e}rifiant que }a<c<b$ (*)
$\text{ }a\leq b\text{ }\Longleftrightarrow a-b\leq 0,\text{ }a,b\in E$ (**)
###### Lemme 1.
Soit $(E,\leq)$ un ensemble totalement ordonné; supposons que
1) La topologie $\tau_{0}$ est séparable.
2) $(E,\leq)$ vérifie la condition $(\ast).$
Alors
(i) $(E,\tau_{j})$ est un espace de fortement de Lindelöf, $j=1,2$ (c’est-à-
dire que tout ouvert de $(E,\tau_{j})$ est un espace de Lindelöf, $j=1,2$).
(ii) Bor$(E,\tau_{0})=Bor(E,\tau_{j}),$ $j=1,2.$
Nous allons montrer le lemme 1, pour $j=1,$ le cas $j=2$ se démontre par une
méthode analogue.
Démonstration de (i).
L’argument qui nous permet de montrer $(i)$ est analogue à celui de [Kell,
p.58,59].
En effet,
_Etape 1:_ Montrons que $(E,\tau_{0})$ a une base dénombrable.
Soit $(a_{n})_{n\in\mathbb{N}}$ une suite dense dans $(E,\tau_{0})$; pour tout
$m,n\in\mathbb{N}$ posons $V_{m,n}=\left\\{x\in E;\text{ }a_{n}>\text{
}x>a_{m}\right\\}$ et $M=\left\\{(m,n)\in\mathbb{N}^{2};\text{
}V_{m,n}\neq\emptyset\right\\}.$ On se propose de montrer que
$(V_{m,n})_{(m,n)\in M}$ forme une base de $(E,\tau_{0}).$
Soit $a,b,c\in E$ tel que $a<c<b;$ comme $(a_{n})_{n\geq 0}$ est une suite
dense dans $E$ et $(E,\leq)$ vérifie la condition $(\ast),$ il existe $m,n\in
E$ tel que $a<a_{m}<c<a_{n}<b$. Il est clair que $c\in
V_{m,n}\subset\left]a,b\right[.$ Nous déduisons que $(V_{m,n})_{(m,n)\in M}$
forme une base de $(E,\tau_{0})$ (comme $E$ vérifie la condition $(\ast),$
$\tau_{0}$ est engendrée par les ouverts sous la forme, $\left]a,b\right[$ ,
$a,b\in E$ avec $a<b$).
Soient maintenant $\left\\{\left]a_{i},b_{i}\right];\text{ }a_{i}<b_{i}\text{
et }i\in I\right\\}$ une famille d’intervalles dans $E$, $U$ $=\underset{i\in
I}{\cup}\left]a_{i},b_{i}\right]$ et $V$ $=\underset{i\in
I}{\cup}\left]a_{i},b_{i}\right[.$ Il s’agit de trouver un sous-ensemble
$I^{\prime}$ dénombrable de $I$ tel que $U$ $=\underset{i\in
I^{\prime}}{\cup}\left]a_{i},b_{i}\right].$ Désignons par $J$ le sous-ensemble
de $I$ tel que $\left\\{b_{j};\text{ }j\in J\text{ }\right\\}\cap
V=\emptyset.$
_Etape 2:_ Montrons que $J$ est dénombrable.
D’après l’étape 1, il existe un sous-ensemble dénombrable $H$ de $J$ , tel que
$\underset{i\in J}{\cup}\left]a_{i},b_{i}\right[=\underset{i\in
H}{\cup}\left]a_{i},b_{i}\right[.$ D’autre part $\left]a_{j},b_{j}\right[\cap$
$\left]a_{j^{\prime}},b_{j^{\prime}}\right[=\emptyset$, si $j\neq j^{\prime},$
$j,j^{\prime}\in J$ (car si $\left]a_{j},b_{j}\right[\cap$
$\left]a_{j^{\prime}},b_{j^{\prime}}\right[\neq\emptyset$, $b_{j}\in V$ ou
$b_{j^{\prime}}\in V)$ par conséquent $J=H$ est un ensemble dénombrable.
En réappliquant l’étape 1, on voit qu’il existe un sous-ensemble dénombrable
$I_{1}$ de $I,$ tel que $V=\underset{i\in
I_{1}}{\cup}\left]a_{i},b_{i}\right[.$
Notons $I_{2}=\left\\{j\in I;\text{ }b_{j}\in V\text{ }\right\\}$ ($I=J\cup
I_{2});$ on a alors $\underset{i\in
I_{2}}{\cup}\left]a_{i},b_{i}\right]\subset V=\underset{i\in
I_{1}}{\cup}\left]a_{i},b_{i}\right[.$
D’après ce qui est précède $U=\left[\underset{i\in
J}{\cup}\left]a_{i},b_{i}\right]\right]\cup\left[\underset{i\in
I_{2}}{\cup}\left]a_{i},b_{i}\right]\right]=\left[\underset{i\in
J}{\cup}\left]a_{i},b_{i}\right]\right]\cup\left[\underset{i\in
I_{1}}{\cup}\left]a_{i},b_{i}\right]\right]=\underset{i\in
I^{\prime}}{\cup}\left]a_{i},b_{i}\right]$, où $I^{\prime}=J\cup I_{1}.$
Si $U=\underset{i\in I}{\cup}\left\\{x\in E;\text{ }x\leq b_{i}\right\\},$ par
un argument analogue à celui du précédent, on montre qu’il existe un sous-
ensemble dénombrable $J_{1}$ de $I,$ tel que $U=\underset{i\in
J_{1}}{\cup}\left\\{x\in E;\text{ }x\leq b_{i}\right\\}.$ Il en résulte que
$(E,\tau_{1})$ est un espace fortement de Lindelöf.$\blacksquare$
Démonstration de (ii).
Comme $(E,\tau_{0})$ est moins fine que $(E,\tau_{1}),$ alors
$Bor(E,\tau_{0})\subset Bor(E,\tau_{1}).$ Pour voir l’inclusion inverse; il
suffit de voir que tout ouvert de $(E,\tau_{1})$ est un borélien de
$(E,\tau_{0})$ (car $(E,\tau_{1})$ est fortement de Lindelöf et
$\left]a,b\right]\in Bor(E,\tau_{0}),$ pour tout $a,b\in E$ avec $a<b$), il en
résulte que $Bor(E,\tau_{1})\subset Bor(E,\tau_{0}),$ d’où
$Bor(E,\tau_{1})=Bor(E,\tau_{0}).\blacksquare$
Soit $(E,\leq)$ un ensemble totalement ordonné; si $E$ admet un maximum, on
note $+\infty=\max_{x\in E}x$ et si $E$ admet un minimum, on note -$\infty$
$=\min_{x\in E}x$.
Dans la suite, on suppose que $-\infty,\infty\notin E$ (dans la cas contraire
on remplace $E$ par $E-\left\\{-\infty,+\infty\right\\}).$
Soient $E$ un ensemble totalement ordonné et $\mu$ une mesure borélienne
positive sur $(E,\tau_{0})$ telle que
$\mu(\left\\{u\in E;\text{ }u>t\right\\})>0\text{, pour tout }t\in E\text{ }$
(0.1)
Soit $t\in E;$ désignons par $f^{t}$ la fonction caractéristique de
$\left\\{u\in E;\text{ }u>t\right\\}$, par $A$ la
sous-$\mathbb{C}^{\ast}$-algèbre de $L^{\infty}(\mu)$ (avec unité) engendrée
par les $f^{t},$ $t\in E$ et par $K$ l’ensemble des caractères sur $A$.
D’après [Godem]$-$[Sto] $K$ est $\sigma(A^{\ast},A)$ compact et $A=C(K)$.
Pour tout $t\in E,$ désignons par $\left[t\right]=\left\\{t^{\prime}\in
E;\text{ }\left\|f^{t}-f^{t^{\prime}}\right\|_{L^{\infty}(E,\mu)}=0\right\\};$
supposons que
$\left\\{\left[t\right];\text{ }t\in E\right\\}\text{ n'est pas
d\'{e}nombrable.}$ (0.2)
Comme $\left\|f^{t}-f^{t^{\prime}}\right\|_{L^{\infty}(E,\mu)}\geq 1$ pour
tout $t,t^{\prime}\in E$ vérifiant que
$\left[t\right]\neq\left[t^{\prime}\right],$ $K$ n’est pas métrisable.
Définissons la fonction $\psi:K\times E\rightarrow\left\\{0,1\right\\},$ par
$\psi(\theta,t)=\theta(f^{t}),\text{ }\theta\in K\text{, }t\in E\text{.}$
Notons $B_{1}=\left\\{\theta\in K\text{; l'application
}t\in(E,\tau_{0})\rightarrow\psi(\theta,t)\text{ est continue}\right\\}.$
Soit $\theta\in K-B_{1};$ on a deux cas:
_Cas 1:_ $\left\\{t\in E;\text{ }\psi(\theta,t)=0\right\\}$ n’est pas fermé:
Soit $h(\theta)$ un point adhérent à $\left\\{t\in E;\text{
}\psi(\theta,t)=0\right\\}$ dans $E$ tel que $\psi(\theta,h(\theta))=1.$
###### Lemme 2.
Le point $h(\theta)$ est unique.
Démonstration.
Supposons que $h^{\prime}(\theta)$ est un point adhérent à $\left\\{t\in
E;\text{ }\psi(\theta,t)=0\right\\}$ tel que
$\psi(\theta,h^{\prime}(\theta))=1$ et $h(\theta)<h^{\prime}(\theta);$
l’ensemble $\left\\{u\in E;\text{ }u<h^{\prime}(\theta)\right\\}$ est un
voisinage de $h(\theta)$ et $h(\theta)$ est adhérent à $\left\\{u\in E;\text{
}\psi(\theta,u)=0\right\\},$ il existe donc $u_{1}\in\left\\{u\in E;\text{
}\psi(\theta,u)=0\right\\}$ tel que $u_{1}<h^{\prime}(\theta);$ ceci est
impossible, car l’application $t\in E\rightarrow\psi(\theta,t)$ est
décroissante (si $t\leq t^{\prime},$ $f^{t}\geq f^{t^{\prime}})$ et
$\psi(\theta,h^{\prime}(\theta))=1$, par conséquent $h(\theta)$ est
unique.$\blacksquare$
_Cas 2:_ $\left\\{t\in E;\psi(\theta,t)=1\right\\}$ n’est pas fermé: On
désigne encore par $h(\theta)$ l’unique point adhérent à $\left\\{t\in
E;\text{ }\psi(\theta,t)=1\right\\}$ dans $E$ tel que
$\psi(\theta,h(\theta))=0$ (par un raisonnement analogue à celui de lemme 2 on
montre que $h(\theta)$ est unique).
Désignons par $B_{2}=\left\\{\theta\in K-B_{1};\text{
}\psi(\theta,h(\theta))=1\right\\}$ et par $B_{3}=\left\\{\theta\in
K-B_{1};\text{ }\psi(\theta,h(\theta))=0\right\\}.$
###### Lemme 3.
Pour tout $\theta\in K-B_{1}$ on a
$h(\theta)=\sup\left\\{t\in E;\text{ }\psi(\theta,t)=1\right\\}.$ (0.3)
Démonstration.
Soit $\theta\in K-B_{1}.$
_Cas 1,_ $\theta\in B_{2}:$ Considérons $v\in\left\\{t\in E;\text{
}\psi(\theta,t)=1\right\\};$ il s’agit de montrer que $v\leq h(\theta)$ .
Supposons que $v>h(\theta)$; l’ensemble $\left\\{u\in E;\text{ }v>u\right\\}$
est un voisinage de $h(\theta)$ et $h(\theta)$ est adhérent à $\left\\{u\in
E;\text{ }\psi(\theta,u)=0\right\\},$ il existe donc $u_{1}\in E$ vérifiant
que $v>u_{1}$ et $\psi(\theta,u_{1})=0$, ceci est impossible car
$\psi(\theta,v)=1.$
_Cas 2,_ $\theta\in B_{3}:$ Remarquons d’abord que si $v\in\left\\{t\in
E;\text{ }\psi(\theta,t)=1\right\\},$ $v\leq h(\theta)$ (car
$\psi(\theta,h(\theta))=0).$
Considérons $a\in E$ tel que $v\leq a,$ $\forall v\in\left\\{t\in E;\text{
}\psi(\theta,t)=1\right\\}$ et montrons que $h(\theta)\leq a.$
Supposons que $h(\theta)>a;$ par la définition de $h(\theta),$ il existe
$v_{1}\in E$ tel que $v_{1}>a$ et $\psi(\theta,v_{1})=1,$ on déduit que
$v_{1}\leq a,$ ceci est impossible, donc $h(\theta)\leq a.\blacksquare$
Par un argument analogue à celui du lemme 3, on montre:
###### Lemme 4.
Pour tout $\theta\in K-B_{1}$ on a
$h(\theta)=\inf\left\\{t\in E;\text{ }\psi(\theta,t)=0\right\\}.$
###### Exemple 1.
Soit $E$ un groupe (non dénombrable) localement compact abélien séparable
$(cf.$[Hew]$)$; supposons que $E$ vérifie la condition $(\ast)$ et que la
topologie de l’ordre coincide avec celle de $E$. Choisissons $\mu=m$ la mesure
de Haar sur $E.$
Montrons que tout ouvert de $E$ non vide est de mesure strictement positive.
Soit $O$ un ouvert non vide de $E;$ pour tout $x\in E,$ $x+O$ est un voisinage
de $x;$ remarquons que $E=\underset{x\in E}{\cup}(x+O);$ d’autre part $E$ est
séparable, d’après l’étape 1 du lemme 1, $E$ possède d’une base dénombrable,
par conséquent il existe une suite dénombrable $(x_{n})_{n\geq 0}$ dans $E,$
telle que $E=\underset{k\geq 0}{\cup}(x_{k}+O),$ il en résulte qu’il existe
$k_{0}\in\mathbb{N}$ vérifiant que $\mu(x_{k_{0}}+O)=\mu(O)>0.$ Comme
$(E,\leq)$ vérifie la condition $(\ast)$ pour tout $a,b\in E$ avec $a<b,$ $\
\mu(\left]a,b\right[)>0,$ donc $\mu$ vérifie les conditions (0.1) et (0.2).
Pour $g\in C(K)\subset L^{\infty}(E,\mu\mathbb{)},$ $\theta\in K$ et $t\in E,$
on définit $g_{t}\in C(K)$ par $g_{t}(u)=g(u-t),$ pour presque tout $u\in E$
et $\theta_{t}\in K$ par $\theta_{t}(r)=\theta(r_{t}),$ $r\in C(K).$
Notons $f$ la fonction caractéristique de $\left\\{t\in E;\text{
}t>0\right\\}.$
###### Lemme 5.
Supposons de plus que $E$ vérifie la condition $(\ast\ast)$. Alors pour tout
$\theta\in$ $B_{2},$ $h(\theta_{u})=h(\theta)-u,$ $\forall u\in E.$
Démonstration.
Comme $E$ vérifie la condition $(\ast\ast)$ on a $f_{t}=f^{t}$, pour tout
$t\in E$. Soient $\theta\in B_{2}$ et $t,u\in E$ tels que
$\psi(\theta_{u},t)=1;$ on a alors
$\psi(\theta_{u},t)=\theta_{u}(f_{t})=\theta\left[(f_{t})_{u}\right]=\theta(f_{t+u})=\psi(\theta,t+u)=1,$
ceci implique que $t+u\leq h(\theta);$ la condition $(\ast\ast)$ entraîne que
$t\leq h(\theta)-u.$ D’après le lemme 3 on a
$h(\theta_{u})=\sup\left\\{t;\text{ }\psi(\theta_{u},t)=1\right\\}\leq
h(\theta)-u.$
Inversement; soit $t\in E$ tel que $\psi(\theta,t)=1;$ pour tout $u\in E$ on a
$\psi(\theta_{u},t-u)=\theta_{u}(f_{t-u})=\theta((f_{t-u})_{u})=\theta(f_{t})=\psi(\theta,t)=1$,
donc $h(\theta_{u})\geq t-u,$ comme $E$ vérifie la condition ($\ast\ast$) on a
$h(\theta_{u})+u\geq t.$ En appliquant le lemme 3 on voit que
$h(\theta_{u})+u\geq h(\theta),$ c’est-à-dire que $h(\theta_{u})\geq
h(\theta)-u,$ d’où $h(\theta_{u})=h(\theta)-u.\blacksquare$
###### Remarque 1.
Dans le lemme précédent on peut remplacer $B_{2}$ par $B_{3}.$
###### Remarque 2.
Sous l’hypothèse du lemme 5, $B_{2}\neq\emptyset.$
En effet,
Supposons que $B_{2}=\emptyset;$ la proposition 8 nous montrera que
$B_{3}=\emptyset,$ donc $K=B_{1}.$ D’autre part le théorème 2 nous montrera
que $B_{1}$ a une base dénombrable, il en résulte que $K$ est métrisable, ce
qui et impossible, car $(E,\mu)$ vérifie la condition (0.2).
###### Exemple 2.
Soient $E=\left[0,1\right]$ et $\mu$ la mesure de Lebesgue sur $E.$
Le corollaire 8, montrera que $K$ s’identifie à
$\left]0,1\right]\times\left\\{1\right\\}\cup\left[0,1\right[\times\left\\{0\right\\}$
qui est un compact de l’intervalle éclaté
$\left[0,1\right]\times\left\\{0,1\right\\}.$
Soit $I$ un ensemble; désignons par $c_{0}(I)$ l’espace des suites (complexes)
$(x_{i})_{i\in I}$ telle que pour tout $\varepsilon>0,$ il existe un sous-
ensemble fini $I_{1}$ de $I$ vérifiant que $\left|x_{i}\right|<\varepsilon,$
pour tout $i\in I-I_{1}.$
Notons $\left\|x\right\|_{c_{0}(I)}=\sup_{i\in I}\left|x_{i}\right|,$ pour
tout $x\in c_{0}(I);$ $(c_{0}(I),\left\|.\right\|)$ est un espace de Banach.
###### Exemple 3.
Soient $E$ un ensemble totalement ordonné et $\mu$ la mesure definie par
$\mu\left\\{a\right\\}=1,$ pour tout $a\in E.$
Pour tout $t\in E$ désignons par $g_{t}\in c_{0}(E)$ la fonction définie par
$g_{t}(y)=\delta_{ty},$ $y\in E$ (le symbole de Kronecker).
Soient $t_{1},....,t_{n}\in E$ tels que $t_{n}<t_{n-1}<...<t_{1}$ et
$\alpha_{1},...,\alpha_{n}\in\mathbb{C}$; si
$g=\underset{}{\overset{}{\underset{k\leq
n}{\mathop{\displaystyle\sum}}\alpha_{k}g_{t_{k}}}}\in c_{0}(E)$ on définit
$f_{g}=\alpha_{0}f^{t_{1}}+\alpha_{1}\mathcal{X}_{\left]t_{2},t_{1}\right]}+...+\alpha_{n-1}\mathcal{X}_{\left]t_{n},t_{n-1}\right]}+\alpha_{n}\mathcal{X}_{\left\\{u;\text{
}u\leq t_{n}\right\\}}$
(remarquons que
$\mathcal{X}_{\left]t_{j},t_{j-1}\right]}=f^{t_{j}}-f^{t_{j-1}}\in C(K)).$
L’application $U:g\in c_{0}(E)\rightarrow f_{g}\in C(K)$ est une isométrie.
Par conséquent $c_{0}(E)$ se plonge isométriquement dans $C(K).$
Dans la suite, $E$ est un ensemble totalement ordonné et $(E,\mu)$ satisfait
les conditions $(\ref{h})$ et (0.2) ($-\infty,+\infty\notin E).$
Soit $C$ un sous-ensemble de $E;$ notons $\mathcal{X}_{C}$ la fonction
caractéristique de $C.$
Désignons par $e$ l’unité de $C(K)$ et par $h_{j}$ la restriction de $h$ à
$B_{j},$ $j=2,3.$
Sur $\left\\{\left[t\right];\text{ }t\in E\right\\}$ on définit la relation
d’ordre suivante: $\left[t\right]<\left[u\right]$ si, $t<u$ et
$\mu(\mathcal{X}_{\left]t,u\right]})>0$ et $\left[t\right]=\left[u\right]$ si,
$\left\|f^{t}-f^{u}\right\|_{L^{\infty}(\mu)}=0.$
Il est clair que $\left\\{\left[t\right];\text{ }t\in E\right\\}$ est
totalement ordonné.
###### Lemme 6.
Soit $t\in$ $E$. Alors sup$\left\\{u\in E;\text{
}u\in\left[t\right]\right\\}=b\in\left[t\right]$ et $\left\\{u\in E;\text{
}u>b\right\\}$ n’est pas férmé (resp. inf$\left\\{u\in E;\text{
}u\in\left[t\right]\right\\}=a\in\left[t\right]$ et $\left\\{u\in E;\text{
}u<a\right\\}$ n’est pas fermé ) si et seulement s’il existe $\theta_{1}\in
B_{2}$ (resp. $\theta_{2}\in B_{2})$ tel que $h($
$\theta_{1})=b\in\left[t\right]$ (resp. $h(\theta_{2})=a\in\left[t\right]).$
Supposons que $\sup\left\\{u\in E;\text{
}u\in\left[t\right]\right\\}=b\in\left[t\right]$ et $\left\\{u\in E;\text{
}u>b\right\\}$ n’est pas fermé; montrons qu’il existe $\theta_{1}\in B_{2}$
tel que $h(\theta_{1})=b.$
On définit la forme linéaire $\theta_{1}$ sur $A=C(K)$ par
$\left\\{\begin{array}[]{c}\theta_{1}(f^{u})=1\text{ si
}\left[u\right]\leq\left[t\right]\\\ \theta_{1}(f^{u})=0,\text{ si
}\left[u\right]>\left[t\right]\end{array}\right\\}$ et $\theta_{1}(e)=1$.
Comme $\left\\{f^{u};\text{ }u\in E\right\\}\cup\left\\{e\right\\}$ forme un
ensemble total dans $C(K)$, la forme linéaire $\theta_{1}$ est bien définie.
Montrons que $\theta_{1}$ est continue sur $A.$
Soit
$g=\underset{j=1}{\overset{n}{\mathop{\displaystyle\sum}}}\alpha_{j}f^{t_{j}}+\alpha_{0}e,$
les $\alpha_{j}\in\mathbb{C}$ et les $t_{j}\in E,$ avec
$\left[t_{i}\right]\neq\left[t_{j}\right]$ si $i\neq j$ , $f^{t_{j}}\neq e$
$\forall 1\leq j\leq n$ et $t_{n}<t_{n-1}<....<t_{1}$ (donc
$\left[t_{n}\right]<\left[t_{n-1}\right]<...<\left[t_{1}\right]).$ Supposons
que $\left[t_{n}\right]\leq\left[t\right]<\left[t_{n-1}\right]$, on a alors
$\theta_{1}(\underset{j=1}{\overset{n}{\mathop{\displaystyle\sum}}}\alpha_{j}f^{t_{j}}+\alpha_{0}e)\underset{}{=\underset{j=1}{\overset{n}{\mathop{\displaystyle\sum}}}\theta_{1}(\alpha_{j}f^{t_{j}}+\alpha_{0}e)=}\alpha_{n}+\alpha_{0}.$
D’autre part
$e=f^{t_{1}}+\underset{j=2}{\overset{n}{\mathop{\displaystyle\sum}}}\mathcal{X}_{\left]t_{j},t_{j-1}\right]}+\mathcal{X}_{\left\\{u\in
E;\text{ }u\leq t_{n}\right\\}},$
$f^{t_{2}}=f^{t_{1}}+\mathcal{X}_{\left]t_{2},t_{1}\right]},....,$
$f^{t_{n}}=f^{t_{1}}+\mathcal{X}_{\left]t_{n},t_{n-1}\right]}+\mathcal{X}_{\left]t_{n-1},t_{n-2}\right]}+...\mathcal{X}_{\left]t_{2},t_{1}\right]}.$
Cela entraîne que
$\displaystyle g$ $\displaystyle=$
$\displaystyle\underset{j=1}{\overset{n}{\mathop{\displaystyle\sum}}}\alpha_{j}f^{t_{j}}+\alpha_{0}e=(\alpha_{0}+\alpha_{1}+...+\alpha_{n})f^{t_{1}}+\left[(\alpha_{0}+\alpha_{2}+...+\alpha_{n})\mathcal{X}_{\left]t_{2},t_{1}\right]}\right]+$
$\displaystyle+\left[(\alpha_{0}+\alpha_{3}+...+\alpha_{n})\mathcal{X}_{\left]t_{3},t_{2}\right]}\right]+...+(\alpha_{0}+\alpha_{n-1}+\alpha_{n})\mathcal{X}_{\left\\{\left]t_{n-1},t_{n-2}\right]\right\\}}+(\alpha_{0}+$
$\displaystyle\alpha_{n})\mathcal{X}_{\left]t_{n},t_{n-1}\right]}+\alpha_{0}\mathcal{X}_{\left\\{u\in
E;\text{ }u\leq t_{n}\right\\}}.$
Notons $\beta_{0}=\alpha_{0}+\alpha_{1}+...+\alpha_{n},$
$\beta_{1}=\alpha_{0}+\alpha_{2}+...+\alpha_{n},$
$\beta_{2}=\alpha_{0}+\alpha_{3}+...+\alpha_{n},...,$
$\beta_{n-1}=\alpha_{0}+\alpha_{n},$ $\beta_{n}=\alpha_{0}.$
Comme $\left[t_{i}\right]\neq\left[t_{j}\right]$, pour tout $i\neq j$ et
$\mu(\left\\{u\in E;\text{ }u\leq t_{t_{n}}\right\\})>0$ (car $f^{t_{n}}\neq
e),$ alors
$\left\|g\right\|_{L^{\infty}(\mu)}=\max(\left|\beta_{0}\right|,...,\left|\beta_{n}\right|).$
Donc
$\left|\theta_{1}(g)\right|=\left|\beta_{n-1}\right|\leq\left\|g\right\|_{L^{\infty}(\mu)}$.
Si $\left[t_{n-1}\right]\leq\left[t\right]<\left[t_{n-2}\right]$ on a
$\theta_{1}(\underset{j=1}{\overset{n}{\mathop{\displaystyle\sum}}}\alpha_{j}f^{t_{j}}+\alpha_{0}e)\underset{}{=\alpha_{0}+\alpha_{n-1}}+\alpha_{n}.$
On conclut que
$\left|\theta_{1}(g)\right|=\left|\beta_{n-2}\right|\leq\left\|g\right\|_{L^{\infty}(\mu)}.$
Si $j<n-1$ et $\left[t_{j}\right]\leq\left[t\right]<\left[t_{j-1}\right],$
$\left|\theta_{1}(g)\right|=\left|\beta_{j-1}\right|\leq\left\|g\right\|_{L^{\infty}(\mu)}$
(on peut également traiter le cas $\left[t\right]\geq\left[t_{1}\right]$ et le
cas $\left[t\right]<\left[t_{n}\right]).$ On en déduit que
$\left|\theta_{1}(g)\right|\leq\left\|g\right\|_{L^{\infty}(\mu)}.$
Montrons que $\theta_{1}$ est un caractère.
Considérons $f=\beta_{n}\mathcal{X}_{\left\\{u\in E;\text{ }u\leq
t_{n}\right\\}}+\beta_{n-1}\mathcal{X}_{\left]t_{n},t_{n-1}\right]}+...+\beta_{1}\mathcal{X}_{\left]t_{2},t_{1}\right]}+\beta_{0}f^{t_{1}}$
et $g=\gamma_{n}\mathcal{X}_{\left\\{u\in E;\text{ }u\leq
t_{n}\right\\}}+\gamma_{n-1}\mathcal{X}_{\left]t_{n},t_{n-1}\right]}+...+\gamma_{1}\mathcal{X}_{\left]t_{2},t_{1}\right]}+\gamma_{0}f^{t_{1}}$
$($où $\left[t_{i}\right]\neq\left[t_{j}\right],\ $si $i\neq j$ ,
$f^{t_{j}}\neq e$ et $t_{n}<t_{n-1}<...<t_{1});$ remarquons que
$fg=\beta_{n}\gamma_{n}\mathcal{X}_{\left\\{u\in E;\text{ }u\leq
t_{n}\right\\}}+\beta_{n-1}\gamma_{n-1}\mathcal{X}_{\left]t_{n},t_{n-1}\right]}+...+\beta_{1}\gamma_{1}\mathcal{X}_{\left]t_{2},t_{1}\right]}+\beta_{0}\gamma_{0}f^{t_{1}}.$
Supposons maintenant
que$\left[t_{j}\right]\leq\left[t\right]<\left[t_{j-1}\right]$,
$j\in\left\\{2,...,n\right\\};$ on a alors $\theta_{1}(f)=\beta_{j-1}$,
$\theta(g)=\gamma_{j-1}$ et $\theta(fg)=\beta_{j-1}\gamma_{j-1},$ c’est-à-dire
que $\theta_{1}(fg)=\theta_{1}(f)\times\theta_{1}(g).$ Si
$\left[t\right]\geq\left[t_{1}\right],$ ou
$\left[t\right]<\left[t_{n}\right],$ par un argument analogue, on montre que
$\theta_{1}(fg)=\theta_{1}(f)\times\theta_{1}(g).$
Observons que $\sup\left\\{v\in E;\text{
}\psi(\theta,v)=1\right\\}=\sup\left\\{v\in E;\text{
}\left[v\right]\leq\left[t\right]\right\\}=\sup\left\\{v\in E;\text{
}\left[v\right]=\left[t\right]\right\\}=\sup\left\\{v\in E;\text{
}v\in\left[t\right]\right\\}=b\in\left[t\right].$ Donc $\psi(\theta_{1},b)=1.$
D’autre part si $u>b$ $\ \left[u\right]>\left[b\right]$, car $\sup\left\\{v\in
E;\text{ }v\in\left[t\right]\right\\}=b,$ comme $\theta_{1}\notin B_{1}$
(remarquons que $b$ est le point de discontinuité de $\theta_{1}$ d’après
l’hypothèse$)$ $\theta_{1}\in B_{2}.$
Inversement; supposons qu’il existe $\theta_{1}\in B_{2}$ tel que
$h(\theta_{1})=b\in\left[t\right];$ montrons que $\sup\left\\{u\in E;\text{
}u\in\left[t\right]\right\\}=b$ et $\left\\{u\in E;\text{ }u>b\right\\}$ n’est
pas fermé.
Il est évident que $h(\theta_{1})$ est le point de discontinuité de
$\theta_{1}$ (par la définition de $B_{2}).$ Montrons que $\sup\left\\{u\in
E;\text{ }u\in\left[t\right]\right\\}=b$
Remarquons d’abord que $\sup\left\\{u\in E;\text{
}u\in\left[b\right]\right\\}\geq b$ . D’autre part
$u\in\left[t\right]=\left[b\right]$ implique que
$\psi(\theta_{1},t)=\psi(\theta_{1},b)=1,$ c’est-à-dire que $t\leq b$ d’après
le lemme 3.
Donc $\sup\left\\{u\in E;\text{ }u\in\left[b\right]\right\\}\leq
b\allowbreak,$ d’où $\sup\left\\{u\in E;\text{
}u\in\left[b\right]\right\\}=b\allowbreak.$
Supposons que $\inf\left\\{u\in E;\text{
}u\in\left[t\right]\right\\}=a\in\left[t\right]$ et $\left\\{u\in E;\text{
}u<a\right\\}$ n’est pas fermé; montrons qu’il existe $\theta_{2}\in B_{3}$ et
$h(\theta_{2})=a.$
On définit la forme linéaire $\theta_{2}$ sur $C(K)$ par
$\left\\{\begin{array}[]{c}\theta_{2}(f^{u})=1\text{ si
}\left[u\right]<\left[t\right]\\\ \theta_{2}(f^{u})=0,\text{ si
}\left[u\right]\geq\left[t\right].\end{array}\right\\}$ et $\theta_{2}(e)=1$.
Par un argument analogue à celui du précédent, on montre que $\theta_{3}\in
B_{3}$ et $h(\theta_{3})=a.$
Inversement; supposons qu’il existe $\theta_{2}\in B_{3}$ tel que
$h(\theta_{2})=a\in\left[t\right].$ Par un argument analogue à celui du
précédent, on montre que $\inf\left\\{u\in E;\text{
}u\in\left[t\right]\right\\}=a$ (en utilisant le lemme 4) et que $\left\\{u\in
E;\text{ }u<a\right\\}$ n’est pas fermé.
###### Remarque 3.
La preuve du lemme 6, nous montre qu’il existe
$\theta^{\prime},\theta^{\prime\prime}\in K$ tel que
$\psi(\theta^{\prime},t)=1$ et $\psi(\theta^{\prime\prime},t)=0,$ $\forall
t\in E$. On pose $h(\theta^{\prime})=+\infty$ et
$h(\theta^{\prime\prime})=-\infty$ et (on remarque que
$\theta^{\prime},\theta^{\prime\prime}\in B_{1}).$
###### Proposition 1.
Supposons que $\ E$ est connexe et $\mu(\left]t,t^{\prime}\right])>0,$ pour
tout $t,t^{\prime}\in E$ vérifiant que $t<t^{\prime}.$ Alors
$h_{j}:B_{j}\rightarrow E$ est surjective, $j=2,3.$
Démonstration.
Comme $E$ est connexe, pour tout $t\in E$ les ensembles $\left\\{u\in E;\text{
}u>t\right\\},\left\\{u\in E;\text{ }u<t\right\\}$ ne sont pas fermés, car ils
sont des ouverts non vides (-$\infty,+\infty\notin E).$ D’autre part,
l’hypothèse entraîne que $\left[t\right]=\left\\{t\right\\},$ pour tout $t\in
E.$ D’après le lemme 6, $h_{j}$ est surjective, $j=2,3.$
On définit l’application $\Phi:K\rightarrow\left\\{0,1\right\\}^{E},$ par
$\Phi(\theta)=(\psi(\theta,t))_{t\in E}$, $\theta\in K;$ il est clair que
$\Phi$ est continue injective, par conséquent $\Phi$ est une homéomorphisme
sur son image, c’est-à-dire que la topologie de $K$ est engendrée par les
ouverts sous la forme, $\left\\{\theta\in K;\text{
}\psi(\theta,t)=1\right\\},$ $t\in E$, ou sous la forme, $\left\\{\theta\in
K;\text{ }\psi(\theta,u)=0\right\\},$ $u\in E$.
###### D finition 1.
On dit que $(E,\leq)$ est un ordre complet (ou $(E,\leq)$ est complet pour
l’ordre) si tout sous-ensemble majorant de $E,$ admet un supremum et tout
sous-ensemble minorant de $E,$ admet un infimum.
###### Remarque 4.
Supposons que $E$ soit un ordre complet; soit $\theta\in
B_{1}-\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\};$ il est clair
que les ensembles $\left\\{t\in E;\text{ }\psi(\theta,t)=1\right\\}$,
$\left\\{v\in E;\text{ }\psi(\theta,v)=0\right\\}$ ne sont pas vides.
Soient $u\in\left\\{v\in E;\text{ }\psi(\theta,v)=1\right\\}$ et
$u^{\prime}\in\left\\{t\in E;\text{ }\psi(\theta,t)=0\right\\}$; on remarque
que $u^{\prime}\geq u,$ donc $h(\theta)=\sup\left\\{t\in E;\text{
}\psi(\theta,t)=1\right\\}$ existe.
Soit $P$ un espace polonais (cf.[Sch(1), p.92]); $B_{1}(P)$ désigne l’espace
des fonctions de permière classe de Baire sur $P$. On munit $B_{1}(P)$ de la
topologie de la convergence simple.
###### D finition 2.
Soit $L$ un compact de Hausdorff; on dit que $L$ est un compact de Rosenthal,
s’il existe un espace polonais $P$ tel que $L$ s’injecte continûment dans
$B_{1}(P)$ (cf.[Gode]).
###### D finition 3.
Soit $X$ un espace de Hausdroff; $X$ est dit analytique, si $X$ est l’image
continue d’un espace polonais(cf.[Sch(1), chap.II,p.96]).
Dans la suite, nous désignons par $K$ le compact défini auparavant.
###### Proposition 2.
Soit $(E,\leq)$ un ensemble totalement ordonné tel que $(E,\tau_{0})$ est
analytique. Alors $K$ est un compact de Rosenthal.
Démonstration.
Comme $(E,\tau_{0})$ est analytique, il existe un espace polonais $P$ et
$H:P\rightarrow(E,\tau_{0})$ une application continue surjective. Soit
$\theta\in B_{2};$ montrons que l’application $\theta\circ H:$ $x\in
P\rightarrow$ $\psi(\theta,H(x))$ $\in\left\\{0,1\right\\}$ est de première
classe de Baire. Remarquons d’abord que
$\left\\{0\right\\},\left\\{1\right\\}$ sont ouverts et fermés dans
$\left\\{0,1\right\\},$ $\left\\{t\in E;\text{
}\psi(\theta,t)=1\right\\}=\left\\{t\in E;\text{ }t\leq h(\theta)\right\\}$
(est un fermé de $(E,\tau_{0}))$ et que $\left\\{t\in E;\text{
}\psi(\theta,t)=0\right\\}=$ $\left\\{t\in E;\text{ }t>h(\theta)\right\\}$
(est un ouvert de $(E,\tau_{0}))$ (d’après (0.3)$).$ Donc $(\theta\circ
H)^{-1}\left\\{0\right\\}=\left\\{x\in P;\text{
}\psi(\theta,H(x))=0\right\\}=\left\\{x\in P;\text{
}H(x)>h(\theta)\right\\}=H^{-1}\left[\left\\{t\in E;\text{
}t>h(\theta)\right\\}\right]$ est un $F_{\sigma}$ de $P$ et $(\theta\circ
H)^{-1}\left\\{1\right\\}=H^{-1}\left[\left\\{t\in E;\text{ }t\leq
h(\theta)\right\\}\right]$ est un $G_{\sigma}$ de $P$ (cf.[Bour-Ros-Tal]$).$
Un raisonnement analogue, nous montre que $\theta\circ H$ est une fonction de
première classe de Baire, pour tout $\theta\in B_{3}.$ D’autre part
$K=B_{1}\cup B_{2}\cup B_{3},$ ceci implique que pour tout $\theta\in K,$
l’application $\theta\circ H$ est de première classe de Baire. Considérons
$\Pi:K\rightarrow B_{1}(P\mathbb{)}$ l’injection définie par
$\Pi(\theta)=\theta\circ H$, $\theta\in K$; $\Pi$ est continue, ce qui
entraîne que $K$ est un compact de Rosenthal.$\blacksquare$
Soit $L$ un compact de Hausdorff; désignons par $M^{+}(L)$ l’espace des
mesures de probabiltés de Radon sur $L.$
###### Proposition 3.
Supposons que $E$ est un espace polonais et la mesure $\mu$ ne charge pas les
points de $E.$ Alors pour tout $\xi\in C(K)^{\ast}$ l’application
$t\in(E,\tau_{0})\rightarrow\xi(f^{t}),$ est de première classe de Baire.
Démonstration.
Il suffit de montrer la proposition 3 pour $\xi$ est dans la boule unité de
$C(K)^{\ast}.$
_Etape 1_ : Montrons que pour toute $g\in L^{1}(E,\mu)$ l’application $t\in
E\rightarrow(g,f^{t})=\mathop{\displaystyle\int}\limits_{E}g(x)f^{t}(x)d\mu(x)$
est continue.
Soient $g\in L^{1}(E,\mu)$ et $(t_{n})_{n\geq 0}$ une suite dans $E$ tels que
$t_{n}\rightarrow t\in(E,\tau_{0})$. Considérons $y\in E$ tel que $y\neq t;$
on se propose de montrer que
$f^{t_{n}}(y)\underset{n\rightarrow\infty}{\rightarrow}f^{t}(y).$
Montrons d’abord qu’il n’existe pas deux sous-suites $(t_{m_{k}})_{k\geq
0}$,$(t_{n_{k}})_{k\geq 0}$ telles que $f^{t_{m_{k}}}(y)=1$ et
$f^{t_{n_{k}}}(y)=0,$ pour tout $k\in\mathbb{N}.$
Supposons qu’il existe deux suites $(t_{m_{k}})_{k\geq 0},(t_{n_{k}})_{k\geq
0}$ vérifiant que $f^{t_{m_{k}}}(y)=1$ et $f^{t_{n_{k}}}(y)=0,$ pour tout
$k\in\mathbb{N};$ ceci entraîne que $t_{n_{k}}$ $\geq y>t_{m_{k}}$ , pour tout
$k\in\mathbb{N}$ , par le passage à la limite on voit que $y=t,$ ce qui est
impossible.
Distinguons deux cas:
_Cas 1_ : Il existe $n_{0}\in\mathbb{N}$ tel que $f^{t_{n}}(y)=1,$ pour tout
$n\geq n_{0}:$ Ce cas entraîne que $y>t_{n}$ $\forall n\geq n_{0},$ donc
$y\geq t,$ comme $y\neq t$ on a $y>t,$ c’est-à-dire que
$f^{t_{n}}(y)\underset{n\rightarrow\infty}{\rightarrow}f^{t}(y)=1.$
_Cas 2_ :__ Il existe $n_{0}\in\mathbb{N}$ tel que $f^{t_{n}}(y)=0,$ pour tout
$n\geq n_{0}:$ Ceci entraîne que $y\leq t_{n},$ $\forall n\geq n_{0},$ par le
passage à la limite, on voit que $y\leq t;$ on déduit que
$f^{t_{n}}(y)\underset{n\rightarrow\infty}{\rightarrow}f^{t}(y)=0.$
Comme la mesure $\mu$ ne charge pas les points de $E$
$(g,f^{t_{n}})=\mathop{\displaystyle\int}\limits_{E-\left\\{t\right\\}}g(y)f^{t_{n}}(y)d\mu(y).$
D’après ce qui est précède et d’après le théorème de convergence dominée, on
voit que
$(g,f^{t_{n}})=\mathop{\displaystyle\int}\limits_{E-\left\\{t\right\\}}g(y)f^{t_{n}}(y)d\mu(y)\underset{n\rightarrow\infty}{\rightarrow}\mathop{\displaystyle\int}\limits_{E-\left\\{t\right\\}}g(y)f^{t}(y)d\mu(y)=(g,f^{t}).\blacksquare$
_Etape 2_ : Soit $\xi$ dans la boule unité de $\left[C(K)\right]^{\ast};$
montrons que l’application $t\rightarrow\xi(f^{t})$ est de première classe de
Baire.
On peut supposer que $\xi\in M^{+}(K)$) (car
$\xi=(u^{+}-u^{-})+i(v^{+}-v^{-})$ et $u^{+},u^{-},v^{+},v^{-}\in M^{+}(K))$.
D’après le théorème de Hahn-Banach, il existe $\xi^{\prime}\in
B_{L^{\infty}(\mu\mathbb{)}^{\ast}}$ qui prolonge $\xi$ avec
$\left\|\xi\right\|=\left\|\xi^{\prime}\right\|;$ par conséquent il existe une
suite généralisée $(g_{i})_{i\in I}$ dans la boule unité de $L^{1}(\mu)$ telle
que $g_{i}\underset{\mathcal{U}}{\rightarrow}\xi^{\prime}$ préfaiblement dans
$\left[L^{\infty}(\mu)\right]^{\ast}$ (où $\mathcal{U}$ est un ultrafiltre non
trivial sur $I\mathbb{)}$. Il existe $\xi_{1},\xi_{2}\in M^{+}(K)$ tels que
$g_{i}^{+}\underset{\mathcal{U}}{\rightarrow}\xi_{1}\in M^{+}(K)$ et
$g_{i}^{-}\underset{\mathcal{U}}{\rightarrow}\xi_{2}\in M^{+}(K),$
$\sigma\left[C(K)^{\ast},C(K)\right]$ (donc $\xi=\xi_{1}-\xi_{2}$ car
$g_{i}=g_{i}^{+}-g_{i}^{-},$ $\forall i\in I).$ D’autre part d’après la
proposition 2, $K$ est un compact de Rosenthal, en appliquant le résultat de
[Gode] on voit que $M^{+}(K)$ est un compact de Rosenthal, ce qui entraîne qu’
il est angelique [Bour-Ros-Tal]. Par conséquent, il existe deux sous-suites
$(g_{j_{k}})_{k\geq 0}$, $(g_{i_{k}})_{k\geq 0}$ de $(g_{i})_{i\in I}$, telles
que $g_{j_{k}}^{+}\underset{k\rightarrow+\infty}{\rightarrow}\xi_{1}$ et
$g_{i_{k}}^{-}\underset{k\rightarrow+\infty}{\rightarrow}\xi_{2},$
préfaiblement, c’est-à-dire que
$h_{k}=g_{j_{k}}^{+}-g_{i_{k}}^{-}\underset{k\rightarrow+\infty}{\rightarrow}\xi_{1}-\xi_{2}=\xi,$
préfaiblement. D’après ce qui est précède on a
$(h_{k},f^{t})\underset{k\rightarrow+\infty}{\rightarrow}\xi(f^{t}),$
$\forall$ $t\in E.$
Finalement, d’après l’étape 1, l’application $t\in E\rightarrow(h_{k},f^{t})$
est continue $\forall k\in\mathbb{N}$, on en déduit que l’application $t\in
E\rightarrow\xi(f^{t})$ est de première classe de Baire.$\blacksquare$
###### Th or m 1.
Le compact $K$ est totalement ordonné (c’est-à-dire que la topologie de $K$
est définie par une relation totalement ordonnée sur $K$).
Démonstration.
On définit sur $K$ la relation d’ordre suivante:
$\forall\theta_{1},\theta_{2}\in
K:\theta_{1}\leq\theta_{2}\Leftrightarrow\psi\left(\theta_{1},t\right)\leq\psi\left(\theta_{2},t\right),\text{
}\forall t\in E\text{.}$
Montrons d’abord que $(K,\leq)$ est totalement ordonné. Fixons
$\theta_{1},\theta_{2}\in K$ tel que $\theta_{1}\neq\theta_{2};$ on peut
supposer qu’il existe $t_{0}\in E$ vérifiant que $\psi(\theta_{1},t_{0})=0$ et
$\psi(\theta_{2},t_{0})=1.$ Montrons que
$\psi(\theta_{1},t)\leq\psi(\theta_{2},t)$ pour tout $t\in E.$
Soit en effet $t\in E;$ si $t<t_{0}$ on a $\psi(\theta_{2},t)=1,$ donc
$\psi(\theta_{1},t)\leq\psi(\theta_{2},t).$
Si $t\geq t_{0}$ on a $\psi(\theta_{1},t)=0,$ cela entraîne que
$\psi(\theta_{1},t)\leq\psi(\theta_{2},t),$ par conséquent
$\theta_{1}\leq\theta_{2}.$
Il reste à montrer que la topologie $K$ et celle de l’ordre sont identiques.
Comme $K$ est compact et $(K,\tau_{0})$ est un espace séparé (voir
l’introduction), il suffit de montrer que la topologie de l’ordre est moins
fine que la topologie de $K.$
Considérons l’ensemble
$Z=\left\\{\alpha\in K;\text{ }\alpha<\theta\right\\}$
$Z$ est un ensemble ouvert de $K$. Pour le voir; soit $\alpha_{0}\in Z;$ il
existe $t\in E$ tel que $\psi(\alpha_{0},t)=0$ et $\psi(\theta,t)=1.$ Posons
$N=\left\\{\beta\in K;\text{ }\psi(\beta,t)=0\right\\}.$
.
Montrons que $N\subset Z$ (remarquons que $\alpha_{0}\in N).$ Soit
$\beta_{0}\in N;$ il clair que $\beta_{0}<\theta$ (car $\psi(\theta,t)=1$,
$\psi(\beta_{0},t)=0$ et $K$ est totalement ordonné), donc $\beta_{0}\in Z$,
par conséquent $N\subset Z.$
Par un argument analogue, on montre que l’ensemble $\left\\{\alpha\in K;\text{
}\alpha>\theta\right\\}$ est un ouvert de $K$.$\blacksquare$
###### D finition 4.
Soit $L$ un compact de Hausdroff; $C(L)$ admet une équivalente
$\tau_{p}$-$Kadec-norme$, s’il existe une équivalente norme $\rho$ sur $C(L)$
telle que la topologie forte et la topologie $\tau_{p}$ coincident sur
$\left\\{f\in C(L);\text{ }\rho(f)=1\right\\}.$
###### Corollaire 1.
L’espace $C(K)$ admet une équivalente $\tau_{p}-Kadec-norme$.
Démonstration.
C’est une conséquence du théorème 1 et le théorème A de [Hay-Jay-Nam-
Rog].$\blacksquare$
###### Corollaire 2.
On a $Bor(C(K),\left\|.\right\|)=Bor(C(K),\tau_{p}).$
Démonstration.
Le corollaire 1, montre qu’il existe sur $C(K)$ une équivalente
$\tau_{p}-Kadec-norme$. et d’après [Jay-Nam-Rog] on a
$Bor(C(K),\left\|.\right\|)=Bor(C(K),\tau_{p}).\blacksquare$
###### Remarque 5.
a) L’espace $(E,\tau_{0})$ est connexe si et seulement si ($E,\leq)$ est
complet pour l’ordre et $E$ vérifie la condition $(\ast)$ [Kell,
remaq.(d),p.58].
b) $h_{j}$:$B_{j}\rightarrow E$ est injective, $j=2,3$ (si
$B_{j}\neq\emptyset).$
Montrons par exemple que $h_{2}$ est injective.
Soit $\theta_{1},\theta_{2}\in B_{2}$ tel que $\theta_{1}\neq\theta_{2},$ il
existe $t\in E$ vérifiant que $\psi(\theta_{1},t)=0$ et $\psi(\theta_{2},t)=1$
(on peut supposer que $\theta_{1}<\theta_{2})$, cela entraîne que
$h(\theta_{1})<t\leq h(\theta_{2}),$ donc $h(\theta_{1})$ $\neq
h(\theta_{2}).\blacksquare$
###### Remarque 6.
Soient $(X,\tau)$ un espace fortement de Lindelöf et $Z$ un sous-espace de
$(X,\tau)$; alors $Z$ est un espace fortement de Lindelöf. En effet,
Soit $(V_{i})_{i\in I}$ une famille d’ouverts de $Z;$ pour tout $i\in I,$ il
existe un ouvert $U_{i}$ de $(X,\tau)$ tel que $V_{i}=Z\cap U_{i}.$ Comme
$(X,\tau)$ est un espace fortement de Lindelöf, il existe un sous-ensemble
dénombrable $I_{1}$ de $I$ tel que $\underset{i\in I}{\cup}U_{i}=$
$\underset{i\in I_{1}}{\cup}U_{i},$ donc $\underset{i\in
I}{\cup}V_{i}=\underset{i\in I}{\cup}Z\cap U_{i}=Z\cap$ $\left[\underset{i\in
I}{\cup}U_{i}\right]=$ $Z\cap\left[\underset{i\in I_{1}}{\cup}U_{i}\right]=$
$\underset{i\in I_{1}}{\cup}Z\cap U_{i}=\underset{i\in
I_{1}}{\cup}V_{i}.\blacksquare$
###### Th or m 2.
Supposons que $(E,\tau_{0})$ est séparable et qu’il vérifie la condition
$(\ast).$ Alors
$\vskip 12.0pt plus 4.0pt minus 4.0ptI)$ $B_{1}$ a une base dénombrable.
$II)$ $K$ est un espace fortement de Lindelöf.
$III)$ $K$ est séparable.
Démonstration de $(I).$
Soit $(a_{n})_{n\geq 0}$ une suite dense dans $(E,\tau_{0});$ pour tout
$m,n\in\mathbb{N}$ notons $V_{m,n}=\left\\{\theta\in B_{1};\text{
}\psi(\theta,a_{m})=0\text{ et }\psi(\theta,a_{n})=1\right\\},$
$U_{m}=\left\\{\theta\in B_{1};\text{ }\psi(\theta,a_{m})=0\right\\}$ et
$W_{n}=\left\\{\theta\in B_{1};\text{ }\psi(\theta,a_{n})=1\right\\}.$
Montrons que les ouverts sous la forme précédente, forment une base
(dénombrable) de $B_{1}$ (on peut supposer que pour tout $m,n\in\mathbb{N}$
$V_{m,n},$ $U_{m}$, $W_{n}$ ne sont pas vides).
Pour cela, soient $t,u\in E$ et $V_{u,t}=\left\\{\theta\in B_{1};\text{
}\psi(\theta,u)=0\text{ et \ }\psi(\theta,t)=1\right\\}\neq\emptyset$.
Montrons d’abord que $u$ est adhérent à $\left\\{x\in E;\text{ }x<u\right\\}$
(pour la topologie $\tau_{0}).$
Soit $a,b\in E$ tel que $u\in\left]a,b\right[;$ comme $(E,\tau_{0})$ vérifie
la condition $(\ast)$, il existe $c\in E$ vérifiant que $a<c<u<b,$ ceci
implique que $c\in\left\\{x\in E;\text{ }x<u\right\\}.$ Donc $u$ est adhérent
à $\left\\{x\in E;\text{ }x<u\right\\}.$
Fixons $\varphi\in V_{u,t}$ ; l’ensemble $\left\\{x\in E;\text{
}\psi(\varphi,x)=0\right\\}$ est un voisinage de $u,$ donc $\left\\{x\in
E\text{; }\psi(\varphi,x)=0\right\\}\cap$ $\left\\{x\in E;\text{
}x<u\right\\}\neq\emptyset,$ par conséquent il existe $m\in\mathbb{N}$ tel que
$\psi(\varphi,a_{m})=0$ et $a_{m}<u$; par un argument analogue, on montre
qu’il existe $n\in\mathbb{N}$ vérifiant que $\psi(\varphi,a_{n})=1$ et
$a_{n}>t.$ D’après ce qui est précède on a $\varphi\in V_{m,n}\subset
V_{u,t}.$
Par un raisonnement analogue à celui du précédent, on montre que pour tout
$\varphi\in\left\\{\theta\in B_{1};\text{ }\psi(\theta,u)=0\right\\}$ (resp.
$\varphi\in\left\\{\theta\in B_{1};\text{ }\psi(\theta,t)=1\right\\}),$ il
existe $m\in\mathbb{N}$ vérifiant que $\varphi\in U_{m}$
$\subset\left\\{\theta\in B_{1};\text{ }\psi(\theta,u)=0\right\\}$ (resp. il
existe $n\in\mathbb{N}$ vérifiant que $\varphi\in W_{n}$
$\subset\left\\{\theta\in B_{1};\text{ }\psi(\theta,t)=1\right\\}).$ Il en
résulte que $B_{1}$ a une base dénombrable.$\blacksquare$
Démonstration de (II).
_Etape 1_ : Montrons que $h_{3}:B_{3}\rightarrow(E,\tau_{1})$ est une
homéorphisme sur son image.
Considérons $t\in E$ et $V_{t}=\left\\{\theta\in B_{3};\text{
}\psi(\theta,t)=1\right\\};$ d’après la définition de $B_{3}$
$V_{t}=\left\\{\theta\in B_{3};\text{ }h(\theta)>t\right\\},$ par conséquent
$h(V_{t})=\left\\{u\in h(B_{3});\text{ }u>t\right\\}$ est un ouvert de
$h(B_{3})\subset(E,\tau_{1}).$ Du même $W_{t}=\left\\{\theta\in B_{3};\text{
}\psi(\theta,t)=0\right\\}=\left\\{\theta\in B_{3};\text{ }h(\theta)\leq
t\right\\},$ donc $h(W_{t})=\left\\{u\in h(B_{3});\text{ }u\leq t\right\\}$
est un ouvert de $h(B_{3})\subset(E,\tau_{1}).$
D’après ce qui est précède, on voit que $V$ est un ouvert de $B_{3}$ si et
seulement si $h(V)$ est un ouvert de
$h(B_{3})\subset(E,\tau_{1})\blacksquare.$
_Etape 2_ : Montrons que $K$ est un espace fortement de Lindelöf.
On sait d’après le lemme 1 que $(E,\tau_{1})$ est un espace fortement de
Lindelöf; donc $(h(B_{3}),\tau_{1})$ est un espace fortement de Lindelöf
(d’après la remarque 6). Ceci implique d’après l’étape 1, que $B_{3}$ est un
espace de Lindelöf. Par un raisonnement analogue, on montre que $B_{2}$ est un
espace fortement de Lindelöf. Finalement, on a $K=B_{1}\cup B_{2}\cup B_{3},$
$B_{1}$ est fortement de Lindelöf (car $B_{1}$ a une base dénombrable d’après
(I)) et $B_{2},B_{3}$ sont fortement de Lindelöf, il en résulte que $K$ est
fortement de Lindelöf.$\blacksquare$
Démonstration de (III).
Soient $W$ un sous-ensemble de $E\times E$ tel que pour tout $(u,t)\in W,$
$V_{u,t}=\left\\{\theta\in K;\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}$ soit fini, $\ M$ le sous-ensemble de $E$ tel que
pour tout $t\in M$ $U_{t}=\left\\{\theta\in K;\text{
}\psi(\theta,t)=0\right\\}$ soit fini et $M^{\prime}$ le sous-ensemble de $E$
tel que pour tout $t\in M^{\prime}$ $U_{t}^{\prime}=\left\\{\theta\in K;\text{
}\psi(\theta,t)=1\right\\}$ soit fini. Comme $K$ est un espace fortement de
Lindelöf (d’après (II)) l’ensemble $D=\left[\underset{(u,t)\in
W}{\cup}V_{u,t}\right]\cup\left[\underset{t\in
M}{\cup}U_{t}\right]\cup\left[\underset{t\in
M^{\prime}}{\cup}U_{t}^{\prime}\right]$ est dénombrable.
D’autre part $(E,\tau_{0})$ a une base dénombrable, donc $h(B_{2})$ et
$h(B_{3})$ sont deux sous-espaces séparables de $(E,\tau_{0})$ (car $h(B_{j})$
a une base dénombrable comme un sous-espace de $(E,\tau_{0})$ $j=2,3).$
Désignons par $D_{1}^{\prime}$ un sous-ensemble dénombrable de $K$ tel que
$h(D_{1}^{\prime})$ soit dense dans $h(B_{2})\subset(E,\tau_{0}),$ par
$D_{2}^{\prime}$ un sous-ensemble dénombrable de $K$ tel que
$h(D_{2}^{\prime})$ soit dense dans $h(B_{3})\subset(E,\tau_{0})$ et par
$D^{\prime\prime}$ un sous-ensemble dénombrable de $B_{1}$ dense dans $B_{1}$.
On se propose de montrer que $C=D\cup D_{1}^{\prime}\cup D_{2}^{\prime}\cup
D^{\prime\prime}$ est dense dans $K.$ Soit en effet $t,u\in E$; posons
$V_{u,t}=\left\\{\theta\in K;\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\},$ $U_{u}=\left\\{\theta\in K;\text{
}\psi(\theta,u)=0\right\\}$ et $U_{t}^{\prime}=\left\\{\theta\in K;\text{
}\psi(\theta,t)=0\right\\}$. Si $(u,t)\in W$ on a $V_{u,t}\subset D,$ donc
$V_{u,t}\cap C\neq\emptyset.$ Si $(u,t)\notin W$, l’ensemble
$\left\\{\theta\in K;\psi(\theta,u)=0\text{ et }\psi(\theta,t)=1\right\\}$
n’est pas fini. Observons que
$\displaystyle V_{u,t}$ $\displaystyle=$ $\displaystyle\left\\{\theta\in
B_{1};\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}\cup\left\\{\theta\in B_{2};\text{
}\psi(\theta,u)=0\text{ et }\psi(\theta,t)=1\right\\}$
$\displaystyle\cup\left\\{\theta\in B_{3};\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}$
_Cas 1_ : $\left\\{\theta\in B_{1};\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}$ n’est pas fini$:$ Comme $D^{\prime\prime}$ est
dense dans $B_{1},$ ceci entraîne que $\left\\{\theta\in B_{1};\text{
}\psi(\theta,u)=0\text{ et }\psi(\theta,t)=1\right\\}\cap
D^{\prime\prime}\neq\emptyset,$ donc $V_{u,t}\cap
D^{\prime\prime}\neq\emptyset,$ c’est-à-dire que $V_{u,t}\cap C\neq\emptyset.$
_Cas 2_ : $\left\\{\theta\in B_{2};\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}$ n’est pas fini: Remarquons que $\left\\{\theta\in
B_{2};\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}=\left\\{\theta\in B_{2};\text{ }t\leq
h(\theta)<u\right\\}$; ceci implique que $\left\\{\theta\in B_{2};\text{
}t<h(\theta)<u\right\\}$ n’est pas fini (car $h_{2}$ est injectived’après la
remarque 5-(b)$)$ donc $\left\\{\theta\in B_{2};\text{
}t<h(\theta)<u\right\\}\cap D_{1}^{\prime}\neq\emptyset;$ comme
$\left\\{\theta\in B_{2};\text{ }t<h(\theta)<u\right\\}\subset V_{u,t},$ on a
$V_{u,t}\cap C\neq\emptyset.$
_Cas 3_ : $\left\\{\theta\in B_{3};\text{ }\psi(\theta,u)=0\text{ et
}\psi(\theta,t)=1\right\\}$ n’est pas fini: Par un argument analogue à celui
du cas 2, on montre que $V_{u,t}\cap C\neq\emptyset.$
D’une façon analogue, on montre que $U_{u}\cap C\neq\emptyset$ et
$U_{t}^{\prime}\cap C\neq\emptyset,$ $\forall t,u\in E,$ par conséquent $K$
est séparable.$\blacksquare$
###### Corollaire 3.
Supposons que $(E,\tau_{0})$ soit séparable et vérifiant la condition
$(\ast)$. Alors $K$ est un compact de Rosenthal
Démonstration.
D’après les théorèmes 1 et 2, $K$ est totalement ordonné et séparable. Nous
appliquons maintenant le résultat de [Osta] nous déduisons que $K$ est
isomorphe à un sous-ensemble de $\left[0,1\right]\times\left\\{0,1\right\\}$,
ce dernier est un compact de Rosenthal, donc $K$ est un compact de Rosenthal
(car tout compact d’un compact de Rosenthal est un compact de
Rosenthal).$\blacksquare$
Posons $\overline{E}=E\cup\left\\{-\infty,+\infty\right\\}.$
###### Proposition 4.
L’application
$h:(K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}\rightarrow(\overline{E},\tau_{0})$
est continue.
Démonstration.
Soit $\alpha\in E\cup\left\\{+\infty\right\\}$ et
$V=\left\\{t\in\overline{E};\text{ }t<\alpha\right\\};$ on se propose de
montrer que $h^{-1}(V)$ est un ouvert de
($K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}$. Pour
cela, soit $\theta\in h^{-1}(V);$ nous distinguons trois cas:
_Cas 1_ , $\theta=\theta^{\prime\prime}\in h^{-1}(V):$ Il existe $\beta\in E$
tel que $\beta<\alpha;$ notons
$C=\left\\{\varphi\in\left[(K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}\right];\text{
}\psi(\varphi,\beta)=0\right\\};$ $C$ est un ouvert de
($K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}$ et
$\theta^{\prime\prime}\in C.$ D’autre part $\varphi\in C$ entraîne que
$h(\varphi)\leq\beta<\alpha,$ ceci implique que $C\subset h^{-1}(V)$.
_Cas 2_ , $\theta\in B_{2}:$ Notons $N=\left\\{t\in E;\text{
}t<\alpha\right\\}$, $N$ est un voisinage de $h(\theta)$ et $h(\theta)$ est
adhérent à $\left\\{u\in E;\text{ }\psi(\theta,u)=0\right\\}$ ; il existe donc
$\gamma\in E$ tel que $\gamma<\alpha$ et $\psi(\theta,\gamma)=0$. Posons
$C=\left\\{\varphi\in\left[(K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}\right];\text{
}\psi(\varphi,\gamma)=0\right\\};$ il est clair que $C$ est ouvert contenant
$\theta$. Montrons que $C\subset h^{-1}(V).$ Soit $\varphi\in C;$ c’est-à-dire
que $\psi(\varphi,\gamma)=0,$ donc $h(\varphi)\leq\gamma<\alpha.$ Il en
résulte que $\varphi\in h^{-1}(V).$
_Cas 3_ , $\theta\in B_{3}$: Posons
$C=\left\\{\varphi\in\left[(K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}\right];\text{
}\psi(\varphi,h(\theta))=0\right\\};$ $C$ est un ouvert contenant $\theta$ et
$C\subset h^{-1}(V).$
Par un argument analogue à celui du précédent on montre que $h^{-1}(V)$ est un
ouvert de
($K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}$ si
$V=\left\\{t\in\overline{E};\text{ }t>\alpha\right\\}$ et $\alpha\in
E\cup\left\\{-\infty\right\\}.\blacksquare$
###### Corollaire 4.
Supposons que $E$ est connexe. Alors $h:K\rightarrow(\overline{E},\tau_{0})$
est continue.
Démonstration.
Il suffit de remarquer que
$B_{1}=\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}$ car $E$ est
connexe.
Par un argument analogue à celui de la proposition 4, on montre
###### Proposition 5.
Supposons que $-\infty,+\infty\in E.$ Alors
$h:(K-B_{1})\cup\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}\rightarrow
E$ est continue.
###### Corollaire 5.
Soit $E$ un ensemble connexe; supposons que $-\infty,+\infty\in E.$ Alors
$h:K\rightarrow E$ est continue.
Désignons par $\tau_{0}^{\prime}$ la topologie de l’ordre de l’ensemble
$J=\left[E\cup\left\\{+\infty\right\\}\right]\times\left\\{1\right\\}\cup\left[E\cup\left\\{-\infty\right\\}\right]\times\left\\{0\right\\};$
remarquons que cet ensemble est totalement ordonné pour l’ordre
lexicographique de $\overline{E}\times\left\\{0,1\right\\}.$
Soit $E_{1}$ un sous-ensemble de $J;$ notons $(E_{1},\tau_{0}^{\prime})$ le
sous-espace topologique $E_{1}$ de $(J,\tau_{0}^{\prime}).$
###### Th or m 3.
Supposons que $E$ est connexe. Alors il existe un isomorphisme (d’ordre) de
$K$ sur
$(\left[h(B_{2}\cup\left\\{\theta^{\prime}\right\\})\times\left\\{1\right\\}\right]\cup\left[h(B_{3}\cup\left\\{\theta^{\prime\prime}\right\\})\times\left\\{0\right\\}\right],\tau_{0}^{\prime}).$
Démonstration.
Remarquons que $B_{1}=\left\\{\theta^{\prime},\theta^{\prime\prime}\right\\}.$
On définit
$T:K\rightarrow(\left[h(B_{2}\cup\left\\{\theta^{\prime}\right\\})\times\left\\{1\right\\}\right]\cup\left[h(B_{3}\cup\left\\{\theta^{\prime\prime}\right\\})\times\left\\{0\right\\}\right],\tau_{0}^{\prime}).$
$T(\theta)=\left\\{\begin{array}[]{c}(h(\theta),1),\text{ si }\theta\in
B_{2}\cup\left\\{\theta^{\prime}\right\\}\\\ (h(\theta),0),\text{ si
}\theta\in
B_{3}\cup\left\\{\theta^{\prime\prime}\right\\}\end{array}\right\\},$
$\theta\in K$. D’après la remarque 5-(b) $T$ est injective, il suffit donc de
montrer que $T$ est continue, car $K$ est un compact de Hausdroff. Pour cela,
soient ($\alpha,\beta)\in J$ et $V=\left\\{(x,y)\in J;\text{
}(x,y)<(\alpha,\beta)\right\\}$ un ouvert de $J.$ Montrons que $T^{-1}(V)$ est
un ouvert de $K$. Nous distinguons deux cas:
_Cas 1_ , $\beta=0:$ Il est évident que
$T^{-1}(V)=h^{-1}(\left\\{t\in\overline{E};\text{ }t<\alpha\right\\})$ qui est
un ouvert de $K,$ d’après le corollaire 4 (car $T(\theta)<(\alpha,0)$ si et
seulement si $h(\theta)<\alpha).$
_Cas 2_ , $\beta=1:$ Soit $\theta\in T^{-1}(V);$ il s’agit de trouver un
voisinage $C$ de $\theta$ dans $T^{-1}(V).$
Distinguons encore trois cas:
_Cas a_ ,__ $h(\theta)<\alpha:$ On pose
$C=h^{-1}(\left\\{t\in\overline{E};\text{ }t<\alpha\right\\})$; comme
$h(\varphi)<\alpha$ entraîne que $T(\varphi)=(h(\varphi),j)<(\alpha,1),$ (
$j\in\left\\{0,1\right\\}),$ $C\subset T^{-1}(V).$ D’autre part d’après le
corollaire 4, $C$ est un ouvert de $K$ ( remarquons que $\theta\in C).$
_Cas b_ , $h(\theta)=\alpha>-\infty:$ Comme $\theta\in T^{-1}(V),$
$\theta\neq\theta^{\prime}.$ On pose $C=\left\\{\varphi\in K;\text{
}\psi(\varphi,h(\theta))=0\right\\};$ $C$ est un ouvert de $K$.
Montrons que $C$ contenant $\theta.$
$\theta\in T^{-1}(V)$ entraîne que $T(\theta)=(h(\theta),j)<(h(\theta),1),$
donc $j=0,$ c’est-à-dire que $\psi(\theta,h(\theta))=0$, par conséquent
$\theta\in C.$
Il reste à montrer que $C\subset T^{-1}(V).$ Soit en effet $\varphi\in C;$
cela signifie que $\psi(\varphi,h(\theta))=0,$ ceci entraîne que
$h(\varphi)\leq h(\theta),$ si $h(\varphi)=h(\theta)=\alpha,$ on a alors
$\psi(\varphi,h(\varphi))=\psi(\varphi,h(\theta))=0,$ c’est-à- dire que
$\varphi\in B_{3}.$ Il en résulte que $T(\varphi)=(h(\varphi),0)<(\alpha,1)$
et $\varphi\in T^{-1}(V).$
Si $h(\varphi)<h(\theta)=\alpha$, on a
$T(\varphi)=$($h(\varphi),j)<(\alpha,1)$ ($j\in\left\\{0,1\right\\}),$ donc
$\varphi\in T^{-1}(V).$ Par conséquent $C\subset T^{-1}(V).$
Soient maintenant $(\alpha,\beta)\in J$ et $V=\left\\{(x,y)\in J;\text{
}(x,y)>(\alpha,\beta)\right\\}$ un ouvert de $J;$ montrons que $T^{-1}(V)$ est
un ouvert de $K.$
_Cas 1_ , $\beta=1:$ Il est facile de voir que
$T^{-1}(V)=h^{-1}(\left\\{t\in\overline{E};\text{ }t>\alpha\right\\})$ qui est
un ouvert de $K,$ d’après le corollaire 4.
_Cas 2_ , $\beta=0:$ Considérons $\theta\in T^{-1}(V);$ nous distinguons trois
cas:
_Cas d,_ $h(\theta)>\alpha$: On choisit
$C=h^{-1}(\left\\{t\in\overline{E};\text{ }t>\alpha\right\\});$ comme
$h(\varphi)>\alpha$ entraîne que $T(\varphi)=(h(\varphi),j)>(\alpha,0),$ alors
$\varphi\in C$ et $C\subset T^{-1}(V)$ (d’après le corollaire 4 , $C$ est un
ouvert de $K).$
_Cas e,_ $h(\theta)=\alpha<+\infty:$ On choisit $C=\left\\{\varphi;\text{
}\psi(\varphi,h(\theta))=1\right\\}($remarquons que
$\theta\neq\theta^{\prime\prime}$ car $\theta\in T^{-1}(V)).$ L’hypothèse
$\theta\in T^{-1}(V)$ entraîne que $T(\theta)=(h(\theta),j)>(h(\theta),0),$
donc $j=1,$ ceci signifie que $\psi(\theta,h(\theta))=1$ ; on déduit que $C$
contenant $\theta$ (il est clair que $C$ est un ouvert de $K)$. Montrons que
$C\subset T^{-1}(V).$ Soit $\varphi\in C;$ si $h(\theta)=h(\varphi)$, ceci
implique que $\psi(\varphi,h(\varphi))=\psi(\varphi,h(\theta))=1$. Donc
$T(\varphi)=((h(\varphi),1)>(\alpha,0),$ on conclut que $\varphi\in
T^{-1}(V).$ Finalement si $h(\theta)<h(\varphi)$ on a $\
T(\varphi)=((h(\varphi),j)>(\alpha,0),$ (car $h(\varphi)>h(\theta)>\alpha),$
donc $\varphi\in T^{-1}(V).$ Il en résulte que $C\subset T^{-1}(V).$
Montrons que $T$ est un isomorphisme d’ordre.
Soit $\theta,\varphi\in K$ tel que $\theta<\varphi;$ il existe $t\in E$
vérifiant que $\psi(\theta,t)=0$ et $\psi(\varphi,t)=1;$ ceci implique que
$h(\varphi)\geq t\geq h(\theta)$ (0.4)
Il est évident que si $h(\theta)<h(\varphi)$ on a
$T(\theta)=(h(\theta),j_{1})<(h(\varphi),j_{2})=T(\varphi)$
($j_{1},j_{2}\in\left\\{0,1\right\\}).$ Si $h(\theta)=h(\varphi),$ d’après
(0.4) $h(\theta)=h(\varphi)=t,$ donc $\theta\in B_{3}$ et $\varphi\in B_{2},$
par conséquent $T(\theta)<T(\varphi).$ Inversement; supposons que
$T(\theta)<T(\varphi);$ ou bien $h(\theta)<h(\varphi)$ (celai implique que
$\theta<\varphi)$ ou bien $h(\theta)=h(\varphi)$, comme
$T(\theta)=(h(\theta),j_{1})<T(\varphi)=(h(\varphi),j_{2}),$ $j_{1}=0$ et
$j_{2}=1.$ Ceci signifie que $\theta\in B_{3}$ et $\varphi\in B_{2},$ on
conclut que $\theta<\varphi$ (car $K$ est totalement ordonné)$.\blacksquare$
###### Corollaire 6.
Supposons que $E$ est connexe et $\mu(\left]t,t^{\prime}\right])>0$, pour tout
$t,t^{\prime}\in E$ vérifiant que $t<t^{\prime}$. Alors $K$ est isomorphe à $\
\left[E\cup\left\\{+\infty\right\\}\right]\times\left\\{1\right\\}\cup\left[E\cup\left\\{-\infty\right\\}\right]\times\left\\{0\right\\}.$
Démonstration.
D’après la proposition 1, $h_{2}$ et $h_{3}$ sont surjectives, ce qui implique
que $T$ (définie dans le théorème 3) est surjective sur $\
\left[E\cup\left\\{+\infty\right\\}\right]\times\left\\{1\right\\}\cup\left[E\cup\left\\{-\infty\right\\}\right]\times\left\\{0\right\\}.\blacksquare$
###### Corollaire 7.
Soient $E$ un groupe abélien; supposons que $E$ est séparable et connexe.
Alors $K$ est isomorphe à $\
\left[E\cup\left\\{+\infty\right\\}\right]\times\left\\{1\right\\}\cup\left[E\cup\left\\{-\infty\right\\}\right]\times\left\\{0\right\\}.$
Démonstration.
Montrons d’abord que $(E,\tau_{0})$ est localement compact.
Choisissons une mesure $\mu^{\prime}$ qui charge tout point de $E$ ; désignons
par $K^{\prime}$ le compact associé à $\mu^{\prime}.$ D’après la proposition
1, la fonction $h_{j}^{\prime}$ est surjective, $j=2,3,$ en utilisant le
corollaire 4, on voit que $h^{\prime}(K^{\prime})=\overline{E}$ est un compact
($h^{\prime}(\theta)=\sup\left\\{t\in E;\text{
}\psi^{\prime}(\theta,t)=1\right\\},$ $\theta\in
K^{\prime}-(B_{1})^{\prime})$.
D’autre part chaque intervalle fermé de $E$ est fermé dans
$(\overline{E},\leq)$. Comme $-\infty,+\infty\notin E$ pour tout $a\in E,$ il
existe $b,c\in E$ vérifiant que $c<a<b$. D’après ce qui est précède
$\left[c,b\right]$ est un voisinage compact de $a,$ cela implique que
$(E,\tau_{0})$ est localement compact.
Comme $(E,\tau_{0})$ est connexe, $E$ vérifie la condition $(\ast)$ [Kell,
remarq.(d),p.58]. L’exemple (1) nons montre que la mesure $\mu=m$ vérifie
l’hypothèse du corollaire 6, donc $K$ est isomorphe à
$\left[E\cup\left\\{+\infty\right\\}\right]$
$\times\left\\{1\right\\}\cup\left[E\cup\left\\{-\infty\right\\}\right]\times\left\\{0\right\\}.\blacksquare$
Supposons que -$\infty,+\infty\in E;$ désignons par $\tau_{0}^{\prime\prime}$
la topologie définie sur
$h(B_{2}\cup\left\\{\theta^{\prime}\right\\})\times\left\\{1\right\\}\cup
h(B_{3}\cup\left\\{\theta^{\prime\prime}\right\\})\times\left\\{0\right\\}$
comme un sous-espace topologique de $E\times\left\\{0,1\right\\}.$
###### Th or m 4.
Supposons que $E$ est connexe et $-\infty,+\infty\in E$. Alors il existe un
isomorphisme de $K$ sur
$(\left[h(B_{2}\cup\left\\{\theta^{\prime}\right\\})\times\left\\{1\right\\}\right]\cup\left[h(B_{3}\cup\left\\{\theta^{\prime\prime}\right\\})\times\left\\{0\right\\}\right],\tau_{0}^{\prime\prime}).$
Démonstration.
Remarquons d’arbord que $\left\\{\theta^{\prime}\right\\}$ et
$\left\\{\theta^{\prime\prime}\right\\}$ sont des points isolés dans $K$ car
$\left\\{\theta^{\prime}\right\\}=\left\\{\theta\in K;\text{
}\psi(\theta,+\infty)=1\right\\}$ et
$\left\\{\theta^{\prime\prime}\right\\}=\left\\{\theta\in K;\text{
}\psi(\theta,-\infty)=0\right\\}.$
La démonstration de ce théorème est analogue à celle du théorème 3 (en
utilisant le corollaire 5) il suffit de rajouter d’après le cas $b$ le cas
suivant;
_Cas c_ , $h(\theta)=\alpha=-\infty:$ C’est-à-dire que
$\theta=\theta^{\prime\prime};$ on choisit
$C=\left\\{\theta^{\prime\prime}\right\\}$ qui est ouvert dans $T^{-1}(V);$
$C$ est ouvert contenant $\theta^{\prime\prime}.$
D’autre part, il faut rajouter d’après le cas e le cas suivant:
_Cas f_ , $h(\theta)=\alpha=+\infty:$ Ce cas entraîne que
$\theta=\theta^{\prime}.$ Soit $C=\left\\{\theta^{\prime}\right\\};$ $C$ est
un ouvert contenant $\theta^{\prime}$ et $C\subset T^{-1}(V).$ $\blacksquare$
###### Corollaire 8.
Supposons que $E$ est connexe, $-\infty,+\infty\in E$ et
$\mu(\left]t,t^{\prime}\right])>0$, pour tout $t,t^{\prime}\in E$ vérifiant
que $t<t^{\prime}$. Alors $K$ est isomorphe à $\
\left[E-\left\\{-\infty\right\\}\right]\times\left\\{1\right\\}\cup\left[E-\left\\{+\infty\right\\}\right]\times\left\\{0\right\\}$
muni de la topologie de l’ordre lexicographique de
$E\times\left\\{0,1\right\\}.$
###### D finition 5.
Soit $X$ un espace de Hausdorff; on dit que $X$ est radonien (cf.[Sch(1),
p.117]), si toute mesure bornée sur $X$ est une mesure de Radon.
###### Proposition 6.
Supposons que
1) $E$ vérifie la condition $(\ast).$
2) $(E,\tau_{0})$ est séparable.
3) $\mu$ est une mesure localement finie, tout ensemble mesurable de mesure
finie de $E$ est réguliére à l’intérieur pour les compacts de $(E,\tau_{0})$
et $\mu$ ne charge pas les points de $E.$
4) $h_{j}$:$B_{j}\rightarrow E$ soit surjective; $j=2,3.$
5) $B_{3}$ est universellement mesurable.
Alors $B_{2}$ n’est pas universellement mesurable.
###### Remarque 7.
La mesure $\mu$ est localement finie, signifie que pour tout $t\in E,$ il
existe un voisinage $V_{t}$ de $t$ tel que $\mu(\overline{V_{t}})<\infty.$
Démonstration de la proposition 6.
_Etape 1:_ Montrons que __ tout compact de $B_{3}$ est métrisable.
Soient $(t_{n})_{n\geq 0}$ une suite dense dans $(E,\tau_{0})$ et $L$ un
compact de $B_{3};$ on définit l’application
$\Delta:L\rightarrow\left\\{0,1\right\\}^{\mathbb{N}}$ par
$\Delta(\theta)=(\psi(\theta,t_{n}))_{n\geq 0},\text{ }\theta\in L.$
Il est clair que $\Delta$ est continue; la preuve de l’étape 1 sera terminée,
si on montre que $\Delta$ est injective.
Pour cela soit, $\theta_{1},\theta_{2}\in L$ tel que
$\Delta(\theta_{1})=\Delta(\theta_{2});$ si $\theta_{1}<\theta_{2},$ il existe
$t_{0}\in E$ vérifiant que $\psi(\theta_{1},t_{0})=0$ et
$\psi(\theta_{2},t_{0})=1,$ donc $h(\theta_{2})>t_{0}\geq h(\theta_{1})$ (car
$B_{3}=\left\\{\theta\in K-B_{1};\text{ }\psi(\theta,h(\theta))=0\right\\}).$
D’autre part $E$ vérifie la condition $(\ast)$, il existe donc
$n\in\mathbb{N}$ tel que $h(\theta_{2})>t_{n}>t_{0}\geq h(\theta_{1}),$ il en
résulte que $\psi(\theta_{1},t_{n})=0$ et $\psi(\theta_{2},t_{n})=1,$ ce qui
est impossible.$\blacksquare$
Par un raisonnement analogue, on montre que tout compact de $B_{2}$ est
métrisable.
_Etape 2:_ Montrons que $(E,\tau_{1})$ est radonien.
D’après l’étape 1 du théorème 2 et l’hypthèse (4) de la proposition 6,
$h_{3}:$ $B_{3}\rightarrow(E,\tau_{1})$ est une homéomorphisme; comme $B_{3}$
est universellement mesurable, $(E,\tau_{1})$ est universellement mesurable.
D’après [Sch(2), th.3.2] toute mesure normale bornée sur $(E,\tau_{1})$ est
une mesure de Radon. Comme $(E,\tau_{1})$ est fortement de Lindelöf (d’après
le lemme 1) toute mesure bornée sur $(E,\tau_{1})$ est normale, donc de Radon,
il en résulte que $(E,\tau_{1})$ est radonien$.\blacksquare$
Supposons maintenant que $B_{2}$ est universellement mesurable; par un
argument analogue à celui de l’étape 2, on montre que $(E,\tau_{2})$ est
radonien.
Notons $i_{j}$ l’application d’identité de
$E:(E,\tau_{0})\rightarrow(E,\tau_{j}),$ $j=1,2;$ $i_{j}$ est borélienne
d’après le lemme 1, $j=1,2$.
Comme tout compact de $(E,\tau_{1})$ est métrisable (d’après l’étape 1), comme
$(E,\tau_{1})$ est radonien (d’après l’étape 2) et $\mu$ vérifie l’hypothèse
(3), d’après [Frem, remaq.c] il existe un compact $L$ de $(E,\tau_{0})$ de
mesure strictement positive tel que la restrication de $i_{1}$ à $L$ soit
continue. Un raisonnement analogue nous permet de trouver un compact
$L^{\prime}$ de $L$ de mesure strictement positive tel que la restriction de
$i_{2}$ à $L^{\prime}$ soit continue.
D’après l’hypothèse $(3)$ de la proposition 6, la mesure $\mu$ ne charge pas
les points de $E,$ donc $L^{\prime}$ n’est pas dénombrable, par conséquent il
existe une suite $(t_{n})_{n\geq 0}$ (infinie) dans $L^{\prime}$ telle que
$t_{n}\rightarrow t_{0}\in L^{\prime}$ pour la topologie $\tau_{0},$ mais
$i_{1}$ et $i_{2}$ sont continues sur $L^{\prime},$ on en déduit que
$t_{n}\rightarrow t_{0}$ dans $(E,\tau_{1})$ et dans $(E,\tau_{2})$; comme
$t_{n}\rightarrow t_{0}$ dans $(E,\tau_{1}),$ pour tout $u<t_{0},$ il existe
$n_{1}\in\mathbb{N}$ tel que pour tout $n\geq n_{1},$
$t_{n}\in\left]u,t_{0}\right]\ $ et comme $t_{n}\rightarrow t_{0}$ dans
$(E,\tau_{2}),$ pour tout $v>t_{0},$ il existe $n_{2}\in\mathbb{N}$ tel que
pour tout $n\geq n_{2},$ $t_{n}\in\left[t_{0},v\right[.$ Il en résulte que
$t_{n}=t_{0},$ pour tout $n\geq\max(n_{1},n_{21}),$ ceci est impossible, car
la suite $(t_{n})_{n\geq 0}$ n’est pas stationnaire.$\blacksquare$
Sous les hypothèses de la proposition 6, on a la proposition suivante:
###### Proposition 7.
La fonction $\psi:(K\times(E,\tau_{0}))\rightarrow\left\\{0,1\right\\}$ n’est
pas borélienne.
Démonstration.
_Etape 1:_ Montrons que $Bor(K)\otimes Bor(E)=Bor(K\times E).$
Il est évident que $Bor(K)\otimes Bor(E)\subset Bor(K\times E).$ Montrons
l’inclusion inverse.
Pour cela, soit $(E_{k})_{k\geq 0}$ une base dénombrable de la topologie de
$(E,\tau_{0})\mathbb{\ }$et $V=\underset{i\in I}{\cup}U_{i}\times W_{i},$ (les
$U_{i}$ sont des ouverts de $K$ et les $W_{i}$ sont des ouverts de $E)$; pour
tout $i\in I,$ il existe un sous-ensemble $M_{i}$ de $\mathbb{N}$
tel que $W_{i}=\underset{k\in M_{i}}{\cup}E_{k},$ donc
$V=\underset{k\in\mathbb{N}}{\cup}\left[(\underset{i\in
Y_{k}}{\cup}U_{i})\times E_{k})\right],$ où $Y_{k}=\left\\{i;\text{ }k\in
M_{i}\right\\}.$ Comme $\underset{i\in Y_{k}}{\cup}U_{i}$ est un borélien de
$K$ (car d’après le théorème 2, $K$ est fortement de Lindelöf), alors
$V\in Bor(K)\otimes Bor(E).$
On en déuit que $Bor(K\times E)\subset Bor(K)\otimes Bor(E),$ d’où
$Bor(K)\otimes Bor(E)=Bor(K\times E).\blacksquare$
_Etape 2:_ Montrons que tout borélien de $K$ est univesellement mesurable.
Comme $K$ est fortement de Lindelöf, toute mesure bornée sur $E$ est normale,
d’autre part $K$ est universellement mesurable, donc toute mesure normale
bornée sur $K$ est une mesure de Radon [Sch(2), th3.2], ceci implique que $K$
est radonien. D’autre part $K$ est complètement régulier et tout borélien est
relativement universellement mesurable dans $K$ $($car $K$ est radonien), donc
d’après [Sch(2), th.3.2] tout borélien de $K$ est universellement
mesurable.$\blacksquare$
Supposons maintetant que $\psi$ est borélienne; observons que $Bor(K)\otimes
Bor(E)$ est engendrée par les ouverts sous la forme $V\times W,$ où $V$ est un
ouvert de $K$ et $W$ est un ouvert de $E.$ D’autre part L’application
$\sigma:K-B_{1}\rightarrow K\times E$ définie par
$\sigma(\theta)=(\theta,h(\theta)),$ $\theta\in K-B_{1}$ est borélienne, car
$Bor(K)\otimes Bor(E)=Bor(K\times E)$ et $\sigma^{-1}(V\times W)=V\cap
h^{-1}(W)$ est un borélien pour tout ouvert $V$ de $K$ et tout ouvert $W$ de
$E,$ ($h$ est continue d’après le corollaire 4$),$ par conséquent
$\psi\circ\sigma$ est borélienne, donc
$(\psi\circ\sigma)^{-1}\left\\{1\right\\}=B_{2}$ est un borélien de $K$ et
($\psi\circ\sigma)^{-1}\left\\{0\right\\}=B_{3}$ est un borélien de $K,$ ceci
est impossible d’après la proposition 6.$\blacksquare$
###### Corollaire 9.
Il existe un compact de Rosenthal $L$ et une fonction
$\delta:L\times\left[0,1\right]\rightarrow\left\\{0,1\right\\}$ tels que
1) $\delta$ n’est pas borélienne.
2) $\delta(.,t)$ $\in C(L),$ pour tout $t\in\left[0,1\right]$ et
$\delta(\theta,.)$ $\in B_{1}(\left[0,1\right]),$ pour tout $\theta\in L.$
Démonstration de (1).
Soient $E=\left[0,1\right]$ (ou $E=\left]0,1\right[)$ et $\mu$ la mesure de
Lebesgue sur $E$; d’après la proposition 2, $L=K$ est un compact de Rosenthal.
Choisissons $\delta=\psi,$ d’après la proposition 7, $\delta$ n’est pas
borélienne.$\blacksquare$
Démonstration de (2).
Par la construction de $\psi,$ il est évident que $\delta(.,t)\in C(L),$ pour
tout $t\in E.$ D’autre part, au cous de la démonstration de la proposition 2
on a montré que l’application $t\in E\rightarrow\delta(\theta,t)$ est de
première classe de Baire, pour tout $\theta\in L.\blacksquare$
###### Remarque 8.
Soient $E$ est un espace compact métrisable et $(X,\mathcal{F})$ un espace
mesuré; supposons que $\delta:E\times X\rightarrow\mathbb{R}$ est une fonction
telle que $\delta(.,x)\in C(E)$ et $\delta(t,.)$ est mesurable, pour tout
$x\in X$ et tout $t\in E.$ Alors $\delta$ est mesurable.
En effet,
On définit l’application $\phi:(X,\mathcal{F})\rightarrow C(E)$ par
$\phi(x)$=$\delta(.,x).$ On se propose de montrer que $\phi$ est mesurable.
Comme $C(E)$ est séparable (car $E$ est un compact métrisable), il suffit de
montrer que $\phi^{-1}(B)$ est mesurable pour toute boule de $C(E).$
Soient $(y_{n})_{n\geq 0}$ une suite dans $E,$ $h\in C(E)$ et $\varepsilon>0;$
on a alors $\phi^{-1}(B(h,\varepsilon))=\left\\{x;\text{
}\left\|\delta(.,x)-h\right\|<\varepsilon\right\\}$
=$\underset{k\in\mathbb{N}}{\cap}\left\\{x;\left|\delta(y_{k},x)-h(y_{k})\right|<\varepsilon\right\\}.$
Comme pour tout $k\in\mathbb{N}$ l’application $x\rightarrow\delta(y_{k},x)$
est mesurable, $\phi^{-1}(B(h,\varepsilon))$ est mesurable. Donc $\phi$ est
mesurable.
D’après ce qui est précède il existe une suite $(\phi_{n})_{n\geq 0}$ de
fonctions étagées à valeurs dans $C(E)$ telle que
$\phi_{n}(x)\rightarrow\phi(x),$ pour tout $x\in X.$ Il en résulte que pour
tout $(t,x)\in E\times X,$ $\phi_{n}(t,x)$
$\underset{n\rightarrow+\infty}{\rightarrow}\delta(t,x).$
Fixons $n\in\mathbb{N}$; il existe $\alpha_{1}^{n},...,\alpha_{m_{n}}^{n}\in
C(E)$ et $D_{1}^{n},...,D_{m_{n}}^{n}$ des ensembles mesurables de $X$ tels
que $\phi_{n}(x)(t)=\underset{j\leq
m_{n}}{\mathop{\displaystyle\sum}}\alpha_{j}^{n}(t)\mathcal{X}_{D_{j}^{n}}(x).$
Puisque l’application $(t,x)\in E\times
X\rightarrow\alpha_{j}^{n}(t)\mathcal{X}_{D_{j}^{n}}(t)$ est mesurable, pour
tout $j\leq m_{n},$ l’application $(t,x)\rightarrow\phi_{n}(x)(t)$ est
mesurable. Par conséquent $\delta$ est mesurable.$\blacksquare$
D’après le corollaire 9 et la remarque 8, on voit la condition que $L$ soit
métrisable est nécessaire ($L$ est un compact de Rosenthal ne suffit pas).
###### Lemme 7.
Soit $E$ un groupe abélien localement compact séparable vérifiant les
conditions ($\ast)$ et ($\ast\ast$). Alors pour tout $\theta\in B-B_{1}$ on a
$\sup\left\\{-t\in E;\text{ }\psi(\theta,t)=0\right\\}=-\inf\left\\{t\in
E;\text{ }\psi(\theta,t)=0\right\\}.$
Démonstration.
Considérons $\theta\in K-B_{1}$ et désignons par $\beta=-\inf\left\\{t\in
E;\text{ }\psi(\theta,t)=0\right\\}$ (d’après le lemme 4 $\beta$ existe). Soit
$t\in E$ tel que $\psi(\theta,t)=0;$ en appliquant la condition $(\ast\ast)$
on voit que $-t\leq\beta,$ pour terminer la preuve, il suffit de montrer que
si $z\in E$ vérifiant $-t\leq z,$ pour tout $t\in E$ avec $\psi(\theta,t)=0$
alors $\beta\leq z.$
Soit un tel $z\in E$; $E$ vérifie la condition ($\ast\ast$), donc $t\geq-z,$
pour tout $t\in E$ tel que $\psi(\theta,t)=0.$ On en déduit que
inf$\left\\{t\in E;\text{ }\psi(\theta,t)=0\right\\}\geq-z,$ c’est-à-dire que
$\beta\leq z$ $($car E vérifie $(\ast\ast)).\blacksquare$
Supposons que $E$ est un groupe abélien localement compact séparable vérifant
la condition ($\ast)$; pour tout $\theta\in K,$ on définit
$\overset{\vee}{\theta}$ par
$\overset{\vee}{\theta}(g)=\theta(\overset{\vee}{g}),$ où
$\overset{\vee}{g(}x)=g(-x),$ $\ x\in E$, $g\in C(K).$ Comme
$\overset{\vee}{(g_{1}g_{2})}=\overset{\vee}{g_{1}}\overset{\vee}{g_{2}},$
$\overset{\vee}{\theta}\in K.$
###### Proposition 8.
Supposons que $E$ est un groupe abélien localement compact séparable vérifiant
les conditons ($\ast)$ et ($\ast\ast$). Alors
(i) $h(\theta)=-h(\overset{\vee}{\theta}),$ pour tout $\theta\in K-B_{1}$.
(ii) $\varphi\in B_{2}$ si et seulement si $\overset{\vee}{\varphi}\in B_{3}.$
(iii) $\overset{\vee}{(\theta^{\prime})}=\theta^{\prime\prime}.$
Démonstration de (i).
Remarquons d’abord que $\overset{\vee}{(f_{-t})}(y)=f_{-t}(-y)=f(t-y).$ Donc
$\overset{\vee}{(f_{-t})}(y)=0,$ si et seulement si $t-y\leq 0$ (c’est-à-dire
que si et seulement si $t\leq y).$ Il en résulte que
$\overset{\vee}{(f_{-t})}(y)=0$ si et seulement si $f_{t}(y)=1,$ presque tout
$y\in E$ et tout $t\in E$ (remarquons que $\mu\left\\{t\right\\}=0),$ donc
$\overset{\vee}{(f_{-t})}=e-f_{t}.$ pour tout $t\in E.$
Soient $\theta\in B_{2}$ et $t\in E;$ comme
$\psi(\overset{\vee}{\theta},-t)=\overset{\vee}{\theta}(f_{-t})=\overset{}{\theta}\overset{}{\overset{\vee}{(f_{-t})}=\theta(e)-\psi(\theta,t)}=1-\psi(\theta,t)$,
on a alors alors
$\psi(\overset{\vee}{\theta},-t)=0\text{ si et seulement si }\psi(\theta,t)=1$
(0.5)
En appliquant (0.5) et le lemmes 3, on voit que
$\displaystyle h(\theta)$ $\displaystyle=$ $\displaystyle\sup\left\\{t\in
E;\text{ }\psi(\theta,t)=1\right\\}=$ $\displaystyle\sup\left\\{t\in E;\text{
}\psi(\overset{\vee}{\theta},-t)=0\right\\}$ $\displaystyle=$
$\displaystyle\sup\left\\{-t\in E;\text{
}\psi(\overset{\vee}{\theta},t)=0\right\\}$ $\displaystyle=$
$\displaystyle-\inf\left\\{t\in E;\text{
}\psi(\overset{\vee}{\theta},t)=0\right\\}$ $\displaystyle=$
$\displaystyle-h(\overset{\vee}{\theta})\text{ (d'apr\\`{e}s les lemmes
\ref{oo} et \ref{hmo}}).$
Un raisonnement analogue nous permet de conclure que
$h(\theta)=-h(\overset{\vee}{\theta})$, pour tout $\theta\in
B_{3}.\blacksquare$
Démonstration de (ii).
D’après ($\ref{n})$ et (i), $\psi(\varphi,h(\varphi))=1$ si et seulement si
$\psi(\overset{\vee}{\varphi},h(\overset{\vee}{\varphi}))=\psi(\overset{\vee}{\varphi},-h(\varphi))=0,$
c’est-à-dire que $\varphi\in B_{2}$ si et seulement si
$\overset{\vee}{\varphi}\in B_{3}.\blacksquare$
Démonstration de (iii).
Soit $t\in E;$ on a vu que $\overset{\vee}{(f_{t})}=e-f_{-t},$ donc
$\overset{\vee}{\theta^{\prime}}(f_{t})=\theta^{\prime}\overset{\vee}{(f_{t})}=\theta^{\prime}(e-f_{-t})=1-1=0.$
Il en résulte que
$\overset{\vee}{\theta^{\prime}}=\theta^{\prime\prime}.\blacksquare$
###### Corollaire 10.
Soit $E$ un groupe abélien localement compact séparable vérifiant les
conditions ($\ast$) et $(\ast\ast).$ Alors $B_{2}$ n’est pas universellement
mesurable.
Démonstration.
D’après le lemme 1, $E$ a une base dénombrable et par hypothèse $E$ est
localement compact, d’après [Sch(1), chap.II,th.6] $E$ est un espace
polonnais. Donc la mesure de Haar vérifie l’hypothèse (3) de la proposition 6
[Sch(1), chap.II,p.92]. Supposons que $B_{2}$ est universellement mesurable;
d’après la proposition 8, $B_{3}$ est universellement mesurable, car
l’application $\theta\in K\rightarrow\overset{}{\overset{\vee}{\theta}\in K}$
est une homéomorphisme. La proposition 6 nous dit que ceci est
impossible.$\blacksquare$
###### Proposition 9.
Sous les hypothèses du corollaire 10, pour tout $\theta\in B_{2}$
l’application $U_{\theta}^{\prime}:E\rightarrow K\times K$ définie par
$U_{\theta}^{\prime}\left(t\right)=(\theta_{t}\overset{\vee}{,(\theta)_{t}})$
n’est pas borélienne.
Démonstration.
Supposons qu’il existe $\theta\in B_{2}$ tel que $U_{\theta}^{\prime}$ soit
une application borélienne; choisissons un sous-ensemble $H$ de $E$ qui
n’appartient pas à la tribu borélienne de $E$ (ceci est possible car $E$ est
un espace polonais, donc
$card\left[Bor(E,\tau_{0})\right]<card\left[\left\\{0,1\right\\}^{E}\right].$
Définissons l’ouvert $V$ de $K$ $\times K$ par
$V=\underset{u\in H}{\cup}\left\\{\varphi\in K;\text{
}\psi(\varphi,-u+h(\theta))=1\right\\}\times\left\\{\varphi\in K;\text{
}\psi(\varphi,-u+h(\overset{\vee}{\theta}))=0\right\\},$
remarquons que $\theta_{t}\in B_{2}$ (car d’après le lemme 5
$\psi(\theta_{t},h(\theta_{t}))=\psi(\theta_{t},h(\theta)-t)=\psi(\theta,h(\theta))=1)$
(d’après la proposition 8 $\overset{\vee}{\theta}\in B_{3}$ , donc
$(\overset{\vee}{\theta})_{t}\in B_{3}),$ par conséquent
$\displaystyle(U_{\theta}^{\prime})^{-1}(V)$ $\displaystyle=$
$\displaystyle\underset{u\in H}{\cup}\left\\{t\in E\text{;
}\psi(\theta_{t},-u+h(\theta))=1\text{ et
}\psi((\overset{\vee}{\theta})_{t}),-u+h(\overset{\vee}{\theta}))=0\right\\}$
$\displaystyle=$ $\displaystyle\left\\{t\in E;\text{
}h(\theta_{t})\geq-u+h(\theta)\text{ et
}h((\overset{\vee}{\theta})_{t})\leq-u+h(\overset{\vee}{\theta})\right\\}\underset{u\in
H}{=\cup}\left\\{u\right\\}$ $\displaystyle=$ $\displaystyle H,$
(car $h(\overset{}{\theta}_{t})=h((\overset{}{\theta})-t$ et
$h((\overset{\vee}{\theta)}_{t})=h((\overset{\vee}{\theta})-t,$ pour tout
$t\in E,$ d’après le lemme 5). Cela implique que $H$ est un ensemble borélien
de $E,$ ce qui est impossible.$\blacksquare$
###### Corollaire 11.
Sous les hypothèses du corollaire 10, la tribu borélienne sur $K\times K$
contient strictement $Bor(K)\otimes Bor(K).$
Démonstration.
Montrons d’abord que les applications $\delta_{1}:t\in
E\rightarrow\rightarrow\theta_{t}\in B_{2}\subset K$, $\delta_{2}:t\in
E\rightarrow(\overset{\vee}{\theta})_{t}\subset K$ sont boréliennes.
Remarquons d’abord que pour tout $u,t\in E$ les ensembles sous la forme,
$\displaystyle(\delta_{1})^{-1}\left[\left\\{\varphi\in K;\text{
}\psi(\varphi,u)=0\right\\}\right]$ (0.6)
$\displaystyle(\delta_{1})^{-1}\left[\left\\{\varphi\in K;\text{
}\psi(\varphi,t)=1\right\\}\right]$
$\displaystyle(\delta_{2}^{-1}\left[\left\\{\varphi\in K;\text{
}\psi(\varphi,u)=0\right\\}\right]$
$\displaystyle(\delta_{2})^{-1}\left[\left\\{\varphi\in K;\text{
}\psi(\varphi,t)=1\right\\}\right]$
sont des boréliens de $E.$ Comme $K$ est fortement de Lindelöf d’après le
théorème 2, $\delta_{1}$ et $\delta_{2}$ sont des applications boréliennes.
Soit $V_{1}$,$V_{2}$ deux ouvrets de $K;$ il est clair que
$(U_{\theta}^{\prime})^{-1}(V_{1}\times
V_{2})=\delta_{1}^{-1}(V_{1})\cap\delta_{2}^{-1}(V_{2})\in Bor(E).$ Il en
résulte que pour tout $S\in Bor(K)\otimes Bor(K)$
$(U_{\theta}^{\prime})^{-1}S$ $\in Bor(E).$
Supposons maintenant que la tribu borélienne sur $K\times K$ est égale à la
tribu produit. D’après ce qui est précède l’application $U_{\theta}^{\prime}$
est borélienne, cela est impossible, d’après la proposition 9.$\blacksquare$
###### Remarque 9.
Si $L$ est un compact métrisable $Bor(L\times L)=Bor(L)\otimes Bor(L).$
L. Lindenstrauss et C. Stegall [Lin-Steg] ont montré qu’il existe un espace de
Banach $X$ qui ne contient pas $\ell^{\infty}$ isomorphiquement et une
application $\sigma:\left\\{0,1\right\\}^{\mathbb{N}}\rightarrow X$
scalairement mesurable, mais $\sigma$ n’est pas faiblemant équivalente à une
fonction mesurable. Dans la proposition suivante, on fournit un autre type
d’exemples. En effet, la fonction considérée (dans la proposition 10) est
localement intégrable au sens de Riemman.
###### Proposition 10.
Il existe un espace de Banach $X$ et une fonction $g:\mathbb{R}\rightarrow X$
tels que
I) $X$ ne contient pas $\ell^{\infty}$ isomorphiquement.
II) $g$ est scalairement mesurable.
III) $g$ ne peut pas ètre faiblement équivalente à une fonction mesurable.
IV) $g$ est localement intégrable au sens de Riemman sur $\mathbb{R}$.
Démonstration de (I).
Considérons $E=\mathbb{R}$ et $X=C(K);$ au cours de la démonstration de la
proposition 3 (étape 2), on a montré que la boule unité de
$X^{\ast}=C(K)^{\ast}$ est angelique, donc $X$ ne contient pas $\ell^{\infty}$
isomorphiquement (remarquons que la boule unité de $(\ell^{\infty})^{\ast}$
n’est pas angelique, d’après [Od-Ros]).
Démonstration de (II).
Soit $g$ :$\mathbb{R}\rightarrow(C(K)\subset L^{\infty}(\mathbb{R})$ définie
par $g(t)=f_{t},$ pour presque tout $t\in\mathbb{R};$ d’après la proposition
3, $g$ est scalairement mesurable.$\blacksquare$
Démonstration de (III).
Supposons qu’il existe une fonction
$h:\mathbb{R}\rightarrow(X,\left\|.\right\|)$ mesurable telle que pour toute
forme linéaire $\xi\in X^{\ast}$ $\xi(g(t))=\xi(h(t)),$ pour presque tout
$t\in\mathbb{R}.$ En particulier $(g(t),u_{n})=(h(t),u_{n})\,,$ pour tout
$n\in\mathbb{N}$ et pour presque tout $t\in\mathbb{R},$ où $(u_{n})_{n\geq 0}$
est une suite dense dans $L^{1}(\mathbb{R}).$ On en déduit que $g=h$ presque-
partout, ce qui est impossible, car $\left\|f_{t}-f_{t^{\prime}}\right\|\geq
1,$ pour tout $t,t^{\prime}$ avec $t\neq t^{\prime}$ (remarquons que pour
presque $t\in\mathbb{R}$ $h(t)$ est à valeurs dans un sous-espace de Banach
séparable de $X).$
Démonstration de (IV).
Soit $a,b\in\mathbb{R}$ tel que $a<b;$ on a alors
$\int\limits_{a}^{b}\mathcal{X}_{\left[t,+\infty\right[}(\omega)dt=\left\\{\begin{array}[]{l}(a\vee\omega)-a,\text{
si }\omega\leq b\\\ b-a,\text{ si }\omega>b\end{array}\right.$
il est facile de voir que pour tout $n\in\mathbb{N}^{\ast}$ et tout $t_{0}=a,$
$t_{1},...,t_{n}=b\in\left[a,b\right]$ avec $t_{0}<t_{1}<...<t_{n}$ on a
$\sum\limits_{k=1}^{n}\mathcal{X}_{\left[t_{k},+\infty\right[}(\omega)(t_{k}-t_{k-1})=\left\\{\begin{array}[]{l}t_{k(\omega)}-a,\text{
o\\`{u} }t_{k(\omega)}\leq\omega<t_{k(\omega)+1},\text{ si
}\omega\in\left[a,b\right]\\\ 0,\text{ si }\omega<a\\\ b-a,\text{ si
}\omega>b\end{array}\right.$
on déduit que
$\left\|\sum\limits_{k=1}^{n}\mathcal{X}_{\left[t_{k},+\infty\right[}(.)(t_{k}-t_{k-1})-\int\limits_{a}^{b}\mathcal{X}_{\left[t,+\infty\right[}(.)dt\right\|_{L^{\infty}(\mathbb{R})}$
$\leq\underset{\omega\in\left[a,b\right]}{\sup}\left|\omega-
t_{k(\omega)}\right|$
$\leq\underset{1\leq k\leq n}{\sup}\left|t_{k}-t_{k-1}\right|.$ Cela implique
que l’application $t\in\mathbb{R}\rightarrow g(t)\in X$ est intégrable au sens
de Riemman sur $\left[a,b\right].\blacksquare$
###### D finition 6.
Soit $(X,\tau)$ un espace topologique séparé.
a) On dit que $(X,\tau)$ est $K-analytique,$ si $X$ est l’image continue d’un
ensemble $K_{\sigma\delta}.$ Rappelons que un ensemble $B$ est
$K_{\sigma\delta}$ dans un espace de Hausdroff $(Z,\tau),$ s’il existe une
suite de compacts $(K_{m,n})_{m,n\in\mathbb{N}}$ dans $(Z,\tau)$ telle que
$B=\underset{m\geq 0}{\cap}\underset{n\geq 0}{\cup}K_{m,n}.$
b) $(X,\tau)$ est souslinien, s’il existe un espace polonais $P$ et une
homéorphisme borélienne $H:P\rightarrow X.$
Soit $D$ un sous-ensemble de $C(K);$ nontons $(D,\tau_{p})$ la topologie
induite de $(C(K),\tau_{p})$ sur $D.$
Désignons par $\tau^{\prime}$ (resp. $\tau^{\prime\prime})$ la topologie
définie sur $C(K)$ telle que $u_{i}\rightarrow u$ dans $(C(K),\tau^{\prime})$
(resp. dans $(C(K),\tau^{\prime\prime}))$ ; si et seulement si
$H_{i}(\theta)\rightarrow H(\theta),$ pour tout $\theta\in B_{2}\cup B_{1}$
(resp. pour tout $\theta\in B_{3}\cup B_{1})$. Remarquons que $\tau^{\prime}$
et $\tau^{\prime\prime}$ sont moins fines que la topologie $\tau_{p}.$
###### Th or m 5.
Soit $E$ un groupe abélien localement compact séparable vérifiant les
conditions $(\ast)$ et $(\ast\ast).$ Alors il existe une
sous-$\mathbb{C}^{\ast}-$a$\lg$èbre $Y_{1}$ de $(C(K)$ et un sous-espace
$Y_{2}$ de $C(K)$ tels que
1 )$(C(K),\tau^{\prime})$ n’est pas universellement mesurable.
2) $Y_{1}$ et $Y_{2}$ sont $\tau_{p}$ fermés et
$(C(K),\tau_{p})=(Y_{1},\tau_{p})\oplus(Y_{2},\tau_{p}).$
3) $(Y_{1},\tau_{p})$ est isomorphe à $(Y_{2},\tau_{p}).$
4$)$ $(Y_{j},\tau_{p})=(Y_{j},\tau^{\prime}),$ $j=1,2.$
5) $(Y_{1},\tau_{p})$ n’est pas $K-analytique.$
6) $(Y_{1},\tau_{p})$ n’est pas souslinien.
7) $Bor(C(K),\tau^{\prime})\otimes Bor(C(K),\tau^{\prime})\neq Bor(C(K)\times
C(K),\tau^{\prime}\times\tau^{\prime})$
8) La projection $P:(C(K),\tau^{\prime})\rightarrow(Y_{1},\tau^{\prime})$
n’est pas borélienne.
9)L’ensemble $\left\\{H:E\rightarrow(C(K),\tau^{\prime});\text{ }H\text{ est
bor\'{e}lienne}\right\\}$ n’est pas un espace vectoriel.
Démonstration de (1).
_Etape 1:_ Soit $\theta\in B_{2}$ ; montrons que $\ \left\\{\theta_{t};t\in
E\right\\}=B_{2}.$
Observons d’abord que d’après la remarque 2, $B_{2}\neq\emptyset.$ Soit
$\varphi\in B_{2};$ il existe $t\in E$ tel que $h(\varphi)=h(\theta_{t})$ (car
$h(\theta_{t})=h(\theta)-t$ d’après le lemme 5 $).$ Comme $\theta_{t}\in
B_{2}$ et $h_{2}$ est injective, d’après la remarque 5 (b) on a
$\varphi=\theta_{t}$.$\blacksquare$
_Etape 2 :_ Montrons qu’il existe une suite dénombrable dans $B_{2}$ qui est
dense dans $K$.
Considérons $(\theta_{n})_{n\geq 0}$ une suite dans $B_{2}$ telle que
$(h(\theta_{n}))_{n\geq 0}$ soit dense dans $E$ $(h_{2}$ est surjective
d’après la remarque 2 et le lemme 5). Soit $t\in E;$ notons
$V=\left\\{\theta\in K;\text{ }\psi(\theta,t)=1\right\\};$ choisissons
$\theta\in K-B_{1}$ tel que $t<h(\theta)$, il existe $\theta_{m}\in B_{2}$
vérifiant que $t<h(\theta_{m}).$ Comme $\psi(\theta_{m},h(\theta_{m}))=1,$
$\theta_{m}\in V.$ Si $U=\left\\{\theta\in K;\text{
}\psi(\theta,u)=0\right\\}$ on choisit $\theta\in K-B_{1}$ tel que
$u>h(\theta),$ il existe $\theta_{m^{\prime}}\in B_{2}$ vérifiant que
$u>h(\theta_{m^{\prime}})>h(\theta).$
Il est clair que $\theta_{m^{\prime}}\in\left\\{\theta\in K;\text{
}\psi(\theta,u)=0\right\\}.$
Supposons que $\theta_{0}\in W=\left\\{\theta\in K;\text{
}\psi(\theta,u)=1\text{ et }\psi(\theta,t)=0\right\\}$; ceci implique que
$u<t,$ comme $E$ vérifie la condition ($\ast)$ et $h_{2}$ est surjective, il
existe $\theta_{1}\in K-B_{1}$ tel que $u<h(\theta_{1})<t.$ On en déduit qu’il
existe $\theta_{n}\in B_{2}$ vérifiant que $u<h(\theta_{n})<t,$ donc
$\theta_{n}\in W$ $.\blacksquare$
D’après l’étape 2, la topologie $\tau^{\prime}$ est séparée.
Désignons par $D=\left\\{f_{t};\text{ }t\in E\right\\}.$
_Etape 3:_ Montrons que ($D,\tau^{\prime})$ est fortement de Lindelöf
($(D,\tau^{\prime})$ comme un sous-espace topologique de
$(C(K),\tau^{\prime})$).
Définissons l’application $\pi:(D,\tau^{\prime})\rightarrow B_{2}$ par
$\pi(f_{t})=\theta_{t},$ $t\in E$ (où $\theta$ est un point fixé de $B_{2})$,
il suffit de montrer que $\pi$ est une homéomorphisme, car $B_{2}$ est un
espace fortement de Lindelöf.
Montrons d’abord que $\pi$ est injective. Pour cela soit $u,v\in E$ tel que
$\theta_{u}=\theta_{v};$ comme $h(\theta_{u})=h(\theta)-u$ et
$h(\theta_{v})=h(\theta)-v,$ alors $u=v.$
Soit $u\in E;$ posons $V_{u}=\left\\{\theta\in B_{2};\text{
}\psi(\theta,u)=0\right\\}$ et $W_{u}=\left\\{\theta\in B_{2};\text{
}\psi(\theta,u)=1\right\\};$ on a alors,
$\pi^{-1}(V_{u})=\left\\{f_{t};\text{
}\psi(\theta_{t},u)=0\right\\}=\left\\{f_{t};\text{
}\theta_{t}(f_{u})=0\right\\}=\left\\{f_{t};\text{
}\theta_{u}(f_{t})=0\right\\}$ est un ouvert de $(D,\tau^{\prime}).$ Du même
on a $\pi^{-1}(W_{u})=\left\\{f_{t};\text{ }\theta_{u}(f_{t})=1\right\\}$ est
un ouvert de ($D,\tau^{\prime}).$
Soient $\theta_{1},\theta_{2}\in B_{1}$ et $t_{1},t_{2}\in E$ $\ ;$ notons
$W_{1}=\left\\{f_{t};\text{ }f_{t}(\theta_{1})=0\right\\}$,
$W_{2}=\left\\{f_{t};\text{ }f_{t}(\theta_{2})=1\right\\}$ ,
$W_{3}=\left\\{f_{t};\text{ }f_{t}(\theta_{t_{1}})=1\right\\}$ ,
$W_{4}=\left\\{f_{t};\text{ }f_{t}(\theta_{t_{2}})=0\right\\}.$
Montrons que $\pi(W_{1})=\left\\{\theta_{t};\text{
}f_{t}(\theta_{1})=1\right\\}$ est un ouvert de $B_{2};$ il suffit de montrer
que $\left\\{\theta_{t};\text{ }f_{t}(\theta_{1})=0\right\\}$ est un fermé de
$B_{2}.$ Soit $(\theta_{t_{i}})_{i\in I}$ une suite généralisée dans
$\left\\{\theta_{t};\text{ }f_{t}(\theta_{1})=0\right\\}$ telle que
$\theta_{t_{i}}\rightarrow\theta_{t};$ comme $h$ est continue d’après la
proposition 4 on a $h(\theta_{t_{i}})\rightarrow h(\theta_{t})$ dans $E$.
D’autre part $h(\theta_{t_{i}})=h(\theta)-t_{i}$ , pour tout $i\in I$ et
$h(\theta_{t})=h(\theta)-t$ ; il en résulte que $t_{i}\rightarrow t$ dans $E$
, ce qui implique que $f_{t_{i}}(\theta_{1})\rightarrow f_{t}(\theta_{1}).$ On
en déduit que $f_{t}(\theta_{1})=0$. Donc $\left\\{\theta_{t};\text{
}f_{t}(\theta_{1})=0\right\\}$ est un fermé de $B_{2}.$ Par un argument
analogue on montre que $\pi(W_{2})$ est un ouvert de $B_{2}.$
Finalement on a $\pi(W_{3})=\left\\{\theta_{t};\text{
}f_{t}(\theta_{t_{1}})=0\right\\}=\left\\{\theta_{t};\text{
}f_{t_{1}}(\theta_{t})=1\right\\}$ et $\pi(W_{4})=\left\\{\theta_{t};\text{
}f_{t}(\theta_{t_{2}})=0\right\\}=\left\\{\theta_{t};\text{
}f_{t_{2}}(\theta_{t})=0\right\\},$ par conséquent $\pi(W_{3})$ et
$\pi(W_{4})$ sont ouverts de $B_{2}.\blacksquare$
Notons $F_{1}$ l’adhérence de $D$ dans $(C(K),\tau^{\prime})$. On définit
l’application $\sigma_{1}:E\rightarrow F_{1}\subset(C(K),\tau^{\prime})$ par
$\sigma(t)=f_{t}$. L’application $\sigma_{1}$ est borélienne, car pour tout
$\theta\in B_{2}\cup B_{1}$ l’application $t\in E\rightarrow
f_{t}(\theta)=\psi(\theta,t)$ est borélienne et ($D$,$\tau^{\prime})$ est
fortement de Lindelöf, d’après l’étape 3.
_Etape 4:_ Montrons que tout compact de $F_{1}$ est métrisable.
Soient $L$ un compact de $F_{1}$ et $(\theta_{n})_{n\geq 0}$ une suite dans
$B_{2}$ dense dans $K$ (une telle suite existe d’après l’étape 2); on définit
l’application $\xi:L\rightarrow\left\\{0,1\right\\}^{\mathbb{N}}$ par
$\xi(g)=(g(\theta_{n}))_{n\geq 0},$ $g\in L.$ il est clair que $\xi$ est
continue injective, par conséquent $L$ est métrisable.$\blacksquare$
Supposons que $(C(K),\tau^{\prime})$ est universellement mesurable.
_Etape 5:_ Montrons que $F_{1}$ est universellement mesurable.
D’après [Sch(2), th.3.2], il suffit de montrer que toute probabilité normale
sur $F_{1}$ est une mesure de Radon. Soit $\nu$ une mesure de probabilité
normale sur $F_{1};$ on définit la mesure $\nu^{\prime}$sur
$(C(K),\tau^{\prime})$ par $\nu^{\prime}(B)=\nu^{\prime}(B\cap F),$ pour tout
borélien $B$ de $(C(K),\tau^{\prime}).$ Montrons que $\nu^{\prime}$ est une
mesure normale sur $(C(K),\tau^{\prime}).$
Pour cela, soit $(U_{i})_{i\in I}$ une suite filtrante croissante d’ouverts
dans $(C(K),\tau^{\prime})$; on alors $\nu^{\prime}(\underset{i\in
I}{\cup}U_{i})=\nu(\underset{i\in I}{(\cup}U_{i})\cap
F_{1})=\nu(\underset{i\in I}{\cup}(U_{i}\cap F_{1}).$ D’autre part $\nu$ est
normale sur $F_{1}$ (muni de la topologie induite) donc
$\nu^{\prime}(\underset{i\in I}{\cup}U_{i})=\nu(\underset{i\in
I}{(\cup}U_{i})\cap F_{1})=\nu(\underset{i\in I}{\cup}(U_{i}\cap
F_{1})=\sup_{i\in I}\nu(U_{i}\cap F_{1})=\sup_{i\in I}\nu^{\prime}(U_{i}).$ Il
en résulte que $\nu^{\prime}$ est une mesure normale sur
$(C(K),\tau^{\prime}).$
Comme $(C(K),\tau^{\prime})$ est universellement mesurable, $\nu^{\prime}$ est
une mesure de Radon. Soit maintenant $B^{\prime}$ un borélien de $F_{1};$
$B^{\prime}$ est un borélien de $(C(K),\tau^{\prime})$ (car la tribu
borélienne de $F_{1}$ est égale à la tribu induite par celle de
$(C(K),\tau^{\prime})),$ donc
$\displaystyle\nu(B^{\prime})$ $\displaystyle=$
$\displaystyle\nu^{\prime}(B^{\prime})=\sup\left\\{\nu^{\prime}(L^{\prime});\text{
}L^{\prime}\text{ est un compact de }B^{\prime}\right\\}=$
$\displaystyle\sup\left\\{\nu(L^{\prime});\text{ }L^{\prime}\text{ est un
compact de }B^{\prime}\right\\}.$
On déduit que $\nu$ est une mesure de Radon.$\blacksquare$
_Etape 6_ : Montrons que $(C(K),\tau^{\prime\prime})$ est universellement
mesurable.
Remarquons que l’application
$g\in(C(K),\tau^{\prime})\rightarrow\overset{\vee}{g}\in(C(K),\tau^{\prime\prime})$
est une homéomorphisme (car si $\theta\in B_{2}\cup B_{1}$
$\overset{\vee}{\theta}\in B_{3}\cup B_{1}$ d’après la proposition 8$)$ donc
$(C(K),\tau^{\prime\prime})$ est universellement mesurable.
Désigons par $F_{2}$ l’adhérence de $\left\\{f_{t};\text{ }t\in E\right\\}$
dans $(C(K),\tau^{\prime\prime});$ par un argument analogue à celui de l’étape
4, on montre que tout compact de $F_{2}$ est métrisable et que l’application
$\sigma_{2}:t\in E\rightarrow f_{t}\in F_{2}$ est borélienne.
Soit $L$ un compact de $(E,\tau_{0})$ de mesure strictement positive; on peut
supposer que $\mu(L)=m(L)=1.$ Considérons $\eta_{1}:L\rightarrow F_{1}$ la
restriction de $\sigma_{1}$ à $L$ et $\nu_{1}=\eta_{1}(m_{L}),$ où $m_{L}$ la
mesure définie par
$m_{L}(C)=m(L\cap C),\text{ }C\in Bor(L).$
_Etape 7:_ Montrons que $\nu_{1}$ est une mesure normale sur $F_{1}.$
Soit $(U_{i})_{i\in I}$ une famille filtrante croissante d’ouverts dans
$F_{1};$ observons que pour tout borélien $B$ de $F_{1}$
$\nu(B)=m(\left\\{t\in L;\text{ }f_{t}\in B\right\\})=m(\left\\{t\in L;\text{
}f_{t}\in B\cap D\right\\}).$
D’après l’étape 3 $(D,\tau^{\prime})$ est fortement de Lindelöf, il existe
donc un sous-ensemble dénombrable $I_{1}$ de $I$ tel que $\underset{i\in
I}{\cup}U_{i}\cap D=\underset{i\in I_{1}}{\cup}U_{i}\cap D.$
Donc
$\displaystyle\nu(\underset{i\in I}{\cup}U_{i})$ $\displaystyle=$
$\displaystyle m(\left\\{t\in L;\text{ }f_{t}\in(\underset{i\in
I}{\cup}U_{i})\cap D\right\\})$ $\displaystyle=$ $\displaystyle m(\left\\{t\in
L;\text{ }f_{t}\in\underset{i\in I}{\cup}U_{i}\cap D\right\\})$
$\displaystyle=$ $\displaystyle m(\left\\{t\in L;\text{
}f_{t}\in\underset{i\in I_{1}}{\cup}U_{i}\cap D\right\\})$ $\displaystyle=$
$\displaystyle m(\left\\{t\in L;\text{ }f_{t}\in\underset{i\in
I_{1}}{(\cup}U_{i})\cap D\right\\})$ $\displaystyle=$
$\displaystyle\nu(\underset{i\in I_{1}}{\cup}U_{i})).$
Comme $I_{1}$ est dénombrable $\nu(\underset{i\in
I_{1}}{\cup}U_{i}))=\sup_{i\in I_{1}}\nu(U_{i})\leq\sup_{i\in I}\nu(U_{i}).$
On en déuit que $\nu(\underset{i\in I}{\cup}U_{i})\leq\sup_{i\in
I}\nu(U_{i})\leq\nu(\underset{i\in I}{\cup}U_{i}),$ c’est-à-dire que $\nu$ est
une mesure normale sur $F_{1}.\blacksquare$
L’étape 5 montre que $F_{1}$ est universellement mesurable et $\nu$ est une
probabilité normale sur $F_{1},$ d’après [Sch(2), th3.2] $\nu$ est une mesure
de radon, par conséquent pour tout $\varepsilon\in\left]0,1\right[,$ il existe
un compact $G_{1}$ de $F_{1}$ tel que $\nu(G_{1})>1-\varepsilon.$ Désignons
par $\eta_{2}$ la restriction de $\sigma_{1}$ à ($\eta_{1})^{-1}(G_{1})\subset
L;$ $\ \eta_{2}$ est une application borélienne à valeurs dans $G_{1}$ et
$m((\eta_{1})^{-1}(G_{1}))=\nu(G_{1})>1-\varepsilon.$ D’autre part d’après
l’étape $4$ $G_{1}$ est métriable, en utilisant le résultat de [Frem] on voit
qu’il existe un compact $L_{1}$ de $(\eta_{1})^{-1}(G_{1})$ tel que
$\mu(L_{1})=m(L_{1})>1-\varepsilon$ et la restriction de $\sigma_{1}$ à
$L_{1}$ soit continue.
Par un argument analogue à celui du précédent (en remplaçant $\sigma_{1}$ par
$\sigma_{2}$ et $L$ par $L_{1})$ on voit qu’il existe un compact $L_{2}$ de
$L_{1}$ telle que $\mu(L_{2})=m(L_{2})>1-\varepsilon$ et la restriction de
$\sigma_{2}$ à $L_{2}$ soit continue.
Soit $(t_{n})_{n\geq 0}$ une suite (non stationnaire) dans $L_{2}$ telle que
$t_{n}\rightarrow t_{0};$ donc
$f_{t_{n}}\underset{n\rightarrow\infty}{\rightarrow}f_{t_{0}}$ dans
$(C(K),\tau^{\prime})$ et dans $(C(K),\tau^{\prime\prime}),$ c’est-à-dire que
l’application $t\in L_{2}\rightarrow f_{t}\in(C(K),\tau_{p})$ est continue,
par conséquent elle est $\tau_{p}-$mesurable.
Comme $Bor(C(K),\tau_{p})=Bor(C(K),\left\|.\right\|)$ d’après le corollaire 2,
l’application $t\in L_{2}\rightarrow f_{t}\in C(K)$ est fortement mesurable,
ceci est impossible car $\left\|f_{t}-f_{t^{\prime}}\right\|\geq 1,$ si $t\neq
t^{\prime}.\blacksquare^{\prime}$
Démonstration de (2).
Soient $Y_{1}$ le sous-espace formé des fonctions paires dans $C(K)\subset
L^{\infty}(E)$ et $Y_{2}$ le sous-espace formé des fonctions impaires dans
$C(K)\subset L^{\infty}(E).$
Observons que si $g\in C(K)$ $\overset{\vee}{g}\in C(K)$ (en effet
$\overset{\vee}{f}_{t}=e-f_{-t},$ $\forall t\in E$). Montrons que $Y_{1}$ est
$\tau_{p}$ fermé dans $C(K).$
Considérons $(H_{i})_{i\in I}$ une suite généralisée dans $Y_{1}$ telle que
$H_{i}\rightarrow H\in(C(K),\tau_{p});$ il s’agit de montrer que $H\in Y_{1}.$
Pour $i\in I$ et $\theta\in K$ on a
$H_{i}(\overset{\vee}{\theta})=\overset{\vee}{H}_{i}(\theta)=h_{i}(\theta),$
par le passage à la limite on voit que $\overset{\vee}{H}(\theta)=H(\theta),$
c’est-à-dire que $\overset{\vee}{H}=H.$ Par un argument analogue on montre que
$Y_{2}$ est $\tau_{p}$ fermé.
Remarquons que l’application $g\in(C(K),\tau_{p})\rightarrow$
$\overset{\vee}{g}\in(C(K),\tau_{p})$ est une homéomorphisme et que pour toute
$g\in C(K)$ on a $g=(g+\overset{\vee}{g})/2+(g-\overset{\vee}{g})/2$
($(g+\overset{\vee}{g})/2\in Y_{1}$ et $(g-\overset{\vee}{g})/2\in Y_{2}),$ ce
qui implique que $(C(K),\tau_{p})=(Y_{1},\tau_{p})\oplus(Y_{2},\tau_{p})$
(donc la projection de $(C(K),\tau_{p})$ sur $(Y_{j},\tau_{p})$ est continue,
$j=1,2).\blacksquare$
Démonstration de (3).
On définit l’opérateur $U:(Y_{1},\tau_{p})\rightarrow(Y_{2},\tau_{p}),$ par
$U(g)(t)=\QATOPD\\{\\}{g(t),\text{ si t}\geq 0}{-g(t),\text{ si }t<0},$ $g\in
Y_{1};$ il est clair que $Ug\in Y_{2},$ pour tout $g\in Y_{1}$. Montrons que
$U$ est continue. Observons que
$\displaystyle U(g)$ $\displaystyle=$ $\displaystyle
g\times\mathcal{X}_{\left\\{t\in E;\text{ }t\geq
0\right\\}}-g\times\mathcal{X}_{\left\\{t\in E;\text{ }t<0\right\\}}=$
$\displaystyle g\times\mathcal{X}_{\left\\{t\in E;\text{
}t>0\right\\}}-g\times\mathcal{X}_{\left\\{t\in E;\text{ }t<0\right\\}}$
$\displaystyle=$ $\displaystyle g\times f-g\times(e-f),\forall g$
$\displaystyle\in$ $\displaystyle Y_{1}$
(où $f=\mathcal{X}_{\left\\{t\in E;\text{ }t>0\right\\}})$. Soit
$(g_{i})_{i\in I}$ une suite généralisée dans $(Y_{1},\tau_{p})$ telle que
$g_{i}\rightarrow g$ dans $(Y_{1},\tau_{p});$ pour tout $\theta\in K$ on a
$U(g_{i})(\theta)=(g_{i}\times
f)(\theta)-\left[g_{i}\times(e-f)\right](\theta)=g_{i}(\theta)f(\theta)-g_{i}(\theta)\times(1-f(\theta))\rightarrow
g(\theta)f(\theta)-g(\theta)\times(1-f(\theta))=U(g)(\theta).$
Montrons que $U^{-1}$ est continue à valeurs dans $(Y_{1},\tau_{p});$ il
suffit de montrer que
$U^{-1}(g)=g\times f-g\times(e-f),\text{ }g\in Y_{2}.$
(l’identification ci-dessus montre que $U^{-1}(g)\in Y_{2},$ pour tout $g\in
Y_{1}$ et par un argument analogue à celui du précédent on montre que $U^{-1}$
est continue).
Pour cela soit $g\in Y_{2};$ comme $f^{2}=f$ et $(e-f)^{2}=e-f$ on a alors
$\displaystyle U(g\times f-g\times(e-f))$ $\displaystyle=$
$\displaystyle\left[g\times f-g\times(e-f)\right]\times f$
$\displaystyle-\left[g\times f-g\times(e-f)\right](e-f)$ $\displaystyle=$
$\displaystyle=$ $\displaystyle\left[2g\times f-g)\right]\times f$
$\displaystyle-\left[g\times f(e-f)-g\times(e-f)\right]$ $\displaystyle=$
$\displaystyle 2g\times f-g\times f$ $\displaystyle-\left[g\times f-g\times
f-g+g\times f\right]$ $\displaystyle=$ $\displaystyle g\times f-\left[g\times
f-g\right]=g.$
Donc $U$ est un isomorphisme.$\blacksquare$
Démonstration de (4).
Montrons que $(Y_{1},\tau^{\prime})=(Y_{1},\tau_{p}).$
Pour $\theta\in B_{2}\cup B_{1}$ et $g\in Y_{1}$ on a
$g(\overset{\vee}{\theta})=(\overset{\vee}{g})(\theta)=g(\theta)$; comme
$\left\\{\overset{\vee}{\theta};\text{ }\theta\in B_{2}\cup
B_{1}\right\\}=B_{3}\cup B_{1}$ d’après la proposition 8, alors
$(Y_{1},\tau^{\prime})=(Y_{1},\tau_{p}).$
Montrons que $(Y_{2},\tau^{\prime})=(Y_{2},\tau_{p}).$
Soit $(g_{i})_{i\in I}$ une suite généralisée dans $Y_{2}$ telle que
$g_{i}(\theta)\rightarrow g(\theta),$ pour tout $\theta\in B_{2}\cup B_{1},$
où $g\in Y_{2}.$ Il s’agit de montrer que $g_{i}\rightarrow g$ dans
$(Y_{2},\tau_{p}).$ Pour tout $i\in I$ on a $U^{-1}(g_{i})=g_{i}\times
f-g_{i}\times(e-f);$ cela implique que $U^{-1}(g_{i})(\theta)\rightarrow
U^{-1}(g)(\theta),$ pour tout $\theta\in
B_{2}\cup\left\\{\theta^{\prime}\right\\}.$ Comme
$(Y_{1},\tau_{p})=(Y_{1},\tau^{\prime})$ $U^{-1}(g_{i})(\theta)\rightarrow
U^{-1}(g)(\theta)$ pour tout $\theta\in K$
(car$(Y_{1},\tau^{\prime})=(Y_{1},\tau_{p})).$ Il en résulte que
$g_{i}\rightarrow g,$ dans $(Y_{2},\tau_{p}).$ Donc
$(Y_{2},\tau^{\prime})=(Y_{2},\tau_{p}).\blacksquare$
Démonstration de (5).
Montrons que $(Y_{1},\tau_{p})$ n’est pas $K-analytique.$
Supposons que $(Y_{1},\tau_{p})$ est $K-analytique$; d’après (3),
$(Y_{2},\tau_{p})$ est $K-analytique$, donc $(C(K),\tau_{p})$ est
$K-analytique$ d’après (2); comme la topologie $\tau^{\prime}$ sur $C(K)$ est
moins fine que $\tau_{p}$, alors $(C(K),\tau^{\prime})$ est $K-analytique.$
D’après [Bress-Si] $(C(K),\tau^{\prime})$ est universellement mesurable, ce
qui est impossible d’après (1)$.\blacksquare$
Démonstration de (6).
Supposons que $(Y_{1},\tau_{p})$ est souslinien; il existe donc un espace
polonnais $P$ et $H:P\rightarrow(Y_{1},\tau_{p})$ une homépmphisme borélienne,
par conséquent
$card\left[Bor(Y_{1},\left\|.\right\|)\right]=card\left[Bor((Y_{1},\tau_{p})\right]=card\left[Bor(P)\right]<card\left[\left\\{0,1\right\\}^{E}\right]$
(car d’après corollaire 2 $Bor(C(K),\tau_{p})=Bor(C(K),\left\|.\right\|).$
Notons $\omega_{t}=$ $f_{t}+\overset{}{\overset{\vee}{(f_{t})}\in Y_{1}}$,
$t\in\allowbreak E;$ d’après ce qui est précède
$card\left[Bor(\left\\{\omega_{t};\text{ }t\in
E\right\\},\left\|.\right\|)\right]\leq
card\left[Bor(Y_{1},\left\|.\right\|)\right]<card\left[\left\\{0,1\right\\}^{E}\right].$
La démonstration sera terminée, si on montre que la topologie de la norme sur
$\left\\{w_{t};\text{ }t\in E\right\\}$ est la topologie discrète. Il suffit
de montrer que $\left\|\omega_{t}-\omega_{t^{\prime}}\right\|\geq 1,$ si
$t\neq t^{\prime}$ (car dans ce cas
$\left\\{\omega_{t}\right\\}=\left\\{\omega_{t^{\prime}};\text{
}\left\|\omega_{t}-\omega_{t^{\prime}}\right\|<1/2\right\\}).$
Soit $t,t^{\prime}\in E$ tel que $t<t^{\prime};$ remarquons que sur
$\left]t,t^{\prime}\right[,$ $f_{t}>f_{t^{\prime}}$ et $f_{-t^{\prime}}\geq
f_{-t}$ presque-partout sur $E$ (car $-t^{\prime}<-t).$ Donc pour presque tout
$y\in\left]t,t^{\prime}\right[$ on a
$\omega_{t}(y)=f_{t}(y)+1-f_{-t}(y)=1+1-f_{-t}(y)=2-f_{-t}(y)$
et
$\omega_{t^{\prime}}(y)=f_{t^{\prime}}(y)+1-f_{-t^{\prime}}(y)=0+1-f_{-t^{\prime}}(y)=1-f_{-t}(y)$
On conclut que $\left\|\omega_{t}-\omega_{t^{\prime}}\right\|\geq 1,$ si
$t\neq t^{\prime}$ (car le cas $f_{-t}(y)=1,$ $f_{-t^{\prime}}(y)=0$ est
exclu)$.\blacksquare$
Démonstration de (7).
Au cours de la démonstration de $(1)$ on a vu que les applications
$\sigma_{1}:t\in E\rightarrow f_{t}\in(C(K),\tau^{\prime}),$ $\sigma_{2}:t\in
E\rightarrow\overset{}{f_{t}\in}(C(K),\tau^{\prime\prime})$ sont boréliennes.
D’autre part comme pour tout $\theta\in B_{2}\cup B_{1}$
$\overset{\vee}{(f_{t})}(\theta)=f_{t}(\overset{\vee}{\theta})$ et
$\overset{\vee}{\theta}\in B_{3}\cup B_{1}$ $($d’après la proposition 8).
$\sigma_{2}$ est borélienne entraîne que l’application $t\in
E\rightarrow\overset{\vee}{(f_{t})}\in(C(K),\tau^{\prime})$ est borélienne.
Supposons maintenant que $Bor(C(K),\tau^{\prime})\otimes
Bor(C(K),\tau^{\prime})=Bor(C(K)\times
C(K),\tau^{\prime}\times\tau^{\prime}).$ Considérons l’application
$\phi_{1}:t\in E\rightarrow(f_{t},\overset{\vee}{(f_{t})})\in C(K)\times C(K)$
et fixons $W_{1},W_{2}$ deux ouverts de $(C(K),\tau^{\prime});$ il est clair
que $\phi_{1}^{-1}(W_{1}\times W_{2})$ est un borélien de $E$, donc pour tout
$V\in Bor(C(K),\tau^{\prime})\otimes Bor(C(K),\tau^{\prime})$
$\phi_{1}^{-1}(V)$ est un borélien de $E.$
D’après ce qui est précède l’application $\phi_{1}:t\in
E\rightarrow(f_{t},\overset{\vee}{(f_{t})})\in(C(K)\times
C(K),\tau^{\prime}\times\tau^{\prime})$ est borélienne.
Définissons une autre application $\phi_{2}:(C(K)\times
C(K),\tau^{\prime}\times\tau^{\prime})\rightarrow(C(K),\tau^{\prime}),$ par
$\phi_{2}(g,u)=g+u,$ $(g,u)\in C(K)\times C(K);$ $\phi_{2}$ est continue, donc
elle est borélienne. Il en résulte que $\phi_{2}\circ\phi_{1}$ est borélienne,
c’est-à-dire que l’appliction $t\in E\rightarrow
w_{t}=f_{t}+\overset{\vee}{(f_{t})}\in(Y_{1},\tau^{\prime})=(Y_{1},\tau_{p})$
est borélienne. D’autre part $Bor(C(K),\tau_{p})=Bor(C(K),\left\|.\right\|),$
d’après le corollaire 2, donc
$Bor(Y_{1},\tau_{p})=Bor(Y_{1},\left\|.\right\|),$ par conséquent $t\in
E\rightarrow\omega_{t}\in Y_{1}$ est fortement mesurable, ceci est impossible,
car au cours de la démonstration de (6), on a vu que
$\left\|\omega_{t}-\omega_{t^{\prime}}\right\|\geq 1,$ si $t\neq
t^{\prime}.\blacksquare$
Démonstration de (8).
Supposons que la projection
$P:(C(K),\tau^{\prime})\rightarrow(Y_{1},\tau^{\prime})=(Y,\tau_{p})$ est
borélienne. Comme l’application $\sigma_{1}:t\in E\rightarrow f_{t}$
$\in(C(K),\tau^{\prime})$ est borélienne, l’application $t\in E\rightarrow
Pf_{t}=w_{t}/2$ $\in(C(K),\tau^{\prime})$ est borélienne, ceci est impossible,
donc $P$ n’est pas borélienne.$\blacksquare$
Démonstration de (9).
Notons $H_{1}=\sigma_{1}:t\in E\rightarrow f_{t}\in(C(K),\tau^{\prime})$ et
$H_{2}:t\in E\rightarrow\overset{\vee}{(f_{t})}\in(C(K),\tau^{\prime});$ au
debut de la preuve de $7,$ on a montré que $S_{1}$ et $S_{2}$ sont borélienne.
et à la fin de la preuve de $7,$ nous montre que $H_{1}+H_{2}$ n’est pas
borélienne.$\blacksquare$
###### Remarque 10.
D’après le corollaire 2 on a $Bor(C(K),\tau_{p})\otimes
Bor(C(K),\tau_{p})=Bor(C(K),\left\|.\right\|)\otimes
Bor(C(K),\left\|.\right\|).$
D’autre part, d’après le résultat de [Tal(1), th.3] on a
$Bor(C(K),\left\|.\right\|)\otimes
Bor(C(K),\left\|.\right\|)=\mathit{Bor(C(K)\times
C(K),}\left\|.\right\|\mathit{).}$ Donc $\mathit{Bor(C(K)\times
C(K),\tau}_{p}\times\tau_{p}\mathit{)\subset}$
$\mathit{Bor(C(K)\times
C(K),}\left\|.\right\|\mathit{)=}Bor(C(K),\left\|.\right\|)\otimes
Bor(C(K),\left\|.\right\|)=Bor(C(K),\tau_{p})\otimes Bor(C(K),\tau_{p}).$
Mais $Bor(C(K),\tau_{p})\otimes
Bor(C(K),\tau_{p})\subset\mathit{Bor(C(K)\times
C(K),\tau}_{p}\mathit{\times\tau}_{p}\mathit{),}$ on déduit que
$Bor(C(K),\tau_{p})\otimes Bor(C(K),\tau_{p})=\mathit{Bor(C(K)\times
C(K),\tau}_{p}\mathit{\times\tau}_{p}\mathit{).}$
Soit $L$ un compact de Hausdroff; dans [Tal(2)] on montre sous l’axiome de
Martine, que $(C(L),\tau_{p})$ est universellement mesurable, dans le lemme
suivant on montre que $(C(L),\tau_{p})$ est universellement mesurable sans
l’axiome de Martine, si $L$ est un compact de Rosenthal.
###### Lemme 8.
Soit $L$ un compact de Rosenthal; alors $(C(L),\tau_{p})$ est universellement
mesurable.
Démonstration.
Il suffit de montrer d’après [Sch(2), th.3.2] que toute mesure de probabilité
normale sur $(C(L),\tau_{p})$ est une mesure de Radon. Soit $\mu$ une
probabilité normale sur $(C(L),\tau_{p});$ on peut supposer que $\mu$ est
portée par la boule unité de $(C(L),\left\|.\right\|).$ Notons $V$ le sous-
espace vectoriel engendré par les éléments de $L$ dans le dual de
$(C(L),\left\|.\right\|);$ pour tout $y\in V$ soit $H_{y}$ l’hyperplan défini
par $y$ dans $C(L),$ c’est-à-dire
$H_{y}=\left\\{f\in C(L);\text{ }(f,y)=y(f)=0\right\\}.$
Considérons l’ensemble $I^{\prime}$ formé des éléménts $y$ de $V$, vérifiant
$\mu(H_{y})=1$ et $E_{\mu}$ est l’intrersection des $H_{y}$ lorsque $y\in
I^{\prime}$ (remarquons que $0\in I).$ Comme $\mu$ est normale on a
$\mu(E_{\mu})=1.$ Pour tout $y\in L$ notons
$\left[y\right]=\left\\{x\in L;\text{ }(f,y)=(f,x),\text{ }\forall f\in
E_{\mu}\right\\};$
pour tout $y\in L$ on définit $\left[y\right](f)=(f,y),$ $f\in E_{\mu}.$
Suppososns que $\left[y\right]=\left[y^{\prime}\right],$ $\mu-$presque-
partout, $(y,y^{\prime}\in L);$ montrons que
$\left[y\right]=\left[y^{\prime}\right]$ partout. En effet, l’hyperplan
$H_{y-y^{\prime}}$ est de mesure est égale à $1,$ donc $H_{y-y^{\prime}}$
contient $E_{\mu},$ par conséquent pour tout $f\in E_{\mu},$
$(f,y)=(f,y^{\prime}).$ Notons $L_{\mu}=\left\\{\left[y\right];\text{ }y\in
L\right\\};$ comme l’application $\Sigma:y\in L\rightarrow\left[y\right]\in
L_{\mu}$ est continue, $L_{\mu}$ est un compact pour la topologie de la
convergence simple, (cette topologie est la topologie produit sur
$\left[-1,+1\right]^{E_{\mu}},$ si $C(L)$ est l’espace des fonctions continues
sur $L$ à valeurs dans $\mathbb{R}).$
On définit l’application $U:L_{\mu}\rightarrow L^{1}(E_{\mu},\mu)$ par
$U(\left[x\right])(f)=f(x),$ pour tout $x\in L.$ D’après ce qui est précède
$U$ est une application injective.
Montrons que $U$ est une application continue. Pour le voir; soit $F$ un fermé
de $L^{1}(E_{\mu},\mu)$ et $\left[x\right]$ est un point adhérent à
$U^{-1}(F);$ il existe une suite généralisée $(\left[x_{i}\right])_{i\in I}$
dans $U^{-1}(F)$ telle que $\left[x_{i}\right]\rightarrow\left[x\right],$
(cette convergence suivant un ultrafiltre non trivial sur $I).$ D’autre part
$L$ est compact, donc il existe $x^{\prime}\in L$ tel que $x_{i}\rightarrow x$
dans $L,$ par la continuité de $\Sigma,$
$\left[x_{i}\right]\rightarrow\left[x^{\prime}\right]$ dans $L_{\mu},$ il en
résulte que $\left[x^{\prime}\right]=\left[x\right].$
$L$ est angelique, car $L$ est un compact de Rosenthal, donc il existe une
sous-suite dénombrable $(x_{i_{n}})_{n\geq 0}$ de $(x_{i})_{i\in I}$ telle que
$x_{i_{n}}\underset{n\rightarrow\infty}{\rightarrow}x;$ comme $\Sigma$ est
continue
$\left[x_{i_{n}}\right]\underset{n\rightarrow\infty}{\rightarrow}\left[x\right]$
dans $L_{\mu}.$
En appliquant le théorème de convergence dominée, on voit que
$U(\left[x_{i_{n}}\right])\underset{n\rightarrow+\infty}{\rightarrow}U(\left[x\right]);$
d’autre part pour tout $n\in\mathbb{N}$ $U(\left[x_{i_{n}}\right])\in F;$ cela
implique que $U(\left[x\right])\in F.$ On en déduit que $U$ est une
homéomorphisme sur son image, ce qui est implique que $L_{\mu}$ est
métrisable.
D’après le raisonnement précédent, on voit que $(C(L_{\mu}),\left\|.\right\|)$
est un espace polonnais, donc
$Bor(C(L_{\mu}),\left\|.\right\|)=Bor(C(L_{\mu}),\tau_{p})$; de l’autre côté
$(C(L_{\mu}),\left\|.\right\|)$ est radonien [Sch(1), chap.II,p.92] et
$(E_{\mu},\tau_{p})$ est un sous-espace fermé de $(C(L_{\mu}),\tau_{p})$, cela
implique que $(E_{\mu},\tau_{p})$ est radonien; il en résulte que $\mu$ est
une mesure de Radon.$\blacksquare$
###### Corollaire 12.
Il existe deux sous-espaces $Z_{1},Z_{2},$ $\tau_{p}$-fermés de
$(C(K),\tau_{p})$ tels que $Z_{1}\cap Z_{2}=\left\\{0\right\\},$
$(Z_{j},\tau^{\prime})$ est universellement mesurables, $j=1,2$ et
$(Z_{1},\tau^{\prime})$ est isomorphe à $(Z_{2},\tau^{\prime}),$ mais
$(Z_{1}\oplus Z_{2},\tau^{\prime})$ n’est pas universellement mesurable.
Démonstration
Soit $Z_{j}=Y_{j},$ $j=1,2;$ d’après le théorème 5 on a
$(Z_{j},\tau^{\prime})=(Z_{j},\tau_{p})$, $Z_{j}$ est $\tau_{p}$ fermé dans
$(C(K),\tau_{p}),$ $j=1,2$ et $(Z_{1},\tau^{\prime})$ est isomorphe à
$(Z_{2},\tau^{\prime})$. D’autre part le lemme 8 montre que $(C(K),\tau_{p})$
est universellement mesurable, donc $(Z_{j},\tau^{\prime})=(Z_{j},\tau_{p})$
est universellment mesurable (car un fermé d’un espace universellement
mesurable est universellement mesurable). Comme $(Z_{1}\oplus
Z_{2},\tau_{1})=(C(K),\tau^{\prime}),$ d’après le théorème 5 $(Z_{1}\oplus
Z_{2},\tau^{\prime})$ n’est pas universellement mesurable.$\blacksquare$
## References
* [Bour-Ros-Tal] J. Bourgain, D. H. Rosenthal and M. Talagrand, Pointwise compact sets of Baire measurable functions, Amer. J. Math. 100, No. 4, 845-886, (1998).
* [Bress-Si] D. W. Bressler and M. Sion, The current theory of analytic sets, Canad. J. Math. 16, 207-230, (1964).
* [Frem] D. H. Fremlin, Measurable functions and almost continuous functions, Manus. Math. 33, 387-405, (1981).
* [Gode] G. Godefroy, Compacts de Rosenthal, Pac. Jour. of Math. Vol. 91, 293-306, (1980).
* [Godem] R. Godement, Sur la théorie des représentation unitaires Ann. Math. t. 53, 68-124, (1951).
* [Hay-Jay-Nam-Rog] R. G. Haydon, J. E. Jayne, I. Namioka and C. A. Rogers, Continuous functions on totally ordered spaces that are compacts in their order topologies, J. Funct. Anal. ,178, No. 1, 23-63, (2000).
* [Hew] E. Hewitt and K. A. Ross, Abstract harmonic analysis I, Springer-Verglas, Berlin-Heidelberg-New-York, (1963).
* [Jay-Nam-Rog] J. E. Jayne, I. Namioka and C. A. Rogers, Continuous functions on product of compact Hausdroff space, Mathematika, 323-330, (1997).
* [Kell] J. L. Kelley, General topology, D.Van Nostrand Company INC, (1955).
* [Sch(1)] Laurent Schwartz, Radon measures on arbitary topological spaces cylindrical measures, Oxford University Press, (1973).
* [Sch(2)] Laurent Schwartz, Certaines propriétés des mesures sur les espaces de Banach, Séminaire Maurey-Schwartz, Exposé No. 23, (1975-1976).
* [Lin-Steg] J. Lindenstrauss and C. Stegall, Exemples of separable spaces which do not contain $\ell_{1}$ and whos duals non separable, Stud. Math. 54, 81-105, (1975-19976).
* [Od-Ros] E. Odell and H. P. Rosnehtal, A double characterization of separable Banach spaces containing $\ell^{1},I$sraël J. Math. 20, 375-384, (1975).
* [Osta] A. J. Ostaszewski, A characterization of compact separable ordered spaces, J. Lond. Math. Soc. II Ser. 7, 758-769, (1974).
* [Raj] M. Raja, Kadec norms and Borel sets in a Banach space, Stud. Math. 136, 1-16, (1999).
* [Sto] M. H. Stone, A general theory of spectra, Proc. Not. Acad. Sci. U. S .A, 26, 280-283, (1940).
* [Tal(1)] M. Talagrand, Est-ce que $\ell^{\infty}$ est un espace mesurable, Bull. Sci. Math. II, serie 103, 255-258, (1979).
* [Tal(2)] M. Talagrand, Solution d’un problème de A.Ionescu-Tulcea, CR. Acad. Sci. Paris, Sér. A, 283, 975-978, (1976).
|
arxiv-papers
| 2012-07-02T10:31:32 |
2024-09-04T02:49:32.490517
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Mohammad Daher and Khalil Saadi",
"submitter": "Mohammad Daher",
"url": "https://arxiv.org/abs/1207.0330"
}
|
1207.0392
|
# Three-intensity decoy state method for device independent quantum key
distribution with basis dependent errors
Xiang-Bin Wang xbwang@mail.tsinghua.edu.cn Department of Physics and State
Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University,
Beijing 100084, China Jinan Institute of Quantum Technology, Shandong Academy
of Information Technology, Jinan, China
###### Abstract
We study the measurement device independent quantum key distribution (MDIQKD)
in practice with limited resource, when there are only 3 different states in
implementing the decoy-state method and when there are basis dependent coding
errors. We present general formulas for the decoy-state method for two-pulse
sources with 3 different states, which can be applied to the recently proposed
MDIQKD with imperfect single-photon source such as the coherent states or the
heralded states from the parametric down conversion. We point out that the
existing result for secure MDIQKD with source coding errors does not always
hold. We find that very accurate source coding is not necessary. In
particular, we loosen the precision of existing result by several magnitude
orders.
###### pacs:
03.67.Dd, 42.81.Gs, 03.67.Hk
## I Introduction
Security for real set-ups of quantum key distribution(QKD)BB84 ; GRTZ02 has
become a major problem in the area in the recent years. The major problems
here include the imperfection of source and the limited efficiency of the
detection device. The decoy state methodILM ; H03 ; wang05 ; LMC05 ; AYKI ;
haya ; peng ; wangyang ; rep ; njp can help to make a set-up with an
imperfect single photon source be as secure as that with a perfect single
photon sourcePNS1 ; PNS .
Besides the source imperfection, the limited detection is another threaten to
the securitylyderson . Theories of the device independent security proofind1
have been proposed to overcome the problem. However, these theories cannot
apply to the existing real set-ups because violation of Bell’s inequality
cannot be strictly demonstrated by existing set-ups.
Very recently, an idea of measurement device independent QKD (MDIQKD) was
proposed based on the idea of entanglement swappingind3 ; ind2 . There, one
can make secure QKD simply by virtual entanglement swapping, i.e., Both Alice
and Bob sends BB84 states to the relay which can be controlled by un-trusted
third party (UTB). After the UTB announced his measurement outcome, Alice and
Bob will post select those bits corresponding to a successful event and
prepared in the same basis for further processing. In the realization, Alice
and Bob can really use entanglement pairsind3 and measure halves of the pair
inside the lab before sending another halves to the UTB. In this way, the
decoy-state method is not necessary even though imperfect entangled pairs
(such as the states generated by the type II parametric down conversion) are
used. Even though there are multi-pair events with small probability, these
events do not affect the security. Alice and Bob only need to check the error
rates of their post selected bits. However, in our existing technologies, high
quality entangled-pair-state generation can not be done efficiently. In the
most matured technology, the generation rate is lower than 1 from 1000 pump
pulses. If we want to obtain a higher key rate, we can choose to directly use
an imperfect single-photon source such as the coherent stateind2 . If we
choose this, we must be careful for two issues. First, we must implement the
decoy-state method for security. Although this has been discussed in Ref.ind2
, calculation formulas for the practical decoy-state implementation with only
a few different states are not given. Second, in this way, the states for
coding are prepared actively. If we cannot guarantee to make exactly the BB84
states, we must take special caution for the security. Although there are
already some results for thistamaki , there are some drawbacks in practical
application of the existing resulttamaki . First, it requests a very accurate
source coding, e.g., a magnitude order of $10^{-7}$ for the state errors for
MDIQKD over a distance longer than 100 kms. Second, the existing conclusion
seem to be not always correct. The existing theorytamaki shows that the
source coding error affects the key rate only through the fidelity between the
density matrices of two bases. According to its conclusion, if the density
operators of states in the two bases are identical, then one can calculate the
key rate as if an ideal realization of MDIQKD were done. In such a case, the
key rate is determined solely by the detected error rate. Consider such a
special case: in the protocol, Alice and Bob can produce the perfect BB84
states in $Z$ basis, $|0_{z}\rangle$ and $|1_{z}\rangle$, but they make big
errors in preparing states in $X$ basis. They actually prepared
$|0_{z}\rangle$ or $|1_{z}\rangle$ whenever they want to prepare states
$|0_{x}\rangle$ or $|1_{x}\rangle$. Given this fact, Eve. or the UTB can
simply measure each pulses in $Z$ basis without causing any additional noise.
Therefore, correct theory should give 0 key rate for this. However, the
existing theory can result in considerable key rate for this case because in
principle one can obtain lots of post selected successful events with small
error rate in MDIQKDtamakie . Such a problem also exists in the MDIQKD
protocol with entangled-pair statesind3 . Although the states out of the labs
are identical for whatever basis, the measurement basis alignment error in
detecting the halves of the pair states inside each labs can undermine the
security. In an extreme example, they make measurement in $Z$ basis perfectly.
But when they want to use $X$ basis, they actually used $Z$ basis. Normally,
users are not likely to make such big mistakes, however, the existing theory
seemed to even allow these mistakes.
Here in this work, we shall first present formulas of 3-state decoy-state
method for the MDIQKD. We then study the problem source coding error. Our
result presented here does not request very accurate source coding as the
existing ones which requests a magnitude order of $10^{-7}$ for the state
errors, while our request for the accuracy is more loosen than this by several
magnitude orders. Based on the idea of constructing virtual BB84 sources, our
result is strict for security.
## II Decoy-state method with only 3 states for MDIQKD
In the protocol, each time a pulse-pair (two-pulse state) is sent to the relay
for detection. The relay is controlled by an un-trusted thirty party (UTP).
The UTP will announce whether the pulse-pair has caused a successful event.
Those bits corresponding to successful events will be post-selected and
further processed for the final key. Since real set-ups only use imperfect
single-photon sources, we need the decoy-state method for security.
We assume Alice (Bob) has three sources, $o_{A},x_{A},y_{A}$
($o_{B},x_{B},y_{B}$) which can only emit three different states
$\rho_{o_{A}}=|0\rangle\langle 0|,\rho_{x_{A}},\rho_{y_{A}}$
($\rho_{o_{B}}=|0\rangle\langle 0|,\rho_{x_{B}},\rho_{y_{B}}$), respectively,
in photon number space. Suppose
$\rho_{x_{A}}=\sum_{k}a_{k}|k\rangle\langle
k|;\,\,\rho_{y_{A}}=\sum_{k}a_{k}^{\prime}|k\rangle\langle k|,\\\
\rho_{x_{B}}=\sum_{k}b_{k}|k\rangle\langle
k|;\,\,\rho_{y_{B}}=\sum_{k}b_{k}^{\prime}|k\rangle\langle k|,$ (1)
and we request the states satisfy the following very important condition:
$\frac{a_{k}^{\prime}}{a_{k}}\geq\frac{a_{2}^{\prime}}{a_{2}}\geq\frac{a_{1}^{\prime}}{a_{1}};\frac{b_{k}^{\prime}}{b_{k}}\geq\frac{b_{2}^{\prime}}{b_{2}}\geq\frac{b_{1}^{\prime}}{b_{1}}$
(2)
for $k\geq 2$. The imperfect sources used in practice such as the coherent
state source, the heralded source out of the parametric-down conversion,
satisfy the above restriction. Given a specific type of source, the above
listed different states have different averaged photon numbers (intensities),
therefore the states can be obtained by controlling the light intensities. At
each time, Alice will randomly select one of her 3 sources to emit a pulse,
and so does Bob. The pulse form Alice and the pulse from Bob form a pulse pair
and are sent to the un-trusted relay. We regard equivalently that each time a
two-pulse source is selected and a pulse pair (one pulse from Alice, one pulse
from Bob) is emitted. There are many different two-pulse sources used in the
protocol. We denote $\alpha\beta$ for the two pulse source when the pulse-pair
is produced by source $\alpha$ at Alice’s side and source $\beta$ at Bob’s
side, $\alpha$ can be one of $\\{o_{A},x_{A},y_{A}\\}$ and $\beta$ can be one
of $\\{o_{B},x_{B},y_{B}\\}$. For example, at a certain time $j$ Alice uses
source $o_{A}$ and Bob uses source $y_{B}$, we say the pulse pair is emitted
by source $o_{A}y_{B}$.
In the protocol, two different bases, $Z$ basis consisting of horizontal
polarization $|H\rangle\langle H|$ and vertical polarization $|V\rangle\langle
V|$, and $X$ basis consisting of $\pi/4$ and $3\pi/4$ polarizations are used.
The density operator in photon number space alone does not describe the state
in the composite space. We shall apply the the decoy-state method analysis in
the same basis (e.g., $Z$ basis or $X$ basis) for pulses from sources
$x_{A},x_{B},y_{A},y_{B}$. Therefore we only need consider the density
operators in the photon number space. For simplicity, we consider pulses from
source prepared in $Z$ basis first.
According to the decoy-state theory, the yield of a certain set of pulse pairs
is defined as source $\alpha\beta$ is defined as the happening rate of a
successful event (announced by the UTP) corresponding to pulse pairs out of
the set. Mathematically, the yield is $n/N$ where $n$ is the number of
successful events happened corresponding to pulse pairs from the set and $N$
is the number of pulse pairs in the set. Obviously, if we regard the pulse
pairs of two-pulse source $\alpha\beta$ as a set, the yield $S_{\alpha\beta}$
for source $\alpha\beta$ is
$S_{\alpha\beta}=\frac{n_{\alpha\beta}}{N_{{\alpha\beta}}}$, where
$n_{\alpha\beta}$ is the number of successful events happened corresponding to
pulse pairs from source $\alpha\beta$ and $N_{\alpha\beta}$ is the number of
times source ${\alpha\beta}$ are used. In the protocol, there are 9 different
two-pulse sources. The yields of these 9 sources can be directly calculated
from the observed experimental data $n_{\alpha\beta}$ and $N_{\alpha\beta}$.
We use capital letter $S_{\alpha\beta}$ for these known values.
We can regard any source as a composite source that consists of many (virtual)
sub-sources, if the source state can be be written in a convex form of
different density operators. For example, two-pulse source $y_{A}y_{B}$
includes a sub-source of pulse pairs of state $\rho_{1}\otimes\rho_{1}$
($\rho_{1}=|1\rangle\langle 1|$) with weight $a_{1}^{\prime}b_{1}^{\prime}$.
This is to say, after we have used source $y_{A}y_{B}$ for $N$ times, we have
actually used sub-source of state $\rho_{1}\otimes\rho_{1}$ for
$a_{1}^{\prime}b_{1}^{\prime}N$ times, asymptotically. Similarly, the source
$x_{A}x_{B}$ also includes a sub-source of state
$\\{\rho_{1}\otimes\rho_{1}\\}$ with weight $a_{1}b_{1}$. These two sub-
sources of state $\rho_{1}\otimes\rho_{1}$ must have the same yield $s_{11}$
because they have the same two-pulse state and the pulse pairs are randomly
mixed. Most generally, denote $s,s^{\prime}$ as the yields of two sets of
pulses, if pulse pairs of these two sets are randomly mixed and all pulses
have the same density operator, then
$s=s^{\prime}$ (3)
asymptotically. This is the elementary assumption of the decoy-state theory.
In the protocol, since each sources are randomly chosen, pulses from each sub-
sources or sources are also randomly mixed. Therefore, the yield of a sub-
source or a source is dependent on the state only, it is independent of which
physical source the pulses are from. Therefore, we can also define the yield
of a certain state: whenever a pulse pair of that state is emitted, the
probability that a successful event happens. Denote
$\Omega_{\alpha\beta}=\rho_{\alpha}\otimes\rho_{\beta}$ (4)
for a two-pulse state. The yield of such a state is also the yield of any
source which produces state $\Omega_{\alpha\beta}$ only, or the yield of a
sub-source from any source, provided that the state of the pulse pairs of the
sub-source is $\Omega_{\alpha\beta}$. Note that, we don’t always know the
value of yield of a state. Because we don’t know which sub-source was used at
which time. We shall use the lower case symbol $s_{\alpha,\beta}$ to denote
the yield of state $\Omega_{\alpha,\beta}$. In general, the yields of a sub-
source (a state), such as $s_{11}$ is not directly known from the experimental
data. But some of them can be deduced from the yields of different real
sources. Define $\rho_{0}=|0\rangle\langle 0|$. According to Eq.(3), if
$\alpha\in\\{0,x_{A},y_{A}\\}$ and $\beta\in\\{0,x_{B},y_{B}\\}$, we have
$s_{\alpha\beta}=S_{\tilde{\alpha}\tilde{\beta}}$ (5)
with the mapping of $\tilde{\alpha}=(o_{A},x_{A},y_{A})$ for
$\alpha=(0,x_{A},y_{A})$, respectively; and
$\tilde{\beta}=(o_{B},x_{B},y_{B})$ for $\beta=(0,x_{B},y_{B})$, respectively.
To understand the meaning of the equation above, we take an example for pulses
from source $y_{A}y_{B}$. By writing the state of this source in the convex
form we immediately know that it includes a sub-source of state
$\rho_{0}\otimes\rho_{y_{B}}$. By observing the results caused by source
$y_{A}y_{B}$ itself we have no way to know the yield of this sub-source
because we don’t know exactly which time source $y_{A}$ emits a vacuum pulse
when we use it. However, the state of this sub-source is the same with the
state of the real source $o_{A}y_{B}$, therefore the yield of any sub-source
of state $\rho_{0}\otimes\rho_{y_{B}}$ must be just the yield of the real
source $o_{A}y_{B}$, which can be directly observed in the experiment.
Mathematically, this is $s_{0y_{B}}=S_{o_{A}y_{B}}$, where the right hand side
is the known value of yield of real source $o_{A}y_{B}$, the left hand side is
the yield of a virtual sub-source from real source $y_{B}y_{B}$.
Our first major task is to deduce $s_{11}$ from the known values, i.e., to
formulate $s_{11}$, the yield of state $|1\rangle\langle
1|\otimes|1\rangle\langle 1|$ in capital-letter symbols
$\\{S_{\alpha\beta}\\}$. We shall use the following convex proposition to do
the calculation.
Denote $S$ to be the yield of a certain source of state $\Omega$. If $\Omega$
has the convex forms of
$\Omega=\sum_{\alpha\beta}c_{\alpha\beta}\Omega_{\alpha\beta}$, we have
$S=\sum_{\alpha,\beta}c_{\alpha\beta}s_{\alpha\beta}.$ (6)
This equation is simply the fact that the total number of successful events
caused by pulses from a certain set is equal to the summation of the numbers
of successful events caused by pulses from each sub-sets.
Consider the convex forms of source $x_{A}x_{B}$ and source $y_{A}y_{B}$.
Explicitly,
$\Omega_{x_{A}x_{B}}=\tilde{c}_{0}\tilde{\Omega}_{0}+a_{1}b_{1}\rho_{1}\otimes\rho_{1}+a_{1}c_{B}\rho_{1}\otimes\rho_{c_{B}}+b_{1}c_{A}\rho_{c_{A}}\otimes\rho_{1}+c_{A}c_{B}\rho_{c_{A}}\otimes\rho_{c_{B}}$
(7)
where
$\tilde{c}_{0}\tilde{\Omega}_{0}=\left(a_{0}\Omega_{0,x}+b_{0}\Omega_{x,0}-a_{0}b_{0}\Omega_{0,0}\right)$,
$c_{A}\rho_{c_{A}}=\left(\sum_{k\geq 2}a_{k}|k\rangle\langle k|\right)$ and
$c_{B}\rho_{c_{B}}=\left(\sum_{k\geq 2}b_{k}|k\rangle\langle k|\right)$.
According to Eq.(6), this leads to
$S_{x_{A}x_{B}}=\tilde{S}_{0}+a_{1}b_{1}s_{11}+a_{1}c_{B}s_{1c_{B}}+b_{1}c_{A}s_{c_{A}1}+c_{A}c_{B}s_{c_{A}c_{B}}$
(8)
and
$\tilde{S}_{0}=a_{0}S_{o_{A}x_{B}}+b_{0}S_{x_{A}o_{B}}-a_{0}b_{0}S_{o_{A}o_{B}}.$
(9)
We also have
$\Omega_{y_{A}y_{B}}=\tilde{c}_{0}^{\prime}\tilde{\Omega}_{0}^{\prime}+a^{\prime}_{1}b^{\prime}_{1}\rho_{1}\otimes\rho_{1}+a^{\prime}_{1}c^{\prime}_{B}\rho_{1}\otimes\rho_{c_{B}^{\prime}}+b^{\prime}_{1}c^{\prime}_{A}\rho_{c_{A}^{\prime}}\otimes\rho_{1}+c^{\prime}_{A}c^{\prime}_{B}\rho_{c_{A}^{\prime}}\otimes\rho_{c_{B}^{\prime}}$
(10)
where
$\tilde{c}_{0}^{\prime}\tilde{\Omega}_{0}^{\prime}=\left(a^{\prime}_{0}\Omega_{0,y_{B}}+b^{\prime}_{0}\Omega_{y_{A},0}-a^{\prime}_{0}b^{\prime}_{0}\Omega_{0,0}\right)$,
$c^{\prime}_{A}\rho_{c_{A}^{\prime}}=\left(\sum_{k\geq
2}a^{\prime}_{k}|k\rangle\langle k|\right)$ and
$c^{\prime}_{B}\rho_{c_{B}^{\prime}}=\left(\sum_{k\geq 2}b_{k}|k\rangle\langle
k|\right)$. According to these, there exists $d_{A}\geq 0$ and $d_{B}\geq 0$
and normalized density operators $\rho_{d_{A}}$ and $\rho_{d_{B}}$ so that
$c_{A}^{\prime}\rho_{c_{A}^{\prime}}=\frac{{a^{\prime}_{2}}}{{a_{2}}}c_{A}\rho_{c_{A}}+d_{A}\rho_{d_{A}};\;c_{B}^{\prime}\rho_{c_{B}^{\prime}}=\frac{{b^{\prime}_{2}}}{{b_{2}}}c_{B}\rho_{c_{B}}+d_{B}\rho_{d_{B}}.$
(11)
Here we have used the condition of Eq.(2). According to the definitions of
$c_{A}\rho_{c_{A}}$ and $c_{A}^{\prime}\rho_{c_{A}^{\prime}}$, we have
$d_{A}\rho_{d_{A}}=c^{\prime}_{A}\rho_{c_{A}^{\prime}}-\frac{{a^{\prime}_{2}}}{{a_{2}}}c_{A}\rho_{c_{A}}=\sum_{k\geq
2}\left(a^{\prime}_{k}-\frac{{a^{\prime}_{2}}}{{a_{2}}}a_{k}\right)|k\rangle\langle
k|$ (12)
Using condition of Eq.(2), we find
$a^{\prime}_{k}-\frac{{a^{\prime}_{2}}}{{a_{2}}}a_{k}=a_{k}\left(\frac{a_{k}^{\prime}}{a_{k}}-\frac{a_{2}^{\prime}}{a_{2}}\right)\geq
0$ for all $k\geq 2$. This proves the first part of Eq.(11). In a similar we
can also prove the second part of Eq.(11). Therefore we have
$\Omega_{y_{A}y_{B}}=\tilde{c}_{0}^{\prime}\tilde{\Omega}_{0}^{\prime}+a^{\prime}_{1}b^{\prime}_{1}\rho_{1}\otimes\rho_{1}$
$+a^{\prime}_{1}\rho_{1}\otimes\left(\frac{{b^{\prime}_{2}}}{{b_{2}}}c_{B}\rho_{c_{B}}+d_{B}\rho_{d_{B}}\right)+b^{\prime}_{1}\left(\frac{{a^{\prime}_{2}}}{{a_{2}}}c_{A}\rho_{c_{A}}+d_{A}\rho_{d_{A}}\right)\otimes\rho_{1}$
$+\left(\frac{{a^{\prime}_{2}}}{{a_{2}}}c_{A}\rho_{c_{A}}+d_{A}\rho_{d_{A}}\right)\otimes\left(\frac{{b^{\prime}_{2}}}{{b_{2}}}c_{B}\rho_{c_{B}}+d_{B}\rho_{d_{B}}\right)$
(13)
which means that
$S_{y_{A}y_{B}}=\tilde{S}_{0}^{\prime}+a^{\prime}_{1}b^{\prime}_{1}s_{11}+\frac{{b_{2}^{\prime}}}{{b_{2}}}a^{\prime}_{1}c_{B}s_{1c_{B}}+\frac{{a_{2}^{\prime}}}{a_{2}}b_{1}^{\prime}c_{A}s_{c_{A}1}+\frac{{b_{2}^{\prime}a_{2}^{\prime}}}{{b_{2}a_{2}}}c_{A}c_{B}s_{c_{A}c_{B}}+\xi$
(14)
where
$\tilde{S}^{\prime}_{0}=a^{\prime}_{0}S_{o_{A}y_{B}}+b^{\prime}_{0}S_{y_{A}o_{B}}-a^{\prime}_{0}b^{\prime}_{0}S_{o_{A}o_{B}}$
(15)
and
$\xi=a^{\prime}_{1}d_{b}s_{1d_{B}}+b^{\prime}_{1}s_{d_{A}1}+c^{\prime}_{A}s_{c_{A}^{\prime}d_{B}}+c_{B}^{\prime}s_{d_{A}c_{B}^{\prime}}\geq
0$. For any sources used in the protocol, we must have either
$K_{a}=\frac{{a^{\prime}_{1}}b^{\prime}_{2}}{a_{1}b_{2}}\leq\frac{a^{\prime}_{2}{b^{\prime}_{1}}}{a_{2}b_{1}}=K_{b}$
or $K_{a}\geq K_{b}.$ Suppose the former one holds. Calculating Eq.(8)$\times
K_{a}-$Eq.(14), we obtain
$s_{11}=\frac{K_{a}(S_{x_{A}x_{B}}-\tilde{S}_{0})-(S_{y_{A}y_{B}}-\tilde{S}_{0}^{\prime})+\zeta_{1}+\zeta_{2}+\xi}{K_{a}a_{1}b_{1}-a_{1}^{\prime}b_{1}^{\prime}}$
where $\tilde{S}_{0}$ and $\tilde{S}^{\prime}_{0}$ are defined by Eq.(9) and
Eq.(15), respectively and
$\zeta_{1}=\left(\frac{a_{2}^{\prime}}{a_{2}}b_{1}^{\prime}-K_{a}b_{1}\right)c_{A}s_{c_{A}1}=\left(K_{b}-K_{a}\right)b_{1}c_{A}s_{c_{A1}}\geq
0$,
$\zeta_{2}=\left(\frac{a_{2}^{\prime}b_{2}^{\prime}}{a_{2}b_{2}}-K_{a}\right)c_{A}c_{B}s_{c_{A}c_{B}}=\left(\frac{a_{1}}{a_{1}^{\prime}}\frac{a_{2}^{\prime}}{a_{2}}-1\right)K_{a}c_{A}c_{B}s_{c_{A}c_{B}}\geq
0$. Note that $\frac{a_{1}}{a_{1}^{\prime}}\frac{a_{2}^{\prime}}{a_{2}}\geq 1$
according to Eq.(2). As shown already, $\xi\geq 0$. Thus we have
$s_{11}\geq\frac{{a_{1}^{\prime}b_{2}^{\prime}}(S_{x_{A}x_{B}}-\tilde{S}_{0})-a_{1}b_{2}(S_{y_{A}y_{B}}-\tilde{S}_{0}^{\prime})}{{a_{1}^{\prime}a_{1}}({b_{2}^{\prime}}b_{1}-b_{2}b_{1}^{\prime})}$
(16)
where $\tilde{S}_{0}$ and $\tilde{S}^{\prime}_{0}$ are defined by Eq.(9) and
Eq.(15), respectively. If $K_{a}\geq K_{b}$ holds, through calculating
Eq.(8)$\times K_{b}-$Eq.(14), we obtain
$s_{11}\geq\frac{{a_{2}^{\prime}b_{1}^{\prime}}(S_{x_{A}x_{B}}-\tilde{S}_{0})-a_{2}b_{1}(S_{y_{A}y_{B}}-\tilde{S}_{0}^{\prime})}{{b_{1}^{\prime}b_{1}}({a_{2}^{\prime}}a_{1}-a_{1}^{\prime}a_{2})}.$
This and Eq.(16) are our major formula for the decoy-state method
implementation for MDIQKD. Note that, this formula always holds for whatever
source that satisfies the condition in Eq.(2). Physical sources such as the
coherent light, the heralded source by the parametric down conversion all meet
the condition. We thus arrive at the major conclusion of this section.
In the protocol, there are two different basis. We denote $s_{11}^{Z}$ and
$s_{11}^{X}$ for yields of single-photon pulse pairs in $Z$ basis and $X$
basis, respectively. Consider those post-selected bits caused by source
$y_{A}y_{B}$ in $Z$ basis. After error test, we know the bit-flip error rate
of this set, say $E^{Z}_{y_{B}y_{B}}$. We also need the phase-flip rate for
the sub-set of bits which are caused by the two-single-photon pulses, say
$E^{ph}_{11}$, which is equal to the flip rate of post selected bits caused by
single-photon in $X$ basis, say $E_{11}^{X}$. We have
$E_{11}^{X}\leq\frac{E_{x_{A}x_{B}}^{X}S_{x_{A}x_{B}}^{X}-a_{0}E_{o_{A}x_{B}}^{X}S_{o_{A}x_{B}}^{X}-b_{0}E_{x_{A}o_{B}}^{X}S_{x_{A}o_{B}}^{X}+a_{0}b_{0}E_{o_{A}o_{B}}^{X}S_{o_{A}o_{B}}^{X}}{a_{1}b_{1}s_{11}^{X}}$
(17)
Here $E_{\alpha\beta}^{X}$ is the error rate for those post selected bits in
$X$ basis, caused by pulses from source $\alpha\beta$; $S_{\alpha\beta}^{X}$
is the yield of source $\alpha\beta$ in $X$ basis. If
$\rho_{x_{A}}=\rho_{x_{B}}$ and $\rho_{y_{A}}=\rho_{y_{B}}$, we simply replace
all $b_{0},b_{1}$ above by $a_{0},a_{1}$. Given this, we can now calculate the
key rate by the well known formula. For example, for those post selected bits
caused by source $y_{A}y_{B}$, it is
$R=a_{1}^{\prime}b_{1}^{\prime}s_{11}^{Z}(1-H(E_{11}^{X}))-fS_{y_{A}y_{B}}H(E_{y_{B}y_{B}}^{Z})$
(18)
where $f$ is the efficiency factor of the error correction method used.
Now we discuss the value of $s_{11}^{X}$ as used in Eq.(17). If we implement
the decoy-state method for different bases separately, we can calculate
$s_{11}^{Z}$ and $s_{11}^{X}$ separately and $s_{11}^{X}$ is known.
We can also choose to implement the decoy-state method only in $Z$ basis. This
is to say, in $X$ basis, we don’t have state $\rho_{y_{A}y_{B}}$, we only have
state $\rho_{x_{A}x_{B}}$. All pulses of state $\rho_{y_{B}y_{B}}$ will be
only prepared in $Z$ basis. The advantage of this is to reduce the basis
mismatch so as to raise the key rate. The value of $s_{11}$ for $X-$basis
pulses can be deduced from that for $Z-$basis. Suppose at each side,
horizontal polarization and vertical polarization have equal probability to be
chosen. For all those single-photon pairs in $Z$ basis, the state in
polarization space is
$\frac{1}{4}\left(\Omega_{11}^{HH}+\Omega_{11}^{VV}+\Omega_{11}^{HV}+\Omega_{11}^{VH}\right)=\frac{1}{4}I$
(19)
where $\Omega_{11}^{PQ}=|P\rangle\langle P|\otimes|Q\rangle\langle Q|$, $P,Q$
indicate the polarization which can be either $H$ or $V$. On the other hand,
for all those two-single-photon pulse pairs prepared in $X$ basis, if the
$\pi/4$ and $3\pi/4$ polarizations are chosen with equal probability, one can
easily find that the density matrix for these single-photon pairs is also
$I/4$. Therefore we conclude
$s_{11}^{Z}=s_{11}^{X}.$ (20)
## III security with basis-dependent coding errors
In practice, there are many imperfections for the real set-ups. For example,
that Eqs.(3,5) only hold asymptotically. The number of pulses is finite hence
these equations do not hold exactly due to the statistical fluctuation. Say,
$s$ and $s^{\prime}$, $s_{\alpha,\beta}$ and $S_{\tilde{,}\tilde{\beta}}$ can
be a bit different. Denote $s_{\alpha\beta}$ and $s_{\alpha\beta}^{\prime}$
for the yields of pulses of states $\rho_{\alpha}\otimes\rho_{\beta}$ from two
different sources. In general we have
$s_{\alpha\beta}=s^{\prime}_{\alpha\beta}(1+\delta_{\alpha\beta})$ (21)
where $\delta_{\alpha\beta}$ is the statistical fluctuation whose value is
among a certain range with a probability exponentially close to 1. The range
can be calculated given the number of pulses of each sub-sources. We can then
seek the worst-case result among the range of $\delta_{\alpha\beta}$. Another
imperfection is the intensity fluctuations. This can also be solved by the way
given inwangyang .
Here we consider the state-dependent coding errors, as studied intamaki .
For clarity, we first consider the normal QKD protocol where Alice sends
pulses and Bob receives and detects them.
The main idea is the to decompose a density operator into convex form and the
concept of virtual sub-sources. The result is enhanced by combining additional
real operation of imperfect phase randomizing.
### III.1 Density operator decomposition, virtual sub-sources, and basis-
dependent error for the normal QKD protocol.
For simplicity, we assume a perfect single-photon source with basis-dependent
coding errors. Say, at a certain time $j$, Alice wants to prepare state
$|0_{jW}\rangle$ or $|1_{jW}\rangle$ in basis $W$ ($W$ can be $Z$ or $X$)
according to her bit value 0 or 1, she actually prepares
$|0^{act}_{jW}\rangle=\cos\theta_{0jW}|0_{jW}\rangle+\sin\theta_{0jW}e^{i\delta_{0jW}}|1_{jW}\rangle$
or
$|1^{act}_{jW}\rangle=\cos\theta_{1jW}|1_{jW}\rangle+\sin\theta_{1jW}e^{i\delta_{1jW}}|0_{jW}\rangle$.
We name this subscribed $\theta$ as error angle. At different times of $j$,
the subscribed values of parameters $\theta$ and $\delta$ can be different and
can be correlated at different times. We set the threshold angles $\theta_{Z}$
and $\theta_{X}$ as
${\rm Max}\\{|\theta_{0jZ}|,|\theta_{1jZ}|\\}\leq\theta_{Z},\;\;{\rm
Max}\\{|\theta_{0jX}|,|\theta_{1jX}|\\}\leq\theta_{X}$ (22)
of course all $\\{|\theta_{0jW}|,|\sin\theta_{1jW}|\\}$ must be rather small,
otherwise no secure final key can be generated. Actually, as shall be shown
latter, our theory also apply to the case that most of these $\theta$ angles
are very small but occasionally the values can be large. In such a case, we
only need to reset the threshold angles as larger than most of
$\\{|\theta_{0jW}|,|\theta_{1jW}|\\}$ so that the threshold values can be
still rather small. For this moment we use Eq.(22). Also, we omit the
subscript $j$ if it does not cause any confusion. Our main idea is to modify
the protocol by randomly producing a wrong state with a certain small
probability. In this way, each single-photon state can be decomposed into a
classical probabilistic mixture of two states, with one of them being ideal
BB84 states. Therefore, there exists a virtual BB84 sub-source in the
protocol, and states generated by that sub-source are perfect BB84 states. By
decomposing the density operator of the BB84 source, $I/2$, one finds that the
yield of such a source is at least half of any other source. Therefore, the
lower bound of fraction of bits caused by the ideal BB84 source can be
calculated with whatever channel loss. With this, the phase flip error rate of
the BB84 sub-source can also be calculated and hence one can obtain the final
key rate.
#### III.1.1 Modified protocol and virtual ideal BB84 sub-sources
We consider the modified protocol as the following:
According to her prepared bit value ($b=0$ or $1$) in $W$ basis, in stead of
preparing state $|0_{W}^{act}\rangle$ (or $|1_{W}^{act}\rangle$), she takes a
probability $1-p_{w}$ to prepare a state $|0^{act}_{W}\rangle$ and a small
probability $p_{w}$ to intentionally prepare a wrong state
$|1^{act}_{W}\rangle$.
Therefore, the density matrix of a pulse corresponding to bit values 0 or 1 in
$Z$ basis is
$\rho_{0}^{Z}=(1-p_{z})|0^{act}_{Z}\rangle\langle
0^{act}_{Z}|+p_{z}|1^{act}_{Z}\rangle\langle 1^{act}_{Z}|$ (23)
or
$\rho_{1}^{Z}=(1-p_{z})|1^{act}_{Z}\rangle\langle
1^{act}_{Z}|+p_{z}|0^{act}_{Z}\rangle\langle 0^{act}_{Z}|,$ (24)
respectively. It is easy to show that, by choosing an appropriate value
$p_{z}$, there exists positive value $\Delta_{z}$ so that the density matrices
of $\rho_{0}^{Z}$ and $\rho_{1}^{Z}$ can be written in the convex forms of
$\rho_{0}^{Z}=\Delta_{z}|0_{Z}\rangle\langle
0_{Z}|+(1-\Delta_{z})\rho_{z0,res}.$ (25)
and
$\rho_{1}^{Z}=\Delta_{z}|1_{Z}\rangle\langle
1_{Z}|+(1-\Delta_{z})\rho_{z1,res}.$ (26)
Here, $\Delta_{z}$ can be rather close to 1 if $\theta_{z}$ is small. For
example, by setting $p_{z}=|\tan\theta_{z}|$, we can take
$\Delta_{z}=\cos^{2}\theta_{z}(1-2\tan\theta_{z})$ (27)
for the above convex forms. Similarly, we find those states for bits 0 or 1 in
$X$ basis can also be decomposed to convex forms of
$\rho_{0}^{X}=\Delta_{x}|0_{X}\rangle\langle
0_{X}|+(1-\Delta_{x})\rho_{x0,res},\;\;\rho_{1}^{X}=\Delta_{x}|1_{X}\rangle\langle
1_{X}|+(1-\Delta_{x})\rho_{x1,res}$ (28)
and we can take
$\Delta_{x}=\cos^{2}\theta_{x}(1-2\tan\theta_{x})$ (29)
by setting
$p_{x}=\tan\theta_{x}.$ (30)
For a pulse sent at any time by Alice, the state can be one of
$\\{\rho_{0}^{Z},\rho_{1}^{Z},\rho_{0}^{X},\rho_{1}^{X}\\}$, depends on the
bit value and the basis she has chosen for that pulse. However, given the
convex forms above, we can now assume different virtual sources. For state
$\rho_{0}^{Z}$, we assume two virtual sources, source $\tilde{z}_{0}$ which
produces state $|0_{Z}\rangle\langle 0_{Z}|$ only; source $z_{0}^{\prime}$
which produces state $\rho_{z0,res}$ only. Say, whenever Alice decides to send
out $\rho_{0}^{Z}$, we assume she uses source $\tilde{z}_{0}$ with probability
$\Delta_{z}$ or uses source $z_{0}^{\prime}$ with probability $1-\Delta_{z}$.
Similarly, we have virtual source $\tilde{z}_{1}$ which only produces state
$|1_{Z}\rangle\langle 1_{Z}|$ and virtual source $z_{1}^{\prime}$ which only
produces state $\rho_{z1,res}$. When Alice decides to send a state
corresponding bit 1 in $Z$ basis, we can equivalently assume that she uses
source $\tilde{z}_{1}$ or source $z_{1}^{\prime}$ with probabilities of
$\Delta_{z}$ and $1-\Delta_{z}$. In the same idea, we also assume virtual sub-
sources $\tilde{x}_{b},x_{b}^{\prime}$ which only produces state
$|b_{X}\rangle\langle b_{X}|$ or $\rho_{xb,res}$, with probabilities of
$\Delta_{x}$ and $1-\Delta_{x}$, and $b=0,1$. If we only use those bits caused
by pulses from virtual sub-sources $\tilde{z}_{b},\tilde{x}_{b}$, it is just
an ideal QKD protocol without any coding error and hence the standard results
apply directly. We call these virtual sub-sources ideal sub-sources because
they produce ideal states as requested by standard BB84 protocol. Also, we
name virtual sub-sources $w_{b}^{\prime}$ as tagged sub-source since we assume
the worst case that Eve can know bit values corresponding to a pulse from any
tagged sub-source. (Here $w$ can be $x$ or $y$ and $b$ can be 0 or 1).
#### III.1.2 Fraction of bits from ideal BB84 source and final key rate
Since these sub-sources are virtual, we don’t know which pulses are from them.
Given a lossy channel, we need to estimate faithfully how many bits are
generated by the ideal sub-source the phase-flip rate for bits from the ideal
sources. Define virtual source $\tilde{w}=\tilde{w}_{0}+\tilde{w}_{1}$, where
$w$ can be either $z$ or $x$. These mean that virtual source $\tilde{z}$ (or
$\tilde{x}$) includes all pulses from ideal BB84 sub-sources in $Z$ (or $X$)
basis. Obviously, density operator of a pulses from such an ideal source is
simply $\tilde{\rho}_{w}=I/2$ . We also regard the two tagged sub-sources
subscribed by 0 or 1 as one composite tagged source $w^{\prime}$, say
$w^{\prime}=w^{\prime}_{0}+w^{\prime}_{1}$. The density operator of a pulse
from such a source in $W$ basis at a certain time $j$ is
$\rho_{w}^{\prime}(j)=\frac{\rho_{w0,res}+\rho_{w1,res}}{2}$. For example, in
$X$ basis, the density operator of a pulse from such a source (source
$x^{\prime}$ ) at time $j$ is
$\rho_{x}^{\prime}(j)=\frac{\rho_{x0,res}+\rho_{x1,res}}{2}$. The state of a
pulse in $X$ basis at time $j$ is
$\rho_{X}(j)=\Delta_{x}I/2+(1-\Delta_{x})\rho_{x}^{\prime}(j).$ (31)
Here $\Delta_{x}$ is independent of time $j$, though $\rho_{x}^{\prime}$ is
dependent on time $j$. This means, whenever there is a pulse in $X$ basis sent
out, it has a probability $\Delta_{x}$ that the ideal source $\tilde{w}$ is
used, a probability $1-\Delta_{x}$ that the tagged source $x^{\prime}$ is
used. To estimate the upper bound of error rate of post selected bits caused
by pulses from source $\tilde{x}$, we need the lower bound of fraction of bits
caused by virtual source $\tilde{x}$ among all post-selected bits in basis
$X$. Note that the density matrix for source $\tilde{x}$ is simply $I/2$,
there always exists a density operator $\bar{\rho}$ so that source $\tilde{x}$
can have the convex form of
$I/2=\frac{1}{2}(\bar{\rho}(j)+\rho_{x}^{\prime}(j)).$ (32)
Here $\bar{\rho}(j)$ is defined as
$\bar{\rho}(j)=\left(\begin{array}[]{cc}c&-d\\\ -b&a\end{array}\right)$ if
$\rho_{x}^{\prime}(j)=\left(\begin{array}[]{cc}a&d\\\ b&c\end{array}\right)$.
This means that we can regard source $\tilde{x}$ as a mixed source consisting
of two parts: source $\tilde{\bar{x}}$ that can only emit $\bar{\rho}(j)$ at
time $j$ and source $\tilde{x}^{\prime}$ that can only emit
$\rho_{x}^{\prime}(j)$ at time $j$. Whenever a pulse is sent out of source
$\tilde{x}$, with half a probability that source $\tilde{x}^{\prime}$ is used,
which generates the same state ($\rho_{x}^{\prime}(j)$) as the tagged source
$x^{\prime}$ does, at any time $j$. Asymptotically, if the total number of
$X$-basis pulses sent out is $N_{x}$, there are
$\tilde{N}_{x}=N_{x}\Delta_{x}$ from ideal source $\tilde{x}$ and
$N_{x}(1-\Delta_{x})$ from tagged source $x^{\prime}$. Denote $\tilde{s}_{x}$,
$\tilde{\bar{s}}_{x}$, $\tilde{s}_{x}^{\prime}$, and $s_{x}^{\prime}$ as the
yield of sources $\tilde{x}$, $\tilde{\bar{x}}$, $\tilde{x}^{\prime}$ and
$x^{\prime}$, respectively. We have
$\tilde{s}_{x}=\frac{1}{2}\tilde{\bar{s}}_{x}+\frac{1}{2}\tilde{s}_{x}^{\prime}\geq\frac{1}{2}s_{x}^{\prime}.$
(33)
Here we have used the following two facts: (1) The yield of any source must be
non-negative, therefore $\tilde{\bar{s}}_{x}\geq 0$; (2) Source
$\tilde{x}^{\prime}$ and source $x^{\prime}$ can only produce the same state
($\rho_{x}^{\prime}(j)$) at any time $j$, they must have the same yield in the
whole protocol. Therefore, among all bits caused by source $X$, the fraction
of bits caused by ideal source $\tilde{x}$ is
$\tilde{\Delta}_{x}=\frac{N_{x}\Delta_{x}\tilde{s}_{x}}{N_{x}\Delta_{x}\tilde{s}_{x}+N_{x}(1-\Delta_{x})s_{x}^{\prime}}\geq\frac{\Delta_{x}/2}{1-\Delta_{x}/2}=\tilde{\Delta}_{x}^{l}.$
(34)
In $Z$ basis, there is also a similar formula. Asymptotically, among all those
post selected bits of basis $W$, the fraction of bits caused by source
$\tilde{w}$ is
$\tilde{\Delta}_{w}\geq\frac{\Delta_{w}}{2-\Delta_{w}}=\frac{\cos^{2}\theta_{w}(1-2\tan\theta_{w})}{\sin^{2}\theta_{w}+\left(\sin\theta_{w}+\cos\theta_{w}\right)^{2}}.$
(35)
Suppose the error rate for all $X$-basis bits is $E^{X}$. Then the error rate
for bits caused by pulses from source $\tilde{x}$ and the phase flip rate of
$Z$-basis bits caused by pulses from source $\tilde{z}$ is
$E^{Z}_{z,ph}=E^{X}_{x}=\frac{E^{X}}{\tilde{\Delta}_{x}}.$ (36)
We have assumed a perfect single-photon source in the above. If we use an
imperfect single-photon source, we need implement the decoy state method. We
have the key rate formula
$R=\tilde{\Delta}_{z}\Delta_{1}(1-H(\frac{E^{X}}{\tilde{\Delta}_{x}\Delta_{1}}))-fH(E)$
(37)
and $\tilde{\Delta}_{x},\;\tilde{\Delta}_{z}$ are given by Eqs.(35), $E$ is
the detected error rate of $Z$-basis bits and $\Delta_{1}$ is the fraction of
single-photon pulses bits in $Z$ basis as post selected.
In the protocol, we request Alice take random flip of her qubits with a small
probability. However, these flipping operations are actually not necessary
physically. In stead of flipping he qubits physically, she can choose to
randomly choosing to flip her classical bit values with the same small
probability. Same with the case of flipping her qubits physically, this will
cause a rise in the error rate. The rise of the bit flip part does not
decrease the final key rate because Alice knows which bits have been flipped.
The rise of the phase flip part is the major factor that causes the final key
dropping. Besides this, there are also factors such as $\tilde{\Delta}_{z}$ in
the key rate formula and $1/\tilde{\Delta}_{x}$ in estimating the phase error.
These also decrease the key rate, but the amount decreased is almost
negligible compared with the factor of phase flip rise. However, all these
does not requests a very accurate source coding. Obviously, one can obtain
final key given the largest source error (i.e., $\sin^{2}\theta_{z}$ )in the
magnitude order of $10^{-4}$. This has already loosened the demand in the
source accuracy, compared with the existing result which requests a magnitude
order of $10^{-7}-10^{-6}$. However, as shall be shown later in our work that
we can further loosen the accuracy to $10^{-2}-10^{-1}$ for the magnitude
order of largest error, by adding phase randomizing operation.
In the study above, we have have set
$\theta_{w}\geq\\{|\theta_{0jW}|,|\theta_{1jW}|\\}$ for all $j$, i.e., error
angles at all individual times must by smaller than the threshold angle. We
can also treat the case most of $|\theta_{0jW}|,|\theta_{1jW}|$ not larger
than $\theta_{w}$ but a small fraction $g_{w}$ of them larger than it. In this
case, we only need to reset $\tilde{\Delta}_{z},\tilde{\Delta}_{x}$ in the key
rate formula Eq.(37) by:
$\tilde{\Delta}_{w}\longrightarrow\frac{1-g_{w}}{1+g_{w}}\tilde{\Delta}_{w}$
(38)
### III.2 Enhanced results with phase randomizing
We can add real physical operations to the protocol in order to further
increase the efficiency. In stead of random flipping to bit values, we can
choose to take a phase randomizing operation to decompose the states into
convex form. Suppose we use the photon polarization space. To each qubits in
$Z$ basis, with half a probability we take an additional unitary operation of
($|H\rangle\longrightarrow|H\rangle$, $|V\rangle\longrightarrow-|V\rangle$);
to each qubit in $X$ basis, with half a probability we take an additional
unitary operation of ($|+\rangle\longrightarrow|+\rangle$,
$|-\rangle\longrightarrow-|-\rangle$). If we can realize such an operation
perfectly, we can obtain convex forms for density operators corresponding to
each bit values in each bases and we can directly use the ideal of virtual
sub-sources to solve the problem. For example, for those pulses corresponding
to bit values 0 and 1in $Z$ basis, we have
$\rho_{0}^{Z}=\cos^{2}\theta_{z0}|0_{Z}\rangle\langle
0_{Z}|+\sin^{2}\theta_{z0}|1_{Z}\rangle\langle
1_{Z}|=\cos^{2}\theta_{z}|0_{Z}\rangle\langle
0_{Z}|+\sin^{2}\theta_{z}\rho_{z0,res}$ (39)
and
$\rho_{1}^{Z}=\cos^{2}\theta_{z1}|1_{Z}\rangle\langle
1_{Z}|+\sin^{2}\theta_{z1}|1_{Z}\rangle\langle
1_{Z}|=\cos^{2}\theta_{z}|1_{Z}\rangle\langle
1_{Z}|+\sin^{2}\theta_{z}\rho_{z1,res}$ (40)
We can regard that there are sub-sources of $z_{0}$ which only emits state
$|0_{Z}\rangle$ and sub-source $z_{1}$ which only emits state $|1_{Z}\rangle$.
Each sub-source will be used with a constant probability
$\cos^{2}\theta_{z}/2$. Density operators for those qubits in $X$ basis can
also be decomposed in
$\rho_{0}^{X}=\cos^{2}\theta_{x}|0_{x}\rangle\langle+|+\sin^{2}\theta_{x}\rho_{x0,res}$
(41)
and
$\rho_{1}^{X}=\cos^{2}\theta_{x}|1_{Z}\rangle\langle
1_{Z}|+\sin^{2}\theta_{x}\rho_{x1,res}$ (42)
We can regard that there are sub-sources of $x_{0}$ which only emits state
$|0_{X}\rangle$ and sub-source $x_{1}$ which only emits state $|1_{x}\rangle$.
Each sub-source will be used with a constant probability
$\cos^{2}\theta_{x}/2$. Therefore pulses from the 4 sub-sources above form the
ideal BB84 states. We can use Eq.(37) for the key rate, but the value
$E_{1}^{X}$ is not over estimate at all, and factors of
$\tilde{\Delta}_{W}=\cos^{2}\theta_{W}$, which is almost 1 if $\theta_{W}$ is
small. In this way, the tolerable largest coding error is in the magnitude
order of $1/10$, if the phase randomization can be realized. What is most
interesting is that we can obtain almost the same good result even though the
phase randomization is a little bit imperfect, through applying results in the
earlier subsection.
In an imperfect phase randomization, to each qubit in $X$ basis, with half a
probability we take an additional unitary operation of
($|0_{X}\rangle\longrightarrow|0_{X}\rangle$,
$|1_{X}\rangle\longrightarrow|1_{X}\rangle-e^{-i\delta_{2}}|+\rangle$). Here
$\delta_{2}$ are errors in the operations, it can be different from time to
time, and can be correlated at different times. We assume the largest values
for $|\delta_{2}|$ is $\delta_{x}$. We can also choose to do phase
randomization for qubts in $Z$-basis, but this is not necessary since the
major factor in efficiency is in tightness of phase flip rate estimation.
Technically, if the phase operation is done in only one basis, the rotation
between the two basis states is negligible. Therefore we can use the above
diagonal form above in $X$-basis for an imperfect phase operation.By the
current matured technology, value $\delta_{x}$ can be controlled below $1/20$.
With these, we obtain the density matrices of qubits in $X$ basis. For a qubit
of bit value 0 in $X$ basis,
$\displaystyle\rho_{0}^{X}=\left(\begin{array}[]{cc}\cos^{2}\theta_{x0}&\sin
2\theta_{x0}(2\sin^{2}\delta_{1}-i\sin\delta)/4\\\ \sin
2\theta_{x0}(2\sin^{2}\delta_{1}+i\sin\delta)/4&\sin^{2}\theta\end{array}\right)$
(45)
This can be directly decomposed in
$\rho_{0}^{X}=\Delta_{x}|0_{X}\rangle\langle
0_{X}|+(1-\Delta_{x})\rho_{x0,res}$ (46)
and $\Delta_{x}=\cos^{2}\theta_{x}-\sin\theta_{x}\sin\delta_{x}/2$. Similarly,
we can also decompose the density matrix for bit value 1 in $X$ basis.
Explicitly,
$\rho_{1}^{X}=\Delta_{x}|1_{X}\rangle\langle
1_{X}|+(1-\Delta_{x})\rho_{x1,res}.$ (47)
Therefore, there exists two virtual ideal sub-sources which emit state
$|+\rangle$ or state $|-\rangle$ only. The fraction of bits caused by pulses
form these two ideal sub-sources among all post-selected $X$ bits is
$\tilde{\Delta}_{x}=\frac{\Delta_{x}}{2-\Delta_{x}}=\frac{2\cos^{2}\theta_{x}-\sin\theta_{x}\sin\delta_{x}/}{4-2\cos^{2}\theta_{x}+\sin\theta_{x}\sin\delta_{x}}.$
(48)
We don’t need to take phase operation to qubits in $Z$ basis. We just take
random flipping to the bit values of $Z$ basis with a small probability as
discussed in the earlier section. We shall still use the key rate formula of
Eq.(37), but the key rate is greatly improved now, because here the phase flip
rate is over estimated only by a negligible amount, i.e., a factor of
$1/\tilde{\Delta}_{AX}$ given by Eq.(48).
### III.3 MDIQKD with source coding errors.
Here we need to convert our results to the case of two-pulse sources. In this
case, both Alice and Bob will send their pulses to the un-trusted third party
(UTB), as has been shown. Neither Alice nor Bob can prepare the coding state
exactly. When anyone of them wants to prepare a state $|b_{W}\rangle$, she
(he) can only prepare a state
$|b^{act}_{W}\rangle=\cos\theta_{bW}|b_{W}\rangle+e^{i\delta_{bW}}\sin\theta_{bW}|\bar{b}_{W}\rangle$
and $b=0,1$, $\bar{b}_{W}=1\oplus b_{W}$. Most generally, Alice and Bob have
different threshold angles, noted as $\theta_{Az},\theta_{Ax}$ for Alice in
$Z$ or $X$ basis; and $\theta_{Bz},\theta_{Bx}$ for Bob in $Z$ or $X$ basis.
In our protocol, we request Bob (Alice) to take a probability $1-p_{Bw}$ (or
$1-p_{Aw}$) to prepare a state $|b^{act}_{W}\rangle$ and probability $p_{Bw}$
($p_{Aw}$) to prepare $|\bar{b}^{act}_{W}\rangle$, if the data of bit value
indicates that he (she) should prepare a state $|b_{W}\rangle$, in basis $W$
(i.e., $Z$ or $X$). By analysis similar to the subsection above, we can also
present the appropriate convex forms and find the ideal sub-sources for Alice
and Bob separately, in both bases. Suppose $\theta_{Aw},\;\theta_{Bw}$ are
threshold angles in basis $W$ for Alice and Bob, respectively. We can set
$p_{\gamma w}=\tan\theta_{\gamma w}$ ($\gamma=A,B$). Then the density
operators at Alice’s side and the one at Bob’s side can be decomposed in
convex forms similar to equation (23,24). We have the decomposition form
$\rho_{b}^{\alpha W}=\Delta_{\alpha w}|b_{W}\rangle\langle
b_{W}|+(1-\Delta_{\alpha w})\rho_{\alpha bw,res}.$ (49)
for a state corresponding to bit value $b$ in basis at side $\alpha=A$ or $B$.
In our notation, as a subscript of $\Delta$, the lower case $w$ can be $x$ or
$z$ if the basis $W$ takes $X$ or $Z$. Here $\Delta_{\alpha
w}=\cos^{2}\theta_{\alpha w}(1-2\tan\theta_{\alpha w})$. Both Alice and Bob
have virtual ideal sub-sources which emit standard BB84 states. Therefore, a
single-photon pair corresponding to bit value $a$ and $b$ at Alice’s side and
Bob’s side in basis $W$ correspond to a two-pulse state
$\rho_{a}^{AW}\otimes\rho_{b}^{BW}=\Delta_{w}^{(2)}|a\rangle\langle
a|\otimes|b\rangle\langle b|+(1-\Delta_{w}^{(2)})\rho_{abw,res}$ (50)
and
$\Delta_{w}^{(2)}=\Delta_{Aw}\Delta_{Bw}$ (51)
where $\Delta_{\gamma w}$ is given by Eqs.(27,29) with $z$ or $x$ replaced by
$\gamma w$ there. We define virtual two-pulse ideal sub-sources
$\\{W_{ab}\\}$, $a,b$ can be 0 or 1. If at a certain time states of both
single-photon pulses are from idea virtual sub-sources and are corresponding
to bit values $a,b$ in basis $W$, we say the pulse-pair is from source
$W_{ab}$, which is a two-pulse ideal virtual sub-source. If at a certain time
the pulses from two sides are in the same basis $W$ but not from any of the
above virtual ideal sub-sources, we regard them as tagged states from source
the tagged source which produce states $\rho_{W,res}^{\prime}$ only.
Therefore, we can regard all single-photon pairs in $Z$ basis as coming form 5
different virtual sources: $Z_{00},Z_{11},Z_{01},Z_{10}$ and
$Z_{res}^{\prime}$, which only emits two-pulse state
$|0_{Z}0_{Z}\rangle,|1_{Z}1_{Z}\rangle,|0_{Z}1_{Z}\rangle,|1_{Z}0_{Z}\rangle$
and state $\rho_{Z,res}^{\prime}=\frac{1}{4}\Sigma_{a,b}\rho_{abW,res}$. We
can also regard the first 4 sources as one composite source, $\tilde{Z}$ which
emits single-photon -pair state of density matrix $I/2$ in $4\times 4$ space
only. We can then regard any single-photon pair in $Z$ basis comes out from
source $\tilde{Z}$ with a probability $\Delta_{z}^{(2)}$, or from source
$Z_{res}^{\prime}$ with a probability $1-\Delta_{z}^{(2)}$. We can find that
the fraction of bits caused by source $\tilde{W}$ among all those post
selected bits in $W$ basis caused by single-photon pairs as
$\tilde{\Delta}_{w}=\frac{\Delta_{w}^{(2)}}{2-\Delta_{w}^{(2)}}$ (52)
Observing the error rate of $X-$basis pairs from the decoy-source,
$E_{x_{A}x_{B}}^{X}$, we can find the upper bound $E_{11}^{X}$, the error rate
of those post-selected bits corresponding to single-photon pairs in $X-$basis
by Eq.(17), and then upper bound the error rate of post selected bits
corresponding to single-photon pairs in $X-$basis bits caused by virtual
source $\tilde{X}$ is
$E_{11,\tilde{X}}\leq E_{11}^{X}/\tilde{\Delta}_{x}$ (53)
where $E_{11}^{X}$ is the error rate for post selected bits in $X$ basis
caused by single-photon pairs, as given by Eq.(17). This is also the
asymptotic phase-flip rate of bits corresponding to two-single-photon pulses
from source $\tilde{Z}$. We can then use the key rate formula of Eq.(37), with
$\tilde{\Delta}_{x}$ and phase-flip rate given above. Finally, we have the
following key rate formula for decoy-state MDIQKD with basis dependent errors:
$R=\tilde{\Delta}_{z}\Delta_{11}^{Z}(1-H(E_{11,\tilde{X}}))-fH(E_{11}^{Z})$
(54)
and
$\Delta_{11}^{Z}=\frac{a_{1}^{\prime}b_{1}^{\prime}s_{11}^{Z}}{S_{y_{A}y_{B}}}$,
$a_{1}^{\prime},b_{1}^{\prime}$ are parameters appeared in the signal states
$\rho_{y_{A}},\rho_{y_{B}}$ as given by Eq.(1), $s_{11}^{Z}$ is given by
Eq.(16), $E_{11}^{Z}$ is the observed error rate for all post-selected bits in
$Z$ basis, $S_{y_{A}y_{B}}$ is the observed yield of two-pulse source
$y_{A}y_{B}$ as defined in Section 1.
We must change the formula if we only implement decoy-state method in $Z$
basis, in preparing $X-$basis bits, we only use source $x_{A}x_{B}$. We need
derive the upper bound of $E_{11,\tilde{Z}}^{ph}$, the phase-flip rate of bits
corresponding to single-photon pairs from source $\tilde{Z}$, which is equal
to $E_{11,\tilde{X}}$. Note that now in general, $s_{11}^{X}\not=s_{11}^{Z}$,
since the polarization states for $Z$ basis and $X$ basis are different. But
the yields from the ideal sources $\tilde{X}$ and $\tilde{Z}$ must be equal.
We have
$s_{11,\tilde{X}}=s_{11,\tilde{Z}}\geq\tilde{\Delta}_{z}^{(2)}s_{11}^{Z}$ (55)
which immediately leads to
$E_{11,\tilde{X}}\leq\frac{E_{11}^{X}}{\tilde{\Delta}_{z}^{(2)}\tilde{\Delta}_{x}^{(2)}}$
(56)
and $\tilde{E}_{11}^{X}$, is given by Eq.(17). With this, the key rate can be
calculated by Eq.(54).
Acknowledgement: We thank Prof. J.W. Pan, C.Z. Peng, and Q. Zhang in USTC for
useful discussions. This work was supported in part by the National Basic
Research Program of China grant No. 2007CB907900 and 2007CB807901, NSFC grant
No. 60725416 and China Hi-Tech program grant No. 2006AA01Z420, and 10000-Plan
of Shandong province.
Note added Applying our formulas for the 3-intensity decoy-state MDI-QKD,
numerical calculation was done recently and a key rate close to the ideal case
was obtained for coherent state sourceqing3 .
## References
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* (3) H. Inamori, N. Lütkenhaus, D. Mayers, European Physical Journal D, 41, 599 (2007), which appeared in the arXiv as quant-ph/0107017; D. Gottesman, H.K. Lo, N. Lütkenhaus, and J. Preskill, Quantum Inf. Comput. 4, 325 (2004).
* (4) W.-Y. Hwang, Phys. Rev. Lett. 91, 057901 (2003).
* (5) X.-B. Wang, Phys. Rev. Lett. 94, 230503 (2005); X.-B. Wang, Phys. Rev. A 72, 012322 (2005).
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* (7) Y. Adachi, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev.Lett. 99, 180503 (2007).
* (8) M. Hayashi, Phys. Rev. A 74, 022306 (2008); ibid 76, 012329 (2007).
* (9) D. Rosenberg em et al., Phys. Rev. Lett. 98, 010503 (2007); T. Schmitt-Manderbach et al., Phys. Rev. Lett. 98, 010504 (2007); Cheng-Zhi Peng et al. Phys. Rev. Lett. 98, 010505 (2007); Z.-L. Yuan, A. W. Sharpe, and A. J. Shields, Appl. Phys. Lett. 90, 011118 (2007); Y. Zhao, B. Qi, X. Ma, H.-K. Lo and L. Qian, Phys. Rev. Lett. 96, 070502 (2006); Y. Zhao, B. Qi, X. Ma, H.-K. Lo, and L. Qian, in Proceedings of IEEE International Symposium on Information Theory, Seattle, 2006, pp. 2094–2098 (IEEE, New York).
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* (11) X.-B. Wang, T. Hiroshima, A. Tomita, and M. Hayashi, Physics Reports 448, 1(2007).
* (12) X.-B. Wang, L. Yang, C.-Z. Peng and J.-W. Pan, New J. Phys. 11, 075006 (2009).
* (13) G. Brassard, N. Lütkenhaus, T. Mor, and B.C. Sanders, Phys. Rev. Lett. 85, 1330 (2000); N. Lütkenhaus, Phys. Rev. A 61, 052304 (2000); N. Lütkenhaus and M. Jahma, New J. Phys. 4, 44 (2002).
* (14) B. Huttner, N. Imoto, N. Gisin, and T. Mor, Phys. Rev. A 51, 1863 (1995); H.P. Yuen, Quantum Semiclassic. Opt. 8, 939 (1996).
* (15) L. Lyderson et al, Nature Photonics, 4, 686(2010); I. Gerhardt et al, Nature Commu. 2, 349 (2011)
* (16) D. Mayers and A. C.-C. Yao, in Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS98) (IEEE Computer Society, Washington, DC, 1998), p. 503; A. Acin et al., Phys. Rev. Lett. 98, 230501 (2007); Scarani V and Renner R Phys. Rev. Lett. 100, 302008 (2008); Scarani V and Renner R 2008 3rd Workshop on Theory of Quantum Computation, Communication and Cryptography (TQC 2008), (University of Tokyo, Tokyo 30 Jan C1 Feb 2008) See also arXiv:0806.0120
* (17) S.L. Braunstein and S. Pirandola, Phys. Rev. Lett. 108, 130502 (2012).
* (18) H.-K. Lo, M. Curty, and B. Qi, Phys. Rev. Lett., 108,130503(2012), K. Tamaki et al, Phys. Rev. A, 2307 (2012).
* (19) K. Tamaki et al, Phys. Rev. A 85, 042307 (2012)
* (20) Being noted by our result, an errata for Ref.tamaki has been issued very recently. See in K. Tamaki et al, arXiv:1111.3413v4.
* (21) Qin-Wang abd Xiang-Bin Wang, arXiv:1305.6480
|
arxiv-papers
| 2012-07-02T14:05:21 |
2024-09-04T02:49:32.507690
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiang-Bin Wang",
"submitter": "Xiang-Bin Wang",
"url": "https://arxiv.org/abs/1207.0392"
}
|
1207.0405
|
# $\mathbb{M}\text{ITRA}$: A Meta-Model for Information Flow in Trust and
Reputation Architectures
Eugen Staab
netRapid GmbH & Co. KG (http://www.netrapid.de/)
eugen.staab@netrapid.de Guillaume Muller
Work partly done at:
Escola Politécnica de São Paulo (http://www.pcs.usp.br/~lti/) and
Commissariat à l’Énergie Atomique (http://www-list.cea.fr/)
guillaume.muller.pro@gmail.com
(August 12, 2010)
###### Abstract
We propose $\mathbb{M}\text{ITRA}$, a meta-model for the information flow in
(computational) trust and reputation architectures. On an abstract level,
$\mathbb{M}\text{ITRA}$ describes the information flow as it is inherent in
prominent trust and reputation models from the literature. We use
$\mathbb{M}\text{ITRA}$ to provide a structured comparison of these models.
This makes it possible to get a clear overview of the complex research area.
Furthermore, by doing so, we identify interesting new approaches for trust and
reputation modeling that so far have not been investigated.
Keywords: Computational Trust, Reputation Systems, Meta Model.
## 1 Introduction
Open and decentralized systems are vulnerable to buggy or malicious agents. To
make these systems robust against malfunction, agents need the ability to
assess the reliability and attitudes of other agents in order to choose
trustworthy interaction partners. To this end, a multitude of trust and
reputation models have been and still are being proposed in the literature. In
short, each such model generally defines all or some of the three following
processes: how evidence about the trustworthiness of an agent is gathered, how
this evidence is combined into a final assessment, and how this final
assessment is used in decision-making.
Currently, it is difficult to get an overview of what has been done in the
area, and what needs to be done. The main reasons for this are: that the
proposed trust and reputation models use no common terminology; that they are
not compatible in their basic structure; and that their respective
contributions are evaluated against different metrics. While not considering
the last point in this article, the first two questions motivate the
introduction of an abstract model that allows researchers to organize their
models in a unified way.
To this end, we propose $\mathbb{M}\text{ITRA}$, a meta-model for the
information flow in computational trust and reputation architectures.
$\mathbb{M}\text{ITRA}$ formalizes and organizes the flow of information
inside and between agents. More precisely, it describes the top-level
processes to gather evidence, and to combine it with information exchanged
with other agents. $\mathbb{M}\text{ITRA}$ makes use of four simple concepts
of information processing, namely the _observation_ , the _evaluation_ , the
_fusion_ and the _filtering_ of information, and abstracts away from numerical
computations. Although being an abstract model, $\mathbb{M}\text{ITRA}$
captures important concepts used by existing trust and reputation models. This
way, $\mathbb{M}\text{ITRA}$ provides a big-picture of the trust and
reputation domain and paves the way for a structured survey of the domain.
The model is useful for the community in at least four respects. First, it
serves as a terminological and structural framework to describe new models.
Secondly, it provides a means for researchers to classify and compare existing
approaches in this domain. In addition to this, $\mathbb{M}\text{ITRA}$ helps
to identify new approaches to model trust and reputation, as we will show in
this article. Finally, it helps newcomers to get a concise overview on the
structure of computational models of trust and reputation.
The remainder of the article is organized as follows. In Sect. 2, we introduce
basic concepts of trust and reputation modeling. We describe the
$\mathbb{M}\text{ITRA}$ model in Sect. 3. Following this, in Sect. 4, we use
$\mathbb{M}\text{ITRA}$ to classify existing models and identify what has not
yet been done in the research field of trust and reputation. Finally, we
review related work in Sect. 5 and we draw conclusions in Sect. 6.
## 2 Basic Concepts
In this section, we describe basic concepts and notations that are used in
this article.
### 2.1 Trust Beliefs/Intentions/Acts and Reputation
observations (_obs_)evaluations (_eval_)trust beliefs (_tb_)trust intentions
(_ti_)trust acts (_ta_) Figure 1: Information Chain in Trust Reasoning.
Following [MC01], we distinguish three concepts that are often confused in the
literature: the _trust belief_ , the _trust intention_ and the _trust act_. In
the same spirit as for the BDI architecture [Rao96], raw observations are at
first evaluated and then used to form trust beliefs, which in turn, are used
to build trust intentions. Finally, these intentions can be used as one
criterion in the decision-making process, eventually leading to a trust or
distrust act. Figure 1 provides a schematic view of the trust information
chain.
A basic trust belief, which we denote with $\text{tb}_{\theta}^{\Gamma}$,
reflects the view of an individual agent $\theta$ on the trustworthiness of
the agents $\Gamma$. In the common case, the set of agents $\Gamma$ contains
only one agent. However, trust beliefs can also reflect how an agent $\theta$
thinks about a group of agents (see also [Hal02, FC08]); the agents belonging
to the group $\Gamma$ have to be similar in some regard and so, experiences
with any of them may to some extent be evidence for the trustworthiness of the
group as a whole. For example, consider several agents being employed by a
certain company; in such a situation, the agents can be judged in their roles
as employees of this company, and their characteristics can be “generalized”
to other agents in the same company [FC08].
There is a second type of trust beliefs, which is also called “reputation”: a
collective assessment of a group of agents about other agents [MFT+08]. Here,
the corresponding trust beliefs are the estimate _by an individual_ of what
could be such a shared opinion of a group of agents $\Theta$ about other
agents $\Gamma$. We denote this collective trust believe by
$\text{tb}_{\Theta}^{\Gamma}$.
Each trust belief is relating to a certain context, for which it is valid. The
context includes a particular task, and the environmental conditions, under
which the trustee is believed (or not) to successfully carry out this task on
behalf of the trustor. We will discuss this concept of context in more detail
later. Now, for given trustor(s) $\Theta$, trustee(s) $\Gamma$ and a certain
context, there can only be one unique trust belief. This makes sure that the
trustor cannot believe that trustee(s) $\Gamma$ are at the same time
trustworthy _and_ untrustworthy concerning a context. Nevertheless, the
various kinds of trust beliefs can still be contradictory. For example, an
agent can believe that the agents that belong to a certain group are usually
untrustworthy, but that a specific agent in this group is well known and
believed to be trustworthy. The reasoning about how to deal with such
situations is typically what occurs in another process, that does not fall
into the scope of this paper.
While trust beliefs are solely estimates about the trustworthiness of other
agents, the process of forming _trust intentions_ incorporates also strategic
considerations or characteristics of the trustee. For example, although a
trustee believes another agent to be trustworthy, he might still be very
pessimistic about relying on this agent. When forming trust intentions, an
agent actually transforms trust beliefs, which are based on the past behavior
of other agents, into its own intended future behavior towards them. This
typically corresponds to computing the “shadow of the future” [Axe84]. Trust
intentions of an agent $\alpha$ towards an agent $\gamma$, derived from sets
of trust beliefs, are denoted by $\text{ti}_{\alpha}^{\gamma}$.
### 2.2 Acquisition of Evidence
Two types of information exist that can be used as input for a trust reasoning
process. An agent can make direct observations about and evaluations of the
behavior of other agents, and it can also receive messages from other agents
that contain observations, evaluations or trust beliefs.
#### 2.2.1 Direct Observations and Evaluations
A trust belief about an agent $\gamma$ is derived from information that
provides evidence for $\gamma$’s trustworthiness. We call such evidence an
_evaluation_ (see Fig. 1). An evaluation is a subjective interpretation of a
set of _direct observations_ about $\gamma$’s behavior towards some other
agent $\delta$. In other words, the process of evaluation decides whether the
observations are evidence for trustworthiness of $\gamma$. We write
$\text{eval}_{\alpha}^{\left<\delta,\gamma\right>}$ to denote an evaluation
that is done by agent $\alpha$ concerning the behavior of agent $\gamma$
towards agent $\delta$. Analogously, the direct observations made by $\alpha$
on an interaction between $\gamma$ and $\delta$ are denoted by
$\text{obs}_{\alpha}^{\left<\delta,\gamma\right>}$.
#### 2.2.2 Communicated Evidence
Besides directly acquired information, an agent can use information received
by messages from other agents. We introduce the notation
$\left(x\right)_{[\theta,\dots,\beta]}^{\alpha}$ for a message $x$ that is
sent by agent $\beta$ to agent $\alpha$ through a path of transmitters $\beta$
(direct sender to $\alpha$), …, $\theta$ (initial sender). We call such a
message _communicated evidence_. To ensure the correctness of the indicated
path, either the final receiver can assess the correctness of this chain of
transmitters (e.g., by spot-checking agents on the path and asking them
whether they have sent the message as is), or mechanisms are put in place to
prove that the message has indeed taken the indicated path and which
intermediary has made which modifications (by means of cryptography, e.g.,
public-key infrastructure and signing).
Direct observations that an agent makes on its own can only be incorrect if
the sensors with which the observations were made are faulty. Communicated
evidence additionally can be wrong when the sender is dishonest or
incompetent.
### 2.3 Context and Uncertainty
Each observation, evaluation, trust belief or trust intention is in general
only valuable if two pieces of information are attached to it:
1. 1.
the context to which the information applies;
2. 2.
a measure of uncertainty of the information itself.
The concept of _context_ is a vital part for evidence-based trust reasoning.
[MC96] give the example that “one would trust one’s doctor to diagnose and
treat one’s illness, but would generally not trust the doctor to fly one on a
commercial airplane”. If some direct observations are made in a certain
context, then this context should be annotated also to the evaluations (and
the trust beliefs and trust intentions) that are based on these observations.
As a result, context annotations should be propagated through the information
chain shown in Fig. 1.
A measure of _uncertainty_ is needed to express how uncertain a piece of
information is believed to be. This is especially important when a trust model
incorporates communicated evidence that may be biased or wrong. Similarly to
the context information, uncertainty should be propagated through the whole
information chain.
## 3 The Meta-Model
Figure 2: Structure of $\mathbb{M}\text{ITRA}$ for an agent $\alpha$. For the
sake of clarity what is not shown in this illustration: agent $\alpha$ can
decide to send any piece of information occurring in the illustration to other
agents.
In this section we present $\mathbb{M}\text{ITRA}$, illustrated in Fig. 2.
This meta-model organizes the different ways of how the information chain of
Fig. 1 can be realized in an agent $\alpha$’s trust model. The agent can send
all pieces of information that occur in this information chain (the square
boxes in the Figure) to other agents; for the sake of clarity, we did not
illustrate such actions in Fig. 2.
On the top level, $\mathbb{M}\text{ITRA}$ divides the trust modeling process
into four consecutive sub-processes:
1. 1.
observation;
2. 2.
evaluation;
3. 3.
fusion;
4. 4.
decision-making.
Let us first exemplify these sub-processes from the perspective of an agent
$\alpha$, which reasons about its trust in an agent $\gamma$. In the
_observation_ process, agent $\alpha$ tries to collect any form of evidence it
can get to assess $\gamma$’s trustworthiness. Agent $\alpha$ judges the direct
observations (either its own or the communicated ones) about $\gamma$’s
behavior in the _evaluation_ process. In the _fusion_ process, $\alpha$ can
use its own evaluations, and the filtered evaluations received from other
agents, to form its trust beliefs about $\gamma$ and an image of other agents’
trust beliefs about $\gamma$. Finally, in the _decision-making_ process,
$\alpha$ builds its trust intentions and applies them in the respective
situations. Below, we describe each sub-process in greater detail and then
have a closer look at the issue of context-sensitivity.
### 3.1 Observation
During observation, agent $\alpha$ uses its sensors to capture information
about other agents and the environment. In network settings for instance,
sensors could be network cards, that can receive or overhear packets from the
network. In the general case, the process of direct observation is
_subjective_ , since different agents may use sensors that differ in certain
respects, for instance in quality.
As mentioned earlier, information can result from direct observation of the
environment or be received as communicated evidence. For an observation
$\text{obs}_{\alpha}^{\left<\delta,\gamma\right>}$, $\alpha$ may be the same
agent as $\delta$; in this case, $\alpha$ observes its own interaction with
some other agent $\gamma$, otherwise $\alpha$ is observing the interactions of
other agents.
Because communicated evidence is received from other agents, $\alpha$ needs to
filter the information. Communicated evidence might be incorrect for different
reasons [CP02, BS08]. First, the communicator can intentionally provide wrong
information, i.e., lie. For instance, the communicator might want to increase
its own reputation or the reputation of an acquaintance; this is usually
called “misleading propaganda”. It can also try to decrease the reputation of
another competing agent, which is usually called “defamation”. Secondly, the
communicator can provide information that is not wrong but leads the receiver
intentionally to wrong conclusions. For instance, it can hide information,
give partial information or give out-of-context information. Finally, the
communicator can unwittingly communicate untrue facts. Note that agent
$\beta$, the last agent who sent the information, needs not to be the
originator of the information [CP02], and therefore might alter the
information if it is not signed. The credibility filter should take all these
aspects into account and filter the information according to how much the
sources of this information are intended to be trusted, more precisely,
whether they are decided to be trusted. In Figure 2, this is indicated by the
“influence”-arrows from “making decisions” to the _credibility filters_ ,
which exactly try to filter out information that does not seem to be credible.
### 3.2 Evaluation
During the process of _evaluation_ , an agent evaluates sets of direct
observations about the behavior of other agents. Such an evaluation estimates
whether the set of observations provides evidence for the agent in question
being trustworthy or untrustworthy. At this stage, it is not yet decided
whether an agent is actually believed to be trustworthy or not; sets of
observations are only examined for their significance in respect to an agent’s
trustworthiness. In many models, this is done by comparing the actual behavior
of an agent to what its behavior was expected to be like. This expected
behavior can, for instance, be determined by formal contracts [Sab02] or
social norms [COTZ00, VM10]. Various representations are used to describe
evaluations: binary (e.g., [SS02]), more fine-grained discrete (e.g.,
$\\{-1,0,+1\\}$ in eBay [eBa09]), or continuous assessments (e.g., [Sab02,
HJS04, VM10]).
As the sets of norms and established (implicit) contracts can be subjective,
an evaluation can be subjective too. Therefore, different agents might
contradictorily interpret the same observations as evidence for trustworthy
and untrustworthy behavior. This is evident in the case of eBay [eBa09] where
each human does the evaluation along his own criteria. As a consequence, an
agent can try to emulate how other agents would evaluate observations, to
eventually emulate their trust beliefs. Therefore, $\mathbb{M}\text{ITRA}$
contains two different ways of evaluation (see Fig. 2): using $\alpha$’s
criteria, which results in evaluations
$\text{eval}_{\alpha}^{\left<\delta,\gamma\right>}$; and the way $\alpha$
thinks $\theta$ would do the evaluation, which results in
$\text{eval}_{\theta}^{\left<\delta,\gamma\right>}$.
For evaluation, it can be vital to know about the causality behind what
happened. Indeed, if the causal relationships are not clear to the evaluating
agent, it can wrongly evaluate a failure to be evidence for untrustworthiness,
although it is actually an _excusable_ failure – or vice versa [CF00, SE07].
### 3.3 Fusion
In a third step, the different evaluations are _fused_ into trust beliefs.
Trust beliefs can be formed based on evaluations of individual agents or
groups of agents, and can be about individuals or groups. Figure 2 shows that
an agent $\alpha$ can fuse evaluations to get its _own trust beliefs_
$\text{tb}_{\alpha}^{\Gamma}$; or to emulate the trust modeling of another
agent $\theta$, in order to estimate this agent’s _$\theta$ ’s trust beliefs_
$\text{tb}_{\theta}^{\Gamma}$. In any case, if $\alpha$ uses evaluations
$\text{eval}_{\beta}^{\left<\cdots\right>}$ received from another agent
$\beta$, it first needs to filter out the “incompatible” subjectivity inherent
in the evaluations: For evaluation, $\beta$ may have applied criteria that
$\alpha$ does not agree with, or – when $\alpha$ is emulating the trust
beliefs of $\theta$ – where $\alpha$ thinks that $\theta$ would not agree
with. In the figure, the check for the match of the applied criteria, and the
potential adjustment of the evaluations, is named _subjectivity filtering_.
The emulation of another agent $\theta$’s trust beliefs is for instance needed
when forming reputation, i.e., trust beliefs that a certain group of agents
$\Theta$ (with $\theta\in\Theta$) would associate to a given trustee. For
this, also trust beliefs received from other agents can be incorporated. A
subjectivity filter should not be applied to these received trust beliefs,
because reputation reflects the subjectivity of different agents. Still, a
credibility filter is applied to avoid a biased reputation estimate.
Furthermore, if an agent forms trust beliefs based on interactions where
itself was not involved, it has to account for the relation between the
interacting agents. Assume, an agent $\alpha$ receives from another agent the
evaluation $\text{eval}_{\theta}^{\left<\delta,\gamma\right>}$, where all four
agents $\alpha$, $\theta$, $\delta$, and $\gamma$ are distinct agents. If
$\alpha$ wants to reason about $\delta$’s trustworthiness towards itself, then
it first needs to apply a _personality filter_. Here, information is filtered
out that contains no evidence for the behavior of $\delta$ towards $\alpha$,
because it is specific for interactions between $\delta$ and $\gamma$. For
example, imagine that the evaluation is about the behavior of a mother
($\delta$) towards her child ($\gamma$), and she behaved very trustworthy.
Then this behavior does not say much about how the mother will behave towards
another unrelated person (e.g., $\alpha$). This shows that trustworthiness is
_directed_ , and that this direction has to be taken into account in trust
reasoning.
Whenever fusing sets of evaluations or trust beliefs into a single trust
belief, it is particularly important to account for the “correlated evidence”
problem [Pea88]. This problem arises when different evaluations or trust
beliefs are based on the same observed interactions of agents. In other words,
the evidence expressed by the evaluations/trust beliefs “overlaps”. If the
reasoning agent is not aware of this overlapping, certain parts of the
evidence will wrongly be amplified and the resulting trust belief be biased.
Many different approaches for fusing evaluations into trust beliefs (e.g.,
[WS07, TPJL06a, RRRJ07]), and trust beliefs into community models (e.g.,
[KSGM03, SFR99]) have been proposed in the literature. Most importantly, each
trust belief should incorporate the two properties mentioned in Sect. 2.3: the
uncertainty, i.e., how strong the belief is, and in which context(s) it
applies.
### 3.4 Decision-Making
The last component in $\mathbb{M}\text{ITRA}$ is the _decision-making_
process, which consists of two steps:
1. 1.
fix trust intentions based on trust beliefs;
2. 2.
apply the trust intentions to make the final decision to act (or not) in
trust.
In the first step, a set of individual and/or collective trust beliefs are
used to _derive_ one or several trust intentions for different contexts. Trust
beliefs can, for instance, be aggregated into trust intentions by simply
averaging over them, or by taking the most “pessimistic” or “optimistic” trust
belief, etc. But also, it is possible to ignore available negative trust
beliefs about another agent $\gamma$, and to decide to act in trust with
$\gamma$ in order to give this agent the opportunity to rethink its behavior
[FC04]. This “advance” in trust, which can in some cases be _forgiveness_ ,
accounts for the dynamics of trust such as “trust begets trust” [BE89].
A trust intention $\text{ti}_{\alpha}^{\gamma}$ has to reflect two things: in
which situations $\alpha$ actually intends to act in trust with $\gamma$
(context), and how strong the intention is (uncertainty). Trust intentions
with these two properties can be used in _decision-making_ in the same way as
other criteria. Although many trust and reputation models from the literature
do not separate the derivation of trust intentions from the process of
decision-making, we strongly argue for a separation of the two processes. The
reason is that the derivation of trust intentions is specific to trust
research, while a trust intention plays in decision making the role of a
context-sensitive criterion in the same way as many other criteria (like
availability of a potential partner). This problem is however studied in depth
in a research area called Multiple Criteria Decision Making (MCDM) [Kal06] and
is not specific to trust.
### 3.5 Context in $\mathbb{M}\text{ITRA}$
In general, _everything_ that can impact the behavior of a trustee and is not
part of the trustee itself, is said to belong to the context [Dey00]. The more
of the available context information is considered by a trust and reputation
model, the better the final decision can be. We propose to arrange the
different facets of context into five classes:
1. 1.
time (points in time or time intervals);
2. 2.
external conditions:
1. (a)
physical conditions,
2. (b)
laws/norms,
3. (c)
other agents nearby;
3. 3.
type of the delegated task;
4. 4.
contract;
5. 5.
information source.
The _information source_ is the one that provides the information, on which
the reasoning is based, e.g., a sensor or another agent. This context facet is
a special case, because it is not linked to a trustee’s behavior. It is
important though, since it allows an agent to have several trust beliefs about
the same trustee, based on different information sources.
Two different contexts can be distinguished: the one in which the information
was taken, and the one in which a decision is made (_decision context_). To
fuse information that comes from different contexts, or use it in decision-
making for different contexts, an agent needs to know how similar the
different contexts are. In the literature, many ways for representing context
information, and similarities between contexts, have been proposed. [KR03]
represent context relations in form of a weighted directed graph. [ŞY07] use
ontologies and a set of rules to represent context information. In a more
general way, [RP07] represent context information as points in a multi-
dimensional space, where each dimension represents one characteristic of the
context (e.g., the point in time, the dollar exchange rate, etc.). The
distance between two points in the context space, which is determined by some
distance metric, states the similarity of the two contexts. These approaches
have in common that an agent was given some similarity metric about different
contexts in advance. $\mathbb{M}\text{ITRA}$ does not assume that, and so, if
the similarity metric cannot be known in advance, an agent needs to learn the
metric.
At which stage information that belongs to different contexts is eventually
fused, is up to the concrete model. However, since during this fusion process
some information is lost, it should be done as late as possible in the
information chain. As a consequence, that principle should generally be used
when processing information.
## 4 Organizing Existing Models
Table 1: Classification with $\mathbb{M}\text{ITRA}$ (Observation). | Observation
---|---
| $obs_{\alpha}^{<\delta,\gamma>}$ | $\left(\text{obs}_{\theta}^{{}\left<\delta,\gamma\right>}\right)_{\left[\beta,\dots\right]}^{\alpha}$ | $\left(\text{eval}_{\theta}^{{}\left<\delta,\gamma\right>}\right)_{\left[\beta,\dots\right]}^{\alpha}$ | $\left(\text{tb}_{\theta}^{\Gamma}\right)_{\left[\beta,\dots\right]}^{\alpha}$ | $\left(\text{tb}_{\Theta}^{\Gamma}\right)_{\left[\beta,\dots\right]}^{\alpha}$
Marsh [Mar94] | | | $\checkmark$ | |
Schillo et al. [SFR99] | | | | $\checkmark$222only considering the case of $|\Gamma|=1$ |
Histos [ZMP99] | | | | $\checkmark$222only considering the case of $|\Gamma|=1$ |
Sporas [ZMP99] | | | $\checkmark$ | |
Beta-Rep. System [JI02] | $\checkmark$ | | $\checkmark$333submitted to a central rating center | | $\checkmark$222only considering the case of $|\Gamma|=1$444provided by a central rating center
Sen and Sajja [SS02] | | | $\checkmark$ | |
EigenTrust [KSG03] | $\checkmark$ | | | | $\checkmark$222only considering the case of $|\Gamma|=1$
Regret [SM03] | | | | $\checkmark$222only considering the case of $|\Gamma|=1$ | $\checkmark$222only considering the case of $|\Gamma|=1$
Secure [CGS+03] | $\checkmark$ | | $\checkmark$ | $\checkmark$222only considering the case of $|\Gamma|=1$ | $\checkmark$111implicitly222only considering the case of $|\Gamma|=1$
Wang and Vassileva [WV03] | $\checkmark$ | | | $\checkmark$222only considering the case of $|\Gamma|=1$ |
PeerTrust [XL04] | $\checkmark$ | | $\checkmark$ | |
Capra and Musolesi [CM06] | $\checkmark$ | | | |
Travos [TPJL06b] | $\checkmark$ | | $\checkmark$ | |
Reece et al. [RRRJ07] | $\checkmark$ | $\checkmark$ | | |
Şensoy and Yolum [ŞY07] | $\checkmark$ | $\checkmark$ | | |
Wang and Singh [WS07] | $\checkmark$111implicitly | | | |
Falcone and Castelfranchi [FC08] | $\checkmark$ | | | |
Liar [VM10] | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$222only considering the case of $|\Gamma|=1$ | $\checkmark$222only considering the case of $|\Gamma|=1$
Vogiatzis et al. [VMC10] | $\checkmark$ | | $\checkmark$ | |
Table 2: Classification with $\mathbb{M}\text{ITRA}$ (Evaluation, Fusion, Decision-Making). | Evaluation | Fusion | Decision-Making
---|---|---|---
| $\text{eval}_{\alpha}^{{}\left<\delta,\gamma\right>}$ | $\text{eval}_{\theta}^{{}\left<\delta,\gamma\right>}$ | $\text{tb}_{\alpha}^{\Gamma}$ | $\text{tb}_{\theta}^{\Gamma}$ | $\text{tb}_{\Theta}^{\Gamma}$ | $\text{ti}_{\alpha}^{\gamma}$
Marsh [Mar94] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | | $\checkmark$
Schillo et al. [SFR99] | $\checkmark$ | | | | $\checkmark$111only considering the case of $|\Gamma|=1$ |
Histos [ZMP99] | | | $\checkmark$111only considering the case of $|\Gamma|=1$ | $\checkmark$111only considering the case of $|\Gamma|=1$ | |
Sporas [ZMP99] | $\checkmark$ | | | | $\checkmark$111only considering the case of $|\Gamma|=1$ |
Beta-Rep. System [JI02] | $\checkmark$ | | | | $\checkmark$111only considering the case of $|\Gamma|=1$222computation performed by central rating center |
Sen and Sajja [SS02] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
EigenTrust [KSG03] | $\checkmark$ | | | | $\checkmark$111only considering the case of $|\Gamma|=1$ |
Regret [SM03] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | $\checkmark$
Secure [CGS+03] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | | $\checkmark$
Wang and Vassileva [WV03] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | | $\checkmark$
PeerTrust [XL04] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | | $\checkmark$
Capra and Musolesi [CM06] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
Travos [TPJL06b] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
Reece et al. [RRRJ07] | | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
Şensoy and Yolum [ŞY07] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
Wang and Singh [WS07] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
Falcone and Castelfranchi [FC08] | $\checkmark$ | | $\checkmark$ | | |
Liar [VM10] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | $\checkmark$111only considering the case of $|\Gamma|=1$ | $\checkmark$111only considering the case of $|\Gamma|=1$ | $\checkmark$
Vogiatzis et al. [VMC10] | $\checkmark$ | | $\checkmark$111only considering the case of $|\Gamma|=1$ | | |
In this section, we exemplify how existing trust and reputation (T&R) models
can be classified by means of $\mathbb{M}\text{ITRA}$. The approach we take
here is to investigate which types of data are used in the considered trust
models. For a small selection of trust models, this is shown in tables 1 and
2. Table 1 lists types of data that belong to the observation process. Table 2
lists the remaining data types, i.e. those in the evaluation, fusion and
decision-making processes. A check-mark indicates whether a trust model
accounts for the respective kind of data. Still, not every model accounts for
the context and uncertainty of the processed information.
Additionally to this classification approach, T&R models can also be described
by specifying:
1. 1.
which filters are applied,
2. 2.
where uncertainty is considered, and
3. 3.
which facets of context are accounted for at which stage.
Since the focus of this article lies on the meta model $\mathbb{M}\text{ITRA}$
itself, we leave it to future work to provide an extensive and comprehensive
classification using $\mathbb{M}\text{ITRA}$. However, as it can already be
seen in the resulting tables, a clear picture emerges of what has been done
(columns with check-marks) and what needs to be done in the T&R modeling
domain – for example, there is no model with no empty columns, i.e., that uses
all available types of information. Furthermore, the column
$\text{eval}_{\theta}^{{}\left<\delta,\gamma\right>}$ is always empty. None of
the here considered trust models111To our knowledge there is no such trust
model in the literature. uses the intermediate step of simulating the
evaluation of another agent to eventually form a collective trust belief
(reputation). However, we believe that this is something what humans do
regularly; for example in form of questions like “What would my mother think
about his behavior?” or “How would my best friend judge this agent’s
behavior?”. This issue deserves more attention as it makes it possible to
evaluate the behavior of other agents in cases where an own opinion on their
behavior is lacking.
## 5 Related Work
Numerous survey or overview papers on trust and reputation models have been
published. Many works start with a basic classification and then enumerate and
describe existing models [SMS05, AG07, JIB07, RHJ04, MHM02]. Opposed to that,
we tried in our work to use a rigorous methodology for the organization of the
state of the art: extract the core structure of prominent trust and reputation
models, in order to get clear and simple differentiation criteria for the
models of the literature.
[KBR05] proposed a generic trust model that integrates several existing trust
models. They show how to map each of these models to their model. However,
they focus on the fusion of evaluations, whereas we examined the overall
structure of trust and reputation modeling.
[CS05] define a functional ontology of reputation, that is used in different
systems [VCSB07, NBSV08] in order to help agents using different trust and
reputation models inter-operate. These approaches try to cope with a situation
where different models exist, whereas we try to propose a unified meta-model
of trust and reputation.
## 6 Conclusion
In this article, we presented $\mathbb{M}\text{ITRA}$, a meta-model for trust
and reputation. The model structures many essential concepts found in the
literature on evidence-based T&R models. The simple and generic structure of
$\mathbb{M}\text{ITRA}$ makes it suitable both for experts to organize their
models in a common way, and for newcomers to easily enter the domain. Although
there is the possibility that a specific T&R model does not comply with
$\mathbb{M}\text{ITRA}$, the fact that the latter was derived from the study
of many existing models argues for its comprehensiveness, and we believe that
its modularity should make it simple to modify in order to encompass new
elements.
Finally, by using $\mathbb{M}\text{ITRA}$, we classified existing T&R models
from the literature. This classification revealed which kinds of information
are commonly used by these models, and which kinds of information are often
neglected. In this way, we found that none of the considered models actually
tries to emulate another agent’s evaluation of a trustee; this emulation would
make it possible to form the reputation of an agent in a new way.
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|
arxiv-papers
| 2012-07-02T14:34:05 |
2024-09-04T02:49:32.517532
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eugen Staab and Guillaume Muller",
"submitter": "Guillaume Muller PhD",
"url": "https://arxiv.org/abs/1207.0405"
}
|
1207.0560
|
# Algorithms for Approximate Minimization of the Difference Between Submodular
Functions, with Applications††thanks: A shorter version of this appeared in
Proc. Uncertainty in Artificial Intelligence (UAI), Catalina Islands, 2012
Rishabh Iyer
Dept. of Electrical Engineering
University of Washington
Seattle, WA-98175, USA Jeff Bilmes
Dept. of Electrical Engineering
University of Washington
Seattle, WA-98175, USA
###### Abstract
We extend the work of Narasimhan and Bilmes [32] for minimizing set functions
representable as a difference between submodular functions. Similar to [32],
our new algorithms are guaranteed to monotonically reduce the objective
function at every step. We empirically and theoretically show that the per-
iteration cost of our algorithms is much less than [32], and our algorithms
can be used to efficiently minimize a difference between submodular functions
under various combinatorial constraints, a problem not previously addressed.
We provide computational bounds and a hardness result on the multiplicative
inapproximability of minimizing the difference between submodular functions.
We show, however, that it is possible to give worst-case additive bounds by
providing a polynomial time computable lower-bound on the minima. Finally we
show how a number of machine learning problems can be modeled as minimizing
the difference between submodular functions. We experimentally show the
validity of our algorithms by testing them on the problem of feature selection
with submodular cost features.
## 1 Introduction
Discrete optimization is important to many areas of machine learning and
recently an ever growing number of problems have been shown to be expressible
as submodular function minimization or maximization (e.g., [21, 25, 27, 30,
29, 31]). The class of submodular functions is indeed special since submodular
function minimization is known to be polynomial time, while submodular
maximization, although NP complete, admits constant factor approximation
algorithms. Let $V=\\{1,2,\cdots,n\\}$ refer a ground set, then
$f:2^{V}\rightarrow\mathbb{R}$ is said to be submodular if for sets
$S,T\subseteq V$, $f(S)+f(T)\geq f(S\cup T)+f(S\cap T)$ (see [11] for details
on submodular, supermodular, and modular functions). Submodular functions have
a diminishing returns property, wherein the gain of an element in the context
of bigger set is lesser than the gain of that element in the context of a
smaller subset. This property occurs naturally in many applications in machine
learning, computer vision, economics, operations research, etc.
In this paper, we address the following problem. Given two submodular
functions $f$ and $g$, and define $v(X)\triangleq f(X)-g(X)$, solve the
following optimization problem:
$\min_{X\subseteq V}[f(X)-g(X)]\equiv\min_{X\subseteq V}[v(X)].$ (1)
A number of machine learning problems involve minimization over a difference
between submodular functions. The following are some examples:
* •
Sensor placement with submodular costs: The problem of choosing sensor
locations $A$ from a given set of possible locations $V$ can be modeled [25,
26] by maximizing the mutual information between the chosen variables $A$ and
the unchosen set $V\backslash A$ (i.e., $f(A)=I(X_{A};X_{V\backslash A})$).
Alternatively, we may wish to maximize the mutual information between a set of
chosen sensors $X_{A}$ and a fixed quantity of interest $C$ (i.e.,
$f(A)=I(X_{A};C)$) under the assumption that the set of features $X_{A}$ are
conditionally independent given $C$ [25]. These objectives are submodular and
thus the problem becomes maximizing a submodular function subject to a
cardinality constraint. Often, however, there are costs $c(A)$ associated with
the locations that naturally have a diminishing returns property. For example,
there is typically a discount when purchasing sensors in bulk. Moreover, there
may be diminished cost for placing a sensor in a particular location given
placement in certain other locations (e.g., the additional equipment needed to
install a sensor in, say, a precarious environment could be re-used for
multiple sensor installations in like environments). Hence, along with
maximizing mutual information, we also want to simultaneously minimize the
cost and this problem can be addressed by minimizing the difference between
submodular functions $f(A)-\lambda c(A)$ for tradeoff parameter $\lambda$.
* •
Discriminatively structured graphical models and neural computation: An
application suggested in [32] and the initial motivation for this problem is
to optimize the EAR criterion to produce a discriminatively structured
graphical model. EAR is basically a difference between two mutual information
functions (i.e., a difference between submodular functions). [32] shows how
classifiers based on discriminative structure using EAR can significantly
outperform classifiers based on generative graphical models. Note also that
the EAR measure is the same as “synergy” in a neural code [3], widely used in
neuroscience.
* •
Feature selection: Given a set of features $X_{1},X_{2},\cdots,X_{|V|}$, the
feature selection problem is to find a small subset of features $X_{A}$ that
work well when used in a pattern classifier. This problem can be modeled as
maximizing the mutual information $I(X_{A};C)$ where $C$ is the class. Note
that $I(X_{A};C)=H(X_{A})-H(X_{A}|C)$ is always a difference between
submodular functions. Under the naïve Bayes model, this function is submodular
[25]. It is not submodular under general classifier models such as support
vector machines (SVMs) or neural networks. Certain features, moreover, might
be cheaper to use given that others are already being computed. For example,
if a subset $S_{i}\subseteq V$ of the features for a particular information
source $i$ are spectral in nature, then once a particular $v\in S_{i}$ is
chosen, the remaining features $S_{i}\setminus\\{v\\}$ may be relatively
inexpensive to compute, due to grouped computational strategies such as the
fast Fourier transform. Therefore, it might be more appropriate to use a
submodular cost model $c(A)$. One such cost model might be
$c(A)=\sum_{i}\sqrt{m(A\cap S_{i})}$ where $m(j)$ would be the cost of
computing feature $j$. Another might be $c(A)=\sum_{i}c_{i}\min(|A\cap
S_{i}|,1)$ where $c_{i}$ is the cost of source $i$. Both offer diminishing
cost for choosing features from the same information source. Such a cost model
could be useful even under the naïve Bayes model, where $I(X_{A};C)$ is
submodular. Feature selection becomes a problem of maximizing
$I(X_{A};C)-\lambda c(A)=H(X_{A})-[H(X_{A}|C)+\lambda c(A)]$, the difference
between two submodular functions.
* •
Probabilistic Inference: A typical instance of probabilistic inference is the
following: We are given a distribution $p(x)\propto\exp(-v(x))$ where
$x\in\\{0,1\\}^{n}$ and $v$ is a pseudo-Boolean function [2]. It is desirable
to compute $\operatorname*{argmax}_{x\in\\{0,1\\}^{n}}p(x)$ which means
minimizing $v(x)$ over $x$, the most-probable explanation (MPE) problem [35].
If $p$ factors with respect to a graphical model of tree-width $k$, then
$v(x)=\sum_{i}v_{i}(x_{C})$ where $C_{i}$ is a bundle of indices such that
$|C|\leq k+1$ and the sets $\mathcal{C}=\\{C_{i}\\}_{i}$ form a junction tree,
and it might be possible to solve inference using dynamic programming. If $k$
is large and/or if hypertree factorization does not hold, then approximate
inference is typically used [40]. On the other hand, defining
$x(X)=\\{x\in\\{0,1\\}^{n}:x_{i}=1\text{ whenever }i\in X\\}$, if the set
function $\bar{v}(X)=v(x(X))$ is submodular, then even if $p$ has large tree-
width, the MPE problem can be solved exactly in polynomial time [18]. This, in
fact, is the basis behind inference in many computer vision models where $v$
is often not only submodular but also has limited sized $|C_{i}|$. For
example, for submodular $v$ and if $|C_{i}|\leq 2$ then graph-cuts can solve
the MPE problem extremely rapidly [24] and even some cases with $v$ non-
submodular [23]. An important challenge is to consider non-submodular $v$ that
can be minimized efficiently and for which there are approximation guarantees,
a problem recently addressed in [19]. On the other hand, if $v$ can be
expressed as a difference between two submodular functions (which it can, see
Lemma 3.1), or if such a decomposition can be computed (which it sometimes
can, see Lemma 3.2), then a procedure to minimize the difference between two
submodular functions offers new ways to solve probabilistic inference. As an
example, a large class of rich higher potentials can be expressed as [13]:
$\displaystyle f(x)=\sum_{C\in\mathcal{C}}w_{C}\prod_{i\in C}x_{i}$ (2)
$\mathcal{C}$ here stands for a set of sets, possibly with higher-order terms
(i.e there exist $C\in\mathcal{C}:|C|>2$). If $w_{C}\leq 0,\forall
C\in\mathcal{C}$, then $f$ is submodular. If $|\mathcal{C}|$ is not large (say
polynomial in $n$), we can efficiently find a decomposition into submodular
components (which will contain the sets $C\in\mathcal{C}:w_{C}\leq 0$) and the
supermodular terms (which contain sets $C\in\mathcal{C}:w_{C}\geq 0$). These
can potentially represent a rich class of potential functions for a number of
applications, particularly in vision.
We note that given a solution to Equation 1, we can also minimize the
difference between two supermodular functions $\min((-g)-(-f))$, maximize the
difference between two submodular functions $\max(-v)=\max(g-f)$, and maximize
the difference between two supermodular functions $\max(-v)=\max((-f)-(-g))$.
Previously, Narasimhan and Bilmes [32] proposed an algorithm inspired by the
convex-concave procedure [41] to address Equation (1). This algorithm
iteratively minimizes a submodular function by replacing the second submodular
function $g$ by it’s modular lower bound. They also show that any set function
can be expressed as a difference between two submodular functions and hence
every set function optimization problem can be reduced to minimizing a
difference between submodular functions. They show that this process converges
to a local minima, however the convergence rate is left as an open question.
In this paper, we first describe tight modular bounds on submodular functions
in Section 2, including lower bounds based on points in the base polytope as
used in [32], and recent upper bounds first described in a result in [17]. In
section 3 Submodular-Supermodular Procedure, we describe the submodular-
supermodular procedure proposed in [32]. We further provide a constructive
procedure for finding the submodular functions $f$ and $g$ for any arbitrary
set function $v$. Although our construction is NP hard in general, we show how
for certain classes of set functions $v$, it is possible to find the
decompositions $f$ and $g$ in polynomial time. In Section 4, we propose two
new algorithms both of which are guaranteed to monotonically reduce the
objective at every iteration and which converge to a local minima. Further we
note that the per-iteration cost of our algorithms is in general much less
than [32], and empirically verify that our algorithms are orders of magnitude
faster on real data. We show that, unlike in [32], our algorithms can be
extended to easily optimize equation (1) under cardinality, knapsack, and
matroid constraints. Moreover, one of our algorithms can actually handle
complex combinatorial constraints, such as spanning trees, matchings, cuts,
etc. Further in Section 5, we give a hardness result that there does not exist
any polynomial time algorithm with any polynomial time multiplicative
approximation guarantees unless P=NP, even when it is easy to find or when we
are given the decomposition $f$ and $g$, thus justifying the need for
heuristic methods to solve this problem. We show, however, that it is possible
to get additive bounds by showing polynomial time computable upper and lower
bound on the optima. We also provide computational bounds for all our
algorithms (including the submodular-supermodular procedure), a problem left
open in [32].
Finally we perform a number of experiments on the feature selection problem
under various cost models, and show how our algorithms used to maximize the
mutual information perform better than greedy selection (which would be near
optimal under the naïve Bayes assumptions) and with less cost.
## 2 Modular Upper and Lower bounds
The Taylor series approximation of a convex function provides a natural way of
providing lower bounds on such a function. In particular the first order
Taylor series approximation of a convex function is a lower bound on the
function, and is linear in $x$ for a given $y$ and hence given a convex
function $\phi$, we have:
$\phi(x)\geq\phi(y)+\langle\nabla\phi(y),x-y\rangle.$ (3)
Surprisingly, any submodular function has both a tight lower [8] and upper
bound [17], unlike strict convexity where there is only a tight first order
lower bound.
### 2.1 Modular Lower Bounds
Recall that for submodular function $f$, the submodular polymatroid, base
polytope and the sub-differential with respect to a set $Y$ [11] are
respectively:
$\displaystyle\mathcal{P}_{f}=\\{x:x(S)\leq f(S),\forall S\subseteq V\\}$ (4)
$\displaystyle\mathcal{B}_{f}=\mathcal{P}_{f}\cap\\{x:x(V)=f(V)\\}$ (5)
$\displaystyle\\!\\!\\!\\!\partial f(Y)=\\{y\in\mathbb{R}^{V}:\forall
X\subseteq V,f(Y)-y(Y)\leq f(X)-y(X)\\}$
The extreme points of this sub-differential are easy to find and characterize,
and can be obtained from a greedy algorithm ([8, 11]) as follows:
###### Theorem 2.1.
([11], Theorem 6.11) A point $y$ is an extreme point of $\partial f(Y)$, iff
there exists a chain $\emptyset=S_{0}\subset S_{1}\subset\cdots\subset S_{n}$
with $Y=S_{j}$ for some $j$, such that $y(S_{i}\setminus
S_{i-1})=y(S_{i})-y(S_{i-1})=f(S_{i})-f(S_{i-1})$.
Let $\sigma$ be a permutation of $V$ and define
$S_{i}^{\sigma}=\\{\sigma(1),\sigma(2),\dots,\sigma(i)\\}$ as $\sigma$’s chain
containing $Y$, meaning $S_{|Y|}^{\sigma}=Y$ (we say that $\sigma$’s chain
contains $Y$). Then we can define a sub-gradient $h^{f}_{Y}$ corresponding to
$f$ as:
$h^{f}_{Y,\sigma}(\sigma(i))=\begin{cases}f(S_{1}^{\sigma})&\text{ if }i=1\\\
f(S_{i}^{\sigma})-f(S_{i-1}^{\sigma})&\text{ otherwise }\end{cases}.$
We get a modular lower bound of $f$ as follows:
$\displaystyle h^{f}_{Y,\sigma}(X)\leq f(X),\forall X\subseteq V,\text{ and
}\forall i,h^{f}_{Y,\sigma}(S_{i}^{\sigma})=f(S_{i}^{\sigma}),$
which is parameterized by a set $Y$ and a permutation $\sigma$. Note
$h(X)=\sum_{i\in X}h(i)$, and $h^{f}_{Y,\sigma}(Y)=f(Y)$. Observe the
similarity to convex functions, where a linear lower bound is parameterized by
a vector $y$.
### 2.2 Modular Upper Bounds
For $f$ submodular, [33] established the following:
$\displaystyle f(Y)\leq f(X)-\sum_{j\in X\backslash Y}f(j|X\backslash
j)+\sum_{j\in Y\backslash X}f(j|X\cap Y),$ $\displaystyle f(Y)\leq
f(X)-\sum_{j\in X\backslash Y}f(j|(X\\!\cup\\!Y)\backslash j)+\sum_{j\in
Y\backslash X}f(j|X)$
Note that $f(A|B)\triangleq f(A\cup B)-f(B)$ is the gain of adding $A$ in the
context of $B$. These upper bounds in fact characterize submodular functions,
in that a function $f$ is a submodular function iff it follows either of the
above bounds. Using the above, two tight modular upper bounds ([17]) can be
defined as follows:
$\displaystyle\\!\\!\\!f(Y)\leq m^{f}_{X,1}(Y)\triangleq
f(X)-\\!\\!\\!\sum_{j\in X\backslash Y}f(j|X\backslash j)+\sum_{j\in
Y\backslash X}f(j|\emptyset),$ $\displaystyle\\!\\!\\!f(Y)\leq
m^{f}_{X,2}(Y)\triangleq f(X)-\\!\\!\\!\sum_{j\in X\backslash
Y}f(j|V\backslash j)+\sum_{j\in Y\backslash X}f(j|X).$
Hence, this yields two tight (at set $X$) modular upper bounds
$m^{f}_{X,1},m^{f}_{X,2}$ for any submodular function $f$. For briefness, when
referring either one we use $m^{f}_{X}$.
## 3 Submodular-Supermodular Procedure
We now review the submodular-supermodular procedure [32] to minimize functions
expressible as a difference between submodular functions (henceforth called DS
functions). Interestingly, any set function can be expressed as a DS function
using suitable submodular functions as shown below. The result was first shown
in [32] using the Lovász extension. We here give a new combinatorial proof,
which avoids Hessians of polyhedral convex functions and which provides a way
of constructing (a non-unique) pair of submodular functions $f$ and $g$ for an
arbitrary set function $v$.
###### Lemma 3.1.
[32] Given any set function $v$, it can be expressed as a DS functions
$v(X)=f(X)-g(X),\forall X\subseteq V$ for some submodular functions $f$ and
$g$.
###### Proof.
Given a set function $v$, we can define $\alpha=\min_{X\subset Y\subseteq
V\setminus j}v(j|X)-v(j|Y)$111We denote $j,X,Y:X\subset Y\subseteq
V\setminus\\{j\\}$ by $X\subset Y\subseteq V\setminus j$.. Clearly $\alpha<0$,
since otherwise $v$ would be submodular. Now consider any (strictly)
submodular function $g$, i.e., one having $\beta=\min_{X\subset Y\subseteq
V\setminus j}g(j|X)-g(j|Y)>0$. Define
$f^{\prime}(X)=v(X)+\frac{|\alpha^{\prime}|}{\beta}g(X)$ with any
$\alpha^{\prime}\leq\alpha$. Now it is easy to see that $f^{\prime}$ is
submodular since $\min_{X\subset Y\subseteq V\setminus
j}f^{\prime}(j|X)-f^{\prime}(j|Y)\geq\alpha+|\alpha^{\prime}|\geq 0$. Hence
$v(X)=f^{\prime}(X)-\frac{|\alpha^{\prime}|}{\beta}g(X)$, is a difference
between two submodular functions. ∎
The above proof requires the computation of $\alpha$ and $\beta$ which has, in
general, exponential complexity. Using the construction above, however, it is
easy to find the decomposition $f$ and $g$ under certain conditions on $v$.
###### Lemma 3.2.
If $\alpha$ or at least a lower bound on $\alpha$ for any set function $v$ can
be computed in polynomial time, functions $f$ and $g$ corresponding to $v$ can
obtained in polynomial time.
###### Proof.
Define $g$ as $g(X)=\sqrt{|X|}$. Then $\beta=\min_{X\subset Y\subseteq
V\setminus j}\sqrt{|X|+1}-\sqrt{|X|}-\sqrt{|Y|+1}+\sqrt{|Y|}=\min_{X\subset
V\setminus
j}\sqrt{|X|+1}-\sqrt{|X|}-\sqrt{|X|+2}+\sqrt{|X|+1}=2\sqrt{n-1}-\sqrt{n}-\sqrt{n-2}$.
The last inequality follows since the smallest difference in gains will occur
at $|X|=n-2$. Hence $\beta$ is easily computed, and given a lower bound on
$\alpha$, from Lemma 3.1 the decomposition can be obtained in polynomial time.
A similar argument holds for $g$ being other concave functions over $|X|$. ∎
Algorithm 1 The submodular-supermodular (SubSup) procedure [32]
1: $X^{0}=\emptyset$ ; $t\leftarrow 0$ ;
2: while not converged (i.e., $(X^{t+1}\neq X^{t})$) do
3: Randomly choose a permutation $\sigma^{t}$ whose chain contains the set
$X^{t}$.
4: $X^{t+1}:=\operatorname*{argmin}_{X}f(X)-h^{g}_{X^{t},\sigma^{t}}(X)$
5: $t\leftarrow t+1$
6: end while
The submodular supermodular (SubSup) procedure is given in Algorithm 1. At
every step of the algorithm, we minimize a submodular function which can be
performed in strongly polynomial time [34, 37] although the best known
complexity is $O(n^{5}\eta+n^{6})$ where $\eta$ is the cost of a function
evaluation. Algorithm 1 is guaranteed to converge to a local minima and
moreover the algorithm monotonically decreases the function objective at every
iteration, as we show below.
###### Lemma 3.3.
[32] Algorithm 1 is guaranteed to decrease the objective function at every
iteration. Further, the algorithm is guaranteed to converge to a local minima
by checking at most $O(n)$ permutations at every iteration.
###### Proof.
The objective reduces at every iteration since:
$\displaystyle f(X^{t+1})-g(X^{t+1})$ $\displaystyle\overset{a}{\leq}$
$\displaystyle f(X^{t+1})-h^{g}_{X^{t},\sigma^{t}}(X^{t+1})$
$\displaystyle\overset{b}{\leq}$ $\displaystyle
f(X^{t})-h^{g}_{X^{t},\sigma^{t}}(X^{t})$ $\displaystyle\overset{c}{=}$
$\displaystyle f(X^{t})-g(X^{t})$
Where (a) follows since $h^{g}_{X^{t},\sigma^{t}}(X^{t+1})\leq g(X^{t+1})$,
and (b) follows since $X^{t+1}$ is the minimizer of
$f(X)-h^{g}_{X^{t},\sigma^{t}}(X)$, and (c) follows since
$h^{g}_{X^{t},\sigma^{t}}(X^{t})=g(X^{t})$ from the tightness of the modular
lower bound.
Further note that, if there is no improvement in the function value by
considering $O(n)$ permutations each with different elements at
$\sigma^{t}(|X^{t}|-1)$ and $\sigma^{t}(|X^{t}|+1)$, then this is equivalent
to a local minima condition on $v$ since
$h^{g}_{X^{t},\sigma^{t}}(S_{|X^{t}|+1}^{\sigma})=f(S_{|X^{t}|+1}^{\sigma})$
and
$h^{g}_{X^{t},\sigma^{t}}(S_{|X^{t}|-1}^{\sigma})=f(S_{|X^{t}|-1}^{\sigma})$.
∎
Algorithm 1 requires performing a submodular function minimization at every
iteration which while polynomial in $n$ is (due to the complexity described
above) not practical for large problem sizes. So while the algorithm reaches a
local minima, it can be costly to find it. A desirable result, therefore,
would be to develop new algorithms for minimizing DS functions, where the new
algorithms have the same properties as the SubSup procedure but are much
faster in practice. We give this in the following sections.
## 4 Alternate algorithms for minimizing DS functions
In this section we propose two new algorithms to minimize DS functions, both
of which are guaranteed to monotonically reduce the objective at every
iteration and converge to local minima. We briefly describe these algorithms
in the subsections below.
### 4.1 The supermodular-submodular (SupSub) procedure
In the submodular-supermodular procedure we iteratively minimized $f(X)-g(X)$
by replacing $g$ by it’s modular lower bound at every iteration. We can
instead replace $f$ by it’s modular upper bound as is done in Algorithm 2,
which leads to the supermodular-submodular procedure.
Algorithm 2 The supermodular-submodular (SupSub) procedure
1: $X^{0}=\emptyset$ ; $t\leftarrow 0$ ;
2: while not converged (i.e., $(X^{t+1}\neq X^{t})$) do
3: $X^{t+1}:=\operatorname*{argmin}_{X}m^{f}_{X^{t}}(X)-g(X)$
4: $t\leftarrow t+1$
5: end while
In the SupSub procedure, at every step we perform submodular maximization
which, although NP complete to solve exactly, admits a number of fast constant
factor approximation algorithms [4, 9]. Notice that we have two modular upper
bounds and hence there are a number of ways we can choose between them. One
way is to run both maximization procedures with the two modular upper bounds
at every iteration in parallel, and choose the one which is better. Here by
better we mean the one in which the function value is lesser. Alternatively we
can alternate between the two modular upper bounds by first maximizing the
expression using the first modular upper bound, and then maximize the
expression using the second modular upper bound. Notice that since we perform
approximate submodular maximization at every iteration, we are not guaranteed
to monotonically reduce the objective value at every iteration. If, however,
we ensure that at every iteration we take the next step only if the objective
$v$ does not increase, we will restore monotonicity at every iteration. Also,
in some cases we converge to local optima as shown in the following theorem.
###### Theorem 4.1.
Both variants of the supermodular-submodular procedure (Algorithm 2)
monotonically reduces the objective value at every iteration. Moreover,
assuming a submodular maximization procedure in line 3 that reaches a local
maxima of $m^{f}_{X^{t}}(X)-g(X)$, then if Algorithm 2 does not improve under
both modular upper bounds then it reaches a local optima of $v$.
###### Proof.
For either modular upper bound, we have:
$\displaystyle f(X^{t+1})-g(X^{t+1})$
$\displaystyle\overset{a}{\leq}m^{f}_{X^{t}}(X^{t+1})-g(X^{t+1})$
$\displaystyle\overset{b}{\leq}m^{f}_{X^{t}}(X^{t})-g(X^{t})$
$\displaystyle\overset{c}{=}f(X^{t})-g(X^{t}),$
where (a) follows since $f(X^{t+1})\leq m^{f}_{X^{t}}(X^{t+1})$, and (b)
follows since we assume that we take the next step only if the objective value
does not increase and (c) follows since $m^{f}_{X^{t}}(X^{t})=f(X^{t})$ from
the tightness of the modular upper bound.
To show that this algorithm converges to a local minima, we assume that the
submodular maximization procedure in line 3 converges to a local maxima. Then
observe that if the objective value does not decrease in an iteration under
both upper bounds, it implies that $m^{f}_{X^{t}}(X^{t})-g(X^{t})$ is already
a local optimum in that (for both upper bounds) we have
$m^{f}_{X^{t}}(X^{t}\cup j)-g(X^{t}\cup j)\geq
m^{f}_{X^{t}}(X^{t})-g(X^{t}),\forall j\notin X^{t}$ and
$m^{f}_{X^{t}}(X^{t}\backslash j)-g(X^{t}\backslash j)\geq
m^{f}_{X^{t}}(X^{t})-g(X^{t}),\forall j\in X^{t}$. Note that
$m^{f}_{X^{t},1}(X^{t}\backslash j)=f(X^{t})-f(j|X^{t}\backslash
j)=f(X^{t}\backslash j)$ and $m^{f}_{X^{t},2}(X^{t}\cup
j)=f(X^{t})+f(j|X^{t})=f(X^{t}\cup j)$ and hence if both modular upper bounds
are at a local optima, it implies
$f(X^{t})-g(X^{t})=m^{f}_{X^{t},1}(X^{t})-g(X^{t})\leq
m^{f}_{X^{t},1}(X^{t}\backslash j)-g(X^{t}\backslash j)=f(X^{t}\backslash
j)-g(X^{t}\backslash j)$. Similarly
$f(X^{t})-g(X^{t})=m^{f}_{X^{t},2}(X^{t})-g(X^{t})\leq
m^{f}_{X^{t},2}(X^{t}\cup j)-g(X^{t}\cup j)=f(X^{t}\cup j)-g(X^{t}\cup j)$.
Hence $X^{t}$ is a local optima for $v(X)=f(X)-g(X)$, since $v(X^{t})\leq
v(X^{t}\cup j)$ and $v(X^{t})\leq v(X^{t}\backslash j)$. ∎
To ensure that we take the largest step at each iteration, we can use the
recently proposed tight (1/2)-approximation algorithm in [4] for unconstrained
non-monotone submodular function maximization — this is the best possible in
polynomial time for the class of submodular functions independent of the P=NP
question. The algorithm is a form of bi-directional randomized greedy
procedure and, most importantly for practical considerations, is linear time
[4]. In practice we just use a combination of a form of a simple greedy
procedure, and the bi-directional randomized algorithm, by picking the best
amongst the two at every iteration. Since the randomized greedy algorithm is
$1/2$ approximate, the combination of the two procedures also will be $1/2$
approximate.
Lastly, note that this algorithm is closely related to a local search
heuristic for submodular maximization [9]. In particular, if instead of using
the greedy algorithm entirely at every iteration, we take only one local step,
we get a local search heuristic. Hence, via the SupSub procedure, we may take
larger steps at every iteration as compared to a local search heuristic.
### 4.2 The modular-modular (ModMod) procedure
The submodular-supermodular procedure and the supermodular-submodular
procedure were obtained by replacing $g$ by it’s modular lower bound and $f$
by it’s modular upper bound respectively. We can however replace both of them
by their respective modular bounds, as is done in Algorithm 3.
Algorithm 3 Modular-Modular (ModMod) procedure
1: $X^{0}=\emptyset$; $t\leftarrow 0$ ;
2: while not converged (i.e., $(X^{t+1}\neq X^{t})$) do
3: Choose a permutation $\sigma^{t}$ whose chain contains the set $X^{t}$.
4:
$X^{t+1}:=\operatorname*{argmin}_{X}m^{f}_{X^{t}}(X)-h^{g}_{X^{t},\sigma^{t}}(X)$
5: $t\leftarrow t+1$
6: end while
In this algorithm at every iteration we minimize only a modular function which
can be done in $O(n)$ time, so this is extremely easy (i.e., select all
negative elements for the smallest minimum, or all non-positive elements for
the largest minimum). Like before, since we have two modular upper bounds, we
can use any of the variants discussed in the subsection above. Moreover, we
are still guaranteed to monotonically decrease the objective at every
iteration and converge to a local minima.
###### Theorem 4.2.
Algorithm 3 monotonically decreases the function value at every iteration. If
the function value does not increase on checking $O(n)$ different permutations
with different elements at adjacent positions and with both modular upper
bounds, then we have reached a local minima of $v$.
###### Proof.
Again we can use similar reasoning as the earlier proofs and observe that:
$\displaystyle f(X^{t+1})-g(X^{t+1})$ $\displaystyle\leq
m^{f}_{X^{t}}(X^{t+1})-h^{g}_{X^{t},\sigma^{t}}(X^{t+1})$ $\displaystyle\leq
m^{f}_{X^{t}}(X^{t})-h^{g}_{X^{t},\sigma^{t}}(X^{t})$
$\displaystyle=f(X^{t})-g(X^{t})$
We see that considering $O(n)$ permutations each with different elements at
$\sigma^{t}(|X^{t}|-1)$ and $\sigma^{t}(|X^{t}|+1)$, we essentially consider
all choices of $g(X^{t}\cup j)$ and $g(X^{t}\backslash j)$, since
$h^{g}_{X^{t},\sigma^{t}}(S_{|X^{t}|+1})=f(S_{|X^{t}|+1})$ and
$h^{g}_{X^{t},\sigma^{t}}(S_{|X^{t}|-1})=f(S_{|X^{t}|-1})$. Since we consider
both modular upper bounds, we correspondingly consider every choice of
$f(X^{t}\cup j)$ and $f(X^{t}\backslash j)$. Note that at convergence we have
that $m^{f}_{X^{t}}(X^{t})-h^{g}_{X^{t},\sigma^{t}}(X^{t})\leq
m^{f}_{X^{t}}(X)-h^{g}_{X^{t},\sigma^{t}}(X),\forall X\subseteq V$ for $O(n)$
different permutations and both modular upper bounds. Correspondingly we are
guaranteed that (since the expression is modular) $\forall j\notin
X^{t},v(j|X^{t})\geq 0$ and $\forall j\in X^{t},v(j|X^{t}\backslash j)\geq 0$,
where $v(X)=f(X)-g(X)$. Hence the algorithm converges to a local minima. ∎
An important question is the choice of the permutation $\sigma^{t}$ at every
iteration $X^{t}$. We observe experimentally that the quality of the algorithm
depends strongly on the choice of permutation. Observe that $f(X)-g(X)\leq
m^{f}_{X^{t}}(X)-h^{g}_{X^{t},\sigma^{t}}(X)$, and
$f(X^{t})-g(X^{t})=m^{f}_{X^{t}}(X^{t})-h^{g}_{X^{t},\sigma^{t}}(X^{t})$.
Hence, we might obtain the greatest local reduction in the value of $v$ by
choosing permutation
$\sigma^{*}\in\operatorname*{argmin}_{\sigma}\min_{X}(m^{f}_{X^{t}}(X)-h^{g}_{X^{t},\sigma^{t}}(X))$,
or the one which maximizes $h^{g}_{X^{t},\sigma^{t}}(X)$. We in fact might
expect that choosing $\sigma^{t}$ ordered according to greatest gains of $g$,
with respect to $X^{t}$, we would achieve greater descent at every iteration.
Another choice is to choose the permutation $\sigma$ based on the ordering of
gains of $v$ (or even $m^{f}_{X^{t}}$). Through the former we are guaranteed
to at least progress as much as the local search heuristic. Indeed, we observe
in practice that the first two of these heuristics performs much better than a
random permutation for both the ModMod and the SubSup procedure, thus
addressing a question raised in [32] about which ordering to use. Practically
for the feature selection problem, the second heuristic seems to work the
best.
### 4.3 Constrained minimization of a difference between submodular functions
In this section we consider the problem of minimizing the difference between
submodular functions subject to constraints. We first note that the problem of
minimizing a submodular function under even simple cardinality constraints in
NP hard and also hard to approximate [38]. Since there does not yet seem to be
a reasonable algorithm for constrained submodular minimization at every
iteration, it is unclear how we would use Algorithm 1. However the problem of
submodular maximization under cardinality, matroid, and knapsack constraints
though NP hard admits a number of constant factor approximation algorithms
[33, 28] and correspondingly the cardinality constraints can be easily
introduced in Algorithm 2. Moreover, since a non-negative modular function can
be easily, directly and even exactly optimized under cardinality, knapsack and
matroid constraints [16], Algorithm 3 can also easily be utilized. In
addition, since problems such as finding the minimum weight spanning tree,
min-cut in a graph, etc., are polynomial time algorithms in a number of cases,
Algorithm 3 can be used when minimizing a non-negative function $v$
expressible as a difference between submodular functions under combinatorial
constraints. If $v$ is non-negative, then so is its modular upper bound, and
then the ModMod procedure can directly be used for this problem — each
iteration minimizes a non-negative modular function subject to combinatorial
constraints which is easy in many cases [16, 15].
## 5 Theoretical results
In this section we analyze the computational and approximation bounds for this
problem. For simplicity we assume that the function $v$ is normalized, i.e
$v(\emptyset)=0$. Hence we assume that $v$ achieves it minima at a negative
value and correspondingly the approximation factor in this case will be less
than $1$.
We note in passing that the results in this section are mostly negative, in
that they demonstrate theoretically how complex a general problem such as
$\min_{X}[f(X)-g(X)]$ is, even for submodular $f$ and $g$. In this paper,
rather than consider these hardness results pessimistically, we think of them
as providing justification for the heuristic procedures given in Section 4 and
[32]. In many cases, inspired heuristics can yield good quality and hence
practically useful algorithms for real-world problems. For example, the ModMod
procedure (Algorithm 3) and even the SupSub procedure (Algorithm 2) can scale
to very large problem sizes, and thus can provide useful new strategies for
the applications listed in Section 1.
### 5.1 Hardness
Observe that the class of DS functions is essentially the class of general set
functions, and hence the problem of finding optimal solutions is NP-hard. This
is not surprising since general set function minimization is inapproximable
and there exist a large class of functions where all (adaptive, possibly
randomized) algorithms perform arbitrarily poorly in polynomial time [39].
Clearly as is evident from Theorem 3.1, even the problem of finding the
submodular functions $f$ and $g$ requires exponential complexity. We moreover
show in the following theorem, however, that this problem is multiplicatively
inapproximable even when the functions $f$ and $g$ are easy to find.
###### Theorem 5.1.
Unless P = NP, there cannot exist any polynomial time approximation algorithm
for $\min_{X}v(X)$ where $v(X)=[f(X)-g(X)]$ is a positive set function and $f$
and $g$ are given submodular functions. In particular, let $n$ be the size of
the problem instance, and $\alpha(n)>0$ be any positive polynomial time
computable function of $n$. If there exists a polynomial-time algorithm which
is guaranteed to find a set
$X^{\prime}:f(X^{\prime})-g(X^{\prime})<\alpha(n)\mbox{OPT}$, where
OPT=$\min_{X}f(X)-g(X)$, then P = NP.
###### Proof.
We prove this by reducing this to the subset sum problem. Given a positive
modular function $m$ and a positive constant $t$, is there a subset
$S\subseteq V$ such that $m(S)=t$? First we choose a random set $C$ (unknown
to the algorithm), and define $t=m(C)$. Define a set function $v$, such that
$v(S)=1$, if $m(S)=t$ and $v(S)=\frac{1}{\alpha(n)}-o(1)$ otherwise. Observe
that $\min_{S}v(S)=\frac{1}{\alpha(n)}-o(1)$, since $\alpha(n)>1$. Note that
$\alpha=\min_{X\subset Y\subseteq V\setminus j}v(j|X)-v(j|Y)\geq
2(\frac{1}{\alpha(n)}-1)$. Hence we can easily compute a lower bound on
$\alpha$ and hence from lemma 3.2 we can directly compute the decomposition
$f$ and $g$. In fact notice that the decomposition is directly computable
since both $\alpha$ and $\beta$ are known.
Now suppose there exists a polynomial time algorithm for this problem with an
approximation factor of $\alpha(n)$. This implies that the algorithm is
guaranteed to find a set $S$, such that $v(S)<1$. Hence this algorithm will
solve the subset sum problem in polynomial time, which is a contradiction
unless P = NP. ∎
In fact we show below that independent of the $P=NP$ question, there cannot
exist a sub-exponential time algorithm for this problem with any constant
factor approximation. The theorem below gives information theoretic hardness
for this problem.
###### Theorem 5.2.
For any $0<\epsilon<1$, there cannot exist any deterministic (or possibly
randomized) algorithm for $\min_{X}[f(X)-g(X)]$ (where $f$ and $g$ are given
submodular functions), that always finds a solution which is at most
$\frac{1}{\epsilon}$ times the optimal, in fewer than $e^{\epsilon^{2}n/8}$
queries.
###### Proof.
For showing this theorem, we use the same proof technique as in [9]. Define
two sets $C$ and $D$, such that $V=C\cup D$ and $|C|=|D|=n/2$. We then define
a set function $v(S)$ which depends only on $k=|S\cap C|$ and $l=|S\cap D|$.
In particular define $v(S)=\frac{1}{\epsilon},\mbox{ if }|k-l|\leq\epsilon n$
and $v(S)=1,\mbox{ if }|k-l|>\epsilon n$. Again, we have a trivial bound on
$\alpha$ here since $v(j|X)\geq\frac{1}{\epsilon}-1$ and $v(j|Y)\leq
1-\frac{1}{\epsilon}$. Hence, $\alpha=\min_{X\subset Y\subseteq V\setminus
j}v(j|X)-v(j|Y)>2|1-\frac{1}{\epsilon}|$. Thus, for this set function, a
decomposition $v=f-g$ can easily be obtained (Lemma 3.2).
Now, let the partition $(C,D)$ be taken uniformly at random and unknown to the
algorithm. The algorithm issues some queries $S$ to the value oracle. Call $S$
“unbalanced” if $|S\cap C|$ differs from $|S\cap D|$ by more than $\epsilon
n$. Recall the Chernoff bounds [1]: Let $Y_{1},Y_{2},\cdots,Y_{t}$ be
independent random variables in $[-1,1]$, such that $\mathbb{E}[Y_{i}]=0$,
then:
$Pr[\sum_{i=1}^{t}Y_{i}>\lambda]\leq 2e^{-\lambda^{2}/2t}.$ (6)
Define $Y_{i}=I(i\in S)[I(i\in C)-I(i\in D)]$. Clearly $Y_{i}\in[-1,1]$, and
we can use the bounds above. Hence for any query $S$, the probability that $S$
is unbalanced is at most $2e^{-\epsilon^{2}n/2}$. Thus, we can see that even
after $e^{\epsilon^{2}n/4}$ number of queries, the probability that the
resulting set is unbalanced is still $2e^{-\epsilon^{2}n/4}$. Hence any
algorithm will query only balanced sets regardless of $C$ and $D$, and
consequently with high probability the algorithm will obtain
$\frac{1}{\epsilon}$ as the minimum, while the actual minimum is $1$. Thus,
such an algorithm will never be able to achieve an approximation factor better
than $\frac{1}{\epsilon}$. ∎
Essentially the theorems above say that even when we are given (or can easily
find) a decomposition such that $v(X)=f(X)-g(X)$, there exist set functions
such that any algorithm (either adaptive or randomized) cannot be approximable
upto any constant factor. It is possible that one could come up with an
information theoretic construction to show this same result for any polynomial
approximation factor. However under the assumption of P$\neq$NP, Theorem 5.1
shows that this problem is inapproximable upto any polynomial factor. Hence
any algorithm trying to find the global optimum for this problem [5, 20] can
only be exponential in the worst case.
Interestingly, the hardness results above holds even when the submodular
functions $f$ and $g$ are monotone. This follows from the following Lemma:
###### Lemma 5.1.
Given (not necessarily monotone) submodular functions $f$ and $g$, there
exists monotone submodular functions $f^{\prime}$ and $g^{\prime}$ such that,
$\displaystyle f(X)-g(X)=f^{\prime}(X)-g^{\prime}(X),\forall X\subseteq V$ (7)
###### Proof.
The proof of this Lemma follows from a simple observation. The decomposition
theorem of [7] shows that any submodular function can be decomposed into a
modular function plus a monotone non-decreasing and totally normalized
polymatroid rank function. Specifically, given submodular $f,g$ we have
$\displaystyle f^{\prime}(X)\triangleq f(X)-\sum_{j\in X}f(j|V\backslash j)$
(8)
and
$\displaystyle g^{\prime}(X)\triangleq g(X)-\sum_{j\in X}g(j|V\backslash j)$
(9)
$f^{\prime},g^{\prime}$ are then totally normalized polymatroid rank
functions. Hence we have: $v(X)=f^{\prime}(X)-g^{\prime}(X)+k(X)$, with
modular $k(X)=\sum_{j\in X}v(j|V\backslash j)$. The idea is then to add $v(j)$
to $f^{\prime}$ if $v(j)\geq 0$ or add it to $g^{\prime}$ other-wise. In
particular, let $V^{+}=\\{j:v(j)\geq 0\\}$ and $V^{-}=\\{j:v(j)<0\\}$. Notice
that $V^{+}\cup V^{-}=V$. Then,
$\displaystyle v(X)=f^{\prime}(X)+k(X\cap V^{+})-\\{g^{\prime}-k(X\cap
V^{-})\\}$ (10)
Notice above that $f^{\prime}(X)+k(X\cap V^{+})$ and $g^{\prime}-k(X\cap
V^{-})$ are both monotone non-decreasing. Hence proved. ∎
This then implies the following corollary.
###### Corollary 5.3.
Given submodular functions $f$ and $g$ such that $v(X)=f(X)-g(X)\geq 0$, the
problem $\min_{X\subseteq V}v(X)$ is inapproximable, even if both $f$ and $g$
are monotone non-decreasing submodular.
### 5.2 Polynomial time lower and upper bounds
Since any submodular function can be decomposed into a modular function plus a
monotone non-decreasing and totally normalized polymatroid rank function [7],
we have: $v(X)=f^{\prime}(X)-g^{\prime}(X)+k(X)$, with modular
$k(X)=\sum_{j\in X}v(j|V\backslash j)$ and $f^{\prime}$ and $g^{\prime}$ being
the totally normalized polymatroid functions.
The algorithms in the previous sections are all based on repeatedly finding
upper bounds for $v$. The following lower bounds directly follow from the
results above.
###### Theorem 5.4.
We have the following two lower bounds on the minimizers of $v(X)=f(X)-g(X)$:
$\displaystyle\min_{X}v(X)$
$\displaystyle\geq\min_{X}f^{\prime}(X)+k(X)-g^{\prime}(V)$
$\displaystyle\min_{X}v(X)$ $\displaystyle\geq
f^{\prime}(\emptyset)-g^{\prime}(V)+\sum_{j\in V}\min(k(j),0)$
###### Proof.
Notice that
$\displaystyle\min_{X}f(X)-g(X)$
$\displaystyle=\min_{X}f^{\prime}(X)-g^{\prime}(X)+k(X)$
$\displaystyle\geq\min_{X}(f^{\prime}(X)+k(X))-\max_{X}g^{\prime}(X)$
$\displaystyle=\min_{X}f^{\prime}(X)+k(X)-g^{\prime}(V)$
To get the second result, we start from the bound above and loosen it as:
$\displaystyle\min_{X}f^{\prime}(X)+k(X)-g^{\prime}(V)$
$\displaystyle\geq\min_{X}f^{\prime}(X)+\min_{X}k(X)-g^{\prime}(V)$
$\displaystyle=f^{\prime}(\emptyset)+\sum_{j\in V}\min(v(j|V\backslash
j),0)-g^{\prime}(V)$ $\displaystyle=f^{\prime}(\emptyset)+\sum_{j\in
V}\min(k(j),0)-g^{\prime}(V)$ (11)
∎
The above lower bounds essentially provide bounds on the minima of the
objective and thus can be used to obtain an additive approximation guarantee.
The algorithms described in this paper are all polynomial time algorithms (as
we show below) and correspondingly from the bounds above we can get an
estimate on how far we are from the optimal.
### 5.3 Computational Bounds
We now provide computational bounds for $\epsilon$-approximate versions of our
algorithms. Note that this was left as an open question in [32]. Finding the
local minimizer of DS functions is PLS complete since it generalizes the
problem of finding the local optimum of the MAX-CUT problem [36]. Note that
this trivially generalizes the MAX-CUT problem since if we set $f(X)=0$ and
$g(X)$ is the cut function, we get the max cut problem. However we show that
an $\epsilon$-approximate version of this algorithm will converge in
polynomial time.
###### Definition 5.1.
An $\epsilon$-approximate version of an iterative monotone non-decreasing
algorithm for minimizing a set function $v$ is defined as a version of that
algorithm, where we proceed to step $t+1$ only if $v(X^{t+1})\leq
v(X^{t})(1+\epsilon)$.
Note that the $\epsilon$-approximate versions of algorithms 1, 2 and 3, are
guaranteed to converge to $\epsilon$-approximate local optima. An
$\epsilon$-approximate local optima of a function $v$ is a set $X$, such that
$v(X\cup j)\geq v(X)(1+\epsilon)$ and $v(X\backslash j)\geq v(X)(1+\epsilon)$.
W.l.o.g., assume that $X^{0}=\emptyset$. Then we have the following
computational bounds:
###### Theorem 5.5.
The $\epsilon$-approximate versions of algorithms 1, 2 and 3 have a worst case
complexity of $O(\frac{\log(|M|/|m|)}{\epsilon}T))$, where
$M=f^{\prime}(\emptyset)+\sum_{j\in V}\min(v(j|V\backslash
j),0)-g^{\prime}(V)$, $m=v(X^{1})$ and $O(T)$ is the complexity of every
iteration of the algorithm (which corresponds to respectively the submodular
minimization, maximization, or modular minimization in algorithms 1, 2 and
3)..
###### Proof.
Observe that $m=v(X^{1})\leq v(X^{0})=0$. Correspondingly if $v(X^{1})=0$, it
implies that the algorithm has converged, and cannot improve (since we are
assuming our algorithms are $\epsilon-$approximate. Hence in this case the
algorithm will converge in one iteration. Consider then the case of $m<0$.
Note also from Theorem 5.4 that $M=f^{\prime}(\emptyset)+\sum_{j\in
V}\min(v(j|V\backslash j),0)-g^{\prime}(V)<0$ and that $\min_{X}f(X)-g(X)\geq
M$. Since we are guaranteed to improve by a factor by at least $1+\epsilon$ at
every iteration we have that in $k$ iterations:
$|m|(1+\epsilon)^{k}\leq|M|\Rightarrow k=O(\frac{\log(|M|/|m|)}{\epsilon})$.
Also since we assume that the complexity at every iteration is $O(T)$ we get
the above result. ∎
Observe that for the algorithms we use, $O(T)$ is strongly polynomial in $n$.
The best strongly polynomial time algorithm for submodular function
minimization is $O(n^{5}\eta+n^{6})$ [34] (the lower bound is currently
unknown). Further the worst case complexity of the greedy algorithm for
maximization is $O(n^{2})$ while the complexity of modular minimization is
just $O(n)$. Note finally that these are worst case complexities and actually
the algorithms run much faster in practice.
(a) SVM
(b) NB
Figure 1: Plot showing the accuracy rates vs. the number of features on the
Mushroom data set.
## 6 Experiments
We test our algorithms on the feature subset selection problem in the
supervised setting. Given a set of features
$X_{V}=\\{X_{1},X_{2},\cdots,X_{|V|}\\}$, we try to find a subset of these
features $A$ which has the most information from the original set $X_{V}$
about a class variable $C$ under constraints on the size or cost of $A$.
Normally the number of features $|V|$ is quite large and thus the training and
testing time depend on $|V|$. In many cases, however, there is a strong
correlation amongst features and not every feature is novel. We can thus
perform training and testing with a much smaller number of features $|A|$
while obtaining (almost) the same error rates.
The question is how to find the most representative set of features $A$. The
mutual information between the chosen set of features and the target class
$C$, $I(X_{A};C)$, captures the relevance of the chosen subset of features. In
most cases the selected features are not independent given the class $C$ so
the naïve Bayes assumption is not applicable, meaning this is not a pure
submodular optimization problem. As mentioned in Section 1, $I(X_{A};C)$ can
be exactly expressed as a difference between submodular functions $H(X_{A})$
and $H(X_{A}|C)$.
### 6.1 Modular Cost Feature Selection
In this subsection, we look at the problem of maximizing
$I(X_{A};C)-\lambda|A|$, as a regularized feature subset selection problem.
Note that a mutual information $I(X_{A};C)$ query can easily be estimated from
the data by just a single sweep through this data. Further we have observed
that using techniques such as Laplace smoothing helps to improve mutual
information estimates without increasing computation. In these experiments,
therefore, we estimate the mutual information directly from the data and run
our algorithms to find the representative subset of features.
We compare our algorithms on two data sets, i.e., the Mushroom data set [14]
and the Adult data set [22] obtained from [10]. The Mushroom data set has 8124
examples with 112 features, while the Adult data set has 32,561 examples with
123 features. In our experiments we considered subsets of features of sizes
between 5%-20% of the total number of features by varying $\lambda$. We tested
the following algorithms for the feature subset selection problem. We
considered two formulations of the mutual information, one under naïve Bayes,
where the conditional entropy $H(X_{A}|C)$ can be written as
$H(X_{A}|C)=\sum_{j\in A}H(X_{i}|C)$ and another where we do not assume such
factorization. We call these two formulations factored and non-factored
respectively. We then considered the simple greedy algorithm, of iteratively
adding features at every step to the factored and non-factored mutual
information, which we call GrF and GrNF respectively. Lastly, we use the new
algorithms presented in this paper on the non-factored mutual information.
We then compare the results of the greedy algorithms with those of the three
algorithms for this problem, using two pattern classifiers based on either a
linear kernel SVM (using [6]) or a naïve Bayes (NB) classifier. We call the
results obtained from the supermodular-submodular heuristic as “SupSub”, the
submodular-supermodular procedure [32] as “SubSup”, and the modular-modular
objective as “ModMod.” In the SubSup procedure, we use the minimum norm point
algorithm [12] for submodular minimization, and in the SubSup procedure, we
use the optimal algorithm of [4] for submodular maximization. We observed that
the three heuristics generally outperformed the two greedy procedures, and
also that GRF can perform quite poorly, thus justifying our claim that the
naïve Bayes assumption can be quite poor. This also shows that although the
greedy algorithm in that case is optimal, the features are correlated given
the class and hence modeling it as a difference between submodular functions
gives the best results. We also observed that the SupSub and ModMod procedures
perform comparably to the SubSup procedure, while the SubSup procedure is
_much_ slower in practice. Comparing the running times, the ModMod and the
SupSub procedure are each a few times slower then the greedy algorithm (ModMod
is slower due computing the modular semigradients), while the SubSup procedure
is around 100 times slower. The SubSup procedure is slower due to general
submodular function minimization which can be quite slow.
The results for the Mushroom data set are shown in Figure 1. We performed a 10
fold cross-validation on the entire data set and observed that when using all
the features SVM gave an accuracy rate of 99.6% while the all-feature NB model
had an accuracy rate of 95.5%. The results for the Adult database are in
Figure 2. In this case with the entire set of features the accuracy rate of
SVM on this data set is 83.9% and NB is 82.3%.
(a) SVM
(b) NB
Figure 2: Plot showing the accuracy rates vs. the number of features on the
Adult data set.
In the mushroom data, the SVM classifier significantly outperforms the NB
classifier and correspondingly GrF performs much worse than the other
algorithms. Also, in most cases the three algorithms outperform GrNF. In the
adult data set, both the SVM and NB perform comparably although SVM
outperforms NB. However in this case also we observe that our algorithms
generally outperform GrF and GrNF.
### 6.2 Submodular cost feature selection
(a) SVM
(b) NB
Figure 3: Plot showing the accuracy rates vs. the cost of features for the
Mushroom data set
(a) SVM
(b) NB
Figure 4: Plot showing the accuracy rates vs. the cost of features for the
Adult data set
We perform synthetic experiments for the feature subset selection problem
under submodular costs. The cost model we consider is
$c(A)=\sum_{i}\sqrt{m(A\cap S_{i})}$. We partitioned $V$ into sets
$\\{S_{i}\\}_{i}$ and chose the modular function $m$ randomly. In this set of
experiments, we compare the accuracy of the classifiers vs. the cost
associated with the choice of features for the algorithms. Recall, with simple
(modular) cardinality costs the greedy algorithms performed decently in
comparison to our algorithms in the adult data set, where the NB assumption is
reasonable. However with submodular costs, the objective is no longer
submodular even under the NB assumption and thus the greedy algorithms perform
much worse. This is unsurprising since the greedy algorithm is approximately
optimal only for monotone submodular functions. This is even more strongly
evident from the results of the mushrooms data-set (Figure 3)
## 7 Discussion
We have introduced new algorithms for optimizing the difference between two
submodular functions, provided new theoretical understanding that provides
some justification for heuristics, have outlined applications that can make
use of our procedures, and have tested in the case of feature selection with
modular and submodular cost features. Our new ModMod procedure is fast at each
iteration and experimentally does about as well as the SupSub and SubSup
procedures. The ModMod procedure, moreover, can also be used under various
combinatorial constraints, and therefore the ModMod procedure may hold the
greatest promise as a practical heuristic. An alternative approach, not yet
evaluated, would be to try the convex-concave procedure [41] on the Lovász
extensions of $f$ and $g$ since subgradients in such case are so easy to
obtain.
Acknowledgments: We thank Andrew Guillory, Manas Joglekar, Stefanie Jegelka,
and the rest of the submodular group at UW for discussions. This material is
based upon work supported by the National Science Foundation under Grant No.
(IIS-1162606), and is also supported by a Google, a Microsoft, and an Intel
research award.
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|
arxiv-papers
| 2012-07-03T01:25:10 |
2024-09-04T02:49:32.538301
|
{
"license": "Public Domain",
"authors": "Rishabh Iyer and Jeff Bilmes",
"submitter": "Rishabh Iyer",
"url": "https://arxiv.org/abs/1207.0560"
}
|
1207.0574
|
# The algebro-geometric solutions for Hunter-Saxton hierarchy
Yu Hou Engui Fan111Corresponding author and e-mail address:
faneg@fudan.edu.cn Peng Zhao
( School of Mathematical Sciences, Institute of Mathematics
and Key Laboratory of Mathematics for Nonlinear Science,
Fudan University, Shanghai 200433, P.R. China)
###### Abstract
This paper is dedicated to provide theta function representation of algebro-
geometric solutions and related crucial quantities for the Hunter-Saxton (HS)
hierarchy through studying a algebro-geometric initial value problem. Our main
tools include the polynomial recursive formalism to derive the HS hierarchy,
the hyperelliptic curve with finite number of genus, the Baker-Akhiezer
functions, the meromorphic function, the Dubrovin-type equations for auxiliary
divisors, and the associated trace formulas. With the help of these tools, the
explicit representations of the Baker-Ahhiezer functions, the meromorphic
function, and the algebro-geometric solutions are obtained for the entire HS
hierarchy.
## 1 Introduction
The Hunter-Saxton (HS) equation
$u_{xxt}=-2uu_{xxx}-4u_{x}u_{xx},$ (1.1)
where $u(x,t)$ is the function of spatial variable $x$ and time variable $t$.
It arises in two different physical contexts in two nonequivalent variational
forms [1], [2]. The first is shown to describe the propagation of weakly
nonlinear orientation waves in a massive nematic liquid crystal director field
[1]-[3]. The second is shown to describe the high frequency limit of the
Camassa-Holm (CH) equation [5], [6], [32]
$u_{t}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}$ (1.2)
which was originally introduced in [5], [6] as model equation for shallow
water waves, and obtained independently in [31] with a bi-hamiltonian
structure.
The HS equation is a completely integrable system with a bi-hamiltonian
structure and hence it possesses a Lax pair, an infinite family of commuting
Hamiltonian flows, as well as an associated sequence of conservation laws,
(Hunter and Zheng [2], Reyes [20]). The inverse scattering solutions have been
obtained by Beals, Sattinger and Szmigielski [19]. Recently, Lenells [23],
[24] and also Khesin and Misiołek [22] pointed out that it describes the
geodesic flow on the homogeneous space related to the Virasoro group. Bressan
and Constantin [25], also Holden [26] constructed a continuous semigroup of
weak, dissipative solutions. Yin [27] proved the local existence of strong
solutions of the periodic HS equation and showed that all strong solutions-
except space independent solutions-blow up in finite time. Gui, Liu and Zhu
[28] studied the wave-breaking phenomena and global existence. Furthermore,
Morozov [29], Sakovich [30] and Reyes [20], [21] investigated (1.1) from a
geometric perspective. However, within the knowledge of the authors, the
algebro-geometric solutions of the entire HS hierarchy are not studied yet.
The main task of this paper focuses on the algebro-geometric solutions of the
whole HS hierarchy in which (1.1) is just the first member. Algebro-geometric
solution, an important feature of integrable system, is a kind of explicit
solutions closely related to the inverse spectral theory [4], [7], [9]-[11].
As a degenerated case of the algebro-geometric solution, the multi-soliton
solution and periodic solution in elliptic function type may be obtained [7],
[8], [33]. A systematic approach, proposed by Gesztesy and Holden to construct
algebro-geometric solutions for integrable equations, has been extended to the
whole (1+1) dimensional integrable hierarchy, such as the AKNS hierarchy, the
CH hierarchy etc. [12]-[15]. Recently, we investigated algebro-geometric
solutions for the Gerdjikov-Ivanov hierarchy, the Degasperis-Procesi hierarchy
and the modified Camassa-Holm hierarchy [16]-[18].
The outline of the present paper is as follows.
In section 2, based on the polynomial recursion formalism, we derive the HS
hierarchy, the associated sequences, and Lax pairs. A hyperelliptic curve
$\mathcal{K}_{n}$ with arithmetic genus $n$ is introduced with the help of the
characteristic polynomial of Lax matrix $V_{n}$ for the stationary HS
hierarchy.
In Section 3, we study a meromorphic function $\phi$ such that $\phi$
satisfies a nonlinear second-order differential equation. Then we study the
properties of the Baker-Akhiezer function $\psi$, and furthermore the
stationary HS equations are decomposed into a system of Dubrovin-type
equations. The stationary trace formulas are obtained for the HS hierarchy.
In Section 4, we present the first set of our results, the explicit theta
function representations of Baker-Akhiezer function, the meromorphic function
and the potentials $u$ for the entire stationary HS hierarchy. Furthermore, we
study the initial value problem on an algebro-geometric curve for the
stationary HS hierarchy.
In Sections 5 and 6, we extend the analysis in Sections 3 and 4 to the time-
dependent case. Each equation in the HS hierarchy is permitted to evolve in
terms of an independent time parameter $t_{r}$. As an initial data we use a
stationary solution of the $n$th equation and then construct a time-dependent
solution of the $r$th equation in the HS hierarchy. The Baker-Akhiezer
function, the meromorphic function, the analogs of the Dubrovin-type
equations, the trace formulas, and the theta function representation in
Section 4 are all extended to the time-dependent case.
## 2 The HS hierarchy
In this section, we derive the HS hierarchy and the corresponding sequence of
zero-curvature pairs by using a polynomial recursion formalism. Moreover, we
introduce the hyperelliptic curve connecting to the stationary HS hierarchy.
Throughout this section, let us we make the following hypothesis.
###### Hypothesis 2.1
In the stationary case we assume that
$\begin{split}&u\in C^{\infty}(\mathbb{R}),\ \ \partial_{x}^{k}u\in
L^{\infty}(\mathbb{R}),\qquad k\in\mathbb{N}_{0}.\end{split}$ (2.1)
In the time-dependent case we suppose
$\begin{split}&u(\cdot,t)\in C^{\infty}(\mathbb{R}),\ \
\partial_{x}^{k}u(\cdot,t)\in L^{\infty}(\mathbb{R}),\quad
k\in\mathbb{N}_{0},~{}~{}t\in\mathbb{R},\\\ &u(x,\cdot),u_{xx}(x,\cdot)\in
C^{1}(\mathbb{R}),\quad x\in\mathbb{R}.\\\ \end{split}$ (2.2)
We start by the polynomial recursion formalism. Define
$\\{f_{l}\\}_{l\in\mathbb{N}_{0}}$, $\\{g_{l}\\}_{l\in\mathbb{N}_{0}}$ and
$\\{h_{l}\\}_{l\in\mathbb{N}_{0}}$ recursively by
$\begin{split}&f_{0}=1,\\\
&f_{l+1,x}=\mathcal{G}(-4u_{xx}f_{l,x}-2u_{xxx}f_{l}),\quad
l\in\mathbb{N}_{0},\\\ &g_{l}=\frac{1}{2}f_{l+1,x},\quad
l\in\mathbb{N}_{0},\\\ &h_{l}=-g_{l+1,x}-u_{xx}f_{l+1},\quad
l\in\mathbb{N}_{0},\\\ \end{split}$ (2.3)
where $\mathcal{G}$ is given by
$\begin{split}&\mathcal{G}:L^{\infty}(\mathbb{R})\rightarrow
L^{\infty}(\mathbb{R}),\\\
&(\mathcal{G}v)(x)=\int_{-\infty}^{x}\int_{-\infty}^{\tau}v(y)~{}dyd\tau,\quad
x\in\mathbb{R},~{}v\in L^{\infty}(\mathbb{R}).\end{split}$ (2.4)
It is easy to see that $\mathcal{G}$ is the resolvent of the one-dimensional
Laplacian operator, that is
$\mathcal{G}=\Big{(}\frac{d^{2}}{dx^{2}}\Big{)}^{-1}.$ (2.5)
Explicitly, one computes
$\begin{split}&f_{0}=1,\\\ &f_{1}=-2u+c_{1},\\\
&f_{2}=\mathcal{G}(4uu_{xx}+2u_{x}^{2})-c_{1}2u+c_{2},\\\ &g_{0}=-u_{x},\\\
&g_{1}=\frac{1}{2}\mathcal{G}(8u_{x}u_{xx}+4uu_{xxx})-c_{1}u_{x},\\\
&h_{0}=-\frac{1}{2}f_{2,xx}-u_{xx}f_{1},\end{split}$ (2.6)
where $\\{c_{l}\\}_{l\in\mathbb{N}_{0}}\subset\mathbb{C}$ are integration
constants and we have used the assumption
$f_{l}(u)|_{u=0}=c_{l},~{}~{}g_{l}(u)|_{u=0}=c_{l},~{}~{}h_{l}(u)|_{u=0}=c_{l},~{}~{}l\in\mathbb{N}.$
(2.7)
Next we introduce the corresponding homogeneous coefficients
$\hat{f}_{l},\hat{g}_{l},$ and $\hat{h}_{l},$ defined through taking $c_{k}=0$
for $k=1,\cdots,l,$
$\begin{split}&\hat{f}_{0}=f_{0}=1,\quad\quad\hat{f}_{l}=f_{l}|_{c_{k}=0,~{}k=1,\ldots,l},\quad
l\in\mathbb{N},\\\
&\hat{g}_{0}=g_{0}=-u_{x},\quad\hat{g}_{l}=g_{l}|_{c_{k}=0,~{}k=1,\ldots,l},\quad
l\in\mathbb{N}.\\\
&\hat{h}_{0}=h_{0},\quad\quad\hat{h}_{l}=h_{l}|_{c_{k}=0,~{}k=1,\ldots,l},\quad
l\in\mathbb{N}.\end{split}$ (2.8)
Hence one can easily conclude that
$f_{l}=\sum_{k=0}^{l}c_{l-k}\hat{f}_{k},\quad
g_{l}=\sum_{k=0}^{l}c_{l-k}\hat{g}_{k},\quad
h_{l}=\sum_{k=0}^{l}c_{l-k}\hat{h}_{k},\quad l\in\mathbb{N}_{0},$ (2.9)
with
$c_{0}=1.$ (2.10)
Now we consider the following $2\times 2$ matrix isospectral problem
$\psi_{x}=U(u,z)\psi=\left(\begin{array}[]{cc}0&1\\\ -z^{-1}u_{xx}&0\\\
\end{array}\right)\psi$ (2.11)
and an auxiliary problem
$\psi_{t_{n}}=V_{n}(z)\psi,$ (2.12)
where $V_{n}(z)$ is defined by
$V_{n}(z)=\left(\begin{array}[]{cc}-G_{n}(z)&F_{n+1}(z)\\\
z^{-1}H_{n}(z)&G_{n}(z)\\\ \end{array}\right)\qquad
z\in\mathbb{C}\setminus\\{0\\},\quad n\in\mathbb{N}_{0},$ (2.13)
assuming $F_{n+1}$, $G_{n}$ and $H_{n}$ to be polynomials of degree $n$ with
$C^{\infty}$ coefficients with respect to $x$. The compatibility condition
between (2.11) and (2.12) yields the stationary zero-curvature equation
$-V_{n,x}+[U,V_{n}]=0,$ (2.14)
that is
$\displaystyle F_{n+1,x}$ $\displaystyle=$ $\displaystyle 2G_{n},$ (2.15)
$\displaystyle H_{n,x}$ $\displaystyle=$ $\displaystyle 2u_{xx}G_{n},$ (2.16)
$\displaystyle zG_{n,x}$ $\displaystyle=$ $\displaystyle-H_{n}-u_{xx}F_{n+1}.$
(2.17)
From (2.15)-(2.17), a direct calculation shows that
$\frac{d}{dx}\mathrm{det}(V_{n}(z,x))=-\frac{1}{z^{2}}\frac{d}{dx}\Big{(}z^{2}G_{n}(z,x)^{2}+zF_{n+1}(z,x)H_{n}(z,x)\Big{)}=0$
(2.18)
and hence $z^{2}G_{n}^{2}+zF_{n+1}H_{n}$ is $x$-independent implying
$z^{2}G_{n}^{2}+zF_{n+1}H_{n}=R_{2n+2},$ (2.19)
where the integration constant $R_{2n+2}$ is a polynomial of degree $2n+2$
with respect to $z$. If $\\{E_{m}\\}_{m=0,\cdots,2n+1}$ denote its zeros, then
$R_{2n+2}(z)=(u_{x}^{2}+h_{0})\prod_{m=0}^{2n+1}(z-E_{m}),\quad
E_{0}=0,~{}\\{E_{m}\\}_{m=1,\cdots,2n+1}\in\mathbb{C}.$ (2.20)
Here we must emphasize that the coefficient $(u_{x}^{2}+h_{0})$ is a constant.
In fact, (2.18) equals
$2z^{2}G_{n}G_{n,x}+zF_{n+1}H_{n,x}+zH_{n}F_{n+1,x}=0.$ (2.21)
Comparing the coefficients of powers $z^{2n+2}$ yields
$2g_{0}g_{0,x}+f_{0}h_{0,x}+h_{0}f_{0,x}=0,$ (2.22)
which together with (2.6) we obtain
$2u_{x}u_{xx}+h_{0,x}=0.$ (2.23)
Hence
$u_{x}^{2}+h_{0}=\partial^{-1}(2u_{x}u_{xx}+h_{0,x})=\mathrm{Constant}.$
(2.24)
For simplicity, we denote it by $a^{2}$. Then $R_{2n+2}(z)$ can be rewritten
as
$R_{2n+2}(z)=a^{2}\prod_{m=0}^{2n+1}(z-E_{m}),\quad
E_{0}=0,~{}\\{E_{m}\\}_{m=1,\cdots,2n+1}\in\mathbb{C}.$ (2.25)
In order to derive the corresponding hyperelliptic curve, we compute the
characteristic polynomial $\mathrm{det}(yI-zV_{n})$ of Lax matrix $zV_{n}$,
$\displaystyle\mathrm{det}(yI-zV_{n})$ $\displaystyle=$ $\displaystyle
y^{2}-z^{2}G_{n}(z)^{2}-F_{n+1}(z)H_{n}(z)$ (2.26) $\displaystyle=$
$\displaystyle y^{2}-R_{2n+2}(z)=0.$
Equation (2.26) naturally leads to the hyperelliptic curve $\mathcal{K}_{n}$,
where
$\mathcal{K}_{n}:\mathcal{F}_{n}(z,y)=y^{2}-R_{2n+2}(z)=0.$ (2.27)
The stationary zero-curvature equation (2.14) implies polynomial recursion
relations (2.3). Introducing the following polynomial $F_{n+1}(z),G_{n}(z)$
and $H_{n}(z)$ with respect to the spectral parameter $z$,
$F_{n+1}(z)=\sum_{l=0}^{n+1}f_{l}z^{n+1-l},$ (2.28)
$G_{n}(z)=\sum_{l=0}^{n}g_{l}z^{n-l},$ (2.29)
$H_{n}(z)=\sum_{l=0}^{n}h_{l}z^{n-l}.$ (2.30)
Inserting (2.28)-(2.30) into (2.15)-(2.17) then yields the recursions
relations (2.3) for $f_{l},$ $l=0,\ldots,n+1,$ and $g_{l}$, $l=0,\ldots,n.$
For fixed $n\in\mathbb{N}_{0}$, by using (2.17), we obtain the recursion for
$h_{l},$ $l=0,\ldots,n-1$ in (2.3) and
$h_{n}=-u_{xx}f_{n+1}.$ (2.31)
Moreover, from (2.16), one infers that
$h_{n,x}-2u_{xx}g_{n}=0,\qquad n\in\mathbb{N}_{0}.$ (2.32)
Hence, insertion of the equation (2.31) and
$f_{n+1,x}-2g_{n}=0$ (2.33)
into (2.32), we derive the stationary HS hierarchy,
$\textrm{s-HS}_{n}(u)=u_{xxx}f_{n+1}(u)+2u_{xx}f_{n+1,x}(u)=0,\quad
n\in\mathbb{N}_{0}.$ (2.34)
Explicitly, the first few equations are as follows
$\begin{split}&\textrm{s-HS}_{0}(u)=-2uu_{xxx}-4u_{x}u_{xx}+c_{1}u_{xxx}=0,\\\
&\textrm{s-HS}_{1}(u)=u_{xxx}\mathcal{G}(4uu_{xx}+2u_{x}^{2})+2u_{xx}\mathcal{G}(8u_{x}u_{xx}+4uu_{xxx})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c_{1}(-2uu_{xxx}-4u_{x}u_{xx})+c_{2}u_{xxx}=0,\\\
&\mathrm{etc}.\end{split}$ (2.35)
By definition, the set of solutions of (2.34) represents the class of algebro-
geometric HS solutions, with $n$ ranging in $\mathbb{N}_{0}$ and $c_{l}$ in
$\mathbb{C},~{}l\in\mathbb{N}$. We call the stationary algebro-geometric HS
solutions $u$ as HS potentials at times.
###### Remark 2.2
Here we emphasize that if $u$ satisfies one of the stationary HS equations in
$(\ref{2.34})$, then it must satisfy infinitely many such equations of order
higher than $n$ for certain choices of integration constants $c_{l}$, this is
a common characteristic of the general integrable soliton equations such as
the KdV, AKNS and CH hierarchies [15].
Next, we introduce the corresponding homogeneous polynomials
$\widehat{F}_{l+1},\widehat{G}_{l},\widehat{H}_{l}$ defined by
$\displaystyle\widehat{F}_{l+1}(z)=F_{l+1}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l+1}\hat{f}_{k}z^{l+1-k},~{}~{}l=0,\ldots,n,$
(2.36)
$\displaystyle\widehat{G}_{l}(z)=G_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{g}_{k}z^{l-k},~{}~{}l=0,\ldots,n,$
(2.37)
$\displaystyle\widehat{H}_{l}(z)=H_{l}(z)|_{c_{k}=0,~{}k=1,\dots,l}=\sum_{k=0}^{l}\hat{h}_{k}z^{l-k},~{}~{}l=0,\ldots,n-1,$
(2.38)
$\displaystyle\widehat{H}_{n}(z)=-u_{xx}\hat{f}_{n+1}+\sum_{k=0}^{n-1}\hat{h}_{k}z^{n-k}.$
(2.39)
Then the corresponding homogeneous formalism of (2.34) are given by
$\textrm{s-}\widehat{\mathrm{HS}}_{n}(u)=\textrm{s-HS}_{n}(u)|_{c_{l}=0,~{}l=1,\dots,n}=0,\qquad
n\in\mathbb{N}_{0}.$ (2.40)
We will end this section by introducing the time-dependent HS hierarchy. This
means that $u$ are now considered as functions of both space and time. We
introduce a deformation parameter $t_{n}\in\mathbb{R}$ in $u$, replacing
$u(x)$ by $u(x,t_{n})$, for each equation in the hierarchy. In addition, we
note that the definitions (2.11), (2.13) and (2.28)-(2.30) of $U,$ $V_{n}$ and
$F_{n+1},G_{n}$ and $H_{n}$ are still apply. Then the compatibility condition
yields the zero-curvature equation
$U_{t_{n}}-V_{n,x}+[U,V_{n}]=0,\qquad n\in\mathbb{N}_{0},$ (2.41)
namely
$\displaystyle-u_{xxt_{n}}-H_{n,x}+2u_{xx}G_{n}=0,$ (2.42) $\displaystyle
F_{n+1,x}=2G_{n},$ (2.43) $\displaystyle zG_{n,x}=-H_{n}-u_{xx}F_{n+1}.$
(2.44)
For fixed $n\in\mathbb{N}$, insertion of the polynomial expressions for
$F_{n+1}$, $G_{n}$ and $H_{n}$ into (2.42)-(2.44), respectively, then we
derive the relations (2.3) for $f_{l}|_{l=0,\ldots,n+1}$,
$g_{l}|_{l=0,\ldots,n}$. $h_{l}|_{l=0,\ldots,n-1}$ and
$h_{n}=-u_{xx}f_{n+1}.$ (2.45)
Moreover, from (2.42), we infer that
$-u_{xxt_{n}}-h_{n,x}+2u_{xx}g_{n}=0,\qquad n\in\mathbb{N}_{0}.$ (2.46)
Hence, together (2.45) and
$f_{n+1,x}=2g_{n},$ (2.47)
(2.46) admits the time-dependent HS hierarchy,
$\displaystyle\mathrm{HS}_{n}(u)=-u_{xxt_{n}}+u_{xxx}f_{n+1}(u)+2u_{xx}f_{n+1,x}(u)=0,$
$\displaystyle\quad(x,t_{n})\in\mathbb{R}^{2},~{}n\in\mathbb{N}_{0}.$ (2.48)
Explicitly, the first few equations are as follows
$\begin{split}&\mathrm{HS}_{0}(u)=-u_{xxt_{0}}-2uu_{xxx}-4u_{xx}u_{x}+c_{1}u_{xxx}=0,\\\
&\mathrm{HS}_{1}(u)=-u_{xxt_{1}}+u_{xxx}\mathcal{G}(4uu_{xx}+2u_{x}^{2})+2u_{xx}\mathcal{G}(8u_{x}u_{xx}+4uu_{xxx})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+c_{1}(-2uu_{xxx}-4u_{x}u_{xx})+c_{2}u_{xxx}=0,\\\
&\mathrm{etc}.\end{split}$ (2.49)
The first equation $\mathrm{HS}_{0}(u)=0$ (with $c_{1}=0$) in the hierarchy
represents the Hunter-Saxton equation discussed in section 1\. Similarly, one
can introduce the corresponding homogeneous HS hierarchy by
$\widehat{\mathrm{HS}}_{n}(u)=\mathrm{HS}_{n}(u)|_{c_{l}=0,~{}l=1,\ldots,n}=0,\qquad
n\in\mathbb{N}_{0}.$ (2.50)
In fact, since the Lenard recursion formalism is almost universally adopted in
the contemporary literature on the integrable soliton equations, it might be
worthwhile to adopt Gesztesy method, an alternative approach using the
polynomial recursion relations.
## 3 The stationary HS formalism
In this section we focus our attention on the stationary case. By using the
polynomial recursion formalism described in section 2, we define a fundamental
meromorphic function $\phi(P,x)$ on a hyperelliptic curve $\mathcal{K}_{n}$.
Moreover, we study the properties of the Baker-Akhiezer function
$\psi(P,x,x_{0})$, Dubrovin-type equations and trace formulas.
We emphasize that the analysis about the stationary case described in section
2 also holds here for the present context.
The hyperelliptic curve $\mathcal{K}_{n}$
$\begin{split}&\mathcal{K}_{n}:\mathcal{F}_{n}(z,y)=y^{2}-R_{2n+2}(z)=0,\\\
&R_{2n+2}(z)=a^{2}\prod_{m=0}^{2n+1}(z-E_{m}),\quad
E_{0}=0,~{}\\{E_{m}\\}_{m=1,\ldots,2n+1}\in\mathbb{C},\end{split}$ (3.1)
which is compactified by joining two points at infinity, $P_{\infty_{\pm}}$,
$P_{\infty_{+}}\neq P_{\infty_{-}}$, but for notational simplicity the
compactification is also denoted by $\mathcal{K}_{n}$. Points $P$ on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}$
are represented as pairs $P=(z,y(P))$, where $y(\cdot)$ is the meromorphic
function on $\mathcal{K}_{n}$ satisfying
$\mathcal{F}_{n}(z,y(P))=0.$
The complex structure on $\mathcal{K}_{n}$ is defined in the usual way by
introducing local coordinates
$\zeta_{Q_{0}}:P\rightarrow(z-z_{0})$
near points $Q_{0}=(z_{0},y(Q_{0}))\in\mathcal{K}_{n}\setminus
P_{0},~{}P_{0}=(0,0),$ which are neither branch nor singular points of
$\mathcal{K}_{n}$; near $P_{0}$, the local coordinates are
$\zeta_{P_{0}}:P\rightarrow z^{1/2};$
near the points $P_{\infty_{\pm}}\in\mathcal{K}_{n}$, the local coordinates
are
$\zeta_{P_{\infty_{\pm}}}:P\rightarrow z^{-1},$
and similarly at branch and singular points of $\mathcal{K}_{n}.$ Hence,
$\mathcal{K}_{n}$ becomes a two-sheeted hyperelliptic Riemann surface of genus
$n\in\mathbb{N}_{0}$ (possibly with a singular affine part) in a standard
manner.
We also notice that fixing the zeros $E_{0}=0,$ $E_{1},\ldots,E_{2n+1}$ of
$R_{2n+2}$ discussed in (3.1) leads to the curve $\mathcal{K}_{n}$ is fixed.
Then the integration constants $c_{1},\ldots,c_{n}$ in $f_{n}$ are uniquely
determined, which is the symmetric functions of $E_{1},\ldots,E_{2n+1}$.
The holomorphic map $\ast,$ changing sheets, is defined by
$\displaystyle\ast:\begin{cases}\mathcal{K}_{n}\rightarrow\mathcal{K}_{n},\\\
P=(z,y_{j}(z))\rightarrow P^{\ast}=(z,y_{j+1(\mathrm{mod}~{}2)}(z)),\quad
j=0,1,\end{cases}$ $\displaystyle
P^{\ast\ast}:=(P^{\ast})^{\ast},\quad\mathrm{etc}.,$ (3.2)
where $y_{j}(z),\,j=0,1$ denote the two branches of $y(P)$ satisfying
$\mathcal{F}_{n}(z,y)=0$.
Finally, positive divisors on $\mathcal{K}_{n}$ of degree $n$ are denoted by
$\mathcal{D}_{P_{1},\ldots,P_{n}}:\begin{cases}\mathcal{K}_{n}\rightarrow\mathbb{N}_{0},\\\
P\rightarrow\mathcal{D}_{P_{1},\ldots,P_{n}}=\begin{cases}\textrm{ $k$ if $P$
occurs $k$ times in $\\{P_{1},\ldots,P_{n}\\},$}\\\ \textrm{ $0$ if
$P\notin$$\\{P_{1},\ldots,P_{n}\\}.$}\end{cases}\end{cases}$ (3.3)
Next, we define the stationary Baker-Akhiezer function $\psi(P,x,x_{0})$ on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\}$ by
$\begin{split}&\psi(P,x,x_{0})=\left(\begin{array}[]{c}\psi_{1}(P,x,x_{0})\\\
\psi_{2}(P,x,x_{0})\\\ \end{array}\right),\\\
&\psi_{x}(P,x,x_{0})=U(u(x),z(P))\psi(P,x,x_{0}),\\\
&zV_{n}(u(x),z(P))\psi(P,x,x_{0})=y(P)\psi(P,x,x_{0}),\\\
&\psi_{1}(P,x_{0},x_{0})=1;\\\
&P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},~{}(x,x_{0})\in\mathbb{R}^{2}.\end{split}$
(3.4)
Closely related to $\psi(P,x,x_{0})$ is the following meromorphic function
$\phi(P,x)$ on $\mathcal{K}_{n}$ defined by
$\phi(P,x)=z\frac{\psi_{1,x}(P,x,x_{0})}{\psi_{1}(P,x,x_{0})},\quad
P\in\mathcal{K}_{n},~{}x\in\mathbb{R}$ (3.5)
with
$\psi_{1}(P,x,x_{0})=\mathrm{exp}\left(z^{-1}\int_{x_{0}}^{x}\phi(P,x^{\prime})~{}dx^{\prime}\right),\quad
P\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\}.$ (3.6)
Then, based on (3.4) and (3.5), a direct calculation shows that
$\displaystyle\phi(P,x)$ $\displaystyle=$
$\displaystyle\frac{y+zG_{n}(z,x)}{F_{n+1}(z,x)}$ (3.7) $\displaystyle=$
$\displaystyle\frac{zH_{n}(z,x)}{y-zG_{n}(z,x)},$
and
$\psi_{2}(P,x,x_{0})=\psi_{1}(P,x,x_{0})\phi(P,x)/z.$ (3.8)
We note that $F_{n+1}$ and $H_{n}$ are polynomials with respect to $z$ of
degree $n+1$ and $n$, respectively. Hence we may write
$F_{n+1}(z)=\prod_{j=0}^{n}(z-\mu_{j}),\quad
H_{n}(z)=h_{0}\prod_{l=1}^{n}(z-\nu_{l}).$ (3.9)
Moreover, defining
$\hat{\mu}_{j}(x)=(\mu_{j}(x),-\mu_{j}(x)G_{n}(\mu_{j}(x),x))\in\mathcal{K}_{n},~{}j=0,\ldots,n,~{}x\in\mathbb{R},$
(3.10)
and
$\hat{\nu}_{l}(x)=(\nu_{l}(x),\nu_{l}(x)G_{n}(\nu_{l}(x),x))\in\mathcal{K}_{n},~{}l=1,\ldots,n,~{}x\in\mathbb{R}.$
(3.11)
Due to assumption (2.1), $u$ is smooth and bounded, and hence $F_{n+1}(z,x)$
and $H_{n}(z,x)$ share the same property. Thus, we infers that
$\mu_{j},\nu_{l}\in C(\mathbb{R}),\quad j=0,\dots,n,~{}l=1,\ldots,n.$ (3.12)
here $\mu_{j},\nu_{l}$ may have appropriate multiplicities.
The branch of $y(\cdot)$ near $P_{\infty_{\pm}}$ is fixed according to
$\underset{|z(P)|\rightarrow\infty\atop P\rightarrow
P_{\infty_{\pm}}}{\mathrm{lim}}\frac{y(P)}{z(P)G_{n}(z(P),x)}=\mp 1.$ (3.13)
Also by (3.7), the divisor $(\phi(P,x))$ of $\phi(P,x)$ is given by
$(\phi(P,x))=\mathcal{D}_{P_{0}\underline{\hat{\nu}}(x)}(P)-\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}(P).$
(3.14)
That means, $P_{0},\hat{\nu}_{1}(x),\ldots,\hat{\nu}_{n}(x)$ are the $n+1$
zeros of $\phi(P,x)$ and $\hat{\mu}_{0}(x),\hat{\mu}_{1}(x),\ldots,$
$\hat{\mu}_{n}(x)$ are its $n+1$ poles. These zeros and poles can be
abbreviated in the following form
$\underline{\hat{\mu}}=\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\},\quad\underline{\hat{\nu}}=\\{\hat{\nu}_{1},\ldots,\hat{\nu}_{n}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}).$
(3.15)
Let us recall the holomorphic map (3),
$\displaystyle\ast:\begin{cases}\mathcal{K}_{n}\rightarrow\mathcal{K}_{n},\\\
P=(z,y_{j}(z))\rightarrow P^{\ast}=(z,y_{j+1(\mathrm{mod}~{}2)}(z)),\quad
j=0,1,\end{cases}$ $\displaystyle
P^{\ast\ast}:=(P^{\ast})^{\ast},\quad\mathrm{etc}.,$ (3.16)
where $y_{j}(z),\,j=0,1$ satisfy $\mathcal{F}_{n}(z,y)=0$, namely
$(y-y_{0}(z))(y-y_{1}(z))=y^{2}-R_{2n+2}(z)=0.$ (3.17)
Hence from (3.17), we can easily get
$\begin{split}&y_{0}+y_{1}=0,\\\ &y_{0}y_{1}=-R_{2n+2}(z),\\\
&y_{0}^{2}+y_{1}^{2}=2R_{2n+2}(z).\\\ \end{split}$ (3.18)
Further properties of $\phi(P,x)$ are summarized as follows.
###### Lemma 3.1
Under the assumption $(\ref{2.1})$, let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},$
and $x\in\mathbb{R}$, and $u$ satisfies the $n$th stationary HS equation
$(\ref{2.34})$. Then
$\phi_{x}(P)+z^{-1}\phi(P)^{2}=-u_{xx},$ (3.19)
$\phi(P)\phi(P^{\ast})=-\frac{zH_{n}(z)}{F_{n+1}(z)},$ (3.20)
$\phi(P)+\phi(P^{\ast})=\frac{2zG_{n}(z)}{F_{n+1}(z)},$ (3.21)
$\phi(P)-\phi(P^{\ast})=\frac{2y}{F_{n+1}(z)}.$ (3.22)
Proof. A direct calculation shows that (3.19) holds. Let us now prove
(3.20)-(3.22). Without loss of generality, let $y_{0}(P)=y(P)$. From (3.7),
(2.19) and (3.18), we arrive at
$\displaystyle\phi(P)\phi(P^{\ast})$ $\displaystyle=$
$\displaystyle\frac{y_{0}+zG_{n}}{F_{n+1}}~{}\times~{}\frac{y_{1}+zG_{n}}{F_{n+1}}$
(3.23) $\displaystyle=$
$\displaystyle\frac{y_{0}y_{1}+(y_{0}+y_{1})zG_{n}+z^{2}G_{n}^{2}}{F_{n+1}^{2}}$
$\displaystyle=$
$\displaystyle\frac{-R_{2n+2}+z^{2}G_{n}^{2}}{F_{n+1}^{2}}=z\frac{-F_{n+1}H_{n}}{F_{n+1}^{2}}$
$\displaystyle=$ $\displaystyle-\frac{zH_{n}}{F_{n+1}},$
$\displaystyle\phi(P)+\phi(P^{\ast})$ $\displaystyle=$
$\displaystyle\frac{y_{0}+zG_{n}}{F_{n+1}}~{}+~{}\frac{y_{1}+zG_{n}}{F_{n+1}}$
(3.24) $\displaystyle=$
$\displaystyle\frac{(y_{0}+y_{1})+2zG_{n}}{F_{n+1}}=\frac{2zG_{n}}{F_{n+1}},$
$\displaystyle\phi(P)-\phi(P^{\ast})$ $\displaystyle=$
$\displaystyle\frac{y_{0}+zG_{n}}{F_{n+1}}~{}-~{}\frac{y_{1}+zG_{n}}{F_{n+1}}$
(3.25) $\displaystyle=$
$\displaystyle\frac{(y_{0}-y_{1})}{F_{n+1}}=\frac{2y_{0}}{F_{n+1}}=\frac{2y}{F_{n+1}}.$
Hence we complete the proof. $\square$
Let us detail the properties of $\psi(P,x,x_{0})$ below.
###### Lemma 3.2
Under the assumption $(\ref{2.1})$, let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},$
$(x,x_{0})\in\mathbb{R}^{2}$, and $u$ satisfies the $n$th stationary HS
equation $(\ref{2.34})$. Then
$\psi_{1}(P,x,x_{0})=\Big{(}\frac{F_{n+1}(z,x)}{F_{n+1}(z,x_{0})}\Big{)}^{1/2}\mathrm{exp}\Big{(}\frac{y}{z}\int_{x_{0}}^{x}F_{n+1}(z,x^{\prime})^{-1}dx^{\prime}\Big{)},$
(3.26)
$\psi_{1}(P,x,x_{0})\psi_{1}(P^{\ast},x,x_{0})=\frac{F_{n+1}(z,x)}{F_{n+1}(z,x_{0})},$
(3.27)
$\psi_{2}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})=-\frac{H_{n}(z,x)}{zF_{n+1}(z,x_{0})},$
(3.28)
$\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})+\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})=2\frac{G_{n}(z,x)}{F_{n+1}(z,x_{0})},$
(3.29)
$\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})-\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})=\frac{-2y}{zF_{n+1}(z,x_{0})}.$
(3.30)
Proof. Equation (3.26) can be proven through the following procedure. Using
(2.15), the expression of $\psi_{1}$, (3.6) and (3.7), we obtain
$\displaystyle\psi_{1}(P,x,x_{0})$ $\displaystyle=$
$\displaystyle\mathrm{exp}\left(z^{-1}\int_{x_{0}}^{x}\frac{y+zG_{n}(z,x^{\prime})}{F_{n+1}(z,x^{\prime})}~{}dx^{\prime}\right)$
$\displaystyle=$
$\displaystyle\mathrm{exp}\left(z^{-1}\int_{x_{0}}^{x}\Big{(}\frac{y}{F_{n+1}(z,x^{\prime})}+\frac{1}{2}\frac{F_{n+1,x^{\prime}}(z,x^{\prime})}{F_{n+1}(z,x^{\prime})}\Big{)}dx^{\prime}\right),$
which implies (3.26). Moreover, (3.6) and (3.8) together with (3.20)-(3.22)
yields
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{1}(P^{\ast},x,x_{0})$ $\displaystyle=$
$\displaystyle\mathrm{exp}\left(z^{-1}\int_{x_{0}}^{x}(\phi(P)+\phi(P^{\ast}))~{}dx^{\prime}\right)$
(3.32) $\displaystyle=$
$\displaystyle\mathrm{exp}\left(z^{-1}\int_{x_{0}}^{x}\frac{2zG_{n}(z,x^{\prime})}{F_{n+1}(z,x^{\prime})}~{}dx^{\prime}\right)$
$\displaystyle=$
$\displaystyle\mathrm{exp}\left(\int_{x_{0}}^{x}\frac{F_{n+1,x^{\prime}}(z,x^{\prime})}{F_{n+1}(z,x^{\prime})}~{}dx^{\prime}\right)$
$\displaystyle=$ $\displaystyle\frac{F_{n+1}(z,x)}{F_{n+1}(z,x_{0})},$
$\displaystyle\psi_{2}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})$ $\displaystyle=$
$\displaystyle
z^{-2}\psi_{1}(P,x,x_{0})\phi(P,x)\psi_{1}(P^{\ast},x,x_{0})\phi(P^{\ast},x)$
(3.33) $\displaystyle=$ $\displaystyle
z^{-2}\frac{F_{n+1}(z,x)}{F_{n+1}(z,x_{0})}\frac{(-zH_{n}(z,x))}{F_{n+1}(z,x)}$
$\displaystyle=$ $\displaystyle-\frac{H_{n}(z,x)}{zF_{n+1}(z,x_{0})},$
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})+\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})$
(3.34)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\psi_{1}(P)\psi_{1}(P^{\ast})\phi(P^{\ast})/z+\psi_{1}(P^{\ast})\psi_{1}(P)\phi(P)/z$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\psi_{1}(P)\psi_{1}(P^{\ast})(\phi(P)+\phi(P^{\ast}))/z$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\frac{F_{n+1}(z,x)}{F_{n+1}(z,x_{0})}\frac{2zG_{n}(z,x)}{zF_{n+1}(z,x)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\frac{2G_{n}(z,x)}{F_{n+1}(z,x_{0})},$
$\displaystyle\psi_{1}(P,x,x_{0})\psi_{2}(P^{\ast},x,x_{0})-\psi_{1}(P^{\ast},x,x_{0})\psi_{2}(P,x,x_{0})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\psi_{1}(P)\psi_{1}(P^{\ast})\phi(P^{\ast})/z+\psi_{1}(P^{\ast})\psi_{1}(P)\phi(P)/z$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}=\psi_{1}(P)\psi_{1}(P^{\ast})(\phi(P^{\ast})-\phi(P))/z$
$\displaystyle=$
$\displaystyle\frac{F_{n+1}(z,x)}{F_{n+1}(z,x_{0})}\frac{-2y}{zF_{n+1}(z,x)}$
(3.35) $\displaystyle=$ $\displaystyle\frac{-2y}{zF_{n+1}(z,x_{0})}.$
Hence (3.27)-(3.30) hold. $\square$
In Lemma 3.2 if we choose
$\psi_{1}(P)=\psi_{1,+},~{}\psi_{1}(P^{\ast})=\psi_{1,-},~{}\psi_{2}(P)=\psi_{2,+},~{}\psi_{2}(P^{\ast})=\psi_{2,-},$
then (3.27)-(3.30) imply
$(\psi_{1,+}\psi_{2,-}-\psi_{1,-}\psi_{2,+})^{2}=(\psi_{1,+}\psi_{2,-}+\psi_{1,-}\psi_{2,+})^{2}-4\psi_{1,+}\psi_{2,-}\psi_{1,-}\psi_{2,+},$
(3.36)
which is equivalent to the basic identity (2.19),
$z^{2}G_{n}^{2}+zF_{n+1}H_{n}=R_{2n+2}$. This fact reveals the relations
between our approach and the algebro-geometric solutions of the HS hierarchy.
###### Remark 3.3
The definition of stationary Baker-Akhiezer function $\psi$ of the HS
hierarchy is analogous to that in the context of KdV or AKNS hierarchies. But
the crucial difference is that $P_{0}$ is a essential singularity of $\psi$ in
the HS hierarchy, which is the same as in CH hierarchy, but different from the
KdV or AKNS hierarchy. This fact will be showed in the asymptotic expansions
of $\psi$ in next section.
Furthermore, we derive Dubrovin-type equations, which are first-order coupled
systems of differential equations and govern the dynamics of the zeros
$\mu_{j}(x)$ and $\nu_{l}(x)$ of $F_{n+1}(z,x)$ and $H_{n}(z,x)$ with respect
to $x$. We recall the affine part of $\mathcal{K}_{n}$ is nonsingular if
$\begin{split}&E_{0}=0,~{}~{}\\{E_{m}\\}_{m=1,\ldots,2n+1}\subset\mathbb{C}\setminus\\{0\\},\\\
&E_{m}\neq E_{m^{\prime}}\quad\textrm{for $m\neq
m^{\prime},m,m^{\prime}=1,\ldots,2n+1$}.\end{split}$ (3.37)
###### Lemma 3.4
Assume that $(\ref{2.1})$ holds and $u$ satisfies the $n$th stationary HS
equation $(\ref{2.34})$.
$(\mathrm{i})$ If the zeros $\\{\mu_{j}(x)\\}_{j=0,\ldots,n}$ of
$F_{n+1}(z,x)$ remain distinct for $x\in\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}$ is an open interval, then
$\\{\mu_{j}(x)\\}_{j=0,\ldots,n}$ satisfy the system of differential
equations,
$\mu_{j,x}=2\frac{y(\hat{\mu}_{j})}{\mu_{j}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}(x)-\mu_{k}(x))^{-1},\quad
j=0,\ldots,n,$ (3.38)
with initial conditions
$\\{\hat{\mu}_{j}(x_{0})\\}_{j=0,\ldots,n}\in\mathcal{K}_{n},$ (3.39)
for some fixed $x_{0}\in\Omega_{\mu}$. The initial value problems
$(\ref{3.38})$, $(\ref{3.39})$ have a unique solution satisfying
$\hat{\mu}_{j}\in C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),\quad
j=0,\ldots,n.$ (3.40)
$(\mathrm{ii})$ If the zeros $\\{\nu_{l}(x)\\}_{l=1,\ldots,n}$ of $H_{n}(z,x)$
remain distinct for $x\in\Omega_{\nu},$ where
$\Omega_{\nu}\subseteq\mathbb{R}$ is an open interval, then
$\\{\nu_{l}(x)\\}_{l=1,\ldots,n}$ satisfy the system of differential
equations,
$\nu_{l,x}=-2\frac{u_{xx}~{}y(\hat{\nu}_{l})}{h_{0}~{}\nu_{l}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq l}^{n}(\nu_{l}(x)-\nu_{k}(x))^{-1},\quad
l=1,\ldots,n,$ (3.41)
with initial conditions
$\\{\hat{\nu}_{l}(x_{0})\\}_{l=1,\ldots,n}\in\mathcal{K}_{n},$ (3.42)
for some fixed $x_{0}\in\Omega_{\nu}$. The initial value problems
$(\ref{3.41})$, $(\ref{3.42})$ have a unique solution satisfying
$\hat{\nu}_{l}\in C^{\infty}(\Omega_{\nu},\mathcal{K}_{n}),\quad
l=1,\ldots,n.$ (3.43)
Proof. For our convenience, let us focus on (3.38) and (3.40), the proof of
(3.41) and (3.43) follows in an identical manner. The derivatives of (3.9)
with respect to $x$ take on
$F_{n+1,x}(\mu_{j})=-\mu_{j,x}\prod_{\scriptstyle k=0\atop\scriptstyle k\neq
j}^{n}(\mu_{j}(x)-\mu_{k}(x)).$ (3.44)
On the other hand, inserting $z=\mu_{j}$ into equation (2.15) leads to
$F_{n+1,x}(\mu_{j})=2G_{n}(\mu_{j})=2\frac{y(\hat{\mu}_{j})}{-\mu_{j}}.$
(3.45)
Comparing (3.44) with (3.45) gives (3.38). The proof of smoothness assertion
(3.40) is analogous to the mCH case in our latest paper [18]. $\square$
Let us now turn to the trace formulas of the HS invariants, which is the
expressions of $f_{l}$ and $h_{l}$ in terms of symmetric functions of the
zeros $\mu_{j}$ and $\nu_{l}$ of $F_{n+1}$ and $H_{n}$, respectively. Here, we
just consider the simplest case.
###### Lemma 3.5
If $(\ref{2.1})$ holds and $u$ satisfies the $n$th stationary HS equation
$(\ref{2.34})$, then
$u=\frac{1}{2}\sum_{j=0}^{n}\mu_{j}-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$ (3.46)
Proof. By comparison of the coefficient of $z^{n}$ of $F_{n+1}$ in (2.28) and
(3.9), taking account into (2.6) yields
$-2u+c_{1}=-\sum_{j=0}^{n}\mu_{j}.$ (3.47)
The constant $c_{1}$ can be determined by a long straightforward calculation
comparing the coefficients of $z^{2n+1}$ in (2.19), which leads to
$c_{1}=-\sum_{m=0}^{2n+1}E_{m}.$ (3.48)
## 4 Stationary algebro-geometric solutions of HS hierarchy
In this section we continue our study of the stationary HS hierarchy, and will
obtain explicit Riemann theta function representations for the meromorphic
function $\phi$, and especially, for the potentials $u$ of the stationary HS
hierarchy.
Let us begin with the asymptotic properties of $\phi$ and $\psi_{j},j=1,2$.
###### Lemma 4.1
Assume that $(\ref{2.1})$ to hold and $u$ satisfies the $n$th stationary HS
equation $(\ref{2.34})$. Moreover, let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}},P_{0}\\},$
$(x,x_{0})\in\mathbb{R}^{2}$. Then
$\phi(P)\underset{\zeta\rightarrow 0}{=}-u_{x}+O(\zeta),\qquad P\rightarrow
P_{\infty_{\pm}},\quad\zeta=z^{-1},$ (4.1) $\phi(P)\underset{\zeta\rightarrow
0}{=}i~{}a\Big{(}\prod_{m=1}^{2n+1}E_{m}\Big{)}^{1/2}f_{n+1}^{-1}\zeta+O(\zeta^{2}),\qquad
P\rightarrow P_{0},\quad\zeta=z^{1/2},$ (4.2)
and
$\displaystyle\psi_{1}(P,x,x_{0})$ $\displaystyle\underset{\zeta\rightarrow
0}{=}$
$\displaystyle\mathrm{exp}\Big{(}(u(x_{0})-u(x))\zeta+O(\zeta^{2})\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}P\rightarrow P_{\infty_{\pm}},~{}\zeta=z^{-1},$
$\displaystyle\psi_{2}(P,x,x_{0})$ $\displaystyle\underset{\zeta\rightarrow
0}{=}$ $\displaystyle
O(\zeta)~{}\mathrm{exp}\Big{(}(u(x_{0})-u(x))\zeta+O(\zeta^{2})\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}P\rightarrow P_{\infty_{\pm}},~{}\zeta=z^{-1},$
$\displaystyle\psi_{1}(P,x,x_{0})$ $\displaystyle\underset{\zeta\rightarrow
0}{=}$
$\displaystyle\mathrm{exp}\Big{(}\frac{i}{\zeta}\int_{x_{0}}^{x}dx^{\prime}~{}a\Big{(}\prod_{m=1}^{2n+1}E_{m}\Big{)}^{1/2}f_{n+1}(x^{\prime})^{-1}+O(1)\Big{)},$
(4.5) $\displaystyle~{}~{}~{}~{}~{}P\rightarrow P_{0},\quad\zeta=z^{1/2},$
$\displaystyle\psi_{2}(P,x,x_{0})$ $\displaystyle\underset{\zeta\rightarrow
0}{=}$ $\displaystyle
O(\zeta^{-1})~{}\mathrm{exp}\Big{(}\frac{i}{\zeta}\int_{x_{0}}^{x}dx^{\prime}~{}a\Big{(}\prod_{m=1}^{2n+1}E_{m}\Big{)}^{1/2}f_{n+1}(x^{\prime})^{-1}+O(1)\Big{)},$
(4.6) $\displaystyle~{}~{}~{}~{}~{}P\rightarrow P_{0},\quad\zeta=z^{1/2}.$
Proof. Under the local coordinates $\zeta=z^{-1}$ near $P_{\infty_{\pm}}$ and
$\zeta=z^{1/2}$ near $P_{0}$, the existence of the asymptotic expansions of
$\phi$ is clear from its explicit expressions in (3.7). Next, we use the
Riccati-type equation (3.19) to compute the explicit expansion coefficients.
Inserting the ansatz
$\phi\underset{z\rightarrow\infty}{=}\phi_{0}+\phi_{1}z^{-1}+O(z^{-2})$ (4.7)
into (3.19) and comparing the powers of $z^{0}$ then yields (4.1). Similarly,
inserting the ansatz
$\phi\underset{z\rightarrow 0}{=}\phi_{1}z^{1/2}+\phi_{2}z+O(z^{3/2})$ (4.8)
into (3.19) and comparing the power of $z^{0}$ then yields (4.2), where we
used (2.31) and
$f_{n+1}h_{n}=-a^{2}\prod_{m=1}^{2n+1}E_{m},$ (4.9)
which can be obtained by (2.19). Finally, expansions (4.3)-(4.6) follow up by
(3.6), (3.8), (4.1) and (4.2). $\square$
###### Remark 4.2
From $(4.5)$ and $(4.6)$, we note the unusual fact that $P_{0}$ is the
essential singularity of $\psi_{j}$, $j=1,2$, this is consistent with Remark
$3.3$. Also the leading-order exponential term $\psi_{j}$, $j=1,2,$ near
$P_{0}$ is $x$-dependent, which makes matters worse. This is in sharp contrast
to standard Baker-Akhiezer functions that typically feature a linear behavior
with respect to $x$ such as $\mathrm{exp}(c(x-x_{0})\zeta^{-1})$ near $P_{0}$.
Let us now introduce the holomorphic differentials $\eta_{l}(P)$ on
$\mathcal{K}_{n}$
$\eta_{l}(P)=\frac{a~{}z^{l-1}}{y(P)}dz,\qquad l=1,\ldots,n,$ (4.10)
and choose a homology basis $\\{a_{j},b_{j}\\}_{j=1}^{n}$ on $\mathcal{K}_{n}$
in such a way that the intersection matrix of the cycles satisfies
$a_{j}\circ b_{k}=\delta_{j,k},\quad a_{j}\circ a_{k}=0,\quad b_{j}\circ
b_{k}=0,\quad j,k=1,\ldots,n.$
Define an invertible matrix $E\in GL(n,\mathbb{C})$ as follows
$\begin{split}&E=(E_{j,k})_{n\times n},\quad E_{j,k}=\int_{a_{k}}\eta_{j},\\\
&\underline{c}(k)=(c_{1}(k),\ldots,c_{n}(k)),\quad
c_{j}(k)=(E^{-1})_{j,k},\end{split}$ (4.11)
and the normalized holomorphic differentials
$\omega_{j}=\sum_{l=1}^{n}c_{j}(l)\eta_{l},\quad\int_{a_{k}}\omega_{j}=\delta_{j,k},\quad\int_{b_{k}}\omega_{j}=\tau_{j,k},\quad
j,k=1,\ldots,n.$ (4.12)
Apparently, the matrix $\tau$ is symmetric and has a positive-definite
imaginary part.
The symmetric function $\Phi_{n}^{(j)}(\bar{\mu})$ and $\Psi_{n+1}(\bar{\mu})$
are defined by
$\Phi_{n}^{(j)}(\bar{\mu})=(-1)^{n}\prod_{\scriptstyle p=0\atop\scriptstyle
p\neq j}^{n}\mu_{p},$ (4.13)
$\Psi_{n+1}(\bar{\mu})=(-1)^{n+1}\prod_{p=0}^{n}\mu_{p}.$ (4.14)
The following result shows that the nonlinearity of the Abel map in the HS
hierarchy. This feature is analogous to CH hierarchy but sharp apposed to
other integrable soliton equations such as KdV and AKNS hierarchies.
###### Theorem 4.3
Assume $(\ref{2.1})$ to hold and suppose that
$\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$ satisfies the stationary Dubrovin
equations $(\ref{3.38})$ on $\Omega_{\mu}$ and remain distinct for
$x\in\Omega_{\mu},$ where $\Omega_{\mu}\subseteq\mathbb{R}$ is an open
interval. Introducing the associated divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$. Then
$\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)})=-\frac{2a}{\Psi_{n+1}(\bar{\mu}(x))}\underline{c}(1),\qquad
x\in\Omega_{\mu}.$ (4.15)
In particular, the Abel map does not linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$ on $\Omega_{\mu}$,
where
$\bar{\mu}(x)=(\mu_{0}(x),\mu_{1}(x),\ldots,\mu_{n}(x))=\mu_{0}(x)\underline{\mu}(x).$
Proof. Is easy to see that
$\frac{1}{\mu_{j}}=\frac{\prod_{\scriptstyle p=0\atop\scriptstyle p\neq
j}^{n}\mu_{p}}{\prod_{p=0}^{n}\mu_{p}}=-\frac{\Phi_{n}^{(j)}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})},\quad
j=1,\ldots,n.$ (4.16)
Let
$\underline{\omega}=(\omega_{1},\ldots,\omega_{n}),$ (4.17)
and choose a appropriate base point $Q_{0}$. Then we arrive at
$\displaystyle\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)})=\partial_{x}\Big{(}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}}\underline{\omega}\Big{)}=\sum_{j=0}^{n}\mu_{j,x}\sum_{k=1}^{n}\underline{c}(k)\frac{a~{}\mu_{j}^{k-1}}{y(\hat{\mu}_{j})}$
$\displaystyle=\sum_{j=0}^{n}\sum_{k=1}^{n}\frac{2a~{}\mu_{j}^{k-1}}{\mu_{j}}\frac{1}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\underline{c}(k)$
$\displaystyle=-\frac{2a}{\Psi_{n+1}(\bar{\mu})}\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\Phi_{n}^{(j)}(\bar{\mu})$
$\displaystyle=-\frac{2a}{\Psi_{n+1}(\bar{\mu})}\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)(U_{n+1}(\bar{\mu}))_{k,j}(U_{n+1}(\bar{\mu}))_{j,1}^{-1}$
(4.18)
$\displaystyle=-\frac{2a}{\Psi_{n+1}(\bar{\mu})}\sum_{k=1}^{n}\underline{c}(k)\delta_{k,1}$
$\displaystyle=-\frac{2a}{\Psi_{n+1}(\bar{\mu})}\underline{c}(1),$
where we used
$(U_{n+1}(\bar{\mu}))=\Big{(}\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq j}^{n}(\mu_{j}-\mu_{l})}\Big{)}_{\scriptstyle
j=0\atop\scriptstyle
k=1}^{n},\quad(U_{n+1}(\bar{\mu}))^{-1}=\Big{(}\Phi_{n}^{(j)}(\bar{\mu})\Big{)}_{j=0}^{n},$
(4.19)
the definition of which is analogous to (E.25) and (E.26) in [15]. $\square$
The analogous results hold for the corresponding divisor
$\mathcal{D}_{\underline{\hat{\nu}}(x)}$ associated with $\phi(P,x)$ can be
obtained in the same way.
Next, we introduce 222 Here we choose the same path of integration from
$Q_{0}$ and $P$ in all integrals in (4.20)
and (4.21).
$\begin{split}&\underline{\widehat{B}}_{Q_{0}}:\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}\rightarrow\mathbb{C}^{n},\\\
&P\mapsto\underline{\widehat{B}}_{Q_{0}}(P)=(\widehat{B}_{Q_{0},1},\ldots,\widehat{B}_{Q_{0},n})\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\begin{cases}\int_{Q_{0}}^{P}\tilde{\omega}_{P_{\infty_{+}},P_{\infty_{-}}}^{(3)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n=1\\\
\Big{(}\int_{Q_{0}}^{P}\eta_{2},\ldots,\int_{Q_{0}}^{P}\eta_{n},\int_{Q_{0}}^{P}\tilde{\omega}_{P_{\infty_{+}},P_{\infty_{-}}}^{(3)}\Big{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}n\geq
2,\end{cases}\end{split}$ (4.20)
where
$\tilde{\omega}_{P_{\infty_{+}},P_{\infty_{-}}}^{(3)}=\frac{a~{}z^{n}}{y(P)}dz$
denotes a differential of the third kind with simple poles at $P_{\infty_{+}}$
and $P_{\infty_{-}}$ and corresponding residues $+1$ and $-1$, respectively.
Moreover,
$\begin{split}&\underline{\hat{\beta}}_{Q_{0}}:\mathrm{Sym}^{n}(\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\})\rightarrow\mathbb{C}^{n},\\\
&\mathcal{D}_{\underline{Q}}\mapsto\underline{\hat{\beta}}_{Q_{0}}(\mathcal{D}_{\underline{Q}})=\sum_{j=1}^{n}\underline{\widehat{B}}_{Q_{0}}(Q_{j}),\\\
&\underline{Q}=\\{Q_{1},\ldots,Q_{n}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}\setminus\\{P_{\infty_{+}},P_{\infty_{-}}\\}).\end{split}$
(4.21)
The following result is a special case of Theorem 4.3, which will be used to
provide the proper change of variables to linear the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$ associated with
$\phi(P,x)$.
###### Theorem 4.4
Assume that $(\ref{2.1})$ holds and the statements of $\mu_{j}$ in Theorem
$4.3$ are all true. Then
$\partial_{x}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}(x)}\eta_{1}=-\frac{2a}{\Psi_{n+1}(\bar{\mu}(x))},\qquad
x\in\Omega_{\mu},$ (4.22)
$\partial_{x}\underline{\hat{\beta}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\begin{cases}2a,~{}~{}~{}~{}~{}~{}~{}~{}n=1,\\\
2a~{}(0,\ldots,0,1),~{}~{}~{}~{}~{}~{}~{}~{}n\geq 2,\end{cases}\quad
x\in\Omega_{\mu}.$ (4.23)
Proof. Equations (4.22) is a special case (4.15) and (4.23) follows from
(4.18). Alternatively, one can follow the same way as shown in Theorem 4.3 to
derive (4.22) and (4.23). $\square$
Let $\theta(\underline{z})$ denote the Riemann theta function associated with
$\mathcal{K}_{n}$ and an appropriately fixed homology basis. We assume
$\mathcal{K}_{n}$ to be nonsingular. Next, choosing a convenient base point
$Q_{0}\in\mathcal{K}_{n}\setminus\\{\hat{\mu}_{0}(x),P_{0}\\}$, the vector of
Riemann constants $\underline{\Xi}_{Q_{0}}$ is given by (A.66) [15], and the
Abel maps $\underline{A}_{Q_{0}}(\cdot)$ and
$\underline{\alpha}_{Q_{0}}(\cdot)$ are defined by
$\begin{split}&\underline{A}_{Q_{0}}:\mathcal{K}_{n}\rightarrow
J(\mathcal{K}_{n})=\mathbb{C}^{n}/L_{n},\\\
&P\mapsto\underline{A}_{Q_{0}}(P)=(\underline{A}_{Q_{0},1}(P),\ldots,\underline{A}_{Q_{0},n}(P))=\left(\int_{Q_{0}}^{P}\omega_{1},\ldots,\int_{Q_{0}}^{P}\omega_{n}\right)(\mathrm{mod}~{}L_{n})\end{split}$
(4.24)
and
$\begin{split}&\underline{\alpha}_{Q_{0}}:\mathrm{Div}(\mathcal{K}_{n})\rightarrow
J(\mathcal{K}_{n}),\\\
&\mathcal{D}\mapsto\underline{\alpha}_{Q_{0}}(\mathcal{D})=\sum_{P\in\mathcal{K}_{n}}\mathcal{D}(P)\underline{A}_{Q_{0}}(P),\end{split}$
(4.25)
where
$L_{n}=\\{\underline{z}\in\mathbb{C}^{n}|~{}\underline{z}=\underline{N}+\tau\underline{M},~{}\underline{N},~{}\underline{M}\in\mathbb{Z}^{n}\\}.$
Let
$\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}(P)=\frac{a}{y}\prod_{j=1}^{n}(z-\lambda_{j})dz$
(4.26)
be the normalized differential of the third kind holomorphic on
$\mathcal{K}_{n}\setminus\\{\hat{\mu}_{0}(x),P_{0}\\}$ with simple poles at
$\hat{\mu}_{0}(x)$ and $P_{0}$ with residues $\pm 1$, respectively, that is,
$\begin{split}&\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(\zeta^{-1}+O(1))d\zeta,\quad\textrm{as
$P\rightarrow\hat{\mu}_{0}(x),$}\\\
&\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}(-\zeta^{-1}+O(1))d\zeta,\quad\textrm{as $P\rightarrow
P_{0},$}\end{split}$ (4.27)
where the local coordinate are given by
$\zeta=z^{-1}~{}~{}\textrm{for $P$ near
$\hat{\mu}_{0}(x)$},\qquad\zeta=z^{1/2}~{}~{}\textrm{for $P$ near $P_{0}$},$
(4.28)
and the constants $\\{\lambda_{j}\\}_{j=1,\ldots,n}$ are determined by the
normalization condition
$\int_{a_{k}}\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}=0,\qquad k=1,\ldots,n.$
Then
$\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}\mathrm{ln}\zeta+e_{0}+O(\zeta),\quad\textrm{as
$P\rightarrow\hat{\mu}_{0}(x),$}$ (4.29)
$\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}(P)\underset{\zeta\rightarrow
0}{=}-\mathrm{ln}\zeta+d_{0}+O(\zeta),\quad\textrm{as $P\rightarrow P_{0},$}$
(4.30)
for some constants $e_{0},d_{0}\in\mathbb{C}$ that arise from the integrals at
their lower limits $Q_{0}$. We also note that
$\underline{A}_{Q_{0}}(P)-\underline{A}_{Q_{0}}(P_{\infty_{\pm}})\underset{\zeta\rightarrow
0}{=}\pm\underline{U}\zeta+O(\zeta^{2}),\quad\textrm{as $P\rightarrow
P_{\infty_{\pm}},$}\quad\underline{U}=\underline{c}(n).$ (4.31)
The following abbreviations are used for our convenience:
$\displaystyle\underline{z}(P,\underline{Q})=\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}),$
$\displaystyle
P\in\mathcal{K}_{n},\,\underline{Q}=(Q_{1},\ldots,Q_{n})\in\mathrm{Sym}^{n}(\mathcal{K}_{n}),$
(4.32)
where $\underline{z}(\cdot,\underline{Q})$ is independent of the choice of
base point $Q_{0}$.
Moreover, from Theorem 4.3 and Theorem 4.4 we note that the Abel map dose not
linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$. However, the change
of variables
$x\mapsto\tilde{x}=\int^{x}dx^{\prime}\Big{(}\frac{2a}{\Psi_{n+1}(\bar{\mu}(x^{\prime}))}\Big{)}$
(4.33)
linearizes the Abel map
$\underline{A}_{Q_{0}}(\mathcal{D}_{\hat{\tilde{\mu}}_{0}(\tilde{x})\underline{\hat{\tilde{\mu}}}(\tilde{x})}),$
$\tilde{\mu}_{j}(\tilde{x})=\mu_{j}(x),j=0,\ldots,n.$ The intricate relation
between the variable $x$ and $\tilde{x}$ is discussed detailedly in Theorem
4.5.
Based on the above all these preparations, let us now give an explicit
representations for the meromorphic function $\phi$ and the stationary HS
solutions $u$ in terms of the Riemann theta function associated with
$\mathcal{K}_{n}$. Here we assume the affine part of $\mathcal{K}_{n}$ to be
nonsingular.
###### Theorem 4.5
Assume that the curve $\mathcal{K}_{n}$ is nonsingular, $(\ref{2.1})$ holds
and $u$ satisfies the $n$th stationary HS equation $(\ref{2.34})$ on $\Omega$.
Moreover, let $P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{0}\\},$ and
$x\in\Omega$, where $\Omega\subseteq\mathbb{R}$ is an open interval. In
addition, suppose that $\mathcal{D}_{\underline{\hat{\mu}}(x)}$, or
equivalently $\mathcal{D}_{\underline{\hat{\nu}}(x)}$ is nonspecial for
$x\in\Omega$. Then, $\phi$ and $u$ have the following representations
$\displaystyle\phi(P,x)$ $\displaystyle=$ $\displaystyle
ia\Big{(}\prod_{m=1}^{2n+1}E_{m}\Big{)}^{1/2}f_{n+1}^{-1}\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x)))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x)))\theta(\underline{z}(P,\underline{\hat{\mu}}(x)))}$
(4.34)
$\displaystyle\times~{}\mathrm{exp}\left(d_{0}-\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}\right),$
$\displaystyle u(x)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=1}^{n}\lambda_{j}$
(4.35)
$\displaystyle-\frac{1}{2}\sum_{j=1}^{n}U_{j}\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x))+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x))+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0}.$
Moreover, let $\mu_{j},$ $j=0,\ldots,n$ be not vanishing on $\Omega$ and
$x,x_{0}\in\Omega.$ Then, we have the following constraint
$\displaystyle 2a(x-x_{0})$ $\displaystyle=$
$\displaystyle-2a\int_{x_{0}}^{x}\frac{dx^{\prime}}{\prod_{k=0}^{n}\mu_{k}(x^{\prime})}\sum_{j=1}^{n}\Big{(}\int_{a_{j}}\tilde{\omega}_{P_{\infty_{+}}P_{\infty_{-}}}^{(3)}\Big{)}c_{j}(1)$
(4.36) $\displaystyle+$
$\displaystyle\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x_{0})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x)))}{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0})))\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x)))}\right)$
and
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)})$
$\displaystyle=$
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0})\underline{\hat{\mu}}(x_{0})})-2a\int_{x_{0}}^{x}\frac{dx^{\prime}}{\Psi_{n+1}(\bar{\mu}(x^{\prime}))}\underline{c}(1)$
(4.37) $\displaystyle=$
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0})\underline{\hat{\mu}}(x_{0})})-\underline{c}(1)(\tilde{x}-\tilde{x}_{0}).$
Proof. First, let us assume
$\mu_{j}(x)\neq\mu_{j^{\prime}}(x),\quad\nu_{k}(x)\neq\nu_{k^{\prime}}(x)\quad\textrm{for
$j\neq j^{\prime},k\neq k^{\prime}$ and $x\in\widetilde{\Omega}$},$ (4.38)
where $\widetilde{\Omega}\subseteq\Omega$. From (3.14),
$\mathcal{D}_{P_{0}\underline{\hat{\nu}}}\sim\mathcal{D}_{\hat{\mu}_{0}\underline{\hat{\mu}}}$,
and $(P_{0})^{\ast}\notin\\{\hat{\nu}_{1},\cdots,\hat{\nu}_{n}\\}$ by
hypothesis, one can use Theorem A.31 [15] to conclude that
$\mathcal{D}_{\underline{\hat{\mu}}}\in\textrm{Sym}^{n}(\mathcal{K}_{n})$ is
nonspecial. This argument is of course symmetric with respect to
$\underline{\hat{\mu}}$ and $\underline{\hat{\nu}}$. Thus,
$\mathcal{D}_{\underline{\hat{\mu}}}$ is nonspecial if and only if
$\mathcal{D}_{\underline{\hat{\nu}}}$ is.
Next, we derive the representations of $\phi$ and $u$ in terms of the Riemann
theta function. A special case of Riemann’s vanishing theorem (Theorem A.26
[15]) yields
$\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{Q}}))=0\quad\textrm{
if and only if $P\in\\{Q_{1},\cdots,Q_{n}\\}$}.$ (4.39)
Therefore, the divisor (3.14) of $\phi(P,x)$ suggests considering expressions
of the following type
$C(x)\frac{\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}))}{\theta(\underline{\Xi}_{Q_{0}}-\underline{A}_{Q_{0}}(P)+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)}))}\mathrm{exp}\Big{(}d_{0}-\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x)P_{0}}^{(3)}\Big{)},$
(4.40)
where $C(x)$ is independent of $P\in\mathcal{K}_{n}$. So, together with the
asymptotic expansion of $\phi(P,x)$ near $P_{0}$ in (4.2), we are able to
obtain (4.34). The representation (4.35) for $u$ on $\widetilde{\Omega}$
follows from trace formula (3.46) and the expression (F.88 [15]) for
$\sum_{j=0}^{n}\mu_{j}$.
To prove the constraint (4.36), one can refs Theorem 4.5 in our latest paper
[18]. Equations (4.37) is clear from (4.15). Finally, the extension of all
results from $x\in\widetilde{\Omega}$ to $x\in\Omega$ follows by the
continuity of $\underline{\alpha}_{Q_{0}}$ and the hypothesis of
$\mathcal{D}_{\underline{\hat{\mu}}(x)}$ being nonspecial for $x\in\Omega$.
$\square$
###### Remark 4.6
The stationary HS solutions $u$ in $(\ref{4.35})$ is a quasi-periodic function
with respect to the new variable $\tilde{x}$ in $(\ref{4.33})$. The Abel map
in $(\ref{4.37})$ linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$ on $\Omega$ with
respect to $\tilde{x}$.
###### Remark 4.7
The similar results to $(\ref{4.36})$ and $(\ref{4.37})$ (i.e. the Abel map
also linearize the divisor $\mathcal{D}_{\underline{\hat{\nu}}(x)}$ on
$\Omega$ with respect to $\bar{x}$) hold for the divisor
$\mathcal{D}_{\underline{\hat{\nu}}(x)}$ associated with $\phi(P,x)$. The
change of variables is
$x\mapsto\bar{x}=\int^{x}dx^{\prime}\Big{(}\frac{1}{\Psi_{n}(\underline{\nu}(x^{\prime}))}\frac{u_{x^{\prime}x^{\prime}}}{h_{0}(x^{\prime})}\Big{)}.$
(4.41)
###### Remark 4.8
Since $\mathcal{D}_{P_{0}\underline{\hat{\nu}}}$ and
$\mathcal{D}_{\hat{\mu}_{0}\underline{\hat{\mu}}}$ are linearly equivalent,
that is
$\underline{A}_{Q_{0}}(\hat{\mu}_{0}(x))+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)})=\underline{A}_{Q_{0}}(P_{0})+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)}).$
(4.42)
Then we infer
$\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\nu}}(x)})=\underline{\Delta}+\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\underline{\hat{\mu}}(x)}),\qquad\underline{\Delta}=\underline{A}_{P_{0}}(\hat{\mu}_{0}(x)).$
(4.43)
Hence
$\underline{z}(P,\underline{\hat{\nu}})=\underline{z}(P,\underline{\hat{\mu}})+\underline{\Delta},\qquad
P\in\mathcal{K}_{n}.$ (4.44)
The representations of $\phi$ and $u$ in $(\ref{4.34})$ and $(\ref{4.35})$ can
be rewritten in terms of $\mathcal{D}_{\underline{\hat{\nu}}(x)}$
respectively.
###### Remark 4.9
We have emphasized in Remark $4.2$ that the Baker-Akhiezer functions $\psi$ in
$(\ref{3.6})$ and $(\ref{3.8})$ for the HS hierarchy enjoy very difference
from standard Baker-Akhiezer functions. Hence, one may not expect the usual
theta function representations of $\psi_{j}$, $j=1,2,$ in terms of ratios of
theta functions times a exponential term including $(x-x_{0})$ multiplying a
meromorphic differential with a pole at the essential singularity of
$\psi_{j}$. However, using the properties of symmetric function and
$(F.89)~{}\cite[cite]{[\@@bibref{}{15}{}{}]}$, we obtain
$\displaystyle F_{n+1}(z)$ $\displaystyle=$ $\displaystyle
z^{n+1}+\sum_{k=0}^{n}\Psi_{n+1-k}(\bar{\mu})z^{k}$ $\displaystyle=$
$\displaystyle z^{n+1}+\sum_{k=1}^{n}\Big{(}\Psi_{n+1-k}(\underline{\lambda})$
$\displaystyle-\sum_{j=1}^{n}c_{j}(k)\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}})+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}})+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0}\Big{)}z^{k}$
(4.45) $\displaystyle=$ $\displaystyle z\prod_{j=1}^{n}(z-\lambda_{j})$
$\displaystyle-\sum_{j=1}^{n}\sum_{k=1}^{n}c_{j}(k)\partial_{\omega_{j}}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}})+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}})+\underline{\omega})}\right)\Big{|}_{\underline{\omega}=0}z^{k},$
and by inserting $(\ref{4.45})$ into $(\ref{3.26})$, we obtain the theta
function representation of $\psi_{1}$. Then, the corresponding theta functions
representation of $\psi_{2}$ follows by $(\ref{3.8})$ and $(\ref{4.34})$.
At the end of this section, we turn to the initial value problem in the
stationary case. We show that the solvability of the Dubrovin equations (3.38)
on $\Omega_{\mu}\subseteq\mathbb{R}$ in fact implies the stationary HS
equation (2.34) on $\Omega_{\mu}$, which amounts to solving the algebro-
geometric initial value problem in the stationary case.
###### Theorem 4.10
Assume that $(\ref{2.1})$ holds and $\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$
satisfies the stationary Dubrovin equations $(\ref{3.38})$ on $\Omega_{\mu}$
and remain distinct and nonzero for $x\in\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}$ is an open interval. Then, $u$ defined by
$u=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=0}^{n}\mu_{j},$
(4.46)
satisfies the $n$th stationary HS equation $(\ref{2.34}),$ that is
$\mathrm{s}\textrm{-}\mathrm{HS}_{n}(u)=0,\quad\textrm{on $\Omega_{\mu}.$}$
(4.47)
Proof. Given the solutions $\hat{\mu}_{j}=(\mu_{j},y(\hat{\mu}_{j}))\in
C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),j=0,\cdots,n$ of (3.38), let us
introduce
$F_{n+1}(z)=\prod_{j=0}^{n}(z-\mu_{j})\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$},$ (4.48)
with $u$ defined by (4.46) up to multiplicative constant. Given $F_{n+1}$ and
$u$, let us denote the polynomial $G_{n}$ by
$G_{n}(z)=\frac{1}{2}F_{n+1,x}(z),\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$},$ (4.49)
and from (4.48), one can see that the degree of $G_{n}$ is $n$ with respect to
$z$. Taking account into (4.48), the Dubrovin equations (3.38) imply
$y(\hat{\mu}_{j})=\frac{1}{2}\mu_{j}\mu_{j,x}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq
j}^{n}(\mu_{j}-\mu_{k})=-\frac{1}{2}\mu_{j}F_{n+1,x}(\mu_{j})=-\mu_{j}G_{n}(\mu_{j}).$
(4.50)
Hence
$R_{2n+2}(\mu_{j})^{2}-\mu_{j}^{2}G_{n}(\mu_{j})^{2}=y(\hat{\mu}_{j})^{2}-\mu_{j}^{2}G_{n}(\mu_{j})^{2}=0,\quad
j=0,\ldots,n.$ (4.51)
Next, let us define a polynomial $H_{n}$ on $\mathbb{C}\times\Omega_{\mu}$
such that
$R_{2n+2}(z)-z^{2}G_{n}(z)^{2}=zF_{n+1}(z)H_{n}(z)$ (4.52)
holds. Such a polynomial $H_{n}$ exists since the left-hand side of (4.52)
vanishes at $z=\mu_{j},~{}j=0,\cdots,n$ by (4.51). We need to determine the
degree of $H_{n}$. By (4.49), we compute
$R_{2n+2}(z)-z^{2}G_{n}(z)^{2}\underset{|z|\rightarrow\infty}{=}h_{0}z^{2n+2}+O(z^{2n+1}),$
(4.53)
with $O(z^{2n+1})$ depending on $x$ by inspection. Therefore, combining
(4.48), (4.49), (4.52) and (4.53), we conclude that $H_{n}$ has degree $n$
with respect to $z$, with the coefficient $h_{0}$ of powers $z^{n}$. Hence, we
may write $H_{n}$ as
$H_{n}(z)=h_{0}\prod_{l=1}^{n}(z-\nu_{l}),\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$}.$ (4.54)
Next, let us consider the polynomial $P_{n}$ by
$P_{n}(z)=H_{n}(z)+u_{xx}F_{n+1}(z)+zG_{n,x}(z).$ (4.55)
Using (4.48), (4.49) and (4.54) we obtain that $P_{n}$ is a polynomial of
degree at most $n$. Differentiating on both sides of (4.52) with respect to
$x$ yields
$2z^{2}G_{n}(z)G_{n,x}(z)+zF_{n+1,x}(z)H_{n}(z)+zF_{n+1}(z)H_{n,x}(z)=0\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$}.$ (4.56)
Multiplying (4.55) by $G_{n}$ and using (4.56), we have
$\displaystyle G_{n}(z)P_{n}(z)$ $\displaystyle=$ $\displaystyle
F_{n+1}(z)(u_{xx}G_{n}(z)-\frac{1}{2}H_{n,x}(z))$ (4.57)
$\displaystyle+~{}(G_{n}(z)-\frac{1}{2}F_{n+1,x}(z))H_{n}(z),$
and hence
$G_{n}(\mu_{j})P_{n}(\mu_{j})=0,\qquad j=1,\ldots,n,$ (4.58)
on $\Omega_{\mu}$ by using (4.49).
Next, let $x\in\widetilde{\Omega}_{\mu}\subseteq\Omega_{\mu}$, where
$\widetilde{\Omega}_{\mu}$ is given by
$\displaystyle\widetilde{\Omega}_{\mu}$ $\displaystyle=$
$\displaystyle\\{x\in\Omega_{\mu}\mid
G(\mu_{j}(x),x)=-\frac{y(\hat{\mu}_{j}(x))}{\mu_{j}(x)}\neq
0,\,j=0,\cdots,n\\}$ (4.59) $\displaystyle=$
$\displaystyle\\{x\in\Omega_{\mu}\mid\mu_{j}(x)\notin\\{E_{m}\\}_{m=0,\cdots,2n+1},j=0,\cdots,n\\},$
Thus, we have
$P_{n}(\mu_{j}(x),x)=0,\quad j=0,\cdots,n,\,~{}x\in\widetilde{\Omega}_{\mu}.$
(4.60)
Since $P_{n}$ is a polynomial of degree at most $n$, (4.60) implies
$P_{n}=0\quad\textrm{on $\mathbb{C}\times\widetilde{\Omega}_{\mu}$},$ (4.61)
So, (2.17) holds, that is,
$zG_{n,x}(z)=-H_{n}(z)-u_{xx}F_{n+1}(z)\quad\textrm{on
$\mathbb{C}\times\widetilde{\Omega}_{\mu}$}.$ (4.62)
Inserting (4.62) and (4.49) into (4.56) yields
$zF_{n+1}(z)(-2u_{xx}G_{n}(z)+H_{n,x}(z))=0,$ (4.63)
namely
$H_{n,x}(z)=2u_{xx}G_{n}(z),\quad\textrm{on
$\mathbb{C}\times\widetilde{\Omega}_{\mu}$}.$ (4.64)
Thus, we obtain the fundamental equations (2.15)-(2.17), and (2.19) on
$\mathbb{C}\times\widetilde{\Omega}_{\mu}$.
In order to extend these results to all $x\in\Omega_{\mu}$, let us consider
the case where $\hat{\mu}_{j}$ admits one of the branch points
$(E_{m_{0}},0)$. Hence, we suppose
$\mu_{j_{1}}(x)\rightarrow E_{m_{0}}\quad\textrm{as $x\rightarrow
x_{0}\in\Omega_{\mu}$},$ (4.65)
for some $j_{1}\in\\{0,\cdots,n\\},\,m_{0}\in\\{1,\cdots,2n+1\\}$. Introducing
$\begin{split}&\zeta_{j_{1}}(x)=\sigma(\mu_{j_{1}}(x)-E_{m_{0}})^{1/2},\quad\sigma=\pm
1,\quad\\\ &\mu_{j_{1}}(x)=E_{m_{0}}+\zeta_{j_{1}}(x)^{2}\end{split}$ (4.66)
for some $x$ in an open interval centered near $x_{0}$, then the Dubrovin
equation (3.38) for $\mu_{j_{1}}$ becomes
$\displaystyle\zeta_{j_{1},x}(x)$ $\displaystyle=$ $\displaystyle
c(\sigma)\frac{a}{E_{m_{0}}}\Big{(}\prod_{\scriptstyle m=0\atop\scriptstyle
m\neq m_{0}}^{2n+1}(E_{m_{0}}-E_{m})\Big{)}^{1/2}$ (4.67)
$\displaystyle\times\prod_{\scriptstyle k=0\atop\scriptstyle k\neq
j_{1}}^{n}(E_{m_{0}}-\mu_{k}(x))^{-1}(1+O(\zeta_{j_{1}}(x)^{2}))$
for some $|c(\sigma)|=1$. Hence (4.61)-(4.64) extend to $\Omega_{\mu}$ by
continuity. Consequently, we obtain relations (2.15)-(2.17) on
$\mathbb{C}\times\Omega_{\mu}$, and can proceed as in Section 2 to see that
$u$ satisfies the stationary HS hierarchy (4.47). $\square$
###### Remark 4.11
The result in Theorem $4.10$ is derived in terms of $u$ and
$\\{\mu_{j}\\}_{j=0,\cdots,n}$, but one can prove the analogous result in
terms of $u$ and $\\{\nu_{l}\\}_{l=1,\cdots,n}$.
###### Remark 4.12
Theorem $4.10$ reveals that given $\mathcal{K}_{n}$ and the initial condition
$(\hat{\mu}_{0}(x_{0}),\hat{\mu}_{1}(x_{0}),\ldots,\hat{\mu}_{n}(x_{0}))$, or
equivalently, the auxiliary divisor
$\mathcal{D}_{\hat{\mu}_{0}(x_{0})\underline{\hat{\mu}}(x_{0})}$ at $x=x_{0}$,
$u$ is uniquely determined in an open neighborhood $\Omega$ of $x_{0}$ by
$(\ref{4.46})$ and satisfies the $n$th stationary HS equation $(\ref{2.34})$.
Conversely, given $\mathcal{K}_{n}$ and $u$ in an open neighborhood $\Omega$
of $x_{0}$, we can construct the corresponding polynomial $F_{n+1}(z,x)$,
$G_{n}(z,x)$ and $H_{n}(z,x)$ for $x\in\Omega$, and then obtain the auxiliary
divisor $\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$ for
$x\in\Omega$ from the zeros of $F_{n+1}(z,x)$ and $(\ref{3.10})$. In that
sense, once the curve $\mathcal{K}_{n}$ is fixed, elements of the isospectral
class of the HS potentials $u$ can be characterized by nonspecial auxiliary
divisor $\mathcal{D}_{\hat{\mu}_{0}(x)\underline{\hat{\mu}}(x)}$.
## 5 The time-dependent HS formalism
In this section, let us go back to the recursive approach detailed in Section
2 and extend the the algebro-geometric analysis of Section 3 to the time-
dependent HS hierarchy.
Throughout this section we assume (LABEL:2.2) to hold.
The time-dependent algebro-geometric initial value problem of the HS hierarchy
is to solve the time-dependent $r$th HS flow with a stationary solution of the
$n$th equation as initial data in the hierarchy. More precisely, given
$n\in\mathbb{N}_{0}$, based on the solution $u^{(0)}$ of the $n$th stationary
HS equation $\textrm{s-HS}_{n}(u^{(0)})=0$ associated with $\mathcal{K}_{n}$
and a set of integration constants
$\\{c_{l}\\}_{l=1,\ldots,n}\subset\mathbb{C}$, we want to build up a solution
$u$ of the $r$th HS flow $\mathrm{HS}_{r}(u)=0$ such that $u(t_{0,r})=u^{(0)}$
for some $t_{0,r}\in\mathbb{R},~{}r\in\mathbb{N}_{0}$.
We employ the notations $\widetilde{V}_{r},$ $\widetilde{F}_{r+1},$
$\widetilde{G}_{r},$ $\widetilde{H}_{r},$ $\tilde{f}_{s}$, $\tilde{g}_{s},$
$\tilde{h}_{s}$ to stand for the time-dependent quantities, which are obtained
in $V_{n},$ $F_{n+1},$ $G_{n},$ $H_{n},$ $f_{l},$ $g_{l},$ $h_{l}$ by
replacing $\\{c_{l}\\}_{l=1,\ldots,n}$ with
$\\{\tilde{c}_{s}\\}_{s=1,\ldots,r}$, where the integration constants
$\\{c_{l}\\}_{l=1,\ldots,n}\subset\mathbb{C}$ in the stationary HS hierarchy
and $\\{\tilde{c}_{s}\\}_{s=1,\ldots,r}\subset\mathbb{C}$ in the time-
dependent HS hierarchy are independent of each other. In addition, we mark the
individual $r$th HS flow by a separate time variable $t_{r}\in\mathbb{R}$.
Let us now provide the time-dependent algebro-geometric initial value problem
as follows
$\displaystyle\begin{split}&\mathrm{HS}_{r}(u)=-u_{xxt_{r}}+u_{xxx}\tilde{f}_{r+1}(u)+2u_{xx}\tilde{f}_{r+1,x}(u)=0,\\\
&u|_{t_{r}=t_{0,r}}=u^{(0)},\end{split}$ (5.1)
$\displaystyle\textrm{s-HS}_{n}(u^{(0)})=u_{xxx}f_{n+1}(u^{(0)})+2u_{xx}f_{n+1,x}(u^{(0)})=0,$
(5.2)
where $t_{0,r}\in\mathbb{R},$ $n,r\in\mathbb{N}_{0}$, $u=u(x,t_{r})$ satisfies
the condition (LABEL:2.2), and the curve $\mathcal{K}_{n}$ is associated with
the initial data $u^{(0)}$ in (5.2). Noticing that the HS flows are
isospectral, we are going a further step and assume that (5.2) holds not only
at $t_{r}=t_{0,r}$, but also at all $t_{r}\in\mathbb{R}$.
Let us now start from the zero-curvature equations (2.41)
$U_{t_{r}}-\widetilde{V}_{r,x}+[U,\widetilde{V}_{r}]=0,$ (5.3)
$-V_{n,x}+[U,V_{n}]=0,$ (5.4)
where
$\begin{split}&U(z)=\left(\begin{array}[]{cc}0&1\\\ -z^{-1}u_{xx}&0\\\
\end{array}\right)\\\
&V_{n}(z)=\left(\begin{array}[]{cc}-G_{n}(z)&F_{n+1}(z)\\\
z^{-1}H_{n}(z)&G_{n}(z)\\\ \end{array}\right)\\\
&\widetilde{V}_{r}(z)=\left(\begin{array}[]{cc}-\widetilde{G}_{r}(z)&\widetilde{F}_{r+1}(z)\\\
z^{-1}\widetilde{H}_{r}(z)&\widetilde{G}_{r}(z)\\\
\end{array}\right)\end{split}$ (5.5)
and
$\displaystyle
F_{n+1}(z)=\sum_{l=0}^{n+1}f_{l}z^{n+1-l}=\prod_{j=0}^{n}(z-\mu_{j}),$ (5.6)
$\displaystyle G_{n}(z)=\sum_{l=0}^{n}g_{l}z^{n-l},$ (5.7) $\displaystyle
H_{n}(z)=\sum_{l=0}^{n}h_{l}z^{n-l}=h_{0}\prod_{l=1}^{n}(z-\nu_{l}),$ (5.8)
$\displaystyle\widetilde{F}_{r+1}(z)=\sum_{s=0}^{r+1}\tilde{f}_{s}z^{r+1-s},$
(5.9) $\displaystyle\widetilde{G}_{r}(z)=\sum_{s=0}^{r}\tilde{g}_{s}z^{r-s},$
(5.10) $\displaystyle\widetilde{H}_{r}(z)=\sum_{s=0}^{r}\tilde{h}_{s}z^{r-s},$
(5.11)
for fixed $n,r\in\mathbb{N}_{0}$. Here $\\{f_{l}\\}_{l=0,\ldots,n+1},$
$\\{g_{l}\\}_{l=0,\ldots,n}$, $\\{h_{l}\\}_{l=0,\ldots,n}$, and
$\\{\tilde{f}_{s}\\}_{s=0,\ldots,r+1},$ $\\{\tilde{g}_{s}\\}_{s=0,\ldots,r}$,
$\\{\tilde{h}_{s}\\}_{s=0,\ldots,r}$, satisfy the relations in (2.3).
Moreover, it is more convenient for us to rewrite the zero-curvature equations
(5.3) and (5.4) as the following forms,
$-u_{xxt_{r}}-\widetilde{H}_{r,x}+2u_{xx}\widetilde{G}_{r}=0,$ (5.12)
$\widetilde{F}_{r+1,x}=2\widetilde{G}_{r},$ (5.13)
$z\widetilde{G}_{r,x}=-\widetilde{H}_{r}-u_{xx}\widetilde{F}_{r+1}$ (5.14)
and
$F_{n+1,x}=2G_{n},$ (5.15) $H_{n,x}=2u_{xx}G_{n},$ (5.16)
$zG_{n,x}=-H_{n}-u_{xx}F_{n+1}.$ (5.17)
From (5.15)-(5.17), we may compute
$\frac{d}{dx}\mathrm{det}(V_{n}(z))=-\frac{1}{z^{2}}\frac{d}{dx}\Big{(}z^{2}G_{n}(z)^{2}+zF_{n+1}(z)H_{n}(z)\Big{)}=0,$
(5.18)
and meanwhile Lemma 5.2 gives
$\frac{d}{dt_{r}}\mathrm{det}(V_{n}(z))=-\frac{1}{z^{2}}\frac{d}{dt_{r}}\Big{(}z^{2}G_{n}(z)^{2}+zF_{n+1}(z)H_{n}(z)\Big{)}=0,$
(5.19)
Hence, $z^{2}G_{n}(z)^{2}+zF_{n+1}(z)H_{n}(z)$ is independent of variables
both $x$ and $t_{r}$, which implies
$z^{2}G_{n}(z)^{2}+zF_{n+1}(z)H_{n}(z)=R_{2n+2}(z).$ (5.20)
This reveals that the fundamental identity (2.19) still holds in the time-
dependent context. Consequently the hyperelliptic curve $\mathcal{K}_{n}$ is
still available by (2.27).
Next, let us introduce the time-dependent Baker-Akhiezer function
$\psi(P,x,x_{0},$ $t_{r},t_{0,r})$ on
$\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}$ by
$\begin{split}&\psi(P,x,x_{0},t_{r},t_{0,r})=\left(\begin{array}[]{c}\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\\\
\psi_{2}(P,x,x_{0},t_{r},t_{0,r})\\\ \end{array}\right),\\\
&\psi_{x}(P,x,x_{0},t_{r},t_{0,r})=U(u(x,t_{r}),z(P))\psi(P,x,x_{0},t_{r},t_{0,r}),\\\
&\psi_{t_{r}}(P,x,x_{0},t_{r},t_{0,r})=\widetilde{V}_{r}(u(x,t_{r}),z(P))\psi(P,x,x_{0},t_{r},t_{0,r}),\\\
&zV_{n}(u(x,t_{r}),z(P))\psi(P,x,x_{0},t_{r},t_{0,r})=y(P)\psi(P,x,x_{0},t_{r},t_{0,r}),\\\
&\psi_{1}(P,x_{0},x_{0},t_{0,r},t_{0,r})=1;\\\
&P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},~{}(x,t_{r})\in\mathbb{R}^{2},\end{split}$
(5.21)
where
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})$ $\displaystyle=$
$\displaystyle\mathrm{exp}\Big{(}\int_{t_{0,r}}^{t_{r}}ds(z^{-1}\widetilde{F}_{r+1}(z,x_{0},s)\phi(P,x_{0},s)$
(5.22)
$\displaystyle-\widetilde{G}_{r}(z,x_{0},s))+z^{-1}\int_{x_{0}}^{x}dx^{\prime}\phi(P,x^{\prime},t_{r})\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\}.$
Closely related to $\psi(P,x,x_{0},t_{r},t_{0,r})$ is the following
meromorphic function $\phi(P,x,t_{r})$ on $\mathcal{K}_{n}$ defined by
$\phi(P,x,t_{r})=z\frac{\psi_{1,x}(P,x,x_{0},t_{r},t_{0,r})}{\psi_{1}(P,x,x_{0},t_{r},t_{0,r})},\quad
P\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},~{}(x,t_{r})\in\mathbb{R}^{2}.$
(5.23)
which implies by (5.21) that
$\displaystyle\phi(P,x,t_{r})$ $\displaystyle=$
$\displaystyle\frac{y+zG_{n}(z,x,t_{r})}{F_{n+1}(z,x,t_{r})}$ (5.24)
$\displaystyle=$ $\displaystyle\frac{zH_{n}(z,x,t_{r})}{y-zG_{n}(z,x,t_{r})},$
and
$\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\phi(P,x,t_{r})/z.$
(5.25)
In analogy to equations (3.10) and (3.11), we define
$\displaystyle\hat{\mu}_{j}(x,t_{r})=(\mu_{j}(x,t_{r}),-\mu_{j}(x,t_{r})G_{n}(\mu_{j}(x,t_{r}),x,t_{r}))\in\mathcal{K}_{n},$
$\displaystyle j=0,\ldots,n,~{}(x,t_{r})\in\mathbb{R}^{2},$ (5.26)
$\displaystyle\hat{\nu}_{l}(x,t_{r})=(\nu_{l}(x,t_{r}),\nu_{l}(x,t_{r})G_{n}(\nu_{l}(x,t_{r}),x,t_{r}))\in\mathcal{K}_{n},$
$\displaystyle l=1,\ldots,n,~{}(x,t_{r})\in\mathbb{R}^{2}.$ (5.27)
The regular properties of $F_{n+1}$, $H_{n}$, $\mu_{j}$ and $\nu_{l}$ are
analogous to those in Section 3 due to assumptions (LABEL:2.2).
From (5.24), the the divisor $(\phi(P,x,t_{r}))$ of $\phi(P,x,t_{r})$ reads
$(\phi(P,x,t_{r}))=\mathcal{D}_{P_{0}\underline{\hat{\nu}}(x,t_{r})}(P)-\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}(P)$
(5.28)
where
$\underline{\hat{\mu}}=\\{\hat{\mu}_{1},\ldots,\hat{\mu}_{n}\\},\quad\underline{\hat{\nu}}=\\{\hat{\nu}_{1},\ldots,\hat{\nu}_{n}\\}\in\mathrm{Sym}^{n}(\mathcal{K}_{n}).$
(5.29)
That means $P_{0},\hat{\nu}_{1}(x,t_{r}),\ldots,\hat{\nu}_{n}(x,t_{r})$ are
the $n+1$ zeros of $\phi(P,x,t_{r})$ and
$\hat{\mu}_{0}(x,t_{r}),\hat{\mu}_{1}(x,t_{r}),\ldots,\hat{\mu}_{n}(x,t_{r})$
its $n+1$ poles.
Further properties of $\phi(P,x,t_{r})$ are summarized as follows.
###### Lemma 5.1
Assume that $(\ref{2.2})$, $(\ref{5.3})$ and $(\ref{5.4})$ hold. Let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},~{}(x,t_{r})\in\mathbb{R}^{2}.$
Then
$\phi_{x}(P)+z^{-1}\phi(P)^{2}=-u_{xx},$ (5.30) $\displaystyle\phi_{t_{r}}(P)$
$\displaystyle=$
$\displaystyle(-z\widetilde{G}_{r}(z)+\widetilde{F}_{r+1}(z)\phi(P))_{x}$
$\displaystyle=$
$\displaystyle\widetilde{H}_{r}(z)+u_{xx}\widetilde{F}_{r+1}(z)+(\widetilde{F}_{r+1}(z)\phi(P))_{x},$
$\phi_{t_{r}}(P)=\widetilde{H}_{r}(z)+2\widetilde{G}_{r}(z)\phi(P)-z^{-1}\widetilde{F}_{r+1}(z)\phi(P)^{2},$
(5.32) $\phi(P)\phi(P^{\ast})=-\frac{zH_{n}(z)}{F_{n+1}(z)},$ (5.33)
$\phi(P)+\phi(P^{\ast})=2\frac{zG_{n}(z)}{F_{n+1}(z)},$ (5.34)
$\phi(P)-\phi(P^{\ast})=\frac{2y}{F_{n+1}(z)}.$ (5.35)
Proof. We just need to prove (5.1) and (5.32). Equations (5.30) and
(5.33)-(5.35) can be proved as in Lemma 3.1. By using (5.21) and (5.23), we
obtain
$\displaystyle\phi_{t_{r}}$ $\displaystyle=$ $\displaystyle
z(\mathrm{ln}\psi_{1})_{xt_{r}}=z(\mathrm{ln}\psi_{1})_{t_{r}x}=z\Big{(}\frac{\psi_{1,t_{r}}}{\psi_{1}}\Big{)}_{x}$
(5.36) $\displaystyle=$ $\displaystyle
z\Big{(}\frac{-\widetilde{G}_{r}\psi_{1}+\widetilde{F}_{r+1}\psi_{2}}{\psi_{1}}\Big{)}_{x}$
$\displaystyle=$
$\displaystyle(-z\widetilde{G}_{r}+\widetilde{F}_{r+1}\phi)_{x},$
which is the fist line of (5.1). Inserting (5.14) into (5.36) yields the
second line of (5.1). Then by the definition of $\phi$ (5.23), one may have
$\displaystyle\phi_{t_{r}}$ $\displaystyle=$ $\displaystyle
z\Big{(}\frac{\psi_{2}}{\psi_{1}}\Big{)}_{t_{r}}=z\Big{(}\frac{\psi_{2,t_{r}}}{\psi_{1}}-\frac{\psi_{2}\psi_{1,t_{r}}}{\psi_{1}^{2}}\Big{)}$
(5.37) $\displaystyle=$ $\displaystyle
z\Big{(}\frac{z^{-1}\widetilde{H}_{r}\psi_{1}+\widetilde{G}_{r}\psi_{2}}{\psi_{1}}-z^{-1}\phi\frac{-\widetilde{G}_{r}\psi_{1}+\widetilde{F}_{r+1}\psi_{2}}{\psi_{1}}\Big{)}$
$\displaystyle=$
$\displaystyle\widetilde{H}_{r}+2\widetilde{G}_{r}\phi-z^{-1}\widetilde{F}_{r+1}\phi^{2},$
which is (5.32). Alternatively, one can insert (5.12)-(5.14) into (5.1) to
obtain (5.32). $\square$
Next we study the time evolution of $F_{n+1}$, $G_{n}$ and $H_{n}$ by using
zero-curvature equations (5.12)-(5.14) and (5.15)-(5.17).
###### Lemma 5.2
Assume that $(\ref{2.2})$ ,$(\ref{5.3})$ and $(\ref{5.4})$ hold. Then
$F_{n+1,t_{r}}=2(G_{n}\widetilde{F}_{r+1}-\widetilde{G}_{r}F_{n+1}),$ (5.38)
$zG_{n,t_{r}}=\widetilde{H}_{r}F_{n+1}-H_{n}\widetilde{F}_{r+1},$ (5.39)
$H_{n,t_{r}}=2(H_{n}\widetilde{G}_{r}-G_{n}\widetilde{H}_{r}).$ (5.40)
Equations $(\ref{5.38})-(\ref{5.40})$ imply
$-V_{n,t_{r}}+[\widetilde{V}_{r},V_{n}]=0.$ (5.41)
Proof. Differentiating both sides of (5.35) with respect to $t_{r}$ leads to
$(\phi(P)-\phi(P^{\ast}))_{t_{r}}=-2yF_{n+1,t_{r}}F_{n+1}^{-2}.$ (5.42)
On the other hand, by (5.32), (5.34) and (5.35), the left-hand side of (5.42)
equals to
$\displaystyle\phi(P)_{t_{r}}-\phi(P^{\ast})_{t_{r}}$ $\displaystyle=$
$\displaystyle
2\widetilde{G}_{r}(\phi(P)-\phi(P^{\ast}))-z^{-1}\widetilde{F}_{r+1}(\phi(P)^{2}-\phi(P^{\ast})^{2})$
(5.43) $\displaystyle=$ $\displaystyle
4y(\widetilde{G}_{r}F_{n+1}-\widetilde{F}_{r+1}G_{n})F_{n+1}^{-2}.$
Combining (5.42) with (5.43) yields (5.38). Similarly, Differentiating both
sides of (5.34) with respect to $t_{r}$ gives
$(\phi(P)+\phi(P^{\ast}))_{t_{r}}=2z(G_{n,t_{r}}F_{n+1}-G_{n}F_{n+1,t_{r}})F_{n+1}^{-2},$
(5.44)
Meanwhile, by (5.32), (5.33) and (5.34), the left-hand side of (5.44) equals
to
$\displaystyle\phi(P)_{t_{r}}+\phi(P^{\ast})_{t_{r}}$ $\displaystyle=$
$\displaystyle
2\widetilde{G}_{r}(\phi(P)+\phi(P^{\ast}))-z^{-1}\widetilde{F}_{r+1}(\phi(P)^{2}+\phi(P^{\ast})^{2})+2\widetilde{H}_{r}$
(5.45) $\displaystyle=$
$\displaystyle-2zG_{n}F_{n+1}^{-2}F_{n+1,t_{r}}+2F_{n+1}^{-1}(\widetilde{H}_{r}F_{n+1}-\widetilde{F}_{r+1}H_{n}).$
Thus, (5.39) clearly follows by (5.44) and (5.45). Hence, insertion of (5.38)
and (5.39) into the differentiation of
$z^{2}G_{n}^{2}+zF_{n+1}H_{n}=R_{2n+2}(z)$ can derive (5.40). Finally, a
direct calculation shows that (5.38)-(5.40) are equivalent to (5.41).
$\square$
Further properties of $\psi$ are summarized as follows.
###### Lemma 5.3
Assume that $(\ref{2.2})$, $(\ref{5.3})$ and $(\ref{5.4})$ hold. Let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},~{}(x,x_{0},t_{r}.t_{0,r})\in\mathbb{R}^{4}.$
Then, we have
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})$ $\displaystyle=$
$\displaystyle\Big{(}\frac{F_{n+1}(z,x,t_{r})}{F_{n+1}(z,x_{0},t_{0,r})}\Big{)}^{1/2}$
(5.46) $\displaystyle\times$
$\displaystyle\mathrm{exp}\Big{(}\frac{y}{z}\int_{t_{0,r}}^{t_{r}}ds\widetilde{F}_{r+1}(z,x_{0},s)F_{n+1}(z,x_{0},s)^{-1}$
$\displaystyle+\frac{y}{z}\int_{x_{0}}^{x}dx^{\prime}F_{n+1}(z,x^{\prime},t_{r})^{-1}\Big{)},$
$\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\psi_{1}(P^{\ast},x,x_{0},t_{r},t_{0,r})=\frac{F_{n+1}(z,x,t_{r})}{F_{n+1}(z,x_{0},t_{0,r})},$
(5.47)
$\psi_{2}(P,x,x_{0},t_{r},t_{0,r})\psi_{2}(P^{\ast},x,x_{0},t_{r},t_{0,r})=-\frac{H_{n}(z,x,t_{r})}{zF_{n+1}(z,x_{0},t_{0,r})},$
(5.48)
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\psi_{2}(P^{\ast},x,x_{0},t_{r},t_{0,r})$
(5.49)
$\displaystyle~{}~{}~{}~{}+\psi_{1}(P^{\ast},x,x_{0},t_{r},t_{0,r})\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=2\frac{G_{n}(z,x,t_{r})}{F_{n+1}(z,x_{0},t_{0,r})},$
$\displaystyle\psi_{1}(P,x,x_{0},t_{r},t_{0,r})\psi_{2}(P^{\ast},x,x_{0},t_{r},t_{0,r})$
(5.50)
$\displaystyle~{}~{}~{}~{}-\psi_{1}(P^{\ast},x,x_{0},t_{r},t_{0,r})\psi_{2}(P,x,x_{0},t_{r},t_{0,r})=-\frac{2y}{zF_{n+1}(z,x_{0},t_{0,r})}.$
Proof. In order to prove (5.46), let us first consider the part of time
variable in the definition (5.22), that is
$\mathrm{exp}\Big{(}\int_{t_{0,r}}^{t_{r}}ds~{}(z^{-1}\widetilde{F}_{r+1}(z,x_{0},s)\phi(P,x_{0},s)-\widetilde{G}_{r}(z,x_{0},s))\Big{)}.$
(5.51)
The integrand in the above integral equals to
$\displaystyle
z^{-1}\widetilde{F}_{r+1}(z,x_{0},s)\phi(P,x_{0},s)-\widetilde{G}_{r}(z,x_{0},s)$
(5.52)
$\displaystyle=z^{-1}\widetilde{F}_{r+1}(z,x_{0},s)\frac{y+zG_{n}(z,x_{0},s)}{F_{n+1}(z,x_{0},s)}-\widetilde{G}_{r}(z,x_{0},s)$
$\displaystyle=\frac{y}{z}\widetilde{F}_{r+1}(z,x_{0},s)F_{n+1}(z,x_{0},s)^{-1}+(\widetilde{F}_{r+1}(z,x_{0},s)G_{n}(z,x_{0},s)$
$\displaystyle~{}~{}~{}~{}-\widetilde{G}_{r}(z,x_{0},s)F_{n+1}(z,x_{0},s))F_{n+1}(z,x_{0},s)^{-1}$
$\displaystyle=\frac{y}{z}\widetilde{F}_{r+1}(z,x_{0},s)F_{n+1}(z,x_{0},s)^{-1}+\frac{1}{2}\frac{F_{n+1,s}(z,x_{0},s)}{F_{n+1}(z,x_{0},s)},$
where we used (5.24) and (5.38). By (5.52), (5.51) reads
$\Big{(}\frac{F_{n+1}(z,x_{0},t_{r})}{F_{n+1}(z,x_{0},t_{0,r})}\Big{)}^{1/2}\mathrm{exp}\Big{(}\frac{y}{z}\int_{t_{0,r}}^{t_{r}}ds\widetilde{F}_{r+1}(z,x_{0},s)F_{n+1}(z,x_{0},s)^{-1}\Big{)}.$
(5.53)
On the other hand, the part of space variable in (5.22) can be written as
$\Big{(}\frac{F_{n+1}(z,x,t_{r})}{F_{n+1}(z,x_{0},t_{r})}\Big{)}^{1/2}\mathrm{exp}\Big{(}\frac{y}{z}\int_{x_{0}}^{x}dx^{\prime}F_{n+1}(z,x^{\prime},t_{r})^{-1}\Big{)},$
(5.54)
which can be proved using the similar procedure to Lemma 3.2. Combining (5.53)
and (5.54) yields (5.46). Evaluating (5.46) at the points $P$ and $P^{\ast}$,
and multiplying the resulting expressions, with noticing
$y(P)+y(P^{\ast})=0,$ (5.55)
leads to (5.47). The remaining statements (5.48)-(5.50) are direct consequence
of (5.25), (5.33)-(5.35) and (5.47). $\square$
In analogy to Lemma 3.4, the dynamics of the zeros
$\\{\mu_{j}(x,t_{r})\\}_{j=0,\ldots,n}$ and
$\\{\nu_{l}(x,t_{r})\\}_{l=1,\ldots,n}$ of $F_{n+1}(z,x,t_{r})$ and
$H_{n}(z,x,t_{r})$ with respect to $x$ and $t_{r}$ are described in terms of
Dubrovin-type equations (see the following Lemma). We assume that the affine
part of $\mathcal{K}_{n}$ to be nonsingular, which implies (3.37) holds in
present context.
###### Lemma 5.4
Assume that $(\ref{2.2})$, $(\ref{5.3})$ and $(\ref{5.4})$ hold.
$(\mathrm{i})$ Suppose that the zeros $\\{\mu_{j}(x,t_{r})\\}_{j=0,\ldots,n}$
of $F_{n+1}(z,x,t_{r})$ remain distinct for $(x,t_{r})\in\Omega_{\mu},$ where
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ is open and connected. Then,
$\\{\mu_{j}(x,t_{r})\\}_{j=0,\ldots,n}$ satisfy the system of differential
equations,
$\mu_{j,x}=2\frac{y(\hat{\mu}_{j})}{\mu_{j}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k})^{-1},\qquad j=0,\ldots,n,$
(5.56)
$\mu_{j,t_{r}}=\frac{2\widetilde{F}_{r+1}(\mu_{j})y(\hat{\mu}_{j})}{\mu_{j}}\prod_{\scriptstyle
k=0\atop\scriptstyle k\neq j}^{n}(\mu_{j}-\mu_{k})^{-1},\qquad j=0,\ldots,n,$
(5.57)
with initial conditions
$\\{\hat{\mu}_{j}(x_{0},t_{0,r})\\}_{j=0,\ldots,n}\in\mathcal{K}_{n},$ (5.58)
for some fixed $(x_{0},t_{0,r})\in\Omega_{\mu}$. The initial value problem
$(\ref{5.57})$, $(\ref{5.58})$ has a unique solution satisfying
$\hat{\mu}_{j}\in C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),\quad
j=0,\ldots,n.$ (5.59)
$\mathrm{(ii)}$ Suppose that the zeros $\\{\nu_{l}(x,t_{r})\\}_{l=1,\ldots,n}$
of $H_{n}(z,x,t_{r})$ remain distinct for $(x,t_{r})\in\Omega_{\nu},$ where
$\Omega_{\nu}\subseteq\mathbb{R}^{2}$ is open and connected. Then,
$\\{\nu_{l}(x,t_{r})\\}_{l=1,\ldots,n}$ satisfy the system of differential
equations,
$\nu_{l,x}=-2\frac{u_{xx}~{}y(\hat{\nu}_{l})}{h_{0}~{}\nu_{l}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq l}^{n}(\nu_{l}-\nu_{k})^{-1},\qquad l=1,\ldots,n,$
(5.60)
$\nu_{l,t_{r}}=\frac{2\widetilde{H}_{r}(\nu_{l})y(\hat{\nu}_{l})}{h_{0}~{}\nu_{l}}\prod_{\scriptstyle
k=1\atop\scriptstyle k\neq l}^{n}(\nu_{l}-\nu_{k})^{-1},\qquad l=1,\ldots,n,$
(5.61)
with initial conditions
$\\{\hat{\nu}_{l}(x_{0},t_{0,r})\\}_{l=1,\ldots,n}\in\mathcal{K}_{n},$ (5.62)
for some fixed $(x_{0},t_{0,r})\in\Omega_{\nu}$. The initial value problem
$(\ref{5.61})$, $(\ref{5.62})$ has a unique solution satisfying
$\hat{\nu}_{l}\in C^{\infty}(\Omega_{\nu},\mathcal{K}_{n}),\quad
l=1,\ldots,n.$ (5.63)
Proof. It suffices to focus on (5.56), (5.57) and (5.59), since the proof
procedure for (5.60), (5.61) and (5.63) is similar.
The proof of (5.56) has been given in Lemma 3.4. We just derive (5.57).
Differentiating on both sides of (5.6) with respect to $t_{r}$ yields
$F_{n+1,t_{r}}(\mu_{j})=-\mu_{j,t_{r}}\prod_{\scriptstyle k=0\atop\scriptstyle
k\neq j}^{n}(\mu_{j}-\mu_{k}).$ (5.64)
On the other hand, inserting $z=\mu_{j}$ into (5.38) and considering (5), we
arrive at
$F_{n+1,t_{r}}(\mu_{j})=2G_{n}(\mu_{j})\widetilde{F}_{r+1}(\mu_{j})=2\frac{y(\hat{\mu}_{j})}{-\mu_{j}}\widetilde{F}_{r+1}(\mu_{j}).$
(5.65)
Combining (5.64) with (5.65) leads to (5.57). The proof of smoothness
assertion (5.59) is analogous to the mCH case in our latest paper [18].
$\square$
Let us now present the $t_{r}$-dependent trace formulas of HS hierarchy, which
are used to construct the algebro-geometric solutions $u$ in section 6. For
simplicity, we just take the simplest case.
###### Lemma 5.5
Assume that $(\ref{2.2})$, $(\ref{5.3})$ and $(\ref{5.4})$ hold. Then, we have
$u=\frac{1}{2}\sum_{j=0}^{n}\mu_{j}-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}.$ (5.66)
Proof. The proof is similar to the corresponding stationary case in Lemma 3.5.
$\square$
## 6 Time-dependent algebro-geometric solutions
In the final section, we extend the results of section 4 from the stationary
HS hierarchy to the time-dependent case. In particular, we obtain Riemann
theta function representations for the Baker-Akhiezer function, the
meromorphic function $\phi$ and the algebro-geometric solutions for the HS
hierarchy.
Let us first consider the asymptotic properties of $\phi$ in the time-
dependent case.
###### Lemma 6.1
Assume that $(\ref{2.2})$,$(\ref{5.3})$ and $(\ref{5.4})$ hold. Let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{\infty_{\pm}},P_{0}\\},$
$(x,t_{r})\in\mathbb{R}^{2}$. Then, we have
$\phi(P)\underset{\zeta\rightarrow 0}{=}-u_{x}+O(\zeta),\qquad P\rightarrow
P_{\infty_{\pm}},\qquad\zeta=z^{-1},$ (6.1) $\phi(P)\underset{\zeta\rightarrow
0}{=}i~{}a\Big{(}\prod_{m=1}^{2n+1}E_{m}\Big{)}^{1/2}f_{n+1}^{-1}\zeta+O(\zeta^{2}),\quad
P\rightarrow P_{0},\quad\zeta=z^{1/2}.$ (6.2)
Proof. The proof is identical to the corresponding stationary case in Lemma
4.1. $\square$
Next, we study the properties of Abel map, which dose not linearize the
divisor $\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$
in the time-dependent HS hierarchy. This is a remarkable difference between
CH, MCH, HS hierarchies and other integrable soliton equations such as KdV and
AKNS hierarchies. For that purpose, we introduce some notations of symmetric
functions.
Let us define
$\begin{split}\mathcal{S}_{k+1}&=\\{\underline{l}=(l_{1},\ldots,l_{k+1})\in\mathbb{N}_{0}^{k+1}|~{}l_{1}<\cdots<l_{k+1}\leq
n\\},\quad k=0,\ldots,n,\\\
\mathcal{T}_{k+1}^{(j)}&=\\{\underline{l}=(l_{1},\ldots,l_{k+1})\in\mathcal{S}_{k+1}|~{}l_{m}\neq
j\\},\quad k=0,\ldots,n-1,~{}j=0,\ldots,n.\end{split}$ (6.3)
The symmetric functions are defined by
$\Psi_{0}(\bar{\mu})=1,\quad\Psi_{k+1}(\bar{\mu})=(-1)^{k+1}\sum_{\underline{l}\in\mathcal{S}_{k+1}}\mu_{l_{1}}\cdots\mu_{l_{k+1}},\quad
k=0,\ldots,n,$ (6.4)
and
$\begin{split}&\Phi_{0}^{(j)}(\bar{\mu})=1,\\\
&\Phi_{k+1}^{(j)}(\bar{\mu})=(-1)^{k+1}\sum_{\underline{l}\in\mathcal{T}_{k+1}^{(j)}}\mu_{l_{1}}\cdots\mu_{l_{k+1}},\\\
&k=0,\ldots,n-1,\quad j=0,\ldots,n,\end{split}$ (6.5)
where $\bar{\mu}=(\mu_{0},\ldots,\mu_{n})\in\mathbb{C}^{n+1}$. The properties
of $\Psi_{k+1}(\bar{\mu})$ and $\Phi_{k+1}^{(j)}(\bar{\mu})$ can be found in
Appendix E [15]. Here we freely use these relations.
Moreover, for the HS hierarchy we have 333$m\wedge n=\mathrm{min}\\{m,n\\}$,
$m\vee n=\mathrm{max}\\{m,n\\}$
$\begin{split}&\widehat{F}_{r+1}(\mu_{j})=\sum_{s=(r-n)\vee
0}^{r+1}\hat{c}_{s}(\underline{E})\Phi_{r+1-s}^{(j)}(\bar{\mu}),\\\
&\widetilde{F}_{r+1}(\mu_{j})=\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\widehat{F}_{s}(\mu_{j})=\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Phi_{k}^{(j)}(\bar{\mu}),\quad
r\in\mathbb{N}_{0},~{}\tilde{c}_{0}=1,\end{split}$ (6.6)
where
$\tilde{d}_{r+1,k}(\underline{E})=\sum_{s=0}^{r+1-k}\tilde{c}_{r+1-k-s}\hat{c}_{s}(\underline{E})\qquad
k=0,\ldots,r+1\wedge n+1.$ (6.7)
###### Theorem 6.2
Assume that $\mathcal{K}_{n}$ is nonsingular and $(\ref{2.2})$ holds. Suppose
that $\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$ satisfies the Dubrovin equations
$(\ref{5.56})$, $(\ref{5.57})$ on $\Omega_{\mu}$ and remain distinct and
$\widetilde{F}_{r+1}(\mu_{j})\neq 0$ for $(x,t_{r})\in\Omega_{\mu}$, where
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ is open and connected. Introducing the
associated divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$, then
$\partial_{x}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})=-\frac{2a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))}\underline{c}(1),\quad(x,t_{r})\in\Omega_{\mu},$
(6.8)
$\displaystyle\partial_{t_{r}}\underline{\alpha}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})$
$\displaystyle=$ $\displaystyle-\frac{2a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))}$
$\displaystyle\times$
$\displaystyle\Big{(}\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Psi_{k}(\bar{\mu}(x,t_{r}))\Big{)}\underline{c}(1)$
$\displaystyle+$ $\displaystyle
2a\Big{(}\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\underline{c}(\ell)\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(x,t_{r})\in\Omega_{\mu}.$
In particular, the Abel map dose not linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$ on
$\Omega_{\mu}$.
Proof. It suffices to prove (6.2), since the proofs of (6.8) has been given in
the stationary context of Theorem 4.3. Let us first give a fundamental
identity (E.17) [15], that is
$\Phi_{k+1}^{(j)}(\bar{\mu})=\mu_{j}\Phi_{k}^{(j)}(\bar{\mu})+\Psi_{k+1}(\bar{\mu}),\quad
k=0,\ldots,n,~{}j=0,\ldots,n.$ (6.10)
Then, together with (6.6) and (4.16), we have
$\displaystyle\frac{\widetilde{F}_{r+1}(\mu_{j})}{\mu_{j}}$ $\displaystyle=$
$\displaystyle\mu_{j}^{-1}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Phi_{m}^{(j)}(\bar{\mu})$
$\displaystyle=$
$\displaystyle\mu_{j}^{-1}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Big{(}\mu_{j}\Phi_{m-1}^{(j)}(\bar{\mu})+\Psi_{m}(\bar{\mu})\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Phi_{m-1}^{(j)}(\bar{\mu})$
$\displaystyle~{}~{}~{}~{}~{}-\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\frac{\Phi_{n}^{(j)}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})}.$
So, using (6), (5.57), (E.9), (E.25) and (E.26) [15], we obtain
$\displaystyle\partial_{t_{r}}\Big{(}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}}\underline{\omega}\Big{)}=\sum_{j=0}^{n}\mu_{j,t_{r}}\sum_{k=1}^{n}\underline{c}(k)\frac{a~{}\mu_{j}^{k-1}}{y(\hat{\mu}_{j})}$
$\displaystyle=2a\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq
j}^{n}(\mu_{j}-\mu_{l})}\frac{\widetilde{F}_{r+1}(\mu_{j})}{\mu_{j}}$
$\displaystyle=2a\sum_{j=0}^{n}\sum_{k=1}^{n}\underline{c}(k)\frac{\mu_{j}^{k-1}}{\prod_{\scriptstyle
l=0\atop\scriptstyle l\neq
j}^{n}(\mu_{j}-\mu_{l})}\Big{(}-\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\frac{\Phi_{n}^{(j)}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})}$
$\displaystyle~{}~{}~{}~{}~{}+\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Phi_{m-1}^{(j)}(\bar{\mu})\Big{)}$
$\displaystyle=$
$\displaystyle-2a\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\frac{\Psi_{m}(\bar{\mu})}{\Psi_{n+1}(\bar{\mu})}\sum_{k=1}^{n}\sum_{j=0}^{n}\underline{c}(k)(U_{n+1}(\bar{\mu}))_{k,j}(U_{n+1}(\bar{\mu}))_{j,1}^{-1}$
(6.12)
$\displaystyle+2a\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\sum_{k=1}^{n}\sum_{j=0}^{n}\underline{c}(k)(U_{n+1}(\bar{\mu}))_{k,j}(U_{n+1}(\bar{\mu}))_{j,n-m+2}^{-1}$
$\displaystyle=$
$\displaystyle-\frac{2a}{\Psi_{n+1}(\bar{\mu})}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\underline{c}(1)$
$\displaystyle+2a\sum_{m=1}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\underline{c}(n-m+2)$
$\displaystyle=$
$\displaystyle-\frac{2a}{\Psi_{n+1}(\bar{\mu})}\sum_{m=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,m}(\underline{E})\Psi_{m}(\bar{\mu})\underline{c}(1)$
$\displaystyle+2a\sum_{m=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-m}(\underline{E})\underline{c}(m).$
Therefore, we complete the proof of (6.2). $\square$
The analogous results hold for the corresponding divisor
$\mathcal{D}_{\underline{\hat{\nu}}(x,t_{r})}$ associated with
$\phi(P,x,t_{r})$.
The following result is a special form of Theorem 6.2, which provides the
constraint condition to linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$
associated with $\phi(P,x,t_{r})$. We recall the definitions of
$\underline{\widehat{B}}_{Q_{0}}$ and $\underline{\hat{\beta}}_{Q_{0}}$ in
(4.20) and (4.21).
###### Theorem 6.3
Assume that $(\ref{2.2})$ holds and the statements of
$\\{\mu_{j}\\}_{j=0,\ldots,n}$ in Theorem $6.2$ are true. Then,
$\partial_{x}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}(x,t_{r})}\eta_{1}=-\frac{2a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))},\qquad(x,t_{r})\in\Omega_{\mu},$
(6.13)
$\partial_{x}\underline{\hat{\beta}}(\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})})=\begin{cases}2a,~{}~{}~{}~{}~{}~{}~{}~{}n=1,\\\
2a(0,\ldots,0,1),~{}~{}~{}~{}~{}~{}~{}~{}n\geq
2,\end{cases}\quad(x,t_{r})\in\Omega_{\mu},$ (6.14)
$\displaystyle\partial_{t_{r}}\sum_{j=0}^{n}\int_{Q_{0}}^{\hat{\mu}_{j}(x,t_{r})}\eta_{1}$
$\displaystyle=$
$\displaystyle-\frac{2a}{\Psi_{n+1}(\bar{\mu}(x,t_{r}))}\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Psi_{k}(\bar{\mu}(x,t_{r}))$
(6.15) $\displaystyle+$ $\displaystyle
2a\tilde{d}_{r+1,n+1}(\underline{E})\delta_{n+1,r+1\wedge
n+1},\quad(x,t_{r})\in\Omega_{\mu},$
$\displaystyle\partial_{t_{r}}\underline{\hat{\beta}}(\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})})$
$\displaystyle~{}~{}=2a\Big{(}\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s+1-n}(\underline{E}),\ldots,\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s+1}(\underline{E}),\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s}(\underline{E}),\Big{)},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\hat{c}_{-l}(\underline{E})=0,~{}l\in\mathbb{N},\quad(x,t_{r})\in\Omega_{\mu}.$
(6.16)
Proof. Equations (6.13) and (6.14) have been proved in the stationary case in
Theorem 4.4. Equations (6.15) and (6.3) follows from (6.12), taking account
into (E.9) [15]. $\square$.
Motivated by Theorem 6.2 and Theorem 6.3, the change of variables
$x\mapsto\tilde{x}=\int^{x}dx^{\prime}\Big{(}\frac{2a}{\Psi_{n+1}(\bar{\mu}(x^{\prime},t_{r}))}\Big{)}$
(6.17)
and
$\displaystyle t_{r}\mapsto\tilde{t}_{r}$ $\displaystyle=$
$\displaystyle\int^{t_{r}}ds\Big{(}\frac{2a}{\Psi_{n+1}(\bar{\mu}(x,s))}\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\Psi_{k}(\bar{\mu}(x,s))$
(6.18)
$\displaystyle~{}~{}~{}~{}~{}-2a\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\frac{\underline{c}(\ell)}{\underline{c}(1)}\Big{)}$
linearizes the Abel map
$\underline{A}_{Q_{0}}(\mathcal{D}_{\hat{\tilde{\mu}}_{0}(\tilde{x},\tilde{t_{r}})\underline{\hat{\tilde{\mu}}}(\tilde{x},\tilde{t}_{r})})$,
$\tilde{\mu}_{j}(\tilde{x},\tilde{t}_{r})=\mu_{j}(x,t_{r})$, $j=0,\ldots,n$.
The intricate relation between the variables $(x,t_{r})$ and
$(\tilde{x},\tilde{t}_{r})$ is detailedly studied in Theorem 6.4.
Next we shall provide an explicit representations of $\phi$ and $u$ in terms
of the Riemann theta function associated with $\mathcal{K}_{n}$, assuming the
affine part of $\mathcal{K}_{n}$ to be nonsingular. Since the Abel map fails
to linearize the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$, one
could argue that it suffices to consider the Dubrovin equations (5.56)-(5.57)
and reconstruct $u$ from the trace formula (5.66). By (4.24)-(4), one of the
principal results reads as follows.
###### Theorem 6.4
Suppose that the curve $\mathcal{K}_{n}$ is nonsingular, $(\ref{2.2})$,
$(\ref{5.3})$ and $(\ref{5.4})$ hold on $\Omega$. Let
$P=(z,y)\in\mathcal{K}_{n}\setminus\\{P_{0}\\}$, and
$(x,t_{r}),(x_{0},t_{0,r})\in\Omega$, where $\Omega\subseteq\mathbb{R}^{2}$ is
open and connected. Moreover, suppose that
$\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})}$, or
$\mathcal{D}_{\underline{\hat{\nu}}(x,t_{r})}$ is nonspecial for
$(x,t_{r})\in\Omega$. Then, $\phi$ and $u$ have the following representations
$\displaystyle\phi(P,x,t_{r})$ $\displaystyle=$ $\displaystyle
ia\Big{(}\prod_{m=1}^{2n+1}E_{m}\Big{)}^{1/2}f_{n+1}^{-1}\frac{\theta(\underline{z}(P,\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P_{0},\underline{\hat{\mu}}(x,t_{r})))}{\theta(\underline{z}(P_{0},\underline{\hat{\nu}}(x,t_{r})))\theta(\underline{z}(P,\underline{\hat{\mu}}(x,t_{r})))}$
(6.19)
$\displaystyle\times~{}\mathrm{exp}\Big{(}d_{0}-\int_{Q_{0}}^{P}\omega_{\hat{\mu}_{0}(x,t_{r})P_{0}}^{(3)}\Big{)},$
$\displaystyle u(x,t_{r})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=1}^{n}\lambda_{j}$
(6.20)
$\displaystyle-\frac{1}{2}\sum_{j=1}^{n}U_{j}\partial_{\omega_{j}}\mathrm{ln}\Big{(}\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r}))+\underline{\omega})}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x,t_{r}))+\underline{\omega})}\Big{)}\Big{|}_{\underline{\omega}=0}$
Moreover, let $\mu_{j}$, $j=0,\ldots,n,$ be nonvanishing on $\Omega$. Then, we
have the following constraint
$\displaystyle
2a(x-x_{0})+2a(t_{r}-t_{0,r})\sum_{s=0}^{r+1}\tilde{c}_{r+1-s}\hat{c}_{s}(\underline{E})$
$\displaystyle=\Big{(}-2a\int_{x_{0}}^{x}\frac{dx^{\prime}}{\prod_{k=0}^{n}\mu_{k}(x^{\prime},t_{r})}-2a\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\int_{t_{0,r}}^{t_{r}}\frac{\Psi_{k}(\bar{\mu}(x_{0},s))}{\Psi_{n+1}(\bar{\mu}(x_{0},s))}ds\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}\times~{}\sum_{j=1}^{n}\Big{(}\int_{a_{j}}\tilde{\omega}_{P_{\infty_{+}}P_{\infty_{-}}}^{(3)}\Big{)}c_{j}(1)$
$\displaystyle~{}~{}~{}~{}~{}+~{}2a(t_{r}-t_{0,r})\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\sum_{j=1}^{n}\Big{(}\int_{a_{j}}\tilde{\omega}_{P_{\infty_{+}}P_{\infty_{-}}}^{(3)}\Big{)}c_{j}(\ell)$
$\displaystyle~{}~{}~{}~{}~{}+~{}\mathrm{ln}\left(\frac{\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x_{0},t_{0,r})))}{\theta(\underline{z}(P_{\infty_{-}},\underline{\hat{\mu}}(x,t_{r})))\theta(\underline{z}(P_{\infty_{+}},\underline{\hat{\mu}}(x_{0},t_{0,r})))}\right),$
(6.21) $(x,t_{r}),(x_{0},t_{0,r})\in\Omega$
with
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})$
$\displaystyle~{}~{}=\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0},t_{r})\underline{\hat{\mu}}(x_{0},t_{r})})-2a\Big{(}\int_{x_{0}}^{x}\frac{dx^{\prime}}{\Psi_{n+1}(\bar{\mu}(x^{\prime},t_{r}))}\Big{)}\underline{c}(1)$
(6.22)
$\displaystyle~{}~{}=\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{0,r})\underline{\hat{\mu}}(x,t_{0,r})})$
$\displaystyle~{}~{}~{}~{}~{}-2a\Big{(}\sum_{k=0}^{(r+1)\wedge(n+1)}\tilde{d}_{r+1,k}(\underline{E})\int_{t_{0,r}}^{t_{r}}\frac{\Psi_{k}(\bar{\mu}(x,s))}{\Psi_{n+1}(\bar{\mu}(x,s))}ds\Big{)}\underline{c}(1)$
$\displaystyle~{}~{}~{}~{}~{}+2a(t_{r}-t_{0,r})\Big{(}\sum_{\ell=1\vee(n+1-r)}^{n+1}\tilde{d}_{r+1,n+2-\ell}(\underline{E})\underline{c}(\ell)\Big{)},$
(6.23) $(x,t_{r}),(x_{0},t_{0,r})\in\Omega.$
Proof. Let us first assume that $\mu_{j}$, $j=0,\ldots,n$, are distinct and
nonvanishing on $\widetilde{\Omega}$ and $\widetilde{F}_{r+1}(\mu_{j})\neq 0$
on $\widetilde{\Omega}$, $j=0,\ldots,n,$ where
$\widetilde{\Omega}\subseteq\Omega$. Then, the representation (6.19) for
$\phi$ on $\widetilde{\Omega}$ follows by combining (5.28), (6.1), (6.2) and
Theorem A.26 [15]. The representation (6.20) for $u$ on $\widetilde{\Omega}$
follows from the trace formulas (5.66) and (F.89) [15]. In fact, since the
proofs of (6.19) and (6.20) are identical to the corresponding stationary
results in Theorem 4.5, which can be extended line by line to the time-
dependent setting, here we omit the corresponding details. The constraint
(6.4) then holds on $\widetilde{\Omega}$ by combining (6.13)-(6.3), and (F.88)
[15]. Equations (6.4) and (6.23) is clear from (6.8) and (6.2). The extension
of all results from $(x,t_{r})\in\widetilde{\Omega}$ to $(x,t_{r})\in\Omega$
then simply follows by the continuity of $\underline{\alpha}_{Q_{0}}$ and the
hypothesis of $\mathcal{D}_{\underline{\hat{\mu}}(x,t_{r})}$ being nonspecial
for $(x,t_{r})\in\Omega$. $\square$
###### Remark 6.5
A closer look at Theorem $6.4$ shows that $(\ref{6.22})$ and $(6.23)$ equal to
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})})$
$\displaystyle=$
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x_{0},t_{r})\underline{\hat{\mu}}(x_{0},t_{r})})-\underline{c}(1)(\tilde{x}-\tilde{x}_{0})$
(6.24) $\displaystyle=$
$\displaystyle\underline{\hat{\alpha}}_{Q_{0}}(\mathcal{D}_{\hat{\mu}_{0}(x,t_{0,r})\underline{\hat{\mu}}(x,t_{0,r})})-\underline{c}(1)(\tilde{t}_{r}-\tilde{t}_{0,r}),$
(6.25)
based on the changing of variables $x\mapsto\tilde{x}$ and
$t_{r}\mapsto\tilde{t}_{r}$ in $(\ref{6.17})$ and $(\ref{6.18})$. Hence, the
Abel map linearizes the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$ on
$\Omega$ with respect to $\tilde{x},\tilde{t}_{r}$. This fact reveals that the
Abel map does not effect the linearization of the divisor
$\mathcal{D}_{\hat{\mu}_{0}(x,t_{r})\underline{\hat{\mu}}(x,t_{r})}$ in the
time-dependent HS case.
###### Remark 6.6
Remark $4.8$ is applicable to the present time-dependent context. Moreover, in
order to obtain the theta function representation of $\psi_{j}$, $j=1,2,$, one
can write $\widetilde{F}_{r+1}$ in terms of $\Psi_{k}(\bar{\mu})$ and use
$(\ref{5.46})$, in analogy to the stationary case studied in Remark $4.9$.
Here we skip the corresponding details.
Let us end this section by providing another principle result about algebro-
geometric initial value problem of HS hierarchy. We will show that the
solvability of the Dubrovin equations (5.56) and (5.57) on
$\Omega_{\mu}\subseteq\mathbb{R}^{2}$ in fact implies (5.3) and (5.4) on
$\Omega_{\mu}$. As pointed out in Remark 4.12, this amounts to solving the
time-dependent algebro-geometric initial value problem (5.1) and (5.2) on
$\Omega_{\mu}$. Recalling definition of $\widetilde{F}_{r+1}(\mu_{j})$
introduced in (6.6), then we may present the following result.
###### Theorem 6.7
Assume that $(\ref{2.2})$ holds and $\\{\hat{\mu}_{j}\\}_{j=0,\ldots,n}$
satisfies the Dubrovin equations $(\ref{5.56})$ and $(\ref{5.57})$ on
$\Omega_{\mu}$ and remain distinct and nonzero for $(x,t_{r})\in\Omega_{\mu}$,
where $\Omega_{\mu}\subseteq\mathbb{R}^{2}$ is open and connected. Moreover,
suppose that $\widetilde{F}_{r+1}(\mu_{j})$ in $(\ref{5.57})$ expressed in
terms of $\mu_{k}$, $k=0,\ldots,n$ by $(\ref{6.6})$. Then $u\in
C^{\infty}(\Omega_{\mu})$ defined by
$u=-\frac{1}{2}\sum_{m=0}^{2n+1}E_{m}+\frac{1}{2}\sum_{j=0}^{n}\mu_{j},$
(6.26)
satisfies the $r$th HS equation $(\ref{5.1})$, that is,
$\mathrm{HS}_{r}(u)=0\quad\textrm{on $\Omega_{\mu}$},$ (6.27)
with initial values satisfying the $n$th stationary HS equation $(5.2)$.
Proof. Given the solutions $\hat{\mu}_{j}=(\mu_{j},y(\hat{\mu}_{j}))\in
C^{\infty}(\Omega_{\mu},\mathcal{K}_{n}),$ $j=0,\cdots,n$ of (5.56) and
(5.57), we introduce polynomials $F_{n+1},G_{n},$ and $H_{n}$ on
$\Omega_{\mu}$, which are exactly the same as in Theorem 4.10 in the
stationary case
$\displaystyle F_{n+1}(z)=\prod_{j=0}^{n}(z-\mu_{j}),$ (6.28) $\displaystyle
G_{n}(z)=\frac{1}{2}F_{n+1,x}(z),$ (6.29) $\displaystyle
zG_{n,x}(z)=-H_{n}(z)-u_{xx}F_{n+1}(z),$ (6.30) $\displaystyle
H_{n,x}(z)=2u_{xx}G_{n}(z),$ (6.31) $\displaystyle
R_{2n+2}(z)=z^{2}G_{n}^{2}(z)+zF_{n+1}(z)H_{n}(z),$ (6.32)
where $t_{r}$ is treated as a parameter. Hence let us focus on the proof of
(5.1).
Let us denote the polynomial $\widetilde{G}_{r}$ and $\widetilde{H}_{r}$ of
degree $r$ by
$\widetilde{G}_{r}(z)=\frac{1}{2}\widetilde{F}_{r+1,x}(z)\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$},$ (6.33)
$\widetilde{H}_{r}(z)=-z\widetilde{G}_{r,x}(z)-u_{xx}\widetilde{F}_{r+1}(z)\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$},$ (6.34)
respectively. Next we want to establish
$F_{n+1,t_{r}}(z)=2(G_{n}(z)\widetilde{F}_{r+1}(z)-F_{n+1}(z)\widetilde{G}_{r}(z))\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$}.$ (6.35)
One computes from (5.56) and (5.57) that
$F_{n+1,x}(z)=-F_{n+1}(z)\sum_{j=0}^{n}\mu_{j,x}(z-\mu_{j})^{-1},$ (6.36)
$F_{n+1,t_{r}}(z)=-F_{n+1}(z)\sum_{j=0}^{n}\widetilde{F}_{r+1}(\mu_{j})\mu_{j,x}(z-\mu_{j})^{-1}.$
(6.37)
Using (6.29) and (6.33) one concludes that (6.35) is equivalent to
$\widetilde{F}_{r+1,x}(z)=\sum_{j=0}^{n}(\widetilde{F}_{r+1}(\mu_{j})-\widetilde{F}_{r+1}(z))\mu_{j,x}(z-\mu_{j})^{-1}.$
(6.38)
Equation (6.38) has been proved in Lemma F.9 [15]. Hence this in turn proves
(6.35).
Next, differentiating (6.29) with respect to $t_{r}$ yields
$F_{n+1,xt_{r}}=2G_{n,t_{r}}.$ (6.39)
On the other hand, the derivative of (6.35) with respect to $x$, taking
account into (6.29), (6.30) and (6.33), we obtain
$\displaystyle F_{n+1,t_{r}x}$ $\displaystyle=$
$\displaystyle-2z^{-1}H_{n}\widetilde{F}_{r+1}-2z^{-1}u_{xx}F_{n+1}\widetilde{F}_{r+1}+2G_{n}\widetilde{F}_{r+1,x}$
(6.40) $\displaystyle-2\widetilde{G}_{r,x}F_{n+1}-4\widetilde{G}_{r}G_{n}.$
Combining (6.34), (6.39) and (6.40) we conclude
$zG_{n,t_{r}}(z)=\widetilde{H}_{r}(z)F_{n+1}(z)-H_{n}(z)\widetilde{F}_{r+1}(z)\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$. }$ (6.41)
Next, differentiating (6.32) with respect to $t_{r}$, inserting the
expressions (6.35) and (6.41) for $F_{n+1,t_{r}}$ and $G_{n,t_{r}}$,
respectively, we obtain
$H_{n,t_{r}}(z)=2(H_{n}(z)\widetilde{G}_{r}(z)-G_{n}(z)\widetilde{H}_{r}(z))\quad\textrm{on
$\mathbb{C}\times\Omega_{\mu}$. }$ (6.42)
Finally, taking the derivative of (6.41) with respect to $x$ and inserting
(6.29), (6.31) and (6.33) for $F_{n+1,x}$, $H_{n,x}$ and
$\widetilde{F}_{r+1,x}$, respectively, yields
$zG_{n,t_{r}x}=F_{n+1}\widetilde{H}_{r,x}+2G_{n}\widetilde{H}_{r}-2u_{xx}G_{n}\widetilde{F}_{r+1}-2H_{n}\widetilde{G}_{r}.$
(6.43)
On the other hand, differentiating (6.30) with respect to $t_{r}$, using
(6.35) and (6.42) for $F_{n+1,t_{r}}$ and $H_{n,t_{r}}$, respectively, leads
to
$zG_{n,xt_{r}}=2G_{n}\widetilde{H}_{r}-2H_{n}\widetilde{G}_{r}-u_{xxt_{r}}F_{n+1}-2u_{xx}(G_{n}\widetilde{F}_{r+1}-\widetilde{G}_{r}F_{n+1})$
(6.44)
Hence, combining (6.43) and (6.44) then yields
$-u_{xxt_{r}}-\widetilde{H}_{r,x}+2u_{xx}\widetilde{G}_{r}=0.$ (6.45)
Thus we proved (5.12)-(5.17) and (5.38)-(5.40) on
$\mathbb{C}\times\Omega_{\mu}$ and hence conclude that $u$ satisfies the $r$th
HS equation (5.1) with initial values satisfying the $n$th stationary HS
equation (5.2) on $\mathbb{C}\times\Omega_{\mu}$. $\square$
###### Remark 6.8
The result in Theorem $6.7$ is presented in terms of $u$ and
$\\{\mu_{j}\\}_{j=0,\cdots,n}$, but of course one can provide the analogous
result in terms of $u$ and $\\{\nu_{l}\\}_{l=1,\cdots,n}$.
The analog of Remark 4.13 directly extends to the current time-dependent HS
hierarchy.
## Acknowledgments
YH and PZ are very grateful to Professor F.Gesztesy for his helps about CH
solutions and relativistic Toda solutions. YH would also like to thank
Professor E.G. Reyes for many valuable suggestions. This work was supported by
grants from the National Science Foundation of China (Project No.10971031),
and the Shanghai Shuguang Tracking Project (Project No.08G
G01).
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|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Hou, Engui Fan and Peng Zhao",
"submitter": "Engui Fan",
"url": "https://arxiv.org/abs/1207.0574"
}
|
1207.0577
|
# Robust Dequantized Compressive Sensing
Ji Liu11footnotemark: 1 ji-liu@cs.wisc.edu Stephen J. Wright22footnotemark: 2
swright@cs.wisc.edu
###### Abstract
We consider the reconstruction problem in compressed sensing in which the
observations are recorded in a finite number of bits. They may thus contain
quantization errors (from being rounded to the nearest representable value)
and saturation errors (from being outside the range of representable values).
Our formulation has an objective of weighted $\ell_{2}$-$\ell_{1}$ type, along
with constraints that account explicitly for quantization and saturation
errors, and is solved with an augmented Lagrangian method. We prove a
consistency result for the recovered solution, stronger than those that have
appeared to date in the literature, showing in particular that asymptotic
consistency can be obtained without oversampling. We present extensive
computational comparisons with formulations proposed previously, and variants
thereof.
###### keywords:
compressive sensing, signal reconstruction, quantization, optimization.
††journal: Applied and Computational Harmonic
Analysislabel1label1footnotetext: Corresponding author.
## 1 Introduction
This paper considers a compressive sensing (CS) system in which the
measurements are represented by a finite number of bits, which we denote by
$B$. By defining a quantization interval $\Delta>0$, and setting
$G:=2^{B-1}\Delta$, we obtain the following values for representable
measurements:
$-G+\frac{\Delta}{2},-G+\frac{3\Delta}{2},\dotsc,-\frac{\Delta}{2},\frac{\Delta}{2},\dotsc,G-\frac{\Delta}{2}.$
(1)
We assume in our model that actual measurements are recorded by rounding to
the nearest value in this set. The recorded observations thus contain (a)
quantization errors, resulting from rounding of the true observation to the
nearest represented number, and (b) saturation errors, when the true
observation lies beyond the range of represented values, namely,
$[-G+\frac{\Delta}{2},G-\frac{\Delta}{2}]$. This setup is seen in some
compressive sensing hardware architectures [see, for example, 15, 20, 19, 21,
9].
Given a sensing matrix $\Phi\in\mathbb{R}^{M\times N}$ and the unknown vector
$x$, the true observations (without noise) would be $\Phi x$. We denote the
recorded observations by the vector $y\in\mathbb{R}^{M}$, whose components
take on the values in (1). We partition $\Phi$ into the following three
submatrices:
* 1.
The saturation parts $\bar{\Phi}_{-}$ and $\bar{\Phi}_{+}$, which correspond
to those recorded measurements that are represented by $-G+{\Delta/2}$ or
$G-{\Delta/2}$, respectively — the two extreme values in (1). We denote the
number of rows in these two matrices combined by $\bar{M}$.
* 2.
The unsaturated part $\tilde{\Phi}\in\mathbb{R}^{\tilde{M}\times N}$, which
corresponds to the measurements that are rounded to non-extreme representable
values.
In some existing analyses [5, 13], the quantization errors are treated as a
random variables following an i.i.d. uniform distribution in the range
$[-{\Delta\over 2},{\Delta\over 2}]$. This assumption makes sense in many
situations (for example, image processing, audio/video processing),
particularly when the quantization interval $\Delta$ is tiny. However, the
assumption of a uniform distribution may not be appropriate when $\Delta$ is
large, or when an inappropriate choice of saturation level $G$ is made. In
this paper, we assume a slightly weaker condition, namely, that the
quantization errors for non-saturated measurements are independent random
variables with zero expectation. (These random variables are of course bounded
uniformly by $\Delta/2$.)
The state-of-the-art formulation to this problem [see 14] is to combine the
basis pursuit model with saturation constraints, as follows:
$\displaystyle\min_{x}~{}$ $\displaystyle\|x\|_{1}$ (2a)
$\displaystyle\mbox{s.t.}\quad\|\tilde{\Phi}x-\tilde{y}\|^{2}$
$\displaystyle\leq\epsilon^{2}\Delta^{2}\;$ ($\ell_{2}$) (2b)
$\displaystyle\bar{\Phi}_{+}x$ $\displaystyle\geq(G-\Delta)\mathbb{1}\;$ ($+$
saturation) (2c) $\displaystyle\bar{\Phi}_{-}x$
$\displaystyle\leq(\Delta-G)\mathbb{1},\;$ ($-$ saturation) (2d)
where $\mathbb{1}$ is a column vector with all entries equal to $1$ and
$\tilde{y}$ is the quantized subvector of the observation vector $y$ that
corresponds to the unsaturated measurements. We refer to this model as “L2 ”
in later discussions. It has been shown that the estimation error arising from
the formulation (2) is bounded by $O(\epsilon\Delta)$ in the $\ell_{2}$ norm
sense [see 14, 6, 13].
The paper proposes a robust model that replaces (2b) with a least-square loss
term in the objective and adds an $\ell_{\infty}$ constraint:
$\displaystyle\min_{x}~{}{1\over 2}\|\tilde{\Phi}x-\tilde{y}\|^{2}$
$\displaystyle+\lambda\Delta\|x\|_{1}$ (3a)
$\displaystyle\mbox{s.t.}\quad\|\tilde{\Phi}x-\tilde{y}\|_{\infty}$
$\displaystyle\leq\Delta/2\;$ ($\ell_{\infty}$) (3b)
$\displaystyle\bar{\Phi}_{+}x$ $\displaystyle\geq(G-\Delta)\mathbb{1}\;$ ($+$
saturation) (3c) $\displaystyle\bar{\Phi}_{-}x$
$\displaystyle\leq(\Delta-G)\mathbb{1}.\;$ ($-$ saturation) (3d)
We refer to this model as LASSO$\infty$ in later discussions. The
$\ell_{\infty}$ constraint (3b) arises from the fact that (unsaturated)
quantization errors are bounded by $\Delta/2$. This constraint may reduce the
feasible region for the recovery problem while retaining feasibility of the
true solution $x^{*}$, thus promoting more robust signal recovery. From the
viewpoint of optimization, the constraint (2b) plays the same role as the
least-square loss term in the objective (3a), when the values of $\epsilon$
and $\lambda$ are related appropriately. However, it will become clear from
our analysis that inclusion of this term in the objective rather than applying
the constraint (2b) can lead a tighter bound on the reconstruction error.
The analysis in this paper shows that when $\Phi$ is a Gaussian ensemble, and
provided that $S\log N=o(M)$ and several mild conditions hold, the estimation
error of for the solution of (3) is bounded by
$\displaystyle\min\left\\{O\left(\sqrt{S(\log
N)/M}\right),O(1)\right\\}\Delta,$
with high probability, where $S$ is the sparsity (the number of nonzero
components in $x^{*}$). This estimate implies that solutions of (3) are, in
the worst case, better than the state-of-the-art model (2), and also better
than the model in which only the $\ell_{\infty}$ constraint (3b) are applied
(in place of the $\ell_{2}$ constraint (2b)), as mentioned by [13]. More
importantly, when the number $\tilde{M}$ of unsaturated measurements goes to
infinity faster than $S\log(N)$, the estimation error for the solution of (3)
vanishes with high probability. (The model (2) does not indicate such an
improvement when more measurements are available.) Although Jacques et al.
[13] show that the estimation error can be eliminated only using an $\ell_{p}$
constraint (in place of the $\ell_{2}$ constraint (2b)) when
$p\rightarrow\infty$, the oversampling condition (that is, the number of
observations required) is more demanding than for our formulation (3).
We use the alternating direction method of multipliers (ADMM) [see 10, 4] to
solve (3). The computational results reported in Section 4 compare the
solution properties for (3) to those for (2) and other formulations. In some
of our examples, we consider choices for the parameter $\lambda$ and
$\epsilon$ that admit the true solution $x^{*}$ as a feasible point with a
specified level of confidence. We find that for these choices of $\lambda$ and
$\epsilon$, the model (3) yields more accurate solutions than the
alternatives, where the signal is sparse and high confidence is desired.
### 1.1 Related Work
There have been several recent works on CS with quantization and saturation.
Laska et al. [14] propose the formulation (2). Jacques et al. [13] replace the
$\ell_{2}$ constraint (2b) by an $\ell_{p}$ constraint ($2\leq p<\infty$) to
handle the oversampling case, and show that values $p$ greater than $2$ lead
to an improvement of factor $1/\sqrt{p+1}$ on the bound of error in the
recovered signal. The model of Zymnis et al. [25] allows Gaussian noise in the
measurements before quantization, and solves the resulting formulation with an
$\ell_{1}$-regularized maximum likelihood formulation. The average distortion
introduced by scalar, vector, and entropy coded quantization of CS is studied
by Dai et al. [8].
The extreme case of 1-bit CS (in which only the sign of the observation is
recorded) has been studied by Gupta et al. [11] and Boufounos and Baraniuk
[3]. In the latter paper, the $\ell_{1}$ norm objective is minimized on the
unit ball, with a sign consistency constraint. The former paper proposes two
algorithms that require at most $O(S\log N)$ measurements to recover the
unknown support of the true signal (though they cannot recover the magnitudes
of the nonzeros reliably).
### 1.2 Notation
We use $\|\cdot\|_{p}$ to denote the $\ell_{p}$ norm, where $1\leq
p\leq\infty$, with $\|\cdot\|$ denoting the $\ell_{2}$ norm. We use $x^{*}$
for the true signal, $\hat{x}$ as the estimated signal (the solution of (3)),
and $h=\hat{x}-x^{*}$ as the difference. As mentioned above, $S$ denotes the
number of nonzero elements of $x^{*}$.
For any $z\in\mathbb{R}^{N}$, we use $z_{i}$ to denote the $i$th component and
$z_{T}$ to denote the subvector corresponding to index set
$T\subset\\{1,2,...,N\\}$. Similarly, we use $\tilde{\Phi}_{T}$ to denote the
column submatrix of $\tilde{\Phi}$ consisting of the columns indexed by $T$.
The cardinality of $T$ is denoted by $|T|$. We use $\tilde{\Phi}_{i}$ to
denote the $i$th column of $\tilde{\Phi}$.
In discussing the dimensions of the problem and how they are related to each
other in the limit (as $N$ and $\tilde{M}$ both approach $\infty$), we make
use of order notation. If $\alpha$ and $\beta$ are both positive quantities
that depend on the dimensions, we write $\alpha=O(\beta)$ if $\alpha$ can be
bounded by a fixed multiple of $\beta$ for all sufficiently large dimensions.
We write $\alpha=o(\beta)$ if for any positive constant $\phi>0$, we have
$\alpha\leq\phi\beta$ for all sufficiently large dimensions. We write
$\alpha=\Omega(\beta)$ if both $\alpha=O(\beta)$ and $\beta=O(\alpha)$.
The projection onto the $\ell_{\infty}$ norm ball with the radius $\lambda$ is
$\mathcal{P}_{\infty}(x,\lambda):=\mbox{sign}(x)\odot\min(|x|,\lambda)$
where $\odot$ denotes componentwise multiplication and $\mbox{sign}(x)$ is the
sign vector of $x$. (The $i$th entry of $\mbox{sign}(x)$ is $1$, $-1$, or $0$
depending on whether $x_{i}$ is positive, negative, or zero, respectively.)
The indicator function $\mathbb{I}_{\Pi}(\cdot)$ for a set $\Pi$ is defined to
be $0$ on $\Pi$ and $\infty$ otherwise.
We partition the sensing matrix $\Phi$ according to saturated and unsaturated
measurements as follows:
$\bar{\Phi}=\left[\begin{matrix}-\bar{\Phi}_{-}\\\
\bar{\Phi}_{+}\end{matrix}\right]~{}\text{and}~{}\Phi=\left[\begin{matrix}\tilde{\Phi}\\\
\bar{\Phi}\end{matrix}\right].$ (4)
The maximum column norm in $\tilde{\Phi}$ is denoted by $f_{\max}$, that is,
$f_{\max}:=\max_{i\in\\{1,2,\dotsc,N\\}}\|\tilde{\Phi}_{i}\|.$ (5)
We define the following quantities associated with a matrix $\Psi$ with $N$
columns:
$\displaystyle\rho^{-}(k,\Psi)$ $\displaystyle:=\min_{|T|\leq
k,h\in\mathbb{R}^{N}}{\|\Psi_{T}h_{T}\|^{2}\over\|h_{T}\|^{2}}$ (6a)
$\displaystyle\rho^{+}(k,\Psi)$ $\displaystyle:=\max_{|T|\leq
k,h\in\mathbb{R}^{N}}{\|\Psi_{T}h_{T}\|^{2}\over\|h_{T}\|^{2}}.$ (6b)
We use the following abbrevations in some places:
$\displaystyle\rho^{-}(k):=\rho^{-}(k,\Phi),\quad$
$\displaystyle{\rho}^{+}(k):={\rho}^{+}(k,\Phi),$
$\displaystyle\tilde{\rho}^{-}(k):={\rho}^{-}(k,\tilde{\Phi}),\quad$
$\displaystyle\tilde{\rho}^{+}(k):={\rho}^{+}(k,\tilde{\Phi}),$
$\displaystyle\bar{\rho}^{-}(k):={\rho}^{-}(k,\bar{\Phi}),\quad$
$\displaystyle\bar{\rho}^{+}(k):={\rho}^{+}(k,\bar{\Phi}).$
Finally, we denote $(z)+:=\max\\{z,0\\}$.
### 1.3 Organization
The ADMM optimization framework for solving (3) is discussed in Section 2.
Section 3 analyzes the properties of the solution of (3) in the worst case and
compares with existing results. Numerical simulations and comparisons of
various formulations are reported in Section 4 and some conclusions are
offered in Section 5. Proofs of the claims in Section 3 appear in the
appendix.
## 2 Algorithm
This section describes the ADMM algorithm for solving (3). For simpler
notation, we combine the saturation constraints as follows:
$\left[\begin{matrix}-\bar{\Phi}_{-}\\\
\bar{\Phi}_{+}\end{matrix}\right]x\geq\left[\begin{matrix}(G-\Delta)\mathbb{1}\\\
(G-\Delta)\mathbb{1}\end{matrix}\right]\;\;\Leftrightarrow\;\;\bar{\Phi}x\geq\bar{y},$
where $\bar{\Phi}$ is defined in (4) and $\bar{y}$ is defined in an obvious
way. To specify ADMM, we introduce auxiliary variables $u$ and $v$, and write
(3) as follows.
$\displaystyle\min_{x}{}$ $\displaystyle{1\over
2}\|\tilde{\Phi}x-\tilde{y}\|^{2}+\lambda\|x\|_{1}$ (7)
$\displaystyle\mbox{s.t.}\quad u$ $\displaystyle=\tilde{\Phi}x-\tilde{y}$
$\displaystyle v$ $\displaystyle=\bar{\Phi}x-\bar{y}$
$\displaystyle\|u\|_{\infty}$ $\displaystyle\leq\Delta/2$ $\displaystyle v$
$\displaystyle\geq\mathbb{0}.$
Introducing Lagrange multipliers $\alpha$ and $\beta$ for the two equality
constraints in (7), we write the augmented Lagrangian for this formulation,
with prox parameter $\theta>0$ as follows:
$\displaystyle L_{A}(x,u,v,\alpha,\beta)$ $\displaystyle={1\over
2}\|\tilde{\Phi}x-\tilde{y}\|^{2}+\lambda\|x\|_{1}+\langle\alpha,u-\tilde{\Phi}x+\tilde{y}\rangle+\langle\beta,v-\bar{\Phi}x+\bar{y}\rangle$
$\displaystyle\quad+{\theta\over
2}\|u-\tilde{\Phi}x+\tilde{y}\|^{2}+{\theta\over
2}\|v-\bar{\Phi}x+\bar{y}\|^{2}+\mathbb{I}_{\|u\|_{\infty}\leq\Delta/2}(u)+\mathbb{I}_{v\geq
0}(v)$
At each iteration of ADMM, we optimize this function with respect to the
primal variables $u$ and $v$ in turn, then update the dual variables $\alpha$
and $\beta$ in a manner similar to gradient descent. The penalty parameter
$\theta$ may be increased before proceeding to the next iteration.
We summarize the ADMM algorithm in Algorithm 1.
Algorithm 1 ADMM for (7)
0: $\tilde{\Phi}$, $\tilde{y}$, $\bar{\Phi}$, $\bar{y}$, $\Delta$, $K$, and
$x$;
1: Initialize $\theta>0$, $\alpha=0$, $\beta=0$, $u=\tilde{\Phi}x-\tilde{y}$,
and $v=\bar{\Phi}x-\bar{y}$;
2: for $k=0:K$ do
3: $u\leftarrow\arg\min_{u}\,L_{A}(x,u,v,\alpha,\beta)$, that is,
$u\leftarrow\mathcal{P}_{\infty}(\tilde{\Phi}x-\tilde{y}-\alpha/\theta,\Delta/2)$;
4: $v\leftarrow\arg\min_{v}\,L_{A}(x,u,v,\alpha,\beta)$, that is,
$v\leftarrow\max(\bar{\Phi}x-\bar{y}-\beta/\theta,0)$;
5: $x\leftarrow\arg\min_{x}\,L_{A}(x,u,v,\alpha,\beta)$;
6: $\alpha\leftarrow\alpha+\theta(u-\tilde{\Phi}x+\tilde{y})$;
7: $\beta\leftarrow\beta+\theta(v-\bar{\Phi}x+\bar{y})$;
8: Possibly increase $\theta$;
9: if stopping criteria is satisfied then
10: break;
11: end if
12: end for
The updates in Steps 3 and 4 have closed-form solutions, as shown. The
function to be minimized in Step 5 consists of an $\|x\|_{1}$ term in
conjunction with a quadratic term in $x$. Many algorithms can be applied to
solve this problem, e.g., the SpaRSA algorithm [23], the accelerated first
order method [18], and the FISTA algorithm [1]. The update strategy for
$\theta$ in Step 7 is flexible. We use the following simple and useful scheme
from He et al. [12] and Boyd et al. [4]:
$\theta:=\left\\{\begin{aligned} &\theta\tau&&\text{if $\|r\|>\mu\|d\|$}\\\
&\theta/\tau&&\text{if $\|r\|<\mu\|d\|$}\\\
&\theta&&\text{otherwise},\end{aligned}\right.$ (8)
where $r$ and $d$ denote the primal and dual residual errors respectively,
specifically,
$r=\left[\begin{matrix}u-\tilde{\Phi}x+\tilde{y}\\\
v-\bar{\Phi}v+\bar{y}\end{matrix}\right]~{}\text{and}~{}d=\theta\left[\begin{matrix}\tilde{\Phi}(x-x_{\mbox{\rm\scriptsize
last}})\\\ \bar{\Phi}(x-x_{\mbox{\rm\scriptsize last}})\end{matrix}\right],$
where $x_{\mbox{\rm\scriptsize last}}$ denotes the previous value of $x$. The
parameters $\mu$ and $\tau$ should be greater than $1$; we used $\mu=10$ and
$\tau=2$. Convergence results for ADMM can be found in [4], for example.
## 3 Analysis
The section analyzes the properties of the solution obtained from our
formulation (3). In Subsection 3.1, we obtain bounds on the norm of the
difference $h$ between the estimator $\hat{x}$ given by (3) and the true
signal $x^{*}$. Our bounds require the true solution $x^{*}$ to be feasible
for the formulation (3); we derive conditions that guarantee that this
condition holds, with a specified probability. In Subsection 3.2, we estimate
the constants that appear in our bounds under certain assumptions, including
an assumption that the full sensing matrix $\Phi$ is Gaussian.
We formalize our assumption about quantization errors as follows.
###### Assumption 1.
The quantization errors $(\tilde{\Phi}x^{*}-\tilde{y})_{i}$,
$i=1,2,\dotsc,\tilde{M}$ are independently distributed with expectation $0$.
(Note that since $\tilde{\Phi}$ and $\tilde{y}$ refer to the unsaturated data,
the quantization error are bounded uniformly by $\Delta/2$.)
### 3.1 Estimation Error Bounds
The following error estimate is our main theorem, proved in the appendix.
###### Theorem 1.
Assume that the true signal $x^{*}$ satisfies
$\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}\leq\lambda\Delta/2,$
(9)
for some value of $\lambda$. Let $s$ be a positive integer in the range
$1,2,\dotsc,N$, and define
$\displaystyle\bar{A}_{0}(\Psi):=$
$\displaystyle{\rho}^{-}(2s,\Psi)-3[{\rho}^{+}(3s,\Psi)-{\rho}^{-}(3s,\Psi)]$
(10a) $\displaystyle\bar{A}_{1}(\Psi):=$ $\displaystyle
4[{\rho}^{+}(3s,\Psi)-{\rho}^{-}(3s,\Psi)],$ (10b)
$\displaystyle\bar{C}_{1}(\Psi):=$ $\displaystyle
4+{\sqrt{10}A_{1}(\Psi)}/{A_{0}(\Psi)},$ (10c)
$\displaystyle\bar{C}_{2}(\Psi):=$ $\displaystyle\sqrt{10/A_{0}(\Psi)}.$ (10d)
We have that for any $T_{0}\subset\\{1,2,...,N\\}$ with $s=|T_{0}|$, if
$A_{0}(\tilde{\Phi})>0$, then
$\displaystyle\|h\|\leq$ $\displaystyle
2\bar{C}_{2}(\tilde{\Phi})^{2}\sqrt{s}\lambda\Delta+\left[\bar{C}_{1}(\tilde{\Phi})/\sqrt{s}\right]\|x^{*}_{T_{0}^{c}}\|_{1}+2.5\bar{C}_{2}(\tilde{\Phi})\sqrt{\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1}},$
(11a) $\displaystyle\|h\|\leq$
$\displaystyle\bar{C}_{2}(\tilde{\Phi})\sqrt{\tilde{M}}\Delta+\left[\bar{C}_{1}(\tilde{\Phi})/\sqrt{s}\right]\|x^{*}_{T_{0}^{c}}\|_{1}.$
(11b)
Suppose that Assumption 1 holds, and let $\pi\in(0,1)$ be given. If we define
$\lambda=\sqrt{2\log{2N/\pi}}f_{\max}$ in (3), then with probability at least
$P=1-\pi$, the inequalities (11a) and (11b) hold.
From the proof in the appendix, one can see that the estimation error bound
(11a) is mainly determined by the least-squares term in the objective (3a),
whereas the estimation error bound (11b) arises from the $L_{\infty}$
constraint (3b).
If we take $T_{0}$ as the support set of $x^{*}$, only the first terms in
(11a) and (11b) remain.
The condition $A_{0}(\tilde{\Phi})>0$ is a sort of restricted isometry (RIP)
condition required in [14]— it assumes reasonable conditioning of column
submatrices of $\tilde{\Phi}$ with $O(S)$ columns. Specifically, the number of
measurements $\tilde{M}$ required to satisfy $\bar{A}_{0}(\tilde{\Phi})>0$ and
RIP are of the same order: $O(S\log(N))$.
### 3.2 Estimating the Constants
Here we discuss the effect of the least-squares term and the $\ell_{\infty}$
constraints by comparing the leading terms on the right-hand sides of (11a)
and (11b). To simplify the comparison, we make the following assumptions.
* (i)
$\Phi$ is a Gaussian random matrix, that is, each entry is i.i.d., drawn from
a standard Gaussian distribution $\mathcal{N}(0,1)$.
* (ii)
the confidence level $P=1-\pi$ is fixed.
* (iii)
$s$ is equal to the sparsity number $S$.
* (iv)
$S\log N=o(M)$.
* (v)
the saturation ratio $\chi:=\bar{M}/M$ is smaller than a small positive
threshold that is defined in Theorem 3.
* (vi)
$T_{0}$ is taken as the support set of $x^{*}$, so that $x^{*}_{T_{0}^{c}}=0$.
Note that (iii) and (iv) together imply that $s=S\ll M$, while (v) implies
that $\tilde{M}=\Omega(M)$.
The discussion following Theorem 3 in Appendix indicates that under these
assumptions, the quantities defined in (10c), (10c), and (5) satisfy the
following estimates:
$\bar{C}_{1}(\tilde{\Phi})=\Omega(1),\quad\bar{C}_{2}(\tilde{\Phi})=\Omega(1/\sqrt{M}),\quad
f_{\max}=\Omega(\sqrt{M}),$
with high probability, for sufficiently high dimensions. Using the estimates
in Theorem 3, with the setting of $\lambda$ from Theorem 1, we have
$\displaystyle\bar{C}_{2}(\tilde{\Phi})^{2}\sqrt{s}\lambda\Delta$
$\displaystyle=O\left({\sqrt{S\log N}f_{\max}\Delta\over
M}\right)=O\left(\sqrt{S\log N\over M}\Delta\right)\rightarrow 0,$ (12a)
$\displaystyle\bar{C}_{2}(\tilde{\Phi})\sqrt{\tilde{M}}\Delta$
$\displaystyle=O\left({\sqrt{\tilde{M}}\Delta\over\sqrt{M}}\right)=O\left(\Delta\right).$
(12b)
By combining the estimation error bounds (11a) and (11b), we have
$\displaystyle\|h\|\leq\min\,\left\\{O\left(\sqrt{S(\log
N)/M}\right),O(1)\right\\}\Delta.$ (13)
In the regime described by assumption (iv), (12a) will be asymptotically
smaller than (12b). The bound in (13) has size $O\left(\sqrt{S(\log
N)/M}\Delta\right)$, consistent with the upper bound of the Dantzig selector
[7] and LASSO [24]333Their bound is $O\left(\sqrt{S(\log N)/M}\sigma\right)$
where $\sigma^{2}$ is the variance of the observation noise which, in the
classical setting for the Dantzig selector and LASSO, is assumed to follow a
Gaussian distribution.. Recall that the estimation error of the formulation
(2) is $O\left(\|\tilde{\Phi}x^{*}-\tilde{y}\|/\sqrt{\tilde{M}}\right)$ [13,
14] under the RIP condition, for the number of measurements defined in (iv).
Since $\|\tilde{\Phi}x^{*}-\tilde{y}\|=O\left(\sqrt{\tilde{M}}\Delta\right)$
[13], this estimate is consistent with the error that would be obtained if we
imposed only the $\ell_{\infty}$ constraint (3b) in our formulation. Note that
it does not converge to zero even all assumptions (i)-(vi) hold. Under the
assumption (iv), the estimation error for (3) will vanish as the dimensions
grow, with probability at least $1-\pi$. By contrast, Jacques et al. [13] do
not account for saturation in their formulation and show that the estimation
error converges to $0$ using an $\ell_{p}$ constraint in place of (2b) when
$p\to\infty$ and oversampling happens — specifically,
$M\geq\Omega\left(\left(S\log(N/S)\right)^{p/2}\right)$. Weaker oversampling
conditions are available using our formulation (3). For example, ${M}=S(\log
N)^{2}$ would produce consistency in our formulation, but not in (2).
## 4 Simulations
This section compares results for five variant formulations. The first one is
our formulation (3), which we refer to as LASSO$\infty$ . We also tried a
variant in which the $\ell_{\infty}$ constraint (3b) was omitted from (3). The
recovery performance for this variant was uniformly worse than for
LASSO$\infty$ , so we do not show it in our figures. (It is, however,
sometimes better than the formulations described below, and uniformly better
than Dantzig .) The remaining four alternatives are based on the following
model, in which the $\ell_{2}$ norm of the residual appears in a constraint
(rather than in the objective) and a constraint of Dantzig type also appears:
$\displaystyle\min_{x}~{}$ $\displaystyle\|x\|_{1}$ (14a)
$\displaystyle\mbox{s.t.}\quad\|\tilde{\Phi}x-\tilde{y}\|^{2}$
$\displaystyle\leq\epsilon^{2}\Delta^{2}\;$ ($\ell_{2}$) (14b)
$\displaystyle\|\tilde{\Phi}x-\tilde{y}\|_{\infty}$
$\displaystyle\leq\Delta/2\;$ ($\ell_{\infty}$) (14c)
$\displaystyle\|\tilde{\Phi}^{T}(\tilde{\Phi}x-\tilde{y})\|_{\infty}$
$\displaystyle\leq\lambda\Delta/2\;$ (Dantzig) (14d)
$\displaystyle\bar{\Phi}_{+}x$ $\displaystyle\geq(G-\Delta)\mathbb{1}\;$ ($+$
saturation) (14e) $\displaystyle\bar{\Phi}_{-}x$
$\displaystyle\leq(\Delta-G)\mathbb{1}.\;$ ($-$ saturation) (14f)
The four formulations are obtained from this model as follows.
* 1.
L$\infty$ : an $\ell_{\infty}$ constraint model that enforces (14c), (14e),
and (14f), but not (14b) or (14d). This model is obtained by letting
$p\to\infty$ in Jacques et al. [13] and adding saturation constraints.
* 2.
L2 : an $\ell_{2}$ constraint model (that is, the state-of-the-art model (2)
[14]) that enforces (14b), (14e), and (14f), but not (14c) or (14d);
* 3.
Dantzig : the Dantzig constraint algorithm with saturation constraints, which
enforces (14d), (14e), and (14f) but not (14b) or (14c);
* 4.
L2Dantzig$\infty$ : the full model defined by (14).
Note that we use the same value of $\lambda$ in (14d) as in (3), since in both
cases they lead to a constraint that the true signal $x^{*}$ satisfies
$\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}\leq\lambda\Delta/2$
with a certain probability; see (14d) and (9). Readers familiar with the
equivalence between LASSO and Dantzig selector [2] may notice that
L2Dantzig$\infty$ has similar theoretical error bounds to LASSO$\infty$ . Our
computational results show that the practical performance of these two
approaches is also similar.
The synthetic data is generated as follows. The measurement matrix
$\tilde{\Phi}\in\mathbb{R}^{M\times N}$ is a Gaussian matrix, each entry being
independently generated from $\mathcal{N}(0,1/R^{2})$, for a given parameter
$R$. The $S$ nonzero elements of $x^{*}$ are in random locations and their
values are drawn from independently from $\mathcal{N}(0,1)$. We use
$\mbox{SNR}=-20\log_{10}(\|\hat{x}-x^{*}\|/\|x^{*}\|)$ as the error metric,
where $\hat{x}$ is the signal recovered from each of the formulations under
consideration. Given values of saturation parameter $G$ and number of bits
$B$, the interval $\Delta$ is defined accordingly as $\Delta=2^{B-1}G$. All
experiments are repeated $30$ times; we report the average performance.
We now describe how the bounds $\lambda$ for (3a) and (14d) and $\epsilon$ for
(14b) were chosen for these experiments. Essentially, $\epsilon$ and $\lambda$
should be chosen so that the constraints (14b) and (14d) admit the true signal
$x^{*}$ with a a high (specified) probability. There is a tradeoff between
tightness of the error estimate and confidence. Larger values of $\epsilon$
and $\lambda$ can give a more confident estimate, since the defined feasible
region includes $x^{*}$ with a higher probability, while smaller values
provide a tighter estimate. Although Lemma 2 suggests how to choose $\lambda$
and [13] show how to determine $\epsilon$, the analysis it not tight,
especially when $M$ and $N$ are not particularly large. We use instead an
approach based on simulation and on making the assumption (not used elsewhere
in the analysis) that the non-saturated quantization errors
$\xi_{i}=(\tilde{\Phi}x^{*}-\tilde{y})_{i}$ are i.i.d. uniform in
$U_{[-\Delta/2,\Delta/2]}$. (As noted earlier, this stronger assumption makes
sense in some settings, and has been used in previous analyses.) We proceed by
generating numerous independent samples of $Z\sim U_{[-\Delta/2,\Delta/2]}$.
Given a confidence level $1-\pi$ (for $\pi>0$), we set $\epsilon$ to the value
for which $\mathbb{P}(Z\geq\epsilon\Delta)=\pi$ is satisfied empirically. A
similar technique is used to determine $\lambda$. When we seek certainty
($\pi=0$, or confidence $P=100\%$), we set $\epsilon$ and $\lambda$ according
to the true solution $x^{*}$, that is,
$\epsilon=\|\tilde{\Phi}x^{*}-\tilde{y}\|/\Delta$ and
$\lambda=2\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}/\Delta$.
To summarize the parameters that are varied in our experiments:
* 1.
$M$ and $N$ are dimensions of $\Phi$,
* 2.
$S$ is sparsity of solution $x^{*}$,
* 3.
$G$ is saturation level,
* 4.
$B$ is number of bits,
* 5.
$R$ is the inverse standard deviation of the elements of $\Phi$, and
* 6.
$P=1-\pi$ denotes the confidence levels, expressed as a percentage.
In Figure 1, we fix the values of $M$, $S$, $G$, $R$, and $P$, choose two
values of $B$: 3 and 5. Plots show the average SNRs (over $30$ trials) of the
solutions $\hat{x}$ recovered from the five models against the dimension $N$.
In this and all subsequent figures, the saturation ratio is defined to be
$\bar{M}/M=(M-\tilde{M})/M$, the fraction of extreme measurements. Our
LASSO$\infty$ formulation and the full model L2Dantzig$\infty$ give the best
recovery performance for small $N$, while for larger $N$, LASSO$\infty$ is
roughly tied with the the L2 model. The L$\infty$ and Dantzig models have
poorer performance, a pattern that we continue to observe in subsequent tests.
Figure 1: Comparison among various models for fixed values $M=300$, $S=10$,
$G=4$, $R=10$, and $P=100\%$, and two values of $B$ (3 and 5, respectively).
The graphs show dimension $N$ (horizontal axis) against SNR (vertical axis)
for values of $N$ between $100$ and $1000$, averaged over $30$ trials for each
combination of parameters.
Figure 2 fixes $N$, $M$, $B$, $G$, $R$, and $P$, and plots SNR as a function
of sparsity level $S$. For all models, the quality of reconstruction decreases
rapidly with $S$. LASSO$\infty$ and L2Dantzig$\infty$ achieve the best results
overall, but are roughly tied with the L2 model for all but the sparsest
signals. The L$\infty$ model is competitive for very sparse signals, while the
Dantzig model lags in performance.
Figure 2: Comparison among various models for $N=500$, $M=300$, $B=4$,
$G=0.4$, $R=10$, and $P=100\%$. The graph shows sparsity level $S$ (horizontal
axis) plotted against SNR (vertical axis), averaged over $30$ trials.
We now examine the effect of number of measurements $M$ on SNR. Figure 3 fixes
$N$, $S$, $G$, $R$, and $P$, and tries two values of $B$: $3$ and $5$,
respectively. Figure 4 fixes $B=4$, and allows $N$ to increase with $M$ in the
fixed ratio $5/4$. These figures indicate that the LASSO$\infty$ and
L2Dantzig$\infty$ models are again roughly tied with the L2 model when the
number of measurements is limited. For larger $M$, our models have a slight
advantage over the L2 and L$\infty$ models, which is more evident when the
quantization intervals are smaller (that is, $B=4$). Another point to note
from Figure 4 is that L$\infty$ outperforms L2 when both $M$ and $N$ are much
larger than the sparsity $S$.
Figure 3: Comparison among various models for fixed values $N=500$, $S=5$,
$G=0.4$, $R=15$, and $P=100\%$, and two values of $B$ ($3$ and $5$). The
graphs show the number of measurements $M$ (horizontal axis) against SNR
(vertical axis) for values of $M$ between $20$ and $300$, averaged over $30$
trials for each combination of parameters. Figure 4: Comparison among various
models for fixed ratio $N/M=5/4$, and fixed values $S=10$, $B=4$, $G=0.4$,
$R=15$, and $P=100\%$. The graph shows the number of measurements $M$
(horizontal axis) against SNR (vertical axis) for values of $M$ between $100$
and $1680$, averaged over $30$ trials for each combination of parameters.
In Figure 5 we examine the effect of the number of bits $B$ on SNR, for fixed
values of $N$, $M$, $S$, $G$, $R$, and $P$. The fidelity of the solution from
all models increases linearly with $B$, with the LASSO$\infty$ ,
L2Dantzig$\infty$ , and L2 models being slightly better than the alternatives.
Figure 5: Comparison among various models for fixed values $N=500$, $M=300$,
$S=10$, $G=0.4$, $R=10$, and $P=100\%$. This graph shows the bit number $B$
(horizontal axis) against SNR (vertical axis), averaged over $30$ trials.
Next we examine the effect on SNR of the confidence level, for fixed values of
$N$, $M$, $B$, $G$, and $R$. In Figure 6, we set $M=300$ and plot results for
two values of $S$: 5 and 15. In Figure 7, we use the same values of $S$, but
set $M=150$ instead. Note first that the confidence level does not affect the
solution of the L$\infty$ model, since this is a deterministic model, so the
reconstruction errors are constant for this model. For the other models, we
generally see degradation as confidence is higher, since the constraints (14b)
and (14d) are looser, so the feasible point that minimizes the objective
$\|\cdot\|_{1}$ is further from the optimum $x^{*}$. Again, we see a clear
advantage for LASSO$\infty$ when the sparsity is low, $M$ is larger, and the
confidence level $P$ is high. For less sparse solutions, the L2 ,
L2Dantzig$\infty$ , and LASSO$\infty$ models have similar or better
performance. In addition, we find that LASSO$\infty$ is more robust to the
choice of confidence parameter than other methods (see also Figure 9),
although this feature of the method is not evident from our theoretical
analysis.
Figure 6: Comparison among various models for fixed values $N=400$, $M=300$,
$B=4$, $G=0.4$, and $R=15$, and sparsity levels $S=5$ and $S=15$. The graphs
show saturation bound $G$ (horizontal axis) against SNR (vertical axis) for
values of $P$ between $0.0001$ and $0.99$, averaged over $30$ trials for each
combination of parameters.
Figure 7: Comparison among various models for fixed values $N=400$, $M=150$,
$B=4$, $G=0.4$, and $R=15$, and sparsity levels $S=5$ and $S=15$. The graphs
show confidence $P$ (horizontal axis) against SNR (vertical axis) for values
of $P$ between $0.0001$ and $0.99$, averaged over $30$ trials for each
combination of parameters.
In Figure 8 we examine the effect of saturation bound $G$ on SNR. We fix $N$,
$M$, $B$, $R$, and $P$, and try two values of $S$: $5$ and $10$. A tradeoff is
evident — the reconstruction performances are not monotonic with $G$. As $G$
increases, the proportion of saturated measurements drops sharply, but the
quantization interval also increases, degrading the quality of the measured
observations. We again note a slight advantage for the LASSO$\infty$ and
L2Dantzig$\infty$ models, with very similar performance by L2 when the
oversampling is lower.
Figure 8: Comparison among various models for fixed values of $N=500$,
$M=150$, $B=4$, $R=15$, $P=100\%$, and two values of $S$: $5$ and $10$. The
graphs show confidence $P$ (horizontal axis) against SNR (left vertical axis)
and saturation ratio (right vertical axis), averaged over $30$ trials for each
combination of parameters.
In Figure 9, we fix $N$, $M$, $S$, $B$, $R$, and tune the value of $G$ to
achieve specified saturation ratios of $2\%$ and $10\%$. We plot SNR against
the confidence level $P$, varied from $0\%$ to $100\%$. Again, we see
generally good performance from the LASSO$\infty$ and L2Dantzig$\infty$
models, with L2 being competitive for less sparse solutions.
Figure 9: Comparison among various models for fixed values of $N=500$,
$M=150$, $S=5$, $B=4$, $R=15$, and two values of saturation ratio: $2\%$ and
$10\%$, which are achieved by tuning the value of $G$. The graphs show
confidence $P$ (horizontal axis) against SNR (vertical axis), averaged over
$30$ trials for each combination of parameters.
Summarizing, we note the following points.
* (a)
Our proposed LASSO$\infty$ formulation gives either best or equal-best
reconstruction performance in most regimes, with a more marked advantage when
the signal is highly sparse and the number of samples is higher.
* (b)
The L2 model has similar performance to the full model, and is even slightly
better than our model for less sparse signals with fewer measurements, since
it is not sensitive to the measurement number as the upper bound suggested by
[14]. Although the inequality in (13) also indicates the estimate error by our
model is bounded by a constant due to the $\ell_{\infty}$ constraint, the
error bound determined by the $\ell_{\infty}$ constraint is not as tight as
the $\ell_{2}$ constraint in general. This fact is evident when we compare the
the L$\infty$ model with the L2 model.
* (c)
The L$\infty$ model performs well (and is competitive with the others) when
the number of unsaturated measurements is relatively large.
* (d)
The L2Dantzig$\infty$ model is competitive with LASSO$\infty$ if $\epsilon$
and $\lambda$ can be determined from the true signal $x^{*}$. Otherwise,
LASSO$\infty$ is more robust to choices of these parameters that do not
require knowledge of the true signals, especially if a high confidence level
is desired.
## 5 Conclusion
We have analyzed a formulation of the reconstruction problem from compressed
sensing in which the measurements are quantized to a finite number of possible
values. Our formulation uses an objective of $\ell_{2}$-$\ell_{1}$ type, along
with explicit constraints that restrict the individual quantization errors to
known intervals. We obtain bounds on the estimation error, and estimate these
bounds for the case in which the sensing matrix is Gaussian. Finally, we prove
the practical utility of our formulation by comparing with an approach that
has been proposed previously, along with some variations on this approach that
attempt to distil the relative importance of different constraints in the
formulation.
## Acknowledgments
The authors acknowledge support of National Science Foundation Grant
DMS-0914524 and a Wisconsin Alumni Research Foundation 2011-12 Fall
Competition Award. The authors are also grateful to the editor and three
referees whose constructive comments on the first version led to improvements
in the manuscript.
## Appendix A
This section contains the proof to a more general form of Theorem 1, developed
via a number of technical lemmas. At the end, we state and prove a result
(Theorem 3) concerning high-probability estimates of the bounds under
additional assumptions on the sensing matrix $\tilde{\Phi}$.
Theorem 1 is a corollary of the following more general result.
###### Theorem 2.
Assume that the true signal $x^{*}$ satisfies
$\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}\leq\lambda\Delta/2,$
(15)
for some value of $\lambda$. Let $s$ and $l$ be positive integers in the range
$1,2,\dotsc,N$, and define
$\displaystyle A_{0}(\Psi):=$
$\displaystyle{\rho}^{-}(s+l,\Psi)-3{\sqrt{s/l}}\left[{\rho}^{+}(s+2l,\Psi)-{\rho}^{-}(s+2l,\Psi)\right]$
(16a) $\displaystyle A_{1}(\Psi):=$ $\displaystyle
4[{\rho}^{+}(s+2l,\Psi)-{\rho}^{-}(s+2l,\Psi)],$ (16b) $\displaystyle
C_{1}(\Psi):=$ $\displaystyle 4+{\sqrt{(1+9s/l)}A_{1}(\Psi)}/{A_{0}(\Psi)},$
(16c) $\displaystyle C_{2}(\Psi):=$
$\displaystyle\sqrt{(1+9s/l)/A_{0}(\Psi)}.$ (16d)
We have that for any $T_{0}\subset\\{1,2,...,N\\}$ with $s=|T_{0}|$, if
$A_{0}(\tilde{\Phi})>0$, then
$\displaystyle\|h\|\leq$
$\displaystyle\frac{6C_{2}(\tilde{\Phi})^{2}\sqrt{s}\lambda\Delta}{\sqrt{1+9s/l}}+\frac{C_{1}(\tilde{\Phi})}{\sqrt{l}}\|x^{*}_{T_{0}^{c}}\|_{1}+2.5{C_{2}(\tilde{\Phi})}\sqrt{\lambda\Delta\|x_{T_{0}^{c}}^{*}\|_{1}},$
(17a) $\displaystyle\|h\|\leq$ $\displaystyle
C_{2}(\tilde{\Phi})\sqrt{\tilde{M}}\Delta+\frac{C_{1}(\tilde{\Phi})}{\sqrt{l}}\|x^{*}_{T_{0}^{c}}\|_{1}.$
(17b)
Suppose that Assumption 1 holds, and let $\pi\in(0,1)$ be given. If we define
$\lambda=\sqrt{2\log{2N/\pi}}f_{\max}$ in (3), then with probability at least
$P=1-\pi$, the inequalities (17a) and (17b) hold.
Theorem 1 can be proven by setting $s=l$ in Theorem 2 and defining
$\bar{C}_{1}(\tilde{\Phi})$ to be $C_{1}(\Psi)$ for $l=s$ and
$\Psi=\tilde{\Phi}$, and similarly for $\bar{C}_{1}(\tilde{\Phi})$,
$\bar{A}_{0}(\tilde{\Phi})$, and $\bar{A}_{1}(\tilde{\Phi})$.
The proof of Theorem 2 essentially follows the standard analysis procedure in
compressive sensing. Some similar lemmas and proofs can be found in Bickel et
al. [2], Candès and Tao [7], Candès [6], Zhang [24], Liu et al. [16, 17]. For
completeness, we include all proofs in the following discussion.
Given the error vector $h=\hat{x}-x^{*}$ and the set $T_{0}$ (with $s$
entries), divide the complementary index set
$T_{0}^{c}:=\\{1,2,...,N\\}\backslash T_{0}$ into a group of subsets $T_{j}$’s
($j=1,2,\dotsc,J$), without intersection, such that $T_{1}$ indicates the
index set of the largest $l$ entries of $h_{T_{0}^{c}}$, $T_{2}$ contains the
next-largest $l$ entries of $h_{T_{0}^{c}}$, and so forth.444The last subset
may contain fewer than $l$ elements.
###### Lemma 1.
We have
$\displaystyle\|\tilde{\Phi}h\|_{\infty}$ $\displaystyle\leq\Delta.$ (18)
###### Proof.
From (3b), and invoking feasibility of $\hat{x}$ and $x^{*}$, we obtain
$\|\tilde{\Phi}h\|_{\infty}=\|\tilde{\Phi}(\hat{x}-x^{*})\|_{\infty}\leq\|\tilde{\Phi}\hat{x}-\tilde{y}\|_{\infty}+\|\tilde{\Phi}x^{*}-\tilde{y}\|_{\infty}\leq\Delta.$
∎
###### Lemma 2.
Suppose that Assumption 1 holds. Given $\pi\in(0,1)$, the choice
$\lambda=\sqrt{2\log{(2N/\pi)}}f_{\max}$ ensures that the true signal $x^{*}$
satisfies (15), that is
$\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}\leq\lambda\Delta/2$
with probability at least $1-\pi$.
###### Proof.
Define the random variable
$Z_{j}=\tilde{\Phi}_{j}^{T}(\tilde{\Phi}x^{*}-\tilde{y})=\tilde{\Phi}_{j}^{T}\xi$,
where $\xi=[\xi_{1},...,\xi_{\tilde{M}}]$ is defined in an obvious way. (Note
that
$\|Z\|_{\infty}=\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}$.)
Since $\mathbb{E}(Z_{j})=0$ (from Assumption 1) and all
$\tilde{\Phi}_{ij}\xi_{i}$’s are in the range
$[-\tilde{\Phi}_{ij}\Delta/2,\tilde{\Phi}_{ij}\Delta/2]$, we use the Hoeffding
inequality to obtain
$\displaystyle\mathbb{P}(Z_{j}>\lambda\Delta/2)=$
$\displaystyle\mathbb{P}(Z_{j}-\mathbb{E}(Z_{j})>\lambda\Delta/2)$
$\displaystyle=$
$\displaystyle\mathbb{P}\left(\sum_{i=1}^{\tilde{M}}\tilde{\Phi}_{ij}\xi_{i}-\mathbb{E}(Z_{j})>\lambda\Delta/2\right)$
$\displaystyle\leq$
$\displaystyle\exp{-2(\lambda\Delta/2)^{2}\over\sum_{i=1}^{\tilde{M}}(\tilde{\Phi}_{ij}\Delta)^{2}}$
$\displaystyle=$ $\displaystyle\exp{-\lambda^{2}\over
2\sum_{i}\tilde{\Phi}_{ij}^{2}}$ $\displaystyle\leq$
$\displaystyle\exp{-\lambda^{2}\over 2f_{\max}^{2}},$
which implies (using the union bound) that
$\displaystyle\mathbb{P}(|Z_{j}|>\lambda\Delta/2)\leq 2\exp{-\lambda^{2}\over
2f_{\max}^{2}}$
$\displaystyle\Rightarrow\mathbb{P}\left(\|Z\|_{\infty}=\max_{j}|Z_{j}|>\lambda\Delta/2\right)\leq
2N\exp{-\lambda^{2}\over 2f_{\max}^{2}}$
$\displaystyle\Rightarrow\mathbb{P}\left(\|Z\|_{\infty}>\sqrt{{1\over
2}\log{2N\over\pi}}f_{\max}\Delta\right)\leq\pi,$
where the last line follows by setting $\lambda$ to the prescribed value. This
completes the proof. ∎
Similar claims with Gaussian (or sub-Guassian) noise assumption to Lemma (2)
can be found in Zhang [24], Liu et al. [17].
###### Lemma 3.
We have
$\|h_{T_{01}^{c}}\|\leq\sum_{j=2}^{J}\|h_{T_{j}}\|\leq\|h_{T_{0}^{c}}\|_{1}/\sqrt{l},$
where $T_{01}=T_{0}\cup T_{1}$.
###### Proof.
First, we have for any $j\geq 1$ that
$\|h_{T_{j+1}}\|^{2}\leq l\|h_{T_{j+1}}\|_{\infty}^{2}\leq
l(\|h_{T_{j}}\|_{1}/l)^{2}=\|h_{T_{j}}\|_{1}^{2}/l,$
because the largest value in $|h_{T_{j+1}}|$ cannot exceed the average value
of the components of $|h_{T_{j}}|$. It follows that
$\|h_{T_{01}^{c}}\|\leq\sum_{j=2}^{J}\|h_{T_{j}}\|\leq\sum_{j=1}^{J-1}\|h_{T_{j}}\|_{1}/\sqrt{l}\leq\|h_{T_{0}^{c}}\|_{1}/\sqrt{l}.$
∎
Similar claims or inequalities to Lemma 3 can be found in Zhang [24], Candès
and Tao [7], Liu et al. [16].
###### Lemma 4.
Assume that (15) holds. We have
$\displaystyle\|h_{T_{0}^{c}}\|_{1}$ $\displaystyle\leq
3\|h_{T_{0}}\|_{1}+4\|x^{*}_{T_{0}^{c}}\|_{1},$ (19a) $\displaystyle\|h\|$
$\displaystyle\leq\sqrt{1+9s/l}\|h_{T_{01}}\|+4\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l}.$
(19b)
###### Proof.
Since $\hat{x}$ is the solution of (3), we have
$\displaystyle 0$ $\displaystyle\geq{1\over
2}\|\tilde{\Phi}\hat{x}-\tilde{y}\|^{2}-{1\over
2}\|\tilde{\Phi}x^{*}-\tilde{y}\|^{2}+\lambda\Delta(\|\hat{x}\|_{1}-\|x^{*}\|_{1})$
$\displaystyle\geq
h^{T}\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})+\lambda\Delta(\|\hat{x}\|_{1}-\|x^{*}\|_{1})\quad\quad(\text{by
convexity of $(1/2)\|\tilde{\Phi}x-\tilde{y}\|^{2}$})$
$\displaystyle=h^{T}\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})+\lambda\Delta(\|\hat{x}_{T_{0}}\|_{1}-\|x^{*}_{T_{0}}\|_{1}+\|\hat{x}_{T_{0}^{c}}\|_{1}-\|x^{*}_{T_{0}^{c}}\|_{1})$
$\displaystyle\geq-\|h\|_{1}\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}+\lambda\Delta(\|\hat{x}_{T_{0}}\|_{1}-\|x^{*}_{T_{0}}\|_{1}+\|\hat{x}_{T_{0}^{c}}\|_{1}-\|x^{*}_{T_{0}^{c}}\|_{1})$
$\displaystyle\geq-\|h\|_{1}\lambda\Delta/2+\lambda\Delta(\|\hat{x}_{T_{0}}\|_{1}-\|x^{*}_{T_{0}}\|_{1}+\|\hat{x}_{T_{0}^{c}}\|_{1}+\|x^{*}_{T_{0}^{c}}\|_{1}-2\|x^{*}_{T_{0}^{c}}\|_{1})~{}~{}~{}~{}(\text{from~{}\eqref{eqn_feasible_extend}})$
$\displaystyle\geq-(\|h_{T_{0}}\|_{1}+\|h_{T_{0}^{c}}\|_{1})\lambda\Delta/2+\lambda\Delta(-\|h_{T_{0}}\|_{1}+\|h_{T_{0}^{c}}\|_{1}-2\|x^{*}_{T_{0}^{c}}\|_{1})$
$\displaystyle={1\over 2}\lambda\Delta\|h_{T_{0}^{c}}\|_{1}-{3\over
2}\lambda\Delta\|h_{T_{0}}\|_{1}-2\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1}.$
It follows that
$3\|h_{T_{0}}\|_{1}+4\|x^{*}_{T_{0}^{c}}\|_{1}\geq\|h_{T_{0}^{c}}\|_{1}$,
proving (19a).
The second inequality (19b) is from
$\displaystyle\|h\|^{2}$
$\displaystyle=\|h_{T_{01}}\|^{2}+\|h_{T_{01}^{c}}\|^{2}$
$\displaystyle\leq\|h_{T_{01}}\|^{2}+\|h_{T_{0}^{c}}\|^{2}_{1}/l~{}~{}~{}~{}(\text{from
Lemma~{}\ref{lem_T01}})$
$\displaystyle\leq\|h_{T_{01}}\|^{2}+(3\|h_{T_{0}}\|_{1}+4\|x^{*}_{T_{0}^{c}}\|_{1})^{2}/l~{}~{}~{}~{}(\text{from
\eqref{eq:lem7.1}})$
$\displaystyle\leq\|h_{T_{01}}\|^{2}+(3\sqrt{s}\|h_{T_{01}}\|+4\|x^{*}_{T_{0}^{c}}\|_{1})^{2}/l$
$\displaystyle=(1+9s/l)\|h_{T_{01}}\|^{2}+24\sqrt{s}/l\|h_{T_{01}}\|\|x^{*}_{T_{0}^{c}}\|_{1}+{16\|x^{*}_{T_{0}^{c}}\|_{1}^{2}/l}$
$\displaystyle\leq\left[\sqrt{1+9s/l}\|h_{T_{01}}\|+4\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l}\right]^{2}.$
∎
###### Lemma 5.
For any matrix $\Psi$ with $N$ columns, and $s,l\leq N$, we have
$\|\Psi h\|^{2}\geq
A_{0}(\Psi)\|h_{T_{01}}\|^{2}-A_{1}(\Psi)\|h_{T_{01}}\|\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l},$
where $A_{0}(\Psi)$ and $A(\Psi)$ are defined in (10a) and (10b) respectively.
###### Proof.
For any $j\geq 2$, we have
$\displaystyle\;\;{|h^{T}_{T_{01}}\Psi_{T_{01}}^{T}\Psi_{T_{j}}h_{T_{j}}|\over\|h_{T_{01}}\|\|h_{T_{j}}\|}$
$\displaystyle=\frac{1}{4}\left|\left\|\Psi_{T_{01}}h_{T_{01}}/\|h_{T_{01}}\|+\Psi_{T_{j}}h_{T_{j}}/\|h_{T_{j}}\|\|^{2}-\|\Psi_{T_{01}}h_{T_{01}}/\|h_{T_{01}}\|-\Psi_{T_{j}}h_{T_{j}}/\|h_{T_{j}}\|\right\|^{2}\right|$
$\displaystyle=\frac{1}{4}\left|\left\|\left[\Psi_{T_{01}}\,:\,\Psi_{T_{j}}\right]\left[\begin{matrix}h_{T_{01}}/\|h_{T_{01}}\|\\\
h_{T_{j}}/\|h_{T_{j}}\|\end{matrix}\right]\right\|^{2}-\left\|\left[\Psi_{T_{01}}\,:\,\Psi_{T_{j}}\right]\left[\begin{matrix}h_{T_{01}}/\|h_{T_{01}}\|\\\
-h_{T_{j}}/\|h_{T_{j}}\|\end{matrix}\right]\right\|^{2}\right|$
$\displaystyle\leq\frac{1}{4}\left({2}{\rho}^{+}({s+2l})-{2}{\rho}^{-}(s+2l)\right)$
$\displaystyle=\frac{1}{2}\left({\rho}^{+}(s+2l)-{\rho}^{-}(s+2l)\right).$
(20)
The inequality above follows from the definitions (6a) and (6b), and the fact
that fact that $h_{T_{01}}/\|h_{T_{01}}\|$ and $h_{T_{j}}/\|h_{T_{j}}\|$ are
$\ell_{2}$-unit vectors, so that
$\left\|\left[\begin{matrix}h_{T_{01}}/\|h_{T_{01}}\|\\\
h_{T_{j}}/\|h_{T_{j}}\|\end{matrix}\right]\right\|^{2}=\left\|\left[\begin{matrix}h_{T_{01}}/\|h_{T_{01}}\|\\\
-h_{T_{j}}/\|h_{T_{j}}\|\end{matrix}\right]\right\|^{2}=2.$
Considering the left side of the claimed inequality, we have
$\displaystyle\;\;\|\Psi h\|^{2}$
$\displaystyle=\|\Psi_{T_{01}}h_{T_{01}}\|^{2}+2h^{T}_{T_{01}}\Psi^{T}_{T_{01}}\Psi_{T_{01}^{c}}h_{T_{01}^{c}}+\|\Psi_{T_{01}^{c}}h_{T_{01}^{c}}\|^{2}$
$\displaystyle\geq\|\Psi_{T_{01}}h_{T_{01}}\|^{2}-2\sum_{j\geq
2}|h^{T}_{T_{01}}\Psi^{T}_{T_{01}}\Psi_{T_{j}}h_{T_{j}}|$
$\displaystyle\geq{\rho}^{-}(s+l)\|h_{T_{01}}\|^{2}-({\rho}^{+}{(s+2l)}-{\rho}^{-}(s+2l))\|h_{T_{01}}\|\sum_{j\geq
2}\|h_{T_{j}}\|~{}~{}~{}~{}(\text{from \eqref{eqn_lem4}})$
$\displaystyle\geq{\rho}^{-}(s+l)\|h_{T_{01}}\|^{2}-({\rho}^{+}(s+2l)-{\rho}^{-}(s+2l))\|h_{T_{01}}\|\|h_{T_{0}^{c}}\|_{1}/\sqrt{l}~{}~{}~{}~{}(\text{from
Lemma~{}\ref{lem_T01}})$
$\displaystyle\geq{\rho}^{-}(s+l)\|h_{T_{01}}\|^{2}-({\rho}^{+}(s+2l)-{\rho}^{-}(s+2l))\|h_{T_{01}}\|(3\|h_{T_{0}}\|_{1}/\sqrt{l}+4\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l})\;\;(\text{from
\eqref{eq:lem7.1}})$
$\displaystyle\geq\left({\rho}^{-}(s+l)-3\sqrt{s/l}({\rho}^{+}(s+2l)-{\rho}^{-}(s+2l))\right)\|h_{T_{01}}\|^{2}-$
$\displaystyle\quad
4({\rho}^{+}(s+2l)-{\rho}^{-}(s+2l))\|x^{*}_{T_{0}^{c}}\|_{1}\|h_{T_{01}}\|/\sqrt{l}~{}~{}~{}~{}(\text{using
$\|h_{T_{0}}\|_{1}\leq\sqrt{s}\|h_{T_{0}}\|\leq\sqrt{s}\|h_{T_{01}}\|$})$
$\displaystyle\geq
A_{0}(\Psi)\|h_{T_{01}}\|^{2}-A_{1}(\Psi)\|h_{T_{01}}\|\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l},$
which completes the proof. ∎
Similar claims or inequalities to (20) can be found in Candès and Tao [7],
Candès [6], Zhang [24].
###### Lemma 6.
Assume that (15) holds. We have
$\displaystyle\|\tilde{\Phi}h\|^{2}\leq$ $\displaystyle{3\over
2}\lambda\Delta\|h\|_{1}\leq
6\sqrt{s}\lambda\Delta\|h_{T_{01}}\|+6\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1},$
(21a) $\displaystyle\|\tilde{\Phi}h\|^{2}\leq$
$\displaystyle\tilde{M}\Delta^{2}.$ (21b)
###### Proof.
Denote the feasible region of (3) as
$F:=\left\\{x~{}|~{}\bar{\Phi}x-\bar{y}\geq
0,~{}\|\tilde{\Phi}x-\tilde{y}\|_{\infty}\leq\Delta/2\right\\}.$
Since $\hat{x}$ is the optimal solution to (3), we have the optimality
condition:
$\displaystyle\tilde{\Phi}^{T}(\tilde{\Phi}\hat{x}-\tilde{y})+\lambda\Delta\partial\|\hat{x}\|_{1}\cap-
N_{F}(\hat{x})\neq\emptyset,$
where $N_{F}(\hat{x})$ denotes the normal cone of $F$ at the point $\hat{x}$
and $\partial\|\hat{x}\|_{1}$ is the subgradient of the function $\|.\|_{1}$
at the point $\hat{x}$. This condition is equivalent to existence of
$g\in\partial\|\hat{x}\|_{1}$ and $n\in N_{F}(\hat{x})$ such that
$\tilde{\Phi}^{T}(\tilde{\Phi}\hat{x}-\tilde{y})+\lambda\Delta g+n=0.$
It follows that
$\displaystyle\quad\tilde{\Phi}^{T}\tilde{\Phi}h+\tilde{\Phi}^{T}(\tilde{\Phi}{x^{*}}-\tilde{y})+\lambda\Delta
g+n=0$ $\displaystyle\Rightarrow
h^{T}\tilde{\Phi}^{T}\tilde{\Phi}h+h^{T}\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})+\lambda\Delta
h^{T}g+h^{T}n=0$
$\displaystyle\Rightarrow\|\tilde{\Phi}h\|^{2}=-h^{T}\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})-\lambda\Delta
h^{T}g-h^{T}n$
$\displaystyle\Rightarrow\|\tilde{\Phi}h\|^{2}\leq-h^{T}\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})-\lambda\Delta
h^{T}g~{}~{}~{}~{}(\text{using $x^{*}\in F$ and so
$-h^{T}n=(x^{*}-\hat{x})^{T}n\leq 0$})$
$\displaystyle\Rightarrow\|\tilde{\Phi}h\|^{2}\leq\|h\|_{1}\|\tilde{\Phi}^{T}(\tilde{\Phi}x^{*}-\tilde{y})\|_{\infty}+\lambda\Delta\|h\|_{1}\|g\|_{\infty}.$
From $\|g\|_{\infty}\leq 1$ and (15), we obtain
$\displaystyle\|\tilde{\Phi}h\|^{2}$
$\displaystyle\leq\lambda\Delta\|h\|_{1}/2+\lambda\Delta\|h\|_{1}$
$\displaystyle={3\over 2}\lambda\Delta\|h\|_{1}$ $\displaystyle={3\over
2}\lambda\Delta(\|h_{T_{0}}\|_{1}+\|h_{T_{0}^{c}}\|_{1})$
$\displaystyle\leq{3\over
2}\lambda\Delta(4\|h_{T_{0}}\|_{1}+4\|x^{*}_{T_{0}^{c}}\|_{1})~{}~{}~{}~{}(\text{from~{}\eqref{eq:lem7.1}})$
$\displaystyle\leq
6\sqrt{s}\lambda\Delta\|h_{T_{0}}\|+6\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1},$
which proves the first inequality.
From (18), the second inequality is obtained by
$\|\tilde{\Phi}h\|^{2}\leq\left(\sqrt{\tilde{M}\|\tilde{\Phi}h\|_{\infty}}\right)^{2}\leq\tilde{M}\Delta^{2}$.
∎
### Proof of Theorem 2
###### Proof.
First, assume that (9) holds. Take $\Psi=\tilde{\Phi}$ in Lemma 5 and apply
(21a). We have
$A_{0}(\tilde{\Phi})\|h_{T_{01}}\|^{2}-({A_{1}({\tilde{\Phi}})/\sqrt{l}})\|x^{*}_{T_{01}^{c}}\|_{1}\|h_{T_{01}}\|\leq\|\tilde{\Phi}h\|^{2}\leq
6\sqrt{s}\lambda\Delta\|h_{T_{01}}\|+6\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1}.$
If follows that
$\displaystyle
A_{0}(\tilde{\Phi})\|h_{T_{01}}\|^{2}-\left(({A_{1}(\tilde{\Phi})/\sqrt{l}})\|x^{*}_{T_{01}^{c}}\|_{1}+6\sqrt{s}\lambda\Delta\right)\|h_{T_{01}}\|\leq
6\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1}.$ (22)
Using $A_{0}(\tilde{\Phi})>0$ (which is assumed in the statement of the
theorem), we recall that for a quadratic inequality $ax^{2}-bx\leq c$ with
$a,b,c>0$, one has
$x\leq{b+\sqrt{b^{2}+4ac}\over 2a}\leq{2b+\sqrt{4ac}\over 2a}={b\over
a}+\sqrt{c\over a}.$ (23)
Hence (22) implies that
$\displaystyle\|h_{T_{01}}\|$ $\displaystyle\leq{1\over
A_{0}(\tilde{\Phi})}\left({(A_{1}(\tilde{\Phi})/\sqrt{l})\|x^{*}_{T_{01}^{c}}\|_{1}}+6\sqrt{s}\lambda\Delta\right)+\sqrt{\frac{\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1}}{A_{0}(\tilde{\Phi})}}$
$\displaystyle={6\sqrt{s}\lambda\Delta\over
A_{0}(\tilde{\Phi})}+{A_{1}(\tilde{\Phi})\over
A_{0}(\tilde{\Phi})\sqrt{l}}\|x^{*}_{T_{0}^{c}}\|_{1}+\sqrt{6\lambda\Delta\over
A_{0}(\tilde{\Phi})}\|x^{*}_{T_{0}^{c}}\|_{1}^{1/2}.$
By invoking (19b), we prove (17a) by
$\displaystyle\|h\|$
$\displaystyle\leq\sqrt{1+9s/l}\|h_{T_{01}}\|+\left(4/\sqrt{l}\right)\|x^{*}_{T_{0}^{c}}\|_{1}$
$\displaystyle\leq{6\sqrt{1+9s/l}\sqrt{s}\lambda\Delta\over
A_{0}(\tilde{\Phi})}+\left(4+\frac{\sqrt{1+9s/l}A_{1}(\tilde{\Phi})}{A_{0}(\tilde{\Phi})}\right)\left(\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l}\right)+\sqrt{(1+9s/l)6\lambda\Delta\over
A_{0}(\tilde{\Phi})}\|x^{*}_{T_{0}^{c}}\|_{1}^{1/2}$
$\displaystyle=6C_{2}(\tilde{\Phi})^{2}\sqrt{s}\lambda\Delta+C_{1}(\tilde{\Phi})\left(\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l}\right)+2.5C_{2}(\tilde{\Phi})\sqrt{\lambda\Delta\|x^{*}_{T_{0}^{c}}\|_{1}}.$
Next we prove (17b). Taking $\Psi=\tilde{\Phi}$ in Lemma 5 and applying (21b),
we have
$\displaystyle
A_{0}(\tilde{\Phi})\|h_{T_{01}}\|^{2}-{\left(A_{1}({\tilde{\Phi}})/\sqrt{l}\right)}\|x^{*}_{T_{01}^{c}}\|_{1}\|h_{T_{01}}\|\leq\|\tilde{\Phi}h\|^{2}\leq\tilde{M}\Delta^{2}.$
Using (23) again, one has
$\|h_{T_{01}}\|\leq\frac{A_{1}(\tilde{\Phi})}{A_{0}(\tilde{\Phi})}\left(\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l}\right)+\frac{\sqrt{\tilde{M}}\Delta}{\sqrt{A_{0}(\tilde{\Phi})}}.$
By invoking (19b), we have
$\displaystyle\|h\|\leq$
$\displaystyle\sqrt{1+9s/l}\|h_{T_{01}}\|+\left(4/\sqrt{l}\right)\|x^{*}_{T_{0}^{c}}\|_{1}$
$\displaystyle\leq$
$\displaystyle\left(4+\frac{\sqrt{1+9s/l}A_{1}(\tilde{\Phi})}{A_{0}(\tilde{\Phi})}\right)\|x^{*}_{T_{0}^{c}}\|_{1}/\sqrt{l}+\sqrt{1+9s/l\over
A_{0}(\tilde{\Phi})}\sqrt{\tilde{M}}\Delta,$
proving (17b).
Note that all claims hold under the assumption that (9) is satisfied. Since
Lemma 2 shows that (9) holds with probability at least $1-\pi$ with taking
$\lambda=\sqrt{2\log(2N/\pi)}f_{\max}$, we conclude that all claims hold with
the same probability. ∎
### High-Probability Estimates of the Estimation Error
For use in these results, we define the quantity
$\chi:={\bar{M}/M}=(M-\tilde{M})/M,$ (24)
which is the fraction of saturated measurements.
###### Theorem 3.
Assume $\Phi\in\mathbb{R}^{M\times N}$ to be a Gaussian random matrix, that
is, each entry is i.i.d. and drawn from a standard Gaussian distribution
$\mathcal{N}(0,1)$. Let $\tilde{\Phi}\in\mathbb{R}^{\tilde{M}\times N}$ be the
submatrix of $\Phi$ taking $\tilde{M}$ rows from $\Phi$, with the remaining
$\bar{M}$ rows being used to form the other submatrix
$\bar{\Phi}\in\mathbb{R}^{\bar{M}\times N}$, as defined in (4). Then by
choosing a threshold $\tau$ sufficiently small, and assuming that $\chi$
satisfies the bound $\chi(1-\log\chi)\leq\tau$, we have for any $k\geq 1$ such
that $k\log N=o(M)$ that, with probability larger than
$1-O\left(\exp(-\Omega(M))\right)$, the following estimates hold:
$\displaystyle\sqrt{{\rho}^{+}(k)}\leq$ $\displaystyle{17\over
16}\sqrt{M}+o\left(\sqrt{M}\right),$ (25a)
$\displaystyle\sqrt{{\rho}^{-}(k)}\geq$ $\displaystyle{15\over
16}\sqrt{M}-o\left(\sqrt{M}\right),$ (25b)
$\displaystyle\sqrt{\tilde{\rho}^{+}(k)}\leq$ $\displaystyle{17\over
16}\sqrt{\tilde{M}}+o\left(\sqrt{M}\right),$ (25c)
$\displaystyle\sqrt{\tilde{\rho}^{-}(k)}\geq$ $\displaystyle{15\over
16}\sqrt{\tilde{M}}-o\left(\sqrt{M}\right).$ (25d)
###### Proof.
From the definition of ${\rho}^{+}(k)$, we have
$\sqrt{{\rho}^{+}(k)}=\max_{|T|\leq
k,T\subset\\{1,2,...,N\\}}\sigma_{\max}(\Phi_{T}),$
where $\sigma_{\max}(\Phi_{T})$ is the maximal singular value of $\Phi_{T}$.
From Vershynin [22, Theorem 5.39], we have for any $t>0$ that
$\sigma_{\max}(\Phi_{T})\leq\sqrt{{M}}+O\left(\sqrt{k}\right)+t$
with probability larger than $1-O\left(\exp(-\Omega(t^{2})\right)$. Since the
number of possible choices for $T$ is
${N\choose k}\leq\left(\frac{eN}{k}\right)^{k},$
we have with probability at least
$1-{N\choose k}O\left(\exp{(-\Omega(t^{2}))}\right)\geq
1-O\left(\exp{(k\log{(eN/k)}}-\Omega(t^{2})\right)$
that
$\displaystyle\sqrt{{\rho}^{+}(k)}$ $\displaystyle=\max_{|T|\leq
k,T\subset\\{1,2,...,N\\}}\sigma_{\max}(\Phi_{T})\leq\sqrt{M}+O\left(\sqrt{k}\right)+t.$
Taking $t=\sqrt{M}/16$, and noting that $k=o(M)$, we obtain the inequality
(25a), with probability at least
$\displaystyle\quad 1-O(\exp{(k\log(eN/k)-\Omega(t^{2}))}$
$\displaystyle=1-O(\exp{(k\log(eN/k)-\Omega(M))}$
$\displaystyle=1-O(\exp{(o(M)-\Omega(M))}$ $\displaystyle\geq
1-O(\exp({-\Omega(M)}))$
The second inequality (25b) can be obtained similarly from
$\min_{|T|\leq
k,T\subset\\{1,2,...,N\\}}\sigma_{\min}(\Phi_{T})\leq\sqrt{M}-O\left(\sqrt{k}\right)-t,$
where $\sigma_{\min}(\Phi_{T})$ is the minimal singular value of $\Phi_{T}$.
(We set $t=\sqrt{M}/16$ as above.)
Next we prove (25c). We have
$\displaystyle\sqrt{\tilde{\rho}^{+}(k)}=$ $\displaystyle\max_{h,|T|\leq
k}\frac{\|\tilde{\Phi}_{T}h_{T}\|}{\|h_{T}\|}\leq\max_{|T|\leq
k,|R|\leq\tilde{M}}\sigma_{\max}(\Phi_{R,T}),$
where $R\subset\\{1,2,\dotsc,M\\}$ and $T\subset\\{1,2,\dotsc,N\\}$ are
subsets of the row and column indices of $\Phi$, respectively, and
$\Phi_{R,T}$ is the submatrix of $\Phi$ consisting of rows in $R$ and columns
in $T$. We now apply the result in Vershynin [22, Theorem 5.39] again: For any
$t>0$, we have
$\sigma_{\max}(\Phi_{R,T})\leq\sqrt{{\tilde{M}}}+O\left(\sqrt{k}\right)+t$
with probability larger than $1-O(\exp{(-\Omega(t^{2}))})$. The number of
possible choices for $R$ is
$\displaystyle{M\choose\bar{M}}\leq\left(\frac{eM}{\bar{M}}\right)^{\bar{M}}$
$\displaystyle=\left(\frac{e}{\chi}\right)^{\chi
M}=\exp(M\chi\log(e/\chi))\leq\exp(\tau M),$
so that the number of possible combinations for $(R,T)$ is bounded as follows:
${M\choose\bar{M}}{N\choose k}\leq\exp\left(\tau M+k\log(eN/k)\right).$
We thus have
$\displaystyle\quad\mathbb{P}\left(\sqrt{\tilde{\rho}^{+}(k)}\leq\sqrt{{\tilde{M}}}+O\left(\sqrt{k}\right)+t\right)$
$\displaystyle\geq\mathbb{P}\left(\max_{|R|\leq\tilde{M},|T|\leq
k}\sigma(\Phi_{R,T})\leq\sqrt{{\tilde{M}}}+O\left(\sqrt{k}\right)+t\right)$
$\displaystyle\geq 1-{M\choose\tilde{M}}{N\choose k}O(e^{-\Omega(t^{2})})$
$\displaystyle=1-{M\choose\bar{M}}{N\choose
k}O(e^{-\Omega(t^{2})})\quad\text{(since $\bar{M}+\tilde{M}=M$)}$
$\displaystyle=1-O\left[\exp\left(\tau
M+k\log({eN}/{k})-\Omega(t^{2})\right)\right].$
Taking $t=\sqrt{\tilde{M}}/16$, and noting again that $k=o(M)$, we obtain the
inequality in (25c). Working further on the probability bound, for this choice
of $t$, we have
$\displaystyle\quad 1-O\left[\exp\left(\tau
M+k\log({eN}/{k})-\Omega(\tilde{M})\right)\right]$
$\displaystyle=1-O\left[\exp\left(\tau
M+k\log({eN}/{k})-\Omega(M)\right)\right]$
$\displaystyle=1-O(\exp(-\Omega(M))),$
where the first equality follows from $\tilde{M}=(1-\chi)M$ and for the second
equality we assume that $\tau$ is chosen small enough to ensure that the
$\Omega(M)$ term in the exponent dominates the $\tau M$ term.
A similar procedure can be used to prove (25d). ∎
We conclude by deriving estimates of $\bar{C}_{1}(\tilde{\Phi})$,
$\bar{C}_{2}(\tilde{\Phi})$, and $f_{\max}$, that are used in the discussion
at the end of Section 3.
From Theorem 3, we have that under assumptions (iii), (iv), and (v), the
quantity $A_{1}(\tilde{\Phi})$ defined in (10b) is bounded as follows:
$\displaystyle\bar{A}_{1}(\tilde{\Phi})$
$\displaystyle=4\left(\sqrt{\tilde{\rho}^{+}(3s)}+\sqrt{\tilde{\rho}^{-}(3s)}\right)\left(\sqrt{\tilde{\rho}^{+}(3s)}-\sqrt{\tilde{\rho}^{-}(3s)}\right)$
$\displaystyle\leq 4\left(2\sqrt{\tilde{M}}+o(\sqrt{M})\right)\left({1\over
8}\sqrt{\tilde{M}}+o\left(\sqrt{M}\right)\right)$
$\displaystyle=\tilde{M}+o(M)=\Omega(M).$
Using $s=l$, the quantity $\bar{A}_{0}(\tilde{\Phi})$ defined in (10a) is
bounded as follows:
$\displaystyle\bar{A}_{0}(\tilde{\Phi})$
$\displaystyle=\tilde{\rho}^{-}(2s)-\frac{3}{4}\bar{A}_{1}(\tilde{\Phi})$
$\displaystyle\geq{15\over 16}\tilde{M}-o(M)-{3\over 4}\tilde{M}-o(M)$
$\displaystyle={3\over 16}\tilde{M}-o(M)$ $\displaystyle=\Omega(M),$
for all sufficiently large dimensions and small saturation ratio $\chi$, since
$\tilde{M}=(1-\chi)M$. Using the estimates above for
$\bar{A}_{0}(\tilde{\Phi})$ and $\bar{A}_{1}(\tilde{\Phi})$, in the
definitions (10c) and (10d), we obtain
$\bar{C}_{1}(\tilde{\Phi})=4+\sqrt{10}{\bar{A}_{1}(\tilde{\Phi})}/{A_{0}(\tilde{\Phi})}=\Omega(1),\quad\bar{C}_{2}(\tilde{\Phi})=\sqrt{10/\bar{A}_{0}(\tilde{\Phi})}=\Omega\left(1/\sqrt{M}\right),$
as claimed. Finally, $f_{\max}$ can be estimated by
$f_{\max}=\sqrt{\tilde{\rho}^{+}(1)}\leq{17\over
16}\sqrt{\tilde{M}}+o(M)=O\left(\sqrt{M}\right).$
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|
arxiv-papers
| 2012-07-03T06:07:13 |
2024-09-04T02:49:32.564508
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ji Liu and Stephen J. Wright",
"submitter": "Ji Liu",
"url": "https://arxiv.org/abs/1207.0577"
}
|
1207.0732
|
# Quantum LDPC Codes Constructed from Point-Line Subsets of the Finite
Projective Plane
Jacob Farinholt Manuscript received Month Day, Year; revised Month Day,
Year.This work was supported by the Naval Surface Warfare Center’s In-house
Laboratory Independent Research (ILIR) and Academic Fellowship (AFP)
programs.J. Farinholt is with the Electromagnetic and Sensor Systems
Department, Naval Surface Warfare Center, Dahlgren Division, Dahlgren, VA
22448 (e-mail:Jacob.Farinholt@navy.mil)
###### Abstract
Due to their fast decoding algorithms, quantum generalizations of low-density
parity check, or LDPC, codes have been investigated as a solution to the
problem of decoherence in fragile quantum states. However, the additional
twisted inner product requirements of quantum stabilizer codes force four-
cycles and eliminate the possibility of randomly generated quantum LDPC codes.
Moreover, the classes of quantum LDPC codes discovered thus far generally have
unknown or small minimum distance, or a fixed rate. This paper presents
several new classes of quantum LDPC codes constructed from finite projective
planes. These codes have rates that increase with the block length $n$ and
minimum weights proportional to $n^{1/2}$.
###### Index Terms:
error correction codes, quantum error correction, finite geometry.
## I Introduction
Classical low-density parity check codes, or LDPC codes, were first discovered
by Gallager [2] in 1960. Later, it was shown that bipartite graphs, called
_Tanner graphs_ , could be used to describe the codes and their actions under
iterative belief propagation decoding algorithms. In 2001, Kou _et al._ [3]
showed that by using finite geometries, many classes of these codes could be
easily generated to have known parameters. Recently, Droms _et al._ [4] and
Castleberry _et al._ [5] showed that LDPC codes constructed from point-line
subsets of finite projective planes could often out-perform those constructed
in [3]. Classical LDPC codes are some of the best known, with rates
asymptotically approaching the Shannon limit [6]. We say a code is an LDPC
code if its parity check matrix is sparse, and its corresponding Tanner graph
has no four-cycles (i.e., any pair of rows in the parity check have no more
than one “1” in common position). In particular, the rows of the parity check
matrix need not all be linearly independent.
The first quantum error correcting codes were discovered by Shor [7],
Calderbank [8], and separately by Steane [9, 10]. Using the stabilizer
formalism introduced by Gottesman [11], it was shown that these corresponding
_stabilizer codes_ could be described using classical parity check matrices
with an added twisted inner product requirement [12]. This twisted inner
product has the unfortunate consequence of forcing four-cycles on a
corresponding Tanner graph. Nevertheless, sparse-graph stabilizer codes with
minimal four-cycles have been suggested as quantum generalizations of
classical LDPC codes. The first such quantum LDPC, or QLDPC, codes were
suggested by Postol in 2001 [13], and many examples were constructed by MacKay
_et al._ [14]. Since then, many other classes of QLDPC codes have been
constructed (see [15, 16] and references therin). However, many of these codes
have unknown or small minimum distance. Tillich and Zemor [15] constructed
fixed-rate QLDPC codes with minimum distances that increase as the square root
of the code length. Aly [16] used finite geometric techniques to create
quantum generalizations of many of the classical LDPC codes constructed in
[3], and suggested that similar codes could be constructed through the use of
projective geometries. In this paper, projective geometries, in particular the
projective plane of order $2^{s}$ and many of its subsets, are used to
construct QLDPC codes with minimum distances proportional to the square root
of the code length, and whose rates also increase with the code length,
providing possibly the best-known rate-increasing QLDPC codes in the current
literature.
This paper is organized as follows. In Section II we introduce quantum
stabilizer codes, classical constructions, and the CSS formalism. In Section
III we introduce finite projective planes and subsets of the plane with
respect to regular hyperovals. Section IV describes methods of constructing
classical self-orthogonal sparse-graph codes from the subsets described in
Section III. Lastly, Section V contains our main results about new classes of
Quantum LDPC codes and their corresponding parameters.
## II Quantum Stabilizer Codes and CSS Codes
The Pauli operators are given by
$\displaystyle I$ $\displaystyle=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\ $
$\displaystyle X=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},$ $\displaystyle Y$
$\displaystyle=\begin{pmatrix}0&-i\\\ i&0\end{pmatrix},\ $ $\displaystyle
Z=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (1)
These operators, along with the scalars $i^{k}$ for $k\in\mathbb{Z}_{4}$, form
the single-qubit _Pauli group_ $\mathcal{P}$. Observe that
$X^{2}=Y^{2}=Z^{2}=I$, and $Y=iXZ$. A pure state quantum bit, or _qubit_ is a
norm-one element of a two-dimensional complex Hilbert space,
$\mathcal{H}_{2}$. It is well-known [9] that correcting errors induced by the
Pauli operators is sufficient to correct arbitrary errors on single qubits.
In the case of multi-qubit states, that is, norm-one elements in
$\mathcal{H}_{2}^{\otimes
n}=\mathcal{H}_{2}\otimes\mathcal{H}_{2}\dots\otimes\mathcal{H}_{2}$, the
$n$-fold tensor product of $\mathcal{H}_{2}$, the corresponding Pauli
operators are $n$-fold tensor products of the operators given in (II), with
corresponding Pauli group $\mathcal{P}_{n}$ on $n$ qubits obtained by
including the scalars $i^{k}$ for $k\in\mathbb{Z}_{4}$.
An arbitrary element $P\in\mathcal{P}_{n}$ is represented as
$P=i^{k}X(a)Z(b),$ (2)
for $k\in\mathbb{Z}_{4}$, and $a=(a_{1},a_{2},\dots,a_{n})$ and
$b=(b_{1},b_{2},\dots,b_{n})$ are both length $n$ binary vectors, with a 1 in
position $i$ of $a$ (resp. $b$) precisely when there is an $X$ (resp. $Z$)
operator acting on qubit $i$. For example, $X(1001)Z(0101)$ should be
interpreted as $X\otimes Z\otimes I\otimes Y$, with the $Y$ at the end since
it is a product of $X$ and $Z$. It has been shown [12] that two elements
$i^{k}X(a)Z(b)$ and $i^{k^{\prime}}X(a^{\prime})Z(b^{\prime})$ of
$\mathcal{P}_{n}$ commute if and only if
$(a|b)*(a^{\prime}|b^{\prime}):=a\cdot b^{\prime}+b\cdot
a^{\prime}=0\pmod{2},$ (3)
where $\cdot$ is the standard inner (dot) product. We call $*$ the _twisted
inner product_.
Observe that the center $C(\mathcal{P}_{n})$ of $\mathcal{P}_{n}$ is given by
$C(\mathcal{P}_{n})=\\{i^{k}I\ |\ k\in\mathbb{Z}_{4}\\}$. Since these elements
are effectively the global phase actions in $\mathcal{P}_{n}$, we can reduce
ourselves to considering the quotient group
$\mathcal{P}_{n}/C(\mathcal{P}_{n})$, the elements of which are equivalence
classes of the form $\\{i^{k}X(a)Z(b)\ |\ k\in\mathbb{Z}_{4}\\}$, for fixed
$a$ and $b$; and we label each equivalence class with its scalar-free element
(e.g. $X(a)Z(b)\equiv\\{i^{k}X(a)Z(b)\ |\ k\in\mathbb{Z}_{4}\\}$). Moreover,
as described in [12], the quotient group $\mathcal{P}_{n}/C(\mathcal{P}_{n})$
is isomorphic to a $2n$-dimensional binary vector space $V_{2n}$ via the map
$X(a)Z(b)\mapsto(a|b),$ (4)
with the commutativity relationship of elements in $\mathcal{P}_{n}$ preserved
by imposing the twisted inner product (3) on the vector space $V_{2n}$.
### II-A Stabilizer Codes
A _stabilizer group_ $\mathcal{S}\subseteq\mathcal{P}_{n}$ is a commutative
subgroup of $\mathcal{P}_{n}$ that does not contain $-I$. By not containing
$-I$, distinct elements of $\mathcal{S}$ are mapped to distinct equivalence
classes in $\mathcal{P}_{n}/C(\mathcal{P}_{n})$. Thus, without loss of
generality, a stabilizer group can be classically represented as a collection
of vectors in $V_{2n}$ with the property that every pair are orthogonal with
respect to the twisted inner product (3).
A _stabilizer code_ $\mathcal{C}(\mathcal{S})$ for stabilizer group
$\mathcal{S}\subseteq\mathcal{P}_{n}$ is defined as the simultaneous $+1$
eigenspace in $\mathcal{H}_{2}^{\otimes n}$ of each element in $\mathcal{S}$.
An error is detected if it anticommutes with any element of the stabilizer
group, thereby producing a nonzero syndrome. Thus, we can classically define a
stabilizer parity check matrix $H(\mathcal{C})$ for the stabilizer code
$\mathcal{C}(\mathcal{S})$ by letting the rows of $H(\mathcal{C})$ be the
vectors corresponding to the generators of $\mathcal{S}$.
Conversely, a binary length-$2n$ matrix $[A|B]$ is a stabilizer parity check
matrix for some stabilizer code if and only if every pair of rows in $[A|B]$
are orthogonal with respect to (3), or equivalently, if and only if
$AB^{t}+BA^{t}=0\pmod{2}.$ (5)
### II-B CSS Codes
A large class of stabilizer codes of particular interest are ones that are
constructed from classical error correcting codes. In particular, if $C_{1}$
and $C_{2}\subseteq C_{1}^{\bot}$ are classical codes with parity check
matrices $H(C_{1})$ and $H(C_{2})$, respectively, then
$\textbf{H}(\mathcal{C})=\begin{pmatrix}H(C_{1})&0\\\ 0&H(C_{2})\end{pmatrix}$
(6)
is a stabilizer parity check for a stabilizer code $\mathcal{C}$. Note that
(3) is satisfied by the fact that $H(C_{1})$ is orthogonal to $H(C_{2})$ by
construction. In particular, if $C_{1}$ and $C_{2}$ are classical
$[n,k_{1},d_{1}]$ and $[n,k_{2},d_{2}]$ codes, respectively, then the
stabilizer code $\mathcal{C}$ encodes $K=k_{1}+k_{2}-n$ qubits into $n$
qubits, and corrects $(D-1)/2$ and fewer arbitrary qubit errors, where
$D=\min(d_{1},d_{2})$ [12]. We call such a code an $[[n,K,D]]$ CSS code, where
CSS are the initials of the discoverers of such codes [8, 10]. Note that it is
standard practice to place the parameters of a quantum code inside double
brackets, as opposed to single brackets in the classical case. A particularly
nice scenario is one in which a classical $[n,k,d]$ code $C$, with parity
check matrix $H(C)$, is dual-containing, in which case
$\textbf{H}(\mathcal{C})=\begin{pmatrix}H(C)&0\\\ 0&H(C)\end{pmatrix}$ (7)
is a stabilizer parity check matrix for an $[[n,2k-n,d]]$ CSS code
$\mathcal{C}$. We call CSS codes constructed in this manner _symmetric_ CSS
codes; otherwise they are called _asymmetric_.
## III Finite Projective Planes and Regular Hyperovals
| | | # of secant lines | # of skew lines
---|---|---|---|---
# of points | # of lines | # of lines | intersecting a non- | intersecting a non-
in $H_{\mathcal{C}}$ | secant to $H_{\mathcal{C}}$ | skew to $H_{\mathcal{C}}$ | hyperoval point | hyperoval point
$q+2$ | $\frac{q^{2}+3q+2}{2}$ | $\frac{q^{2}-q}{2}$ | $\frac{q+2}{2}$ | $\frac{q}{2}$
TABLE I: Number of points and lines with respect to the hyperoval
$H_{\mathcal{C}}$ and non-hyperoval points in $PG(2,q)$.
A _finite projective plane_ $PG(2,q)$ is a finite collection of points, along
with subsets of points (lines) satisfying:
1. 1.
Any two distinct points determine a unique line.
2. 2.
Any two distinct lines determine a unique point.
3. 3.
There exist four points, no three of which are colinear.
Note that the second axiom implies that there are no parallel lines in this
geometry. See Figure 1 for an example of a finite projective plane.
Figure 1: The Fano Plane, $PG(2,2)$, is the simplest example of a finite
projective plane.
The value $q$ is called the _order_ of the projective plane, and the following
properties of $PG(2,q)$ can be determined [17]:
1. 1.
Every line contains $q+1$ distinct points.
2. 2.
Every point is incident with $q+1$ distinct lines.
3. 3.
There are exactly $q^{2}+q+1$ points and $q^{2}+q+1$ lines in the plane.
When $q$ is a prime power, then points and lines of $PG(2,q)$ can be
represented as 1- and 2-dimensional subspaces, respectively, of the
3-dimensional vector space $V_{3}(q)$ over $\mathbb{F}_{q}$. Points are
equivalence classes of the form $[x,y,z]\equiv\\{(cx,cy,cz)\ |\
c\in\mathbb{F}_{q}-\\{0\\}\\}$, for $x$, $y$, and $z$ in $\mathbb{F}_{q}$.
While lines are 2-dimensional subspaces, each can be uniquely represented by
its dual in the 1-dimensional subspace. Thus, to distinguish lines from
points, we label points in brackets, e.g. $[x,y,z]$, and lines in perentheses,
e.g. $(a,b,c)$.
### III-A Conics, Hyperovals, and Subsets of lines
A _conic_ $\mathcal{C}$ is a set of $q+1$ points in $PG(2,q)$ whose
coordinates satisfy a non-degenerate quadradic equation, that is,
$\mathcal{C}:=\\{[x,y,z]:ax^{2}+by^{2}+cz^{2}+fyz+gzx+hxy=0\\},$ (8)
for some $a,b,c,f,g,h,\in\mathbb{F}_{q}$. Conics have the property that no
three-point subsets are colinear. As such, every line must be either _skew_
(i.e. intersects $\mathcal{C}$ at no points), _tangent_ (i.e. intersects
$\mathcal{C}$ at one point), or _secant_ (i.e. intersects $\mathcal{C}$ at
exactly two points). A well-known result in projective geometry is that when
$q=2^{s}$, all of the tangent lines are concurrent at a point outside the
conic, called the _nucleus_. If the nucleus is added to the conic, we obtain a
_regular hyperoval_ , $H_{\mathcal{C}}$ (hereafter called simply a hyperoval).
Lines tangent to the conic are then secant to the hyperoval, and hence all
lines are either secant or skew to $H_{\mathcal{C}}$. Table I lists the number
of points and lines with respect to a hyperoval.
### III-B Incidence Stuctures and Parity Checks
The _incidence matrix_ $M_{\pi}$ for $\pi=PG(2,q)$ is constructed by letting
the columns correspond to points, and rows correspond to lines, with
$M_{\pi_{i,j}}=1$ when line $i$ contains point $j$, and $M_{\pi_{i,j}}=0$
otherwise. This matrix is sparse by construction. It was shown in [18] that
the rank of this matrix is $\begin{pmatrix}p+1\\\ 2\end{pmatrix}^{s}+1$, where
$q=p^{s}$. In particular, when $p=2$, this reduces to $3^{s}+1$.
The incidence matrix $M_{\pi}$ is well-suited to act as a parity check matrix
for a classical LDPC code, as it is sparse, and any two rows have exactly one
“1” in common position. However, since this matrix is not self-dual, it must
be adapted by adding a column of all ones, called the _unit vector_ , or
$u$-vector, to it, denoting the new matrix by $M_{\pi}^{\prime}$. Note that
since every row of $M_{\pi}$ has odd weight, adding this vector does not
affect the rank. Since $M_{\pi}^{\prime}$ is self-dual, sparse, and any two
rows have exactly two “1”s in common position, it can be used in the
construction of a parity check matrix for a quantum LDPC code.
We can likewise consider classical LDPC parity check matrices constructed by
the incidence structures of subsets of points and lines in $PG(2,q)$. Such
classical codes have been recently studied [4, 5], but again, these incidence
matrices must be adapted to make them self-orthogonal in order to use them to
construct parity checks for QLDPC codes.
## IV Classical Self-Orthogonal LDPC Codes
Here we construct self-orthogonal parity check matrices for classical LDPC
codes constructed from the incidence matrices from point-line subsets of
$PG(2,q)$. In particular, we assume that $q=2^{s}$ so that we can use the
subset structure based on hyperovals. We then study the properties of the
corresponding classical codes. The results of this section are summarized in
Table II.
### IV-A All Points and All Lines, $M_{\pi}^{\prime}$
We again let $M_{\pi}$ be the incidence matrix for $\pi=PG(2,q)$ for
$q=2^{s}$, and let $M_{\pi}^{\prime}$ be the concatenation of $M_{\pi}$ with
the $u$-vector, i.e. $M_{\pi}^{\prime}=[M_{\pi}|\mathbf{1}]$. This matrix has
$4^{s}+2^{s}+2$ columns and $4^{s}+2^{s}+1$ rows. As discussed in Section
III-B, the rank of this matrix is $3^{s}+1$.
###### Proposition IV.1
The matrix $M_{\pi}^{\prime}$ is a parity check for a classical
$[4^{s}+2^{s}+2,\ 4^{s}-3^{s}+2^{s}+1,\ 2^{s}+2]$ LDPC code.
###### Proof:
The length and dimension are obvious. To prove minimum distance, we have two
cases:
* $A)$:
The last bit of a minimum weight codeword is a 0. In this case, the codeword
will have the same weight as a minimum weight codeword for a code with
classical parity check matrix given by $M_{\pi}$, which is known to be
$2^{s}+2$ [3].
* $B)$:
The last bit $c_{n}$ of a minimum weight codeword $c$ is a 1. Each additional
1-bit in $c$ will cause $c$ to be orthogonal to $2^{s}+1$ rows of
$M_{\pi}^{\prime}$. Thus, in addition to $c_{n}$, the codeword $c$ needs at
least $m$ additional 1-bits, where $m$ is the smallest integer such that
$m(2^{s}+1)\geq 4^{s}+2^{s}+1$, which is determined to be $2^{s}+1$. Thus, if
$c_{n}=1$, then the weight of $c$ is lower bounded by $2^{s}+2$, completing
the proof.
∎
Note that, we can alternatively prove results on minimum weight from a graph-
theoretic approach. In particular, $M_{\pi}^{\prime}$ can be graphically
viewed as the incidence matrix for a projective plane $\pi=PG(2,q)$ with an
additional point $u$ through which every line intersects. Then a codeword is
graphically viewed as a collection of points in this “extended projective
plane” $\pi^{\prime}$, through which every line intersects an even number of
times, since this would correspond to a vector having an even number of 1-bits
in common position with each row of the parity check, and hence orthogonal to
each row. In particular, if the codeword has a 0 at the $u$-column, then we
reduce ourselves to studying a collection of points $\mathcal{S}$ in $\pi$
satisfying this. However, if the codeword has a 1 at the $u$-column, then the
collection of points contains the point $u$ through which every line
intersects. Then the remaining points $\mathcal{S}$ are all in $\pi$ such that
every line intersects $\mathcal{S}$ an odd number of times. The weight of the
codeword is given by $|\mathcal{S}|$ or $|\mathcal{S}|+1$ depending on whether
the codeword has a 0 or a 1 at the end, respectively. Thus, we can determine
the minimum weight of a code by finding a minimal set of points $\mathcal{S}$
in $PG(2,q)$ in each case. Such an approach will be used for many of the
proofs in the following constructions.
### IV-B Skew Lines and Non-Hyperoval Points, $H(C_{sk})$
| $n$ | $k$ | $d$
---|---|---|---
$C_{\pi}^{\prime}$ | $4^{s}+2^{s}+2$ | $4^{s}-3^{s}+2^{s}+1$ | $2^{s}+2$
| | $4^{s}-3^{s}-1\leq k$ | $2^{s-1}+1$
$C_{sk}$ | $4^{s}$ | $\leq 4^{s}-3^{s}+2^{s}$ | $\leq d\leq 2^{s}$
| | | $2^{s-1}+2$
$C_{seA}$ | $4^{s}+2^{s}+2$ | $4^{s}-3^{s}+2^{s}+1$ | $\leq d\leq 2^{s}+2$
$C_{se}$ | $4^{s}$ | $4^{s}-3^{s}+2^{s}+1$ | $2^{s-1}+2\leq d\leq 2^{s}+2$
TABLE II: Summary of classical LDPC codes constructed from $PG(2,2^{s})$ and
subsets with respect to a hyperoval.
Suppose we restrict ourselves to the incidence structure formed by only the
lines skew to a hyperoval, along with the $u$-vector. In such a case, we can
find a weight one codeword by letting the last bit be 0, and letting
$\mathcal{S}$ consist of only a hyperoval point. Since skew lines intersect
the hyperoval nowhere, $\mathcal{S}$ corresponds to a codeword. A code of
minimum weight 1 is useless, so we will instead delete the columns
corresponding to hyperoval points. Thus, we let $H(C_{sk})$ be the incidence
matrix whose rows correspond to skew lines and whose columns correspond to
non-hyperoval points in$PG(2,q)$, concatenated with the $u$-vector. This
matrix will have $\frac{q^{2}-q}{2}$ rows and $q^{2}$ columns.
###### Proposition IV.2
The matrix $H(C_{sk})$ is a parity check matrix for a classical
$[4^{s},k_{sk},d_{sk}]$ LDPC code $C_{sk}$, where $4^{s}-3^{s}-1\leq
k_{sk}\leq 4^{s}-3^{s}+2^{s}$ and $2^{s-1}+1\leq d_{sk}\leq 2^{s}$.
###### Proof:
We know that $\dim(M_{\pi}^{\prime})=3^{s}+1$. Since skew lines never
intersect hyperoval points, removal of the columns corresponding to hyperoval
points does not change the dimension. In [5] it was shown that removing rows
corresponding to lines secant to the conic will not affect the dimension.
Removing the lines tangent to the conic (i.e. the remaining lines secant to
the hyperoval) gives us the matrix $H(C_{sk})$, but may possibly decrease the
dimension. Thus $3^{s}-2^{s}\leq\dim(H(C_{sk}))\leq 3^{s}+1$, from which we
determine the bounds on the code dimension.
To prove minimum distance, we again consider two cases, namely, if the last
bit of a codeword is a 0 or a 1.
* $A)$:
Suppose the codeword has a 0 at the $u$-column. Then we view the codeword as a
collection of non-hyperoval points $\mathcal{S}$ in $PG(2,q)$ such that any
skew line intersects $\mathcal{S}$ in an even number of places. Let $p$ be a
point in $\mathcal{S}$. Then it must be intersected by $q/2$ skew lines. Then
for each of these skew lines, $\mathcal{S}$ must have an additional point
through which the line intersects. Thus, $|\mathcal{S}|\geq\frac{q}{2}+1$.
* $B)$:
Suppose the codeword has a 1 at the $u$-column. Then we view the codeword as a
collection of non-hyperoval points $\mathcal{S}$ in $PG(2,q)$ such that any
skew line intersects $\mathcal{S}$ in an odd number of places, namely, at
least once. Since any two lines in $PG(2,q)$ intersect at exactly one point,
we obtain a minimum when we choose $\mathcal{S}$ to be a line. Since hyperoval
points are removed, secant lines have the fewest number of non-hyperoval
points in $PG(2,q)$, namely $q-1$. Thus, if the codeword has a 1 in the
$u$-column, then it must have weight at least $|\mathcal{S}|+1=q$. Since
$\mathcal{S}$ was found constructively, codewords of such weight do exist, and
hence this acts as our upper bound on the minimum weight.
∎
### IV-C Secant Lines and All Points, $H(C_{seA})$
We now consider classical LDPC codes whose incidence matrices are constructed
by removing the rows in $M_{\pi}^{\prime}$ corresponding to lines skew to a
given hyperoval, leaving only the rows corresponding to the lines secant to
the hyperoval. This matrix, denoted $H(C_{seA})$, will have $q^{2}+q+2$
columns and $(q^{2}+3q+2)/2$ rows.
###### Proposition IV.3
The matrix $H(C_{seA})$ is a parity check matrix for a classical
$[4^{s}+2^{s}+2,\ 4^{s}-3^{s}+2^{s}+1,\ d_{seA}]$ LDPC code $C_{seA}$, where
$2^{s-1}+2\leq d_{seA}\leq 2^{s}+2$.
###### Proof:
We know $\dim(M_{\pi}^{\prime})=3^{s}+1$. It is also known [5] that removing
the rows corresponding to skew lines (thereby giving us $H(C_{seA})$) does not
change the rank of this matrix. Thus $\dim(H(C_{seA}))=3^{s}+1$. Since the
length $n$ is $4^{s}+2^{s}+2$, we solve for $\dim(C_{seA})=n-\dim(H(C_{seA}))$
to obtain our result.
To prove the bounds on the minimum distance, we again have the following two
cases:
* $A)$:
Suppose the codeword has a 0 at the $u$-column. Then we view the codeword as a
collection of points $\mathcal{S}$ in $PG(2,q)$ such that any secant line
intersects $\mathcal{S}$ in an even number of places. Let $p$ be a point in
$\mathcal{S}$. Then it must be intersected by $(q+2)/2$ secant lines. Then for
each of these secant lines, $\mathcal{S}$ must have an additional point
through which the line intersects. Thus,
$|\mathcal{S}|\geq\frac{q+2}{2}+1=\frac{q}{2}+2=2^{s-1}+2$.
* $B)$:
Suppose the codeword has a 1 at the $u$-column. Then it corresponds to a
collection of points $\mathcal{S}$ in $PG(2,q)$ such that every line secant to
the hyperoval intersects it an odd number of times, namely, at least once.
Since each line in $PG(2,q)$ intersects all other lines exactly once, we
obtain a minimum when we choose the points in $\mathcal{S}$ to be a collection
of $q+1$ points that make up a line. Thus, if a codeword has a 1 at the
$u$-column, then it must have weight at least $|\mathcal{S}|+1=q+2=2^{s}+2$.
Since $\mathcal{S}$ was found constructively, codewords of such weight do
exist, and hence this acts as the upper bound on the minimum weight.
∎
### IV-D Secant Lines and Non-Hyperoval Points, $H(C_{se})$
While removing hyperoval points was not necessary to obtain good classical
codes constructed from secant lines, we nevertheless remove them here for
reasons that will become apparent later. Let $H(C_{se})$ be the matrix formed
by removing from $M_{\pi}^{\prime}$ the columns corresponding to hyperoval
points and the rows corresponding to lines skew to the hyperoval. This matrix
will have $(q^{2}+3q+2)/2$ rows and $q^{2}$ columns.
###### Proposition IV.4
The matrix $H(C_{se})$ is a parity check matrix for a classical $[4^{s},\
4^{s}-3^{s}+2^{s}+1,\ d_{se}]$ LDPC code $C_{se}$, where $2^{s-1}+2\leq
d_{se}\leq 2^{s}+2$.
###### Proof:
We know $\dim(M_{\pi})=3^{s}+1$. In [5] it was shown that columns
corresponding to hyperoval points and rows corresponding to skew lines can be
removed without affecting the dimension. Since the weight of each row of the
resulting matrix is odd, we can add the $u$-vector, to obtain the matrix
$H(C_{se})$, without affecting the rank. Thus, $\dim(H(C_{se}))=3^{s}+1$, from
which we determine the dimension of $C_{se}$.
To prove the minimum distance, we have the following two cases:
* $A)$:
Suppose a codeword has a 0 at the $u$-column. Then it corresponds to a
collection of non-hyperoval points $\mathcal{S}$ in $PG(2,q)$ such that every
secant line intersects $\mathcal{S}$ in an even number of points. Let
$p\in\mathcal{S}$. Since it is intersected by $(q+2)/2$ secant lines, we must
have at least one additional point in $\mathcal{S}$ for each line to
intersect. Thus $|\mathcal{S}|\geq\frac{q}{2}+2=2^{s-1}+2$.
* $B)$:
Suppose a codeword has a 1 at the $u$-column. The proof continues in a similar
fashion to that of Proposition IV.3. A set of points $\mathcal{S}$ in
$PG(2,q)$ corresponding to a codeword with a $1$ at the $u$-column will be
minimal when it intersects every secant line exactly once in non-hyperoval
points, which occurs when $\mathcal{S}$ is a collection of $q+1$ points
corresponding to a skew line. Thus, $|\mathcal{S}|+1=2^{s}+2$. Again, since
such codewords do exist, this weight acts as an upper bound on the minimum
weight of the code.
∎
## V Quantum LDPC Codes
Using the results from Section IV, we construct parity check matrices for
QLDPC codes using the asymmetric and symmetric CSS constructions (6) and (7).
### V-A Symmectric QLDPC Codes from All Points and All Lines
The symmetric QLDPC codes $\mathcal{C}_{\pi}$ constructed here have as a
parity check matrix $H(\mathcal{C}_{\pi})$ of the form
$H(\mathcal{C}_{\pi})=\begin{bmatrix}M_{\pi}^{\prime}&0\\\
0&M_{\pi}^{\prime}\end{bmatrix},$ (9)
where $M_{\pi}^{\prime}$ is as defined in Section IV-A. Recall that
$M_{\pi}^{\prime}$ is self-orthogonal by construction, and is a parity check
matrix for a classical LDPC code, and hence $H(\mathcal{C}_{\pi})$ is well-
defined.
###### Theorem V.1
Given a finite projective plane $\pi=PG(2,2^{s})$ for some positive integer
$s$, the matrix $H(\mathcal{C}_{\pi})$ in (9) is a parity check matrix for an
$[[n,\ 2k-n,\ D]]$ QLDPC code $\mathcal{C}_{\pi}$, where $n=4^{s}+2^{s}+2$,
$k=4^{s}-3^{s}+2^{s}+1$, and $D=2^{s}+2$.
###### Proof:
This follows immediately from Proposition IV.1 and the properties of symmetric
CSS codes. ∎
Observe that this code has a rate $\frac{2k-n}{n}$ that rapidly increases with
the length, and a minimum distance that increases on the order $\sqrt{n}$. The
number of stabilizers used for error correction, given by the number of rows
in the parity check, is almost $2n$.
### V-B Asymmetric QLDPC Codes
The asymmetric QLDPC codes $\mathcal{C}_{asym}$ constructed here have as a
parity check matrix $H(\mathcal{C}_{asym})$ of the form
$H(\mathcal{C}_{asym})=\begin{bmatrix}H(C_{sk})&0\\\
0&H(C_{se})\end{bmatrix},$ (10)
where $H(C_{sk})$ and $H(C_{se})$ are as defined in Sections IV-B and IV-D,
respectively. Note that $H(C_{sk})$ and $H(C_{se})$ are orthogonal by
construction, and both have the same block length. Moreover, each is a parity
check for a classical LDPC code, making $H(\mathcal{C}_{asym})$ a well-defined
parity check for a QLDPC code.
###### Theorem V.2
Given a finite projective plane $PG(2,2^{s})$ for some positive integer $s$,
the matrix $H(\mathcal{C}_{asym})$ in (10) is a parity check matrix for a
$[[4^{s},K,D]]$ QLDPC code $\mathcal{C}_{asym}$, where $4^{s}-2\cdot
3^{s}+2\leq K\leq 4^{s}-2\cdot 3^{s}+2^{s}-1$, and $D\geq 2^{s-1}+1$.
###### Proof:
The length is determined by observing that $H(C_{sk})$ and $H(C_{se})$ are
parity checks for classical codes of length $4^{s}$. We obtain the dimension
$K$ and minimum distance $D$ from Propositions IV.2 and IV.4, and the fact
that $K=\dim(\mathcal{C}_{asym})=k_{sk}+k_{se}-n$, and minimum distance $D$ is
bounded below by $\min(d_{sk},d_{se})$, where $k_{sk}$, $k_{se}$, $d_{sk}$ and
$d_{se}$ are respectively the dimensions and minimum distances of the
classical codes $C_{sk}$ and $C_{se}$, respectively. ∎
While the bound on the dimension may be coarse, it is important to observe
that the rate of these codes nevertheless increases rapidly with the code
length $n$. The code will have only $n+\sqrt{n}+1$ parity checks, and very few
four-cycles.
### V-C Symmectric QLDPC Codes from Skew Lines
The parity check matrix $H(\mathcal{C}_{symSK})$ for the symmetric QLDPC codes
$\mathcal{C}_{symSK}$ constructed here have the form
$H(\mathcal{C}_{symSK})=\begin{bmatrix}H(C_{sk})&0\\\
0&H(C_{sk})\end{bmatrix},$ (11)
where $H(C_{sk})$ is the self-orthogonal classical LDPC parity check matrix
defined in Section IV-B.
###### Theorem V.3
Given a finite projective plane $PG(2,2^{s})$ for some positive integer $s$,
the matrix $H(\mathcal{C}_{symSK})$ defined in (11) is a parity check matrix
for a $[[4^{s},\ 2k_{sk}-4^{s},\ D]]$ QLDPC code $\mathcal{C}_{symSK}$, where
$4^{s}-3^{s}-1\leq k_{sk}\leq 4^{s}-3^{s}+2^{s}$, and $D\geq 2^{s-1}+1$.
###### Proof:
This follows immediately from Proposition IV.2 and the properties of symmetric
CSS codes. ∎
Although the bound on the dimension of these codes is not as tight as that of
the asymmetric codes described in Section V-B, these codes nevertheless have a
fast rate that increases with the length $n$. The bound on the minimum
distance is the same as for the asymmetric codes, showing that these, too,
describe fast-rate QLDPC codes with good minimum distance. The codes will have
$n-\sqrt{n}$ parity checks and few four-cycles.
Code | CSS Type | Length | Dimension | Minimum | Number of
---|---|---|---|---|---
| | | | Distance | Stabilizers
| | | | (Lower Bound) |
$C_{\pi}$ | Symmetric | $4^{s}+2^{s}+2$ | $4^{s}-2\cdot 3^{s}+2^{s}$ | $2^{s}+2$ | $2^{2s+1}+2^{s+1}+2$
$C_{asym}$ | Asymmetric | $4^{s}$ | $4^{s}-2\cdot 3^{s}+2\leq K$ | $2^{s-1}+1$ | $4^{s}+2^{s}+1$
| | | $\leq 4^{s}-2\cdot 3^{s}+2^{s}-1$ | |
$C_{symSK}$ | Symmetric | $4^{s}$ | $4^{s}-2\cdot 3^{s}-2\leq K$ | $2^{s-1}+1$ | $4^{s}-2^{s}$
| | | $\leq 4^{s}-2\cdot 3^{s}+2^{s+1}$ | |
$C_{symSE}$ | Symmetric | $4^{s}+2^{s}+2$ | $4^{s}-2\cdot 3^{s}+2^{s}$ | $2^{s-1}+2$ | $4^{s}+3\cdot 2^{s}+2$
TABLE III: QLDPC Code Parameters for Parity Checks Constructed from
$PG(2,2^{s})$
### V-D Symmetric QLDPC Codes from Secant Lines
Two different classical LDPC codes constructed from secant lines were
discussed in Section IV, namely those whose parity checks $H(C_{seA})$ were
constructed from all points, and those whose parity checks $H(C_{se})$ had the
columns corresponding to hyperoval points removed. Although $H(C_{se})$ is
orthogonal to $H(C_{sk})$, it is not self-orthogonal, while $H(C_{seA})$ is.
Thus, the parity check matrix $H(\mathcal{C}_{symSE})$ for the symmetric QLDPC
codes $\mathcal{C}_{symSE}$ constructed here have the form
$H(\mathcal{C}_{symSE})=\begin{bmatrix}H(C_{seA})&0\\\
0&H(C_{seA})\end{bmatrix}.$ (12)
###### Theorem V.4
Given a finite projective plane $PG(2,2^{s})$, for some positive integer $s$,
the matrix $H(\mathcal{C}_{symSE})$ in (12) is a parity check matrix for an
$[[n,2k_{seA}-n,D]]$ QLDPC code $\mathcal{C}_{symSE}$, where
$n=4^{s}+2^{s}+2$, $k_{seA}=4^{s}-3^{s}+2^{s}+1$, and $D\geq 2^{s-1}+2$.
###### Proof:
Here $k_{seA}=\dim(C_{seA})$. The rest follows from Proposition IV.3 and the
properties of symmetric CSS codes. ∎
Note that the length and dimension (and therefore the rate) of this code is
the same as that of $\mathcal{C}_{\pi}$, while the minimum distance and number
of stabilizers is roughly half that of $\mathcal{C}_{\pi}$.
## VI Conclusion
Table III gives a summary of the results for each of the QLDPC codes discussed
in this report.
While many of the parameters are not exact for many of these codes, the bounds
nevertheless indicate that each of these codes are at least comparable to most
quantum LDPC codes in the literature. In fact, as previously mentioned, many
of these parameters are completely unknown for the other quantum LDPC codes.
While further research is necessary to determine exact values of minimum
distance and dimension of most of these codes, it is nevertheless established
that the projective plane is a very useful tool in the construction of quantum
low-density parity check codes. Similar techniques can also be used to
construct QLDPC codes from $PG(m,p^{s})$ for $m>2$ and/or $p$ an odd prime.
Note that in the case of $PG(2,p^{s})$ where $p$ is an odd prime, the
$u$-column is not necessarily linearly dependent on the columns of the
corresponding incidence matrix, making it much more difficult to determine
dimensions of corresponding QLDPC codes. However, this is resolved if in
addition to concatenating the $u$-column to the incidence matrix of the
projective plane, you also concatenate an identity matrix to it. This would
change the parameters in a known manner. Additionally, if $p$ is odd, then the
point-line subsets should be taken with respect to a conic, as we lose the
regular hyperoval structure present when $p$ is even.
## Acknowledgment
The author would like to thank James Troupe, Keye Martin, Keith Mellinger, and
Geir Agnarsson for useful discussions.
## References
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* [15] J.-P. Tillich and G. Zémor, “Quantum ldpc codes with positive rate and minimum distance proportional to n 1/2,” in _Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 2_ , ser. ISIT’09. Piscataway, NJ, USA: IEEE Press, 2009, pp. 799–803. [Online]. Available: http://dl.acm.org/citation.cfm?id=1701275.1701299
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|
arxiv-papers
| 2012-07-03T16:06:02 |
2024-09-04T02:49:32.589218
|
{
"license": "Public Domain",
"authors": "Jacob Farinholt",
"submitter": "Jacob Farinholt",
"url": "https://arxiv.org/abs/1207.0732"
}
|
1207.0805
|
arxiv-papers
| 2012-07-03T14:32:20 |
2024-09-04T02:49:32.600160
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G.Geethu Lakshmi",
"submitter": "Pallavali Radha Krishna Reddy",
"url": "https://arxiv.org/abs/1207.0805"
}
|
|
1207.0878
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2012-180 LHCb-PAPER-2012-017 4 July 2012
Measurement of the $\kern 3.73305pt\overline{\kern-3.73305ptB}{}^{0}_{s}$
effective lifetime in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ final state
The LHCb collaboration†††Authors are listed on the following pages.
The effective lifetime of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson in the decay mode $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ is measured using 1.0 fb-1 of data collected in $pp$
collisions at $\sqrt{s}=7$ TeV with the LHCb detector. The result is $1.700\pm
0.040\pm 0.026\,\mathrm{ps}$ where the first uncertainty is statistical and
the second systematic. As the final state is $C\\!P$-odd, and $C\\!P$
violation in this mode is measured to be small, the lifetime measurement can
be translated into a measurement of the decay width of the heavy $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate, $\Gamma_{\rm
H}=0.588\pm 0.014\pm 0.009\,\rm{ps}^{-1}$.
Submitted to Physics Review Letters
LHCb collaboration
R. Aaij38, C. Abellan Beteta33,n, A. Adametz11, B. Adeva34, M. Adinolfi43, C.
Adrover6, A. Affolder49, Z. Ajaltouni5, J. Albrecht35, F. Alessio35, M.
Alexander48, S. Ali38, G. Alkhazov27, P. Alvarez Cartelle34, A.A. Alves Jr22,
S. Amato2, Y. Amhis36, L. Anderlini17, J. Anderson37, R.B. Appleby51, O.
Aquines Gutierrez10, F. Archilli18,35, A. Artamonov 32, M. Artuso53,35, E.
Aslanides6, G. Auriemma22,m, S. Bachmann11, J.J. Back45, V. Balagura28,35, W.
Baldini16, R.J. Barlow51, C. Barschel35, S. Barsuk7, W. Barter44, A. Bates48,
C. Bauer10, Th. Bauer38, A. Bay36, J. Beddow48, I. Bediaga1, S. Belogurov28,
K. Belous32, I. Belyaev28, E. Ben-Haim8, M. Benayoun8, G. Bencivenni18, S.
Benson47, J. Benton43, R. Bernet37, M.-O. Bettler17, M. van Beuzekom38, A.
Bien11, S. Bifani12, T. Bird51, A. Bizzeti17,h, P.M. Bjørnstad51, T. Blake35,
F. Blanc36, C. Blanks50, J. Blouw11, S. Blusk53, A. Bobrov31, V. Bocci22, A.
Bondar31, N. Bondar27, W. Bonivento15, S. Borghi48,51, A. Borgia53, T.J.V.
Bowcock49, C. Bozzi16, T. Brambach9, J. van den Brand39, J. Bressieux36, D.
Brett51, M. Britsch10, T. Britton53, N.H. Brook43, H. Brown49, A. Büchler-
Germann37, I. Burducea26, A. Bursche37, J. Buytaert35, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez33,n, A. Camboni33, P. Campana18,35, A.
Carbone14, G. Carboni21,k, R. Cardinale19,i,35, A. Cardini15, L. Carson50, K.
Carvalho Akiba2, G. Casse49, M. Cattaneo35, Ch. Cauet9, M. Charles52, Ph.
Charpentier35, P. Chen3,36, N. Chiapolini37, M. Chrzaszcz 23, K. Ciba35, X.
Cid Vidal34, G. Ciezarek50, P.E.L. Clarke47, M. Clemencic35, H.V. Cliff44, J.
Closier35, C. Coca26, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins35, A.
Comerma-Montells33, A. Contu52, A. Cook43, M. Coombes43, G. Corti35, B.
Couturier35, G.A. Cowan36, D. Craik45, S. Cunliffe50, R. Currie47, C.
D’Ambrosio35, P. David8, P.N.Y. David38, I. De Bonis4, K. De Bruyn38, S. De
Capua21,k, M. De Cian37, J.M. De Miranda1, L. De Paula2, P. De Simone18, D.
Decamp4, M. Deckenhoff9, H. Degaudenzi36,35, L. Del Buono8, C. Deplano15, D.
Derkach14,35, O. Deschamps5, F. Dettori39, J. Dickens44, H. Dijkstra35, P.
Diniz Batista1, F. Domingo Bonal33,n, S. Donleavy49, F. Dordei11, A. Dosil
Suárez34, D. Dossett45, A. Dovbnya40, F. Dupertuis36, R. Dzhelyadin32, A.
Dziurda23, A. Dzyuba27, S. Easo46, U. Egede50, V. Egorychev28, S. Eidelman31,
D. van Eijk38, F. Eisele11, S. Eisenhardt47, R. Ekelhof9, L. Eklund48, I. El
Rifai5, Ch. Elsasser37, D. Elsby42, D. Esperante Pereira34, A.
Falabella16,e,14, C. Färber11, G. Fardell47, C. Farinelli38, S. Farry12, V.
Fave36, V. Fernandez Albor34, F. Ferreira Rodrigues1, M. Ferro-Luzzi35, S.
Filippov30, C. Fitzpatrick47, M. Fontana10, F. Fontanelli19,i, R. Forty35, O.
Francisco2, M. Frank35, C. Frei35, M. Frosini17,f, S. Furcas20, A. Gallas
Torreira34, D. Galli14,c, M. Gandelman2, P. Gandini52, Y. Gao3, J-C.
Garnier35, J. Garofoli53, J. Garra Tico44, L. Garrido33, D. Gascon33, C.
Gaspar35, R. Gauld52, E. Gersabeck11, M. Gersabeck35, T. Gershon45,35, Ph.
Ghez4, V. Gibson44, V.V. Gligorov35, C. Göbel54, D. Golubkov28, A.
Golutvin50,28,35, A. Gomes2, H. Gordon52, M. Grabalosa Gándara33, R. Graciani
Diaz33, L.A. Granado Cardoso35, E. Graugés33, G. Graziani17, A. Grecu26, E.
Greening52, S. Gregson44, O. Grünberg55, B. Gui53, E. Gushchin30, Yu. Guz32,
T. Gys35, C. Hadjivasiliou53, G. Haefeli36, C. Haen35, S.C. Haines44, S.
Hall50, T. Hampson43, S. Hansmann-Menzemer11, N. Harnew52, S.T. Harnew43, J.
Harrison51, P.F. Harrison45, T. Hartmann55, J. He7, V. Heijne38, K.
Hennessy49, P. Henrard5, J.A. Hernando Morata34, E. van Herwijnen35, E.
Hicks49, M. Hoballah5, P. Hopchev4, W. Hulsbergen38, P. Hunt52, T. Huse49,
R.S. Huston12, D. Hutchcroft49, D. Hynds48, V. Iakovenko41, P. Ilten12, J.
Imong43, R. Jacobsson35, A. Jaeger11, M. Jahjah Hussein5, E. Jans38, F.
Jansen38, P. Jaton36, B. Jean-Marie7, F. Jing3, M. John52, D. Johnson52, C.R.
Jones44, B. Jost35, M. Kaballo9, S. Kandybei40, M. Karacson35, T.M. Karbach9,
J. Keaveney12, I.R. Kenyon42, U. Kerzel35, T. Ketel39, A. Keune36, B. Khanji6,
Y.M. Kim47, M. Knecht36, O. Kochebina7, I. Komarov29, R.F. Koopman39, P.
Koppenburg38, M. Korolev29, A. Kozlinskiy38, L. Kravchuk30, K. Kreplin11, M.
Kreps45, G. Krocker11, P. Krokovny31, F. Kruse9, M. Kucharczyk20,23,35,j, V.
Kudryavtsev31, T. Kvaratskheliya28,35, V.N. La Thi36, D. Lacarrere35, G.
Lafferty51, A. Lai15, D. Lambert47, R.W. Lambert39, E. Lanciotti35, G.
Lanfranchi18, C. Langenbruch35, T. Latham45, C. Lazzeroni42, R. Le Gac6, J.
van Leerdam38, J.-P. Lees4, R. Lefèvre5, A. Leflat29,35, J. Lefrançois7, O.
Leroy6, T. Lesiak23, L. Li3, Y. Li3, L. Li Gioi5, M. Lieng9, M. Liles49, R.
Lindner35, C. Linn11, B. Liu3, G. Liu35, J. von Loeben20, J.H. Lopes2, E.
Lopez Asamar33, N. Lopez-March36, H. Lu3, J. Luisier36, A. Mac Raighne48, F.
Machefert7, I.V. Machikhiliyan4,28, F. Maciuc10, O. Maev27,35, J. Magnin1, S.
Malde52, R.M.D. Mamunur35, G. Manca15,d, G. Mancinelli6, N. Mangiafave44, U.
Marconi14, R. Märki36, J. Marks11, G. Martellotti22, A. Martens8, L. Martin52,
A. Martín Sánchez7, M. Martinelli38, D. Martinez Santos35, A. Massafferri1, Z.
Mathe12, C. Matteuzzi20, M. Matveev27, E. Maurice6, A. Mazurov16,30,35, J.
McCarthy42, G. McGregor51, R. McNulty12, M. Meissner11, M. Merk38, J. Merkel9,
D.A. Milanes13, M.-N. Minard4, J. Molina Rodriguez54, S. Monteil5, D. Moran12,
P. Morawski23, R. Mountain53, I. Mous38, F. Muheim47, K. Müller37, R.
Muresan26, B. Muryn24, B. Muster36, J. Mylroie-Smith49, P. Naik43, T.
Nakada36, R. Nandakumar46, I. Nasteva1, M. Needham47, N. Neufeld35, A.D.
Nguyen36, C. Nguyen-Mau36,o, M. Nicol7, V. Niess5, N. Nikitin29, T. Nikodem11,
A. Nomerotski52,35, A. Novoselov32, A. Oblakowska-Mucha24, V. Obraztsov32, S.
Oggero38, S. Ogilvy48, O. Okhrimenko41, R. Oldeman15,d,35, M. Orlandea26, J.M.
Otalora Goicochea2, P. Owen50, B.K. Pal53, A. Palano13,b, M. Palutan18, J.
Panman35, A. Papanestis46, M. Pappagallo48, C. Parkes51, C.J. Parkinson50, G.
Passaleva17, G.D. Patel49, M. Patel50, G.N. Patrick46, C. Patrignani19,i, C.
Pavel-Nicorescu26, A. Pazos Alvarez34, A. Pellegrino38, G. Penso22,l, M. Pepe
Altarelli35, S. Perazzini14,c, D.L. Perego20,j, E. Perez Trigo34, A. Pérez-
Calero Yzquierdo33, P. Perret5, M. Perrin-Terrin6, G. Pessina20, A.
Petrolini19,i, A. Phan53, E. Picatoste Olloqui33, B. Pie Valls33, B.
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G. Polok23, A. Poluektov45,31, E. Polycarpo2, D. Popov10, B. Popovici26, C.
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Raniuk40, N. Rauschmayr35, G. Raven39, S. Redford52, M.M. Reid45, A.C. dos
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S. Roiser35, V. Romanovsky32, A. Romero Vidal34, M. Rosello33,n, J.
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Zvyagin35.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Roma Tor Vergata, Roma, Italy
22Sezione INFN di Roma La Sapienza, Roma, Italy
23Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
24AGH University of Science and Technology, Kraków, Poland
25Soltan Institute for Nuclear Studies, Warsaw, Poland
26Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
27Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
28Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
29Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
30Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
31Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
32Institute for High Energy Physics (IHEP), Protvino, Russia
33Universitat de Barcelona, Barcelona, Spain
34Universidad de Santiago de Compostela, Santiago de Compostela, Spain
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
37Physik-Institut, Universität Zürich, Zürich, Switzerland
38Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
39Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
40NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
41Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
42University of Birmingham, Birmingham, United Kingdom
43H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
44Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
45Department of Physics, University of Warwick, Coventry, United Kingdom
46STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
47School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
48School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
49Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
50Imperial College London, London, United Kingdom
51School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
52Department of Physics, University of Oxford, Oxford, United Kingdom
53Syracuse University, Syracuse, NY, United States
54Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
55Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
The decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$, $f_{0}(980)\\!\rightarrow\pi^{+}\pi^{-}$, discovered by
LHCb [1] at close to the predicted rate [2], is important for $C\\!P$
violation [3] and lifetime studies. In this Letter, we make a precise
determination of the lifetime. The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ final state is $C\\!P$-odd, and in the absence of $C\\!P$
violation, can be produced only by the decay of the heavy ($\rm{H}$), and not
by the light ($\rm{L}$), $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass eigenstate [4]. As the
measured $C\\!P$ violation in this final state is small [5], a measurement of
the effective lifetime, $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}$, can be translated into a measurement of the decay width,
$\Gamma_{\rm H}$. This helps to determine the decay width difference,
$\Delta\Gamma_{\rm s}=\Gamma_{\rm L}-\Gamma_{\rm H}$, a number of considerable
interest for studies of physics beyond the Standard Model (SM) [6,
*Lenz:2012az, *Bobeth:2011st]. Furthermore, this measurement can be used as a
constraint in the fit that determines the mixing-induced $C\\!P$-violating
phase in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays,
$\phi_{s}$, using the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final states, and
thus improve the accuracy of the $\phi_{s}$ determination [9, 5]. In the SM,
if sub-leading penguin contributions are neglected,
$\phi_{s}=-2\arg\left[\frac{V_{ts}^{\phantom{*}}V_{tb}^{*}}{V_{cs}^{\phantom{*}}V_{cb}^{*}}\right]$,
where the $V_{ij}$ are the Cabibbo-Kobayashi-Maskawa matrix elements, which
has a value of $-0.036\,^{+0.0016}_{-0.0015}$ rad [10]. Note that the LHCb
measurement of $\phi_{s}$ [5] corresponds to a limit on $\cos\phi_{s}$ greater
than 0.99 at 95% confidence level, consistent with the SM prediction.
The decay time evolution for the sum of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, via the
$b\\!\rightarrow c\overline{}cs$ tree amplitude, to a $C\\!P$-odd final state,
$f_{-}$, is given by [11, *Bigi:2000yz]
$\Gamma\left(B^{0}_{s}\rightarrow f_{-}\right)+\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow
f_{-}\right)=\frac{\cal N}{2}e^{-\Gamma_{\rm
s}t}\,\Bigg{\\{}e^{\Delta\Gamma_{\rm
s}t/2}(1+\cos\phi_{s})+e^{-\Delta\Gamma_{\rm
s}t/2}(1-\cos\phi_{s})\Bigg{\\}}\,,$ (1)
where ${\cal N}$ is a time-independent normalisation factor and $\Gamma_{\rm
s}$ is the average decay width. We measure the effective lifetime by
describing the decay time distribution with a single exponential function
$\Gamma\left(B^{0}_{s}\rightarrow f_{-}\right)+\Gamma\left(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow f_{-}\right)={\cal
N}e^{-t/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}}.$ (2)
Our procedure involves measuring the lifetime with respect to the well
measured $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime, in the
decay mode $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\\!\rightarrow K^{-}\pi^{+}$ (the
inclusion of charge conjugate modes is implied throughout this Letter). In
this ratio, the systematic uncertainties largely cancel.
The data sample consists of 1.0 fb-1 of integrated luminosity collected with
the LHCb detector [13] in $pp$ collisions at the LHC with 7
$\mathrm{\,Te\kern-1.00006ptV}$ centre-of-mass energy. The detector is a
single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$,
designed for the study of particles containing $b$ or $c$ quarks. The detector
includes a high precision tracking system consisting of a silicon-strip vertex
detector surrounding the $pp$ interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet and three stations of silicon-
strip detectors and straw drift-tubes placed downstream. Charged hadrons are
identified using two ring-imaging Cherenkov (RICH) detectors. Muons are
identified by a muon system composed of alternating layers of iron and
multiwire proportional chambers. The trigger consists of a hardware stage,
based on information from the calorimeter and muon systems, followed by a
software stage that applies a full event reconstruction. The simulated events
used in this analysis are generated using Pythia 6.4 [14] with a specific LHCb
configuration [15], where decays of hadronic particles are described by EvtGen
[16], and the LHCb detector simulation [17] based on Geant4 [18,
*Agostinelli:2002hh].
The selection criteria we use for this analysis are the same as those used to
measure $\phi_{s}$ in $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays [20]. Events are triggered by a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
decay, requiring two identified muons with opposite charge, transverse
momentum greater than 500 MeV (we work in units where $c=\hbar=1$), invariant
mass within 120 MeV of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass
[21], and form a vertex with a fit $\chi^{2}$ less than 16.
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ candidates are
first selected by pairing an opposite sign pion combination with a
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate that has a dimuon
invariant mass from -48 MeV to +43 MeV from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [21]. The pions are
required to be identified positively in the RICH detector, have a minimum
distance of approach with respect to the primary vertex (impact parameter) of
greater than 9 standard deviation significance, have a transverse momentum
greater than 250 MeV and fit to a common vertex with the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ with a $\chi^{2}$ less than 16.
Furthermore, the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
candidate must have a vertex with a fit $\chi^{2}$ less than 10, flight
distance from production to decay vertex greater than 1.5 mm and the angle
between the combined momentum vector of the decay products and the vector
formed from the positions of the primary and the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decay vertices (pointing
angle) is required to be consistent with zero. Events satisfying this
preselection are then further filtered using requirements determined using a
Boosted Decision Tree (BDT) [22, *Hocker:2007ht]. The BDT uses nine variables
to differentiate signal from background: the identification quality of each
muon, the probability that each pion comes from the primary vertex, the
transverse momentum of each pion, the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ vertex fit quality, flight
distance from production to decay vertex and pointing angle. It is trained
with simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ signal events and two background samples from data, the
first with like-sign pions with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{\pm}\pi^{\pm}$ mass within $\pm 50$ MeV of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass and the second from the
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ upper mass sideband
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ mass between
200 and 250 MeV above the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mass.
As the effective $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime is measured relative to that of the decay $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, we use the same
trigger, preselection and BDT to select ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ events, except for the hadron identification that is
applied independently of the BDT. The selected $\pi^{+}\pi^{-}$ and
$K^{-}\pi^{+}$ invariant mass distributions, for candidates with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass within $\pm
20$ MeV of the respective $B$ mass peaks are shown in Fig. 1. The background
distributions shown are determined by fitting the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass distribution
in bins of $\pi^{+}\pi^{-}$ ($K^{-}\pi^{+}$) mass. Further selections of $\pm
90$ $\mathrm{\,Me\kern-1.00006ptV}$ around the $f_{0}(980)$ mass and $\pm 100$
$\mathrm{\,Me\kern-1.00006ptV}$ around the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass are applied. The
$f_{0}(980)$ selection results in a $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ sample that is greater than 99.4% $C\\!P$-odd at 95%
confidence level [24].
Figure 1: Invariant mass distributions of selected (a) $\pi^{+}\pi^{-}$ and
(b) $K^{-}\pi^{+}$ combinations (solid histograms) for events within $\pm 20$
MeV of the respective $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ mass peaks. Backgrounds
(dashed histograms) are determined by fitting the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass in bins of
$\pi^{+}\pi^{-}$ ($K^{-}\pi^{+}$) mass. Regions between the arrows are used in
the subsequent analysis.
The analysis exploits the fact that the kinematic properties of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ decay are very similar to those of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. We can select
$B$ mesons in either channel using identical kinematic constraints and hence
the decay time acceptance introduced by the trigger, reconstruction and
selection requirements should almost cancel in the ratio of the decay time
distributions. Therefore, we can determine the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime, $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}$, relative to the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ lifetime,
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$, from the variation of the ratio
of the $B$ meson yields with decay time
$R(t)=R(0)e^{-t(1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}-1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
0.99998pt\overline{\kern-0.99998ptK}{}^{*0}})}=R(0)e^{-t\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}}\,\mathrm{,}$ (3)
where the width difference $\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}=1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}-1/\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$.
We test the cancellation of acceptance effects using simulated $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ events. Both the
acceptances themselves and also the ratio exhibit the same behaviour. Due to
the selection requirements, they are equal to 0 at $t=0$, after which there is
a sharp increase, followed by a slow variation for $t$ greater then 1 ps.
Based on this, we only use events with $t$ greater than 1 ps in the analysis.
To good approximation, the acceptance ratio is linear between 1 and 7 ps, with
a slope of $a=0.0125\pm 0.0036$ ps-1 (see Fig. 2). We use this slope as a
correction to Eq. 3 when fitting the measured decay time ratio
$R(t)=R_{0}(1+at)e^{-t\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}}.$ (4)
Figure 2: Ratio of decay time acceptances between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays obtained from
simulation. The solid (blue) line shows the result of a linear fit.
Differences between the decay time resolutions of the decay modes could affect
the decay time ratio. To measure the decay time resolution, we use prompt
events containing a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ meson. Such
events are found using a dimuon trigger, plus two opposite-charged tracks with
similar selection criteria as for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ (${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$) events, apart from any decay time biasing requirements
such as impact parameters and $B$ flight distance, additionally including that
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$) mass be within
$\pm 20$ $\mathrm{\,Me\kern-1.00006ptV}$ of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ ($\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) mass. To describe the decay time
distribution of these events, we use a triple Gaussian function with a common
mean, and two long lived components, modelled by exponential functions
convolved with the triple Gaussian function. The events are dominated by zero
lifetime background with the long lived components comprising less than 5% of
the events. We find the average effective decay time resolution for $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays to be $41.0\pm
0.9$ fs and $44.1\pm 0.2$ fs respectively, where the uncertainties are
statistical only. This difference was found not to bias the decay time ratio
using simulated experiments.
In order to determine the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime, we determine the yield of $B$ mesons for both
decay modes using unbinned maximum likelihood fits to the $B$ mass
distributions in 15 bins of decay time of equal width between 1 and 7 ps. We
perform a $\chi^{2}$ fit to the ratio of the yields as a function of decay
time and determine the relative lifetime according to Eq. 4. We obtain the
signal and peaking background shape parameters by fitting the time-integrated
dataset. In each decay time bin, we use these shapes and determine the
combinatorial background parameters from the upper mass sidebands,
$5450<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0})<5600$ MeV and
$5450<m({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<5550$ MeV. With this approach,
the combinatorial backgrounds are re-evaluated in each bin and we make no
assumptions on the shape of the background decay time distributions. This
method was tested with high statistics simulated experiments and found to be
unbiased.
Figure 3: Invariant mass distributions of selected (a)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$ and (b)
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{-}\pi^{+}$ candidates. The
solid (blue) curves show the total fits, the long dashed (purple) curves show
the respective $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signals, and the
dotted (gray) curve shows the combinatorial background. In (a) the short
dashed (blue-green) curve shows the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ background and the dash dotted (green) curve shows the
$\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$$K^{-}$ $\pi^{+}$ reflection. In (b) the short dashed (red) curve near
5370 MeV shows the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{-}\pi^{+}$ background.
The time-integrated fits to the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mass spectra are shown in Fig. 3.
The signal distributions are described by the sum of two Crystal Ball
functions [25] with common means and resolutions for the Gaussian core, but
different parameters describing the tails
$f(m;\mu,\sigma,n_{l,r},\alpha_{l,r})=\begin{cases}\left(\frac{n_{l}}{|\alpha_{l}|}\right)^{n_{l}}\cdot\exp\left(\frac{-|\alpha_{l}|^{2}}{2}\right)\cdot\left(\frac{n_{l}}{|\alpha_{l}|}-|\alpha_{l}|-\frac{|m-\mu|}{\sigma}\right)^{-n_{l}},&\text{if
$\frac{m-\mu}{\sigma}\leq-\alpha_{l}$,}\\\
\left(\frac{n_{r}}{|\alpha_{r}|}\right)^{n_{r}}\cdot\exp\left(\frac{-|\alpha_{r}|^{2}}{2}\right)\cdot\left(\frac{n_{r}}{|\alpha_{r}|}-|\alpha_{r}|-\frac{|m-\mu|}{\sigma}\right)^{-n_{r}},&\text{if
$\frac{m-\mu}{\sigma}\geq\alpha_{r}$,}\\\
\exp(\frac{-(m-\mu)^{2}}{2\sigma^{2}}),&\text{otherwise,}\end{cases}$ (5)
where $\mu$ is the mean and $\sigma$ the width of the core, while $n_{l,r}$
are the exponent of the left and right tails, and $\alpha_{l,r}$ are the left
and right transition points between the core and tails. The left hand tail
accounts for final state radiation and interactions with matter, while the
right hand tail describes non-Gaussian detector effects only seen with
increased statistics. The combinatorial backgrounds are described by
exponential functions. All parameters are determined from data. There are
$4040\pm 75$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $131\,920\pm 400$ $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal decays. The
decay time distributions, determined using fits to the invariant mass
distributions in bins of decay time as described above, are shown in Fig. 4.
These are made by placing the fitted signal yields at the average $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay time within the
bin rather than at the centre of the decay time bin. This procedure corrects
for the exponential decrease of the decay time distributions across the bin.
The subsequent decay time ratio distribution is shown in Fig. 5, and the
fitted reciprocal lifetime difference is
$\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=-0.070\pm
0.014\,\mathrm{ps}^{-1}$, where the uncertainty is statistical only. Taking
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ to be the mean $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime $1.519\pm 0.007$ ps [21],
we determine $\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}}=1.700\pm 0.040\,\mathrm{ps}$.
Figure 4: Decay time distributions for (a) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and (b) $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. In (b) the error
bars are smaller than the points. Figure 5: Decay time ratio between $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$, and the fit for
$\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}$.
Sources of systematic uncertainty on the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime are investigated and listed in Table 1. We first
investigate our assumptions about the signal and combinatorial background mass
shapes. The relative change of the determined $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ lifetime between fits with double Crystal Ball functions and
double Gaussian functions for the signal models is 0.001 ps, and between fits
with exponential functions and straight lines for the combinatorial background
models is 0.010 ps. The different particle identification criteria used to
select $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)\rightarrow\mu^{+}\mu^{-}\pi^{+}\pi^{-}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow\mu^{+}\mu^{-}K^{-}\pi^{+}$
decays could affect the acceptance cancellation between the modes. In order to
investigate this effect, we loosen and tighten the particle identification
selection for the kaon, modifying the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal yield by $+2$%
and $-20$% respectively, and repeat the analysis. The larger difference with
respect to the default selection, 0.007 ps, is assigned as a systematic
uncertainty. We also assign half of the relative change between the fit
without the acceptance correction and the default fit, 0.018 ps, as a
systematic uncertainty. Potential statistical biases of our method were
evaluated with simulated experiments using similar sample sizes to those in
data. An average bias of 0.012 ps is seen and included as a systematic
uncertainty. The observed bias vanishes in simulated experiments with large
sample sizes. As a cross-check, the analysis is performed with various decay
time bin widths and fit ranges, and consistent results are obtained. The
possible $C\\!P$-even component, limited to be less than 0.6% at 95%
confidence level [24], introduces a 0.001 ps systematic uncertainty. Using the
PDG value for the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime
[21] as input requires the propagation of its error as a systematic
uncertainty. All the contributions are added in quadrature and yield a total
systematic uncertainty on the lifetime of 0.026 ps (1.5%). Thus the effective
lifetime of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final
state in $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ decays, when
describing the decay time distribution as a single exponential is
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=1.700\pm 0.040\pm
0.026\,{\rm ps}\,.$ (6)
Table 1: Summary of systematic uncertainties on the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ effective lifetime. Source | Uncertainty (ps)
---|---
Signal mass shape | $0.001$
Background mass shape | $0.010$
Kaon identification | $0.007$
Acceptance | $0.018$
Statistical bias | $0.012$
$C\\!P$-even component | $0.001$
$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime [21] | $0.009$
Sum in quadrature | $0.026$
Given that $\phi_{s}$ is measured to be small, and the decay is given by a
pure $b\\!\rightarrow c\overline{}cs$ tree amplitude, we may interpret the
inverse of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ effective lifetime as a measurement of $\Gamma_{\rm H}$ with
an additional source of systematic uncertainty due to a possible non-zero
value of $\phi_{s}$. For $\cos\phi_{s}=0.99$, $\Gamma_{\rm s}=0.6580$ ps-1 and
$\Delta\Gamma_{\rm s}=0.116$ ps-1 [5],
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}$ changes by 0.002
ps. This is added in quadrature to the systematic uncertainties on
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}$ to obtain the final
systematic uncertainty on $\Gamma_{\rm H}$.
In summary, the effective lifetime of the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ meson in the $C\\!P$-odd
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$ final state has been
measured with respect to the well measured $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ lifetime in the final state
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. The analysis exploits the
kinematic similarities between the $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}f_{0}(980)$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays to determine
an effective lifetime of
$\tau_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=1.700\pm 0.040\pm
0.026\,\mathrm{ps},$
corresponding to a width difference of
$\Delta_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}}=-0.070\pm 0.014\pm
0.001\,\mathrm{ps}^{-1},$
where the uncertainties are statistical and systematic respectively. This
result is consistent with, and more precise than, the previous measurement of
$1.70\,^{+0.12}_{-0.11}\pm 0.03$ ps from CDF [26]. Interpreting this as the
lifetime of the heavy $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
eigenstate, we obtain
$\Gamma_{\rm H}=0.588\pm 0.014\pm 0.009\,\mathrm{ps}^{-1}.$
This value of $\Gamma_{\rm H}$ is consistent with the value $0.600\pm 0.013$
ps-1, calculated from the values of $\Gamma_{\rm s}$ and $\Delta\Gamma_{\rm
s}$ in Ref. [5].
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
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|
arxiv-papers
| 2012-07-04T02:22:37 |
2024-09-04T02:49:32.607044
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb Collaboration: R. Aaij, C. Abellan Beteta, A. Adametz, B. Adeva,\n M. Adinolfi, C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio,\n M. Alexander, S. Ali, G. Alkhazov, P. Alvarez Cartelle, A. A. Alves Jr, S.\n Amato, Y. Amhis, L. Anderlini, J. Anderson, R. B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J. J. Back, V. Balagura, W. Baldini, R. J. Barlow, C. Barschel,\n S. Barsuk, W. Barter, A. Bates, C. Bauer, Th. Bauer, A. Bay, J. Beddow, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P. M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, C. Blanks, J. Blouw, S. Blusk, A. Bobrov, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T. J. V. Bowcock, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N. H. Brook, H. Brown, A. B\\\"uchler-Germann, I. Burducea, A. Bursche, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, L. Carson, K.\n Carvalho Akiba, G. Casse, M. Cattaneo, Ch. Cauet, M. Charles, Ph.\n Charpentier, P. Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G.\n Ciezarek, P. E. L. Clarke, M. Clemencic, H. V. Cliff, J. Closier, C. Coca, V.\n Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, G. Corti, B. Couturier, G. A. Cowan, D. Craik, S. Cunliffe,\n R. Currie, C. D'Ambrosio, P. David, P. N. Y. David, I. De Bonis, K. De Bruyn,\n S. De Capua, M. De Cian, J. M. De Miranda, L. De Paula, P. De Simone, D.\n Decamp, M. Deckenhoff, H. Degaudenzi, L. Del Buono, C. Deplano, D. Derkach,\n O. Deschamps, F. Dettori, J. Dickens, H. Dijkstra, P. Diniz Batista, F.\n Domingo Bonal, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A.\n Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, F. Eisele, S. Eisenhardt, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, D. Esperante\n Pereira, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V.\n Fave, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov,\n C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank,\n C. Frei, M. Frosini, S. Furcas, A. Gallas Torreira, D. Galli, M. Gandelman,\n P. Gandini, Y. Gao, J-C. Garnier, J. Garofoli, J. Garra Tico, L. Garrido, D.\n Gascon, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, V. Gibson, V. V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L. A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S. C. Haines, S. Hall, T. Hampson, S. Hansmann-Menzemer, N.\n Harnew, S. T. Harnew, J. Harrison, P. F. Harrison, T. Hartmann, J. He, V.\n Heijne, K. Hennessy, P. Henrard, J. A. Hernando Morata, E. van Herwijnen, E.\n Hicks, M. Hoballah, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, R. S.\n Huston, D. Hutchcroft, D. Hynds, V. Iakovenko, P. Ilten, J. Imong, R.\n Jacobsson, A. Jaeger, M. Jahjah Hussein, E. Jans, F. Jansen, P. Jaton, B.\n Jean-Marie, F. Jing, M. John, D. Johnson, C. R. Jones, B. Jost, M. Kaballo,\n S. Kandybei, M. Karacson, T. M. Karbach, J. Keaveney, I. R. Kenyon, U.\n Kerzel, T. Ketel, A. Keune, B. Khanji, Y. M. Kim, M. Knecht, O. Kochebina, I.\n Komarov, R. F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L.\n Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M.\n Kucharczyk, V. Kudryavtsev, T. Kvaratskheliya, V. N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R. W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, O. Leroy, T. Lesiak, L. Li, Y.\n Li, L. Li Gioi, M. Lieng, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, J.\n von Loeben, J. H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, J. Luisier,\n A. Mac Raighne, F. Machefert, I. V. Machikhiliyan, F. Maciuc, O. Maev, J.\n Magnin, S. Malde, R. M. D. Mamunur, G. Manca, G. Mancinelli, N. Mangiafave,\n U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, L. Martin, A.\n Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, A. Massafferri, Z.\n Mathe, C. Matteuzzi, M. Matveev, E. Maurice, A. Mazurov, J. McCarthy, G.\n McGregor, R. McNulty, M. Meissner, M. Merk, J. Merkel, D. A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, R. Mountain,\n I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, J.\n Mylroie-Smith, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N.\n Neufeld, A. D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, N. Nikitin, T.\n Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J. M. Otalora\n Goicochea, P. Owen, B. K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C. J. Parkinson, G. Passaleva, G. D.\n Patel, M. Patel, G. N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D. L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, A. Petrolini, A. Phan, E. Picatoste Olloqui, B.\n Pie Valls, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus,\n F. Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, V. Pugatch, A. Puig Navarro, W. Qian,\n J. H. Rademacker, B. Rakotomiaramanana, M. S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Redford, M. M. Reid, A. C. dos Reis, S. Ricciardi,\n A. Richards, K. Rinnert, D. A. Roa Romero, P. Robbe, E. Rodrigues, F.\n Rodrigues, P. Rodriguez Perez, G. J. Rogers, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rosello, J. Rouvinet, T. Ruf, H. Ruiz, G. Sabatino, J. J.\n Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann, B. Sanmartin\n Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, R. Santinelli, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, S. Schleich, M. Schlupp, M.\n Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R.\n Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, T. Skwarnicki, N. A. Smith, E.\n Smith, M. Smith, K. Sobczak, F. J. P. Soler, A. Solomin, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U. Straumann, V.\n K. Subbiah, S. Swientek, M. Szczekowski, P. Szczypka, T. Szumlak, S.\n T'Jampens, M. Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J.\n van Tilburg, V. Tisserand, M. Tobin, S. Tolk, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M. T. Tran, A. Tsaregorodtsev, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, U. Uwer, V. Vagnoni, G. Valenti, R. Vazquez Gomez, P.\n Vazquez Regueiro, S. Vecchi, J. J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, I. Videau, D. Vieira, X. Vilasis-Cardona, J. Visniakov,\n A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss},\n H. Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D. R. Ward, N. K.\n Watson, A. D. Webber, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, L.\n Wiggers, G. Wilkinson, M. P. Williams, M. Williams, F. F. Wilson, J. Wishahi,\n M. Witek, W. Witzeling, S. A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, F.\n Xing, Z. Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W. C. Zhang, Y. Zhang, A. Zhelezov, L. Zhong,\n A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1207.0878"
}
|
1207.0913
|
# Estimating Node Influenceability in Social Networks
Rong-Hua Li Jeffrey Xu Yu Zechao Shang The Chinese University of Hong Kong
The Chinese University of Hong Kong The Chinese University of Hong Kong
###### Abstract
Influence analysis is a fundamental problem in social network analysis and
mining. The important applications of the influence analysis in social network
include influence maximization for viral marketing, finding the most
influential nodes, online advertising, etc. For many of these applications, it
is crucial to evaluate the influenceability of a node. In this paper, we study
the problem of evaluating influenceability of nodes in social network based on
the widely used influence spread model, namely, the independent cascade model.
Since this problem is #P-complete, most existing work is based on Naive Monte-
Carlo (NMC) sampling. However, the NMC estimator typically results in a large
variance, which significantly reduces its effectiveness. To overcome this
problem, we propose two families of new estimators based on the idea of
stratified sampling. We first present two basic stratified sampling (BSS)
estimators, namely BSS-I estimator and BSS-II estimator, which partition the
entire population into $2^{r}$ and $r+1$ strata by choosing $r$ edges
respectively. Second, to further reduce the variance, we find that both BSS-I
and BSS-II estimators can be recursively performed on each stratum, thus we
propose two recursive stratified sampling (RSS) estimators, namely RSS-I
estimator and RSS-II estimator. Theoretically, all of our estimators are shown
to be unbiased and their variances are significantly smaller than the variance
of the NMC estimator. Finally, our extensive experimental results on both
synthetic and real datasets demonstrate the efficiency and accuracy of our new
estimators.
###### keywords:
Influenceability, influence networks, independent cascade model, Stratified
sampling, uncertain graph
R-H. Li, Jeffrey. Yu, Z. Shang. Estimating Node Influenceability in Social
Networks.
The work was supported by grants of the Research Grants Council of the Hong
Kong SAR, China No. 419008 and 419109.
Author’s addresses: Rong-Hua Li, Jeffery Xu Yu, and Zechao Shang, Department
of System Engineering and Engineering Management, The Chinese University of
Hong Kong, Sha Tin, N.T., Hong Kong.
## 1 Introduction
Large scale online social networks (OSNs) such as Facebook and Twitter have
become increasingly popular in the last years. Users in OSNs are able to share
thoughts, activities, photos, and other information with their friends. As a
result, the OSNs become an important medium for information dissemination and
influence spread. A fundamental problem in such OSNs is to analyze and study
the social influence among users [Tang et al. (2009)]. Important applications
of influence analysis in OSNs include influence maximization for viral marking
[Kempe et al. (2003), Chen et al. (2010)], finding the most influential nodes
[Liu et al. (2009), Lappas et al. (2010)], online advertising, etc.
Especially, the influence maximization problem has recently attracted
tremendous attention in research community [Leskovec et al. (2007), Chen et
al. (2009), Chen et al. (2010), Goyal et al. (2011)]. For many of these
applications, a very important step is to accurately evaluate the
influenceability of a node in OSNs.
The influenceability evaluation problem is based on influence spread in a
network. Generally, the influence spread in a network can be modeled as a
stochastic cascade model. In the literature, a widely used cascade mode is the
independent cascade (IC) model. In the IC model, each node $i$ has a single
chance to influence his/her neighbor $j$ with a probability $p_{ij}$, and such
“influence event” is independent of the other “influence events” over other
nodes. Due to the independent property, the IC model can be represented by the
probabilistic graph model, where each edge in the graph is associated with a
probability and the existence of an edge is independent of any other edges
[Potamias et al. (2010)]. In this paper, we focus on the IC model and assume
that the influence probabilities of all the edges in a social network are
given in advance111Learning the influence probabilities is out of scope of
this paper. In the literature, there are some studies, such as [Goyal et al.
(2010)], on learning the influence probabilities in social network.. In
addition, we use the probabilistic graph model to represent the IC model.
This problem is equivalent to calculate the expected number of nodes in
$\mathcal{G}$ that are reachable from $s$, which is known to be #P-complete
[Chen et al. (2010)]. The existing algorithms for this problem are based on
naive Monte-Carlo sampling estimator (NMC) [Kempe et al. (2003), Kempe et al.
(2005), Chen et al. (2009)]. However, NMC may result in a large variance,
which significantly reduces its effectiveness. We will discuss this issue in
detail in Section 3.
Given the IC model and a seed node $s$, the influenceability evaluation
problem is to compute the expected influence spread by the seed node $s$. This
problem is equivalent to calculate the expected number of nodes in a
probabilistic graph $\mathcal{G}$ that are reachable from $s$, which is known
to be #P-complete [Chen et al. (2010)]. As a result, there is no hope to
exactly evaluate the influenceability in polynomial time unless P=#P. The
existing algorithm for this problem is based on Naive Monte-Carlo sampling
[Kempe et al. (2003), Kempe et al. (2005), Chen et al. (2009)]. As our
analysis given in Section 3, the Naive Monte-Carlo (NMC) estimator leads to a
large variance, and thus it significantly reduces the effectiveness of the
estimator. Theoretically, the NMC estimator can achieve arbitrarily close
approximation to the exact value of the influenceability. However, this
requires a large number of samples. Since performing a Monte-Carlo estimation
needs to flip $m$ coins to determine all the $m$ edges of the network, the NMC
estimator is extremely expensive to get a meaningful approximation of the
influenceability in large networks. Consequently, the key issue to accelerate
the NMC estimator is to reduce the number of samples that are needed to
achieve a good accuracy.
In order to reduce the number of samples used in the NMC estimator, one
potential solution is to reduce its variance. In this paper, we propose two
types of the Monte-Carlo estimator, namely type-I estimator and type-II
estimator, based on the idea of stratified sampling. All of our proposed
estimators are shown to be unbiased and their variance are significantly
smaller than the variance of the NMC estimator. To the best of our knowledge,
this is the first work that addresses and studies the variance problem in NMC
for influenceability evaluation problem.
To develop new type-I estimators, we devise an exact divide-and-conquer
enumeration algorithm. Our exact algorithm starts by enumerating $r$ edges,
thus resulting in $2^{r}$ cases. Then, for each case the algorithm recursively
enumerates another $r$ edges. The recursion will terminate after all the $m$
edges are enumerated. This exact algorithm has exponential time complexity to
evaluate node’s influenceability. Based on the exact algorithm, we propose a
basic stratified sampling (BSS) estimator, namely BSS-I estimator, to estimate
a node’s influenceability. In particular, we first select $r$ edges and
determine their statuses (existence or inexistence). Obviously, this process
generates $2^{r}$ cases. Then, we let each case be a stratum, and draw samples
separately from each stratum. By carefully allocating the sample size for each
stratum, we prove that the variance of the BSS-I estimator is smaller than the
variance of the NMC estimator. Interestingly, we find that our BSS-I estimator
can be recursively performed in each stratum, and thereby we propose a
recursive stratified sampling estimator, namely RSS-I estimator. Since the
RSS-I estimator recursively reduces the variance in each stratum, its variance
is significantly smaller than the variance of the BSS-I estimator. It is
important to note that both BSS-I and RSS-I estimators have the same time
complexity as the NMC estimator.
In addition to the type-I estimators (BSS-I and RSS-I), we further develop two
type-II estimators based on a new stratification method. The new
stratification method partitions the population into $r+1$ strata by picking
$r$ edges. In the first stratum which is denoted by stratum $0$, we set the
statuses of all the $r$ edges to “$0$”, which denotes the edge inexistence. In
the $i$-th ($1\leq i\leq r$) stratum, we set the statuses of all the first
$i-1$ edges to “$0$”, the $i$-th edge to “$1$”, which signifies the edge
existence, and the rest $r-i$ edges to “$*$”, which denotes the status of the
edge to be determined. Based on such stratification approach, we propose a
basic stratified sampling estimator, namely BSS-II estimator. Similar to the
idea of the RSS-I estimator, we develop a recursive stratified sampling
estimator based on BSS-II estimator, namely RSS-II estimator. We conduct
extensive experimental studies on both synthetic and real datasets, and we
show that both RSS-I and RSS-II estimators reduce the variance of the NMC
estimator significantly.
Note that the stratification approach in both type-I and type-II estimators
are based on the $r$ selected edges. Thus, an edge-selection strategy may
significantly affect the performance of the estimators. In this paper, we
present two edge-selection strategies for the proposed estimators: random
edge-selection and Breadth-First-Search (BFS) edge-selection. The random edge-
selection is to pick $r$ unsampled edges randomly for stratification, while
the BFS edge-selection picks $r$ unsampled edges according to their BFS
visiting order (the BFS starts from the seed node $s$). In our experiments, we
show that an estimator with the BFS edge-selection strategy significantly
outperforms the same estimator with the random edge-selection strategy.
Besides the influenceability estimation problem in social networks, our
proposed estimation methods can be applied in many other application domains.
For example, consider an application in a communication network with link
failure. Given a router $s$, it needs to count the expected number of hosts in
the network that are reachable from $s$. Such count assists network resource
planing, and is also useful for network resource estimation, for example in
P2P networks. Our proposed algorithms can provide accurate estimators for such
application domains. In addition, our influenceability estimation methods can
be directly used to the so-called influence function evaluation problem [Kempe
et al. (2003)], in which the seed is not only one node but a set of nodes. We
can solve this problem by adding a virtual node $s$ and link it to the set of
seed nodes. Finally, our proposed stratified sampling estimators are very
general, and can be easily used to handle uncertain graph mining problems,
such as network reliability estimation [Rubino (1999)], shortest path
[Potamias et al. (2010)], and reachability computation problem [Jin et al.
(2011b)].
The rest of this paper is organized as follows. We give the problem statement
in Section 2, and introduce the Naive Monte-Carlo estimator in Section 3. We
propose the type-I and type-II estimators in Section 4 and Section 5,
respectively. Extensive experimental studies are reported in Section 6.
Section 7 discusses the related work and Section 8 concludes this work.
## 2 Problem Statement
We consider a social network $G=(V,E)$, where $V$ denotes a set of nodes and
$E$ denotes a set of directed edges between the nodes. Let $n=|V|$ and $m=|E|$
be the number of nodes and edges in $G$, respectively. In a social network,
users (nodes) can perform actions, and the actions can propagate over the
network. For example, in Twitter, an action denotes a user posts a tweet, and
the action propagation denotes the event that the same tweet is re-posted
(retweeted) by his/her followers. In this paper, we adopt the independent
cascade (IC) model [Kempe et al. (2003), Kempe et al. (2005)] to model such
action propagation process. In the IC model, every edge $(u,v)$ is associated
with an influence probability $p_{uv}$ (Fig. 1(a)), which represents the
probability that a node $v$ performs an action followed by the same action
taken by its adjacent node $u$. We refer to a social network $G$ with
influence probabilities as an influence network denoted by
$\mathcal{G}=(V,E,P)$, where the set $P$ represents the set of influence
probabilities. We call a node an active node if it performs an action.
The propagation process of the IC model unfolds in discrete steps. More
precisely, we assume that a node $v$ follows a node $u$, and at step $t$ node
$u$ performs an action $\alpha$ and node $v$ does not. Then, node $u$ is given
a single chance to influence node $v$, and it succeeds with probability
$p_{uv}$. This probability is independent of other nodes that attempt to
influence node $v$. If node $u$ succeeds, then node $v$ will perform action
$\alpha$ at step $t+1$. In other words, node $v$ is influenced by node $u$ at
step $t+1$. It is important to note that whether $u$ succeeds or not, it
cannot make any attempts to influence $v$ again. The process terminates when
there is no new node can be influenced.
The IC model can be initiated by a single node $s$ such that the node performs
an action before any other nodes in $V\backslash\\{s\\}$. The seed node $s$
models the source of influence, and it can spread across the network following
the IC model. The propagation process is a stochastic process, after the
process terminates, the number of active nodes is a random variable.
Therefore, we take the expectation of this random variable to measure the
_influence spread_ of $s$, and it is denoted as $F_{s}(\mathcal{G})$. We refer
to the expected influence spread of $s$ (i.e. $F_{s}(\mathcal{G})$) as the
influenceability of node $s$. In this paper, we aim to evaluate the
influenceability $F_{s}(\mathcal{G})$ given a seed node $s$. In the following
subsection, we will give a formal definition of $F_{s}(\mathcal{G})$ based on
the probabilistic graph model [Potamias et al. (2010)].
(a) Influence network $\mathcal{G}$
(b) The source node $s=v_{5}$
(c) Possible graph $G_{1}$ with probability $0.000007056$, and
$f_{s}(G_{1})=3$
(d) Possible graph $G_{2}$ with probability $0.00003704$, and $f_{s}(G_{2})=5$
Figure 1: A Simple Influence Network
### 2.1 Influenceability Evaluation
Based on the IC model, an influence network $\mathcal{G}=(V,E,P)$ is
represented by the probabilistic graph model [Potamias et al. (2010)], where
the existence of an edge is independent of any other edges.
Given an influence network $\mathcal{G}=(V,E,P)$, we denote a possible graph
$G_{P}=(V_{P},E_{P})$ which can obtained by sampling each edge $e$ in
$\mathcal{G}$ according to the influence probability $p_{e}$ associated with
the edge $e$ ($p_{e}\in P$). Here, we have $V=V_{P}$, $E_{P}\subseteq E$, and
the possible graph $G_{P}$ has the probability $\Pr[G_{P}]$, which is given by
$\Pr[G_{P}]=\prod\limits_{e\in E_{P}}{p_{e}}\prod\limits_{e\in E\backslash
E_{P}}{(1-p_{e})}.$ (1)
The total number of such possible graphs is $2^{m}$, where $m$ is the number
of edges in ${\cal G}$. For example, in Fig. 1(a), the influence network
$\mathcal{G}$ has $2^{10}$ possible graphs and the possible graph $G_{1}$
(Fig. 1(c)) and $G_{2}$ (Fig. 1(d)) have probability $\Pr[G_{1}]=0.000007056$
and $\Pr[G_{2}]=0.00003704$, respectively.
According to the IC model, given a seed node $s$, the influenceability of $s$,
denoted by $F_{s}(\mathcal{G})$, is the expected influence spread over all the
possible graphs of $\mathcal{G}$. Therefore, based on the probabilistic graph
model, the influenceability $F_{s}(\mathcal{G})$ can be given by
$F_{s}(\mathcal{G})=\sum\limits_{G_{P}\in\Omega}{\Pr[G_{P}]f_{s}(G_{P})},$ (2)
where $\Omega$ denotes the set of all possible graphs of $\mathcal{G}$, and
$f_{s}(G_{P})$ is the number of nodes that are reachable from the seed node
$s$ in the possible graph $G_{P}$. Note that $f_{s}(G_{P})$ is a random
variable and its expectation is $F_{s}(\mathcal{G})$, i.e.
$F_{s}(\mathcal{G})=\mathbb{E}[f_{s}(G_{P})]$.
As an example, consider the source node $s=v_{5}$ in the influence network
${\cal G}$ in Fig. 1(a). $F_{s}(\mathcal{G})$ can be computed by enumerating
all $2^{10}$ possible graphs, $G_{P}$, and computing the corresponding
$\Pr[G_{P}]$ and $f_{s}(G_{P})$. For instance, from Fig. 1(c) and Fig. 1(d),
we have $f_{s}(G_{1})$ $=3$ and $f_{s}(G_{2})=5$. In this example, the exact
$F_{s}(\mathcal{G})$ is $0.46123456$.
Equipped with the definition of $F_{s}(\mathcal{G})$, we describe the
influenceability evaluation problem as follows.
Problem Statement: Given an influence network $\mathcal{G}$ and a seed node
$s$, the influenceability evaluation problem is to compute the
influenceability $F_{s}(\mathcal{G})$ (Eq. (2)).
It is important to note that the influenceability evaluation problem is known
to be #P-complete [Chen et al. (2010)], even for the very special influence
network where the influence probabilities of all edges are equivalent. There
is no hope to exactly evaluate the influenceability in polynomial time unless
P = #P. Given the hardness of this problem, in this paper, our goal is to
develop an efficient and accurate approximate algorithm to evaluate
$F_{s}(\mathcal{G})$ given a seed node $s$.
An important metric for evaluating the accuracy of an approximate algorithm is
the mean squared error (MSE), which is denoted by
$\mathbb{E}[(\hat{F}_{s}(\mathcal{G})-F_{s}(\mathcal{G}))^{2}]$, where
$\hat{F}_{s}(\mathcal{G})$ denotes an estimator of $F_{s}(\mathcal{G})$ by the
approximate algorithm. By the so-called variance-bias decomposition [Jin et
al. (2011b)], this metric can be decomposed into two parts.
$\mathbb{E}[(\hat{F}_{s}(\mathcal{G})-F_{s}(\mathcal{G}))^{2}]=Var(\hat{F}_{s}(\mathcal{G}))+[\mathbb{E}(\hat{F}_{s}(\mathcal{G})-F_{s}(\mathcal{G}))]^{2},$
(3)
where $\mathbb{E}(\hat{F}_{s}(\mathcal{G}))$ and
$Var(\hat{F}_{s}(\mathcal{G}))$ denote the expectation and variance of the
estimator $\hat{F}_{s}(\mathcal{G})$, respectively. If an estimator is
unbiased, then the second term in Eq. (3) will be canceled out. Therefore, the
variance of the unbiased estimator becomes the only indicator for evaluating
the accuracy of the estimator.
## 3 Naive Monte-Carlo
In this section, we introduce the Naive Monte-Carlo (NMC) sampling for
estimating the influenceability $F_{s}(\mathcal{G})$ given a seed node $s$,
which is the only existing algorithm used in the influence maximization
literature [Kempe et al. (2003), Leskovec et al. (2007), Chen et al. (2009),
Chen et al. (2010)]. This method first samples $N$ possible graphs
$G_{1},G_{2},\cdots,G_{N}$ of $\mathcal{G}$ according to the influence
probabilities $P$, and then calculates the number of reachable nodes from the
seed node $s$ in each possible graph $G_{i}$, $i=1,2,\cdots,N$, i.e.,
$f_{s}(G_{i})$. Finally, the NMC estimator $\hat{F}_{NMC}$ is given below.
$\hat{F}_{NMC}=\frac{{\sum\limits_{i=1}^{N}{f_{s}(G_{i})}}}{N}.$ (4)
The NMC estimator is an unbiased estimator of $F_{s}(\mathcal{G})$, such that
$\mathbb{E}(\hat{F}_{NMC})=F_{s}(\mathcal{G})$. The variance of the NMC
estimator is given as follows.
$\begin{array}[]{l}Var(\hat{F}_{NMC})=\frac{{\mathbb{E}[f_{s}(G)^{2}]-(\mathbb{E}[f_{s}(G)])^{2}}}{N}\\\
\quad\quad\quad\quad\quad\;\;\;=\frac{{\sum\limits_{G_{P}\in\Omega}{\Pr[G_{P}]f_{s}(G)^{2}}-F_{s}(\mathcal{G_{P}})^{2}}}{N}.\end{array}$
(5)
Notice that exactly computing the variance $Var(\hat{F}_{{\sl NMC}})$ is
extremely expensive, because we have to enumerate all the possible graphs to
determine it, whose time complexity is exponential. In practice, we resort to
an unbiased estimator of $Var(\hat{F}_{NMC})$ to evaluate the accuracy of the
estimator $\hat{F}_{NMC}$ [Jin et al. (2011b)]. In this case, an unbiased
estimator of $Var(\hat{F}_{NMC})$ is given by the following equation.
$\widetilde{Var}(\hat{F}_{NMC})=\frac{{\sum\limits_{i=1}^{N}{(f_{s}(G_{i})-\hat{F}_{NMC})^{2}}}}{{N-1}}=\frac{{\sum\limits_{i=1}^{N}{f_{s}(G_{i})^{2}}-N\hat{F}_{NMC}^{2}}}{{N-1}}.$
(6)
According to Eq. (6), $\widetilde{Var}(\hat{F}_{NMC})$ may be very large,
because the value of $f_{s}(G_{i})$ falls into the interval $[0,n-1]$, which
may result in $\widetilde{Var}(\hat{F}_{NMC})$ as large as $O(n^{2})$. Here,
$n$ is the number of nodes in ${\cal G}$. For example, assume $f_{s}(G_{i})=0$
for $i=1,\cdots,N/2$ and $f_{s}(G_{i})=n-1$ for $i=N/2+1,\cdots,N$, then
$\widetilde{Var}(\hat{F}_{NMC})$ equals to $N(n-1)^{2}/4(N-1)=O(n^{2})$.
Therefore, the key issue that we address in this paper is to design more
accurate estimators than the NMC estimator for estimating the influenceability
$F_{s}(\mathcal{G})$.
The NMC algorithm is described in Algorithm 1. The algorithm works in $N$
iterations (line 2-5). In each iteration, the NMC algorithm needs to generate
a possible graph by tossing $m$ biased coins for $m$ edges in ${\cal G}$,
which takes $O(m)$ time complexity (line 3). Then, the algorithm invokes a BFS
algorithm to calculate the number of reachable nodes from $s$, which again
causes $O(m)$ time complexity (line 4). As a result, the time complexity of
the NMC algorithm is $O(Nm)$.
Input: | Influence network $\mathcal{G}$, sample size $N$, and the seed node $s$.
---|---
Output: The NMC estimator $\hat{F}_{NMC}$.
1: $\hat{F}_{NMC}\leftarrow 0$;
2: for $i=1$ to $N$ do
3: Flip $m$ biased coins to generate a possible graph $G_{i}$;
4: Compute $f_{s}(G_{i})$ by the BFS algorithm;
5: $\hat{F}_{NMC}\leftarrow\hat{F}_{NMC}+f_{s}(G_{i})$;
6: $\hat{F}_{NMC}\leftarrow\hat{F}_{NMC}/N$;
7: return $\hat{F}_{NMC}$;
ALGORITHM 1 NMC ($\mathcal{G}$, $N$, $s$)
## 4 New Type-I Estimators
In this section, we first introduce an exact algorithm for computing the
influenceability $F_{s}(\mathcal{G})$, which will guide us to design the new
estimators. We will propose two new estimators based on the idea of stratified
sampling [Thompson (2002)]. Both estimators are shown to be unbiased, and
their variance is significantly smaller than the variance of the NMC
estimator. We refer to the two estimators as the type-I estimators.
### 4.1 An exact algorithm
We introduce an exact divide-and-conquer enumeration algorithm to evaluate the
influenceability for a given influence network $\mathcal{G}=(V,E,P)$ with $n$
nodes and $m$ edges. The main idea of our exact algorithm is described as
follows. First, the algorithm divides the entire probability space $\Omega$
(all the possible graphs) into $2^{r}$ different subspaces by randomly
enumerating $r$ ($r<m$) edges that have not been enumerated. Note that $r$ is
a small number (eg. $r=5$). In each subspace, the exact algorithm recursively
enumerates another $r$ edges, and this process will terminate until all the
edges are enumerated. The partition method of the exact algorithm is described
in Table 4.1. In Table 4.1, “$0$”, “$1$”, and “$*$” denote the statuses of
inexistence, existence, and not-yet-enumerated, for the edges, respectively.
Each case from $1$, $2$, $\cdots$, to $r$ corresponds to a subspace. And
$\Omega_{i}$, for $i=1,2,\cdots 2^{r}$, denotes the probability space of the
case $i$, which represents the set of all possible graphs in the case $i$.
Probability space partition in the exact algorithm Edges $e_{1}$ $e_{2}$
$e_{3}$ $\cdots$ $e_{r}$ $e_{r+1}$ $\cdots$ $e_{m}$ Prob. Space Case 1 0 0 0
$\cdots$ 0 $*$ $\cdots$ $*$ $\Omega_{1}$ Case 2 1 0 0 $\cdots$ 0 $*$ $\cdots$
$*$ $\Omega_{2}$ Case 3 0 1 0 $\cdots$ 0 $*$ $\cdots$ $*$ $\Omega_{3}$
$\;\;\;\cdots$ $\cdots$ $\cdots$ Case $2^{r}$ 1 1 1 $\cdots$ 1 $*$ $\cdots$
$*$ $\Omega_{2^{r}}$
To clarify our algorithm, let $T=(e_{1},e_{2},\cdots,e_{r})$ be the set of
selected $r$ edges, and $X_{i}=(X_{i,1},X_{i,2},\cdots,X_{i,r})$ be the status
vector corresponding to the selected $r$ edges under the case $i$, where
$X_{i,j}=0$ signifies that the edge $e_{j}$ does no exist , and $X_{i,j}=1$
otherwise. For example, for case $1$ in Table 4.1, the status vector is
$X_{1}=(0,0,\cdots,0)$, which means that all the selected $r$ edges do not
exist. In other words, all the possible graphs in $\Omega_{1}$ do not include
the edges in $T$. The probability of a possible graph in case $i$ is given by
$\pi_{i}=\Pr[G_{P}\in\Omega_{i}]=\prod\limits_{e_{j}\in T\wedge
X_{i,j}=1}{p_{j}}\prod\limits_{e_{j}\in T\wedge X_{i,j}=0}{(1-p_{j})}.$ (7)
In addition, let $A_{1}$ be the set of edges that have been enumerated, and
$A_{2}$ be the set of edges that have not been enumerated, such that
$A_{1}\cup A_{2}=E$, and $A_{1}\cap A_{2}=\emptyset$. Then, the
influenceability of the node $s$ under the case $i$ is defined as
$F_{s}(\mathcal{G}(A_{1},A_{2},X_{i}))=\sum\limits_{G_{P}\in\Omega_{i}}{f_{s}(G_{P})\frac{{\Pr[G_{P}]}}{{\pi_{i}}}},$
(8)
where $\mathcal{G}(A_{1},A_{2},X_{i})$ denotes the set of possible graphs in
the case $i$, i.e. $\Omega_{i}$. According to Eq. (8),
$F_{s}(\mathcal{G}(A_{1},A_{2},X_{i}))$ denotes the expected spread over all
the possible graphs in $\Omega_{i}$, and ${\Pr[G_{P}]}/{\pi_{i}}$ is the
probability of a possible graph $G_{P}$ conditioning on it exists in
$\Omega_{i}$. It is worth of noting that
$F_{s}(\mathcal{G})=F_{s}(\mathcal{G}(\emptyset,E,\emptyset))$. Based on Eq.
(8), we have the following theorem.
###### Theorem 4.1.
Let $F_{s}(\mathcal{G}(A_{1},A_{2},X_{i}))$ be the influenceability of the
node $s$ under the case $i$ as defined in Eq. (8), and $T$ be a set of $r$
edges randomly selected from $A_{2}$. For any $T$, we have $2^{r}$ cases, and
let $Y_{j}$ ($j=1,\cdots,2^{r}$) be the corresponding status vector. Then, we
have
$F_{s}(\mathcal{G}(A_{1},A_{2},X_{i}))=\sum\nolimits_{j=1}^{2^{r}}{\pi_{j}F_{s}(\mathcal{G}(A_{1}\cup
T,A_{2}\backslash T,[X_{i},Y_{j}]))},$ (9)
where $[X_{i},Y_{j}]$ is a new status vector generated by appending $Y_{j}$ to
$X_{i}$.
Based on Theorem 4.1, we develop a recursive enumeration algorithm described
in Algorithm 2. Algorithm 2 first partitions the entire probability space
$\Omega$ into $2^{r}$ subspaces, and then the same procedure will be
recursively performed on each subspace based on Theorem 4.1 (line 9-17 in
Algorithm 2. The algorithm terminates until all the edges are enumerated. The
influenceability $F_{s}(\mathcal{G})$ can be computed by invoking EXACT
($\mathcal{G},\emptyset,E,\emptyset,s$).
Input: | Influence network $\mathcal{G}$, the set of edges that have been
---|---
| enumerated $A_{1}$, the Set of edges that have not been
| enumerated $A_{2}$, sample size $N$, and the seed node $s$.
Output: | The exact value of $F_{s}(\mathcal{G})$
1: if $A_{2}=\emptyset$ then
2: Compute $f_{s}(G(V,A_{1},A_{2},X))$ by the BFS algorithm;
3: return $f_{s}(G(V,A_{1},A_{2},X))$;
4: else
5: if $|A_{2}|<r$ then
6: $l\leftarrow|A_{2}|$;
7: else
8: $l\leftarrow r$;
9: Select $l$ edges from $A_{2}$ randomly;
10: Let $T$ be the set of selected edges;
11: $F\leftarrow 0$;
12: for $i=1$ to $2^{l}$ do
13: Let $X_{i}$ be the status vector of set $T$ under the case $i$;
14: Compute $\pi_{i}$ by Eq. (7);
15: Append $X_{i}$ to $X$;
16: $u_{i}\leftarrow$ EXACT ($\mathcal{G}$, $A_{1}\cup T$, $A_{2}\backslash
T$, $X$, $s$);
17: $F\leftarrow F+\pi_{i}u_{i}$;
18: return $F$;
ALGORITHM 2 EXACT ($\mathcal{G}$, $A_{1}$, $A_{2}$, $X$, $s$)
The enumeration procedure given in Algorithm 2 can be characterized by a full
$2^{r}$-ary tree which is depicted in Fig. 2. Note that, to simplify our
analysis, here we assume that $r$ is divisible by $m$. In the tree, each node
represents a probability space that consists of a set of possible graphs. For
example, the root node denotes the probability space that includes the set of
all possible graphs, and each leaf node denotes the probability space that
includes only one possible graph. Each internal node has $2^{r}$ children, and
each child corresponds to a case described in Table 4.1. To compute
$F_{s}(\mathcal{G})$, we need to traverse all the nodes in the tree. Because
the number of nodes in the tree is $O(2^{m})$, the time complexity of
Algorithm 2 is $O(2^{m})$. Therefore, the exact algorithm only works on small
networks due to the nature of #P-complete of the influenceability evaluation
problem. In the following, we will develop two types of efficient
approximation algorithms for evaluating the influenceability.
Figure 2: The Enumeration Tree of the Exact Algorithm.
### 4.2 Basic stratified sampling estimator (I)
As discussed in Section 3, the NMC estimator leads to a large variance. To
reduce the variance, we propose a new stratified sampling estimator for
influenceability evaluation. We call this new estimator the basic stratified
sampling (BSS) estimator, because it servers as the basis for designing
recursive stratified sampling (RSS) estimator which will be described in
Section 4.3. To distinguish the type-II estimators which will be introduced in
Section 5, we refer to the new estimators presented in this section as the
type-I estimators. Specifically, we refer to the type-I BSS and RSS estimator
as the BSS-I and RSS-I estimator, respectively.
Unlike the NMC sampler which draws a sample (a possible graph) from the entire
population (all the possible graphs), the stratified sampling [Thompson
(2002)] first divides the population into $M$ disjoint groups, which are
called _strata_ , and then independently picks separate samples from these
groups. Stratified sampling is a commonly used technique for reducing variance
[Thompson (2002)] in sampling design. There are two key techniques in
stratified sampling: _stratification_ , which is a process for partitioning
the entire population into disjoint strata, and _sample allocation_ , which is
a procedure to determine the sample size that needs to be drawn from each
stratum. Below, we will introduce our stratification and sample allocation
method.
Stratification: Our idea of stratification is based on the exact algorithm
described in the previous subsection. First, we choose $r$ edges and determine
their statuses ($0/1$), where $r$ is a small number. Recall that this process
generates $2^{r}$ various cases as shown in Table 4.1, and thereby it
partitions the set of possible graphs $\Omega$ into $2^{r}$ subsets
$\Omega_{1},\cdots,\Omega_{2^{r}}$. Second, we let each subset be a stratum.
This is because $\Omega_{1},\cdots,\Omega_{2^{r}}$ are disjoint sets and
$\Omega=\bigcup\nolimits_{i=1}^{2^{r}}{\Omega_{i}}$, thus each case is indeed
a valid stratum. It is worth of mentioning that our stratification process
corresponds to the top two layers in the enumeration tree (Fig. 2), the root
node denotes the entire population, and each child represents a stratum. The
stratification process is depicted in Table 4.2.
Stratum design of the BSS-I/RSS-I estimator Edges $e_{1}$ $e_{2}$ $e_{3}$
$\cdots$ $e_{r}$ $e_{r+1}$ $\cdots$ $e_{m}$ Prob. space Stratum 1 0 0 0
$\cdots$ 0 $*$ $\cdots$ $*$ $\Omega_{1}$ Stratum 2 1 0 0 $\cdots$ 0 $*$
$\cdots$ $*$ $\Omega_{2}$ Stratum 3 0 1 0 $\cdots$ 0 $*$ $\cdots$ $*$
$\Omega_{3}$ $\;\;\;\cdots$ $\cdots$ $\cdots$ Stratum $2^{r}$ 1 1 1 $\cdots$ 1
$*$ $\cdots$ $*$ $\Omega_{2^{r}}$
In our stratification approach, a question that arises is how to select the
$r$ edges for stratification. As shown in our experiments, the edge-selection
strategy for choosing $r$ edges significantly affects the performance of the
estimator. One straightforward strategy is to randomly pick $r$ edges from the
edge set $E$. We refer to this edge selection strategy as the random edge-
selection strategy. With this strategy, the selected $r$ edges may not have
direct contributions for computing the influenceability. For example, in Fig.
1(b), for the source node $s=v_{5}$, assume $r=2$ and the selected edges are
$\\{v_{1}\to v_{2},v_{6}\to v_{2}\\}$. The edges $\\{v_{1}\to v_{2},v_{6}\to
v_{2}\\}$ have no direct contributions for calculating the influenceability
$F_{s}(\mathcal{G})$. This may reduce the performance of the BSS-I estimator.
For avoiding such a problem, we introduce another heuristic edge-selection
strategy based on the BFS visiting order of the edges. To estimate
$F_{s}(\mathcal{G})$, we first perform a BFS algorithm starting from the node
$s$ to obtain the first $r$ edges according to the BFS visiting order of the
edges. Then, we use these $r$ edges for stratification. We refer to such edge-
selection strategy as the BFS edge-selection strategy. Consider the same
example in Fig. 1(b), assume $r=2$, the first $r$ edges are $\\{v_{5}\to
v_{3},v_{5}\to v_{6}\\}$. Then, we partition the population into 4 strata
according to the statuses of these two edges. Obviously, according to the BFS
edge-selection strategy, the selected edges have direct contribution to
calculate the influenceability. In our experiments, we find that the
performance of the BSS-I estimator with BFS edge-selection strategy is
significantly better than the performance of the BSS-I estimator with random
edge-selection strategy.
The BSS-I estimator: Let $N$ be the total number of samples, $N_{i}$ be the
number of samples drawn from the stratum $i$ ($i=1,2,\cdots,2^{r}$), and
$G_{i,j}$ ($j=1,2,\cdots,N_{i}$) be a possible graph sampled from the stratum
$i$. Then, the BSS-I estimator is given as follows.
$\hat{F}_{BSSI}=\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\frac{1}{{N_{i}}}\sum\nolimits_{j=1}^{N_{i}}{f_{s}(G_{i,j})}},$
(10)
where $\pi_{i}$ is defined in Eq. (7). The following theorem shows that
$\hat{F}_{BSSI}$ is an unbiased estimator of the influenceability
$F_{s}(\mathcal{G})$.
###### Theorem 4.2.
$F_{s}(\mathcal{G})=\mathbb{E}(\hat{F}_{BSSI})$.
###### Proof 4.3.
We prove it by the following equalities.
$\begin{array}[]{l}\mathbb{E}(\hat{F}_{BSSI})=\mathbb{E}(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\frac{1}{{N_{i}}}\sum\nolimits_{j=1}^{N_{i}}{f_{s}(G_{i,j})}})\\\
\quad\;\quad\quad\quad\;=\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mathbb{E}(f_{s}(G_{i,j}))}\\\
\quad\;\quad\quad\quad\;=\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\sum\nolimits_{G_{P}\in\Omega_{i}}{f_{s}(G_{P})\frac{{\Pr[G_{P}]}}{{\pi_{i}}}}}\\\
\quad\;\quad\quad\quad\;=\sum\nolimits_{G_{P}\in\Omega}{\Pr[G_{P}]f_{s}(G_{P})}\\\
\quad\;\quad\quad\quad\;=F_{s}(\mathcal{G})\\\ \end{array}$
Let $\sigma_{i}$ be the variance of the sample in the stratum $i$. Since the
samples are independently drawn by the basic stratified sampling algorithm,
thus the variance of the BSS-I estimator is given by
$Var(\hat{F}_{BSSI})=\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}^{2}\frac{{\sigma_{i}}}{{N_{i}}}},$
(11)
where $\pi_{i}$ is given in Eq. (7).
Sample allocation: As discussed above, the BSS-I estimator is unbiased and the
variance of the BSS-I estimator depends on the sample size of all the strata,
i.e., $N_{i}$, for $i=1,2,\cdots 2^{r}$. Thus, a question that arises is how
to allocate the sample size for each stratum $i$ ($i=1,2,\cdots,2^{r}$) to
minimize the variance of the BSS-I estimator, i.e. $Var(\hat{F}_{BSSI})$.
Formally, the sample allocation problem is formulated as follows.
$\begin{array}[]{l}\min\;Var(\hat{F}_{BSSI})=\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}^{2}\frac{{\sigma_{i}}}{{N_{i}}}}\\\
s.t.\quad\quad\sum\nolimits_{i=1}^{2^{r}}{N_{i}}=N.\\\ \end{array}$ (12)
By applying the Lagrangian method, we can derive the optimal sample allocation
as given by
$N_{i}={N\pi_{i}\sqrt{\sigma_{i}}}/{\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\sqrt{\sigma_{i}}}},$
(13)
for $i=1,\cdots,2^{r}$. From Eq. (13), the optimal allocation needs to know
the variance of the sample in each stratum, i.e. $\sigma_{i}$, for
$i=1,\cdots,2^{r}$. However, such variances are unavailable in our problem.
Interestingly, we find that, if the sample size of the stratum $i$ is
allocated to $\pi_{i}N$, then the variance of the BSS-I estimator will be
smaller than the variance of the NMC estimator. We have the following theorem.
###### Theorem 4.4.
If $N_{i}=\pi_{i}N$, then $Var(\hat{F}_{BSSI})\leq Var(\hat{F}_{{\sl NMC}})$.
###### Proof 4.5.
If $N_{i}=\pi_{i}N$, then we have
$Var(\hat{F}_{BSSI})=\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\frac{{\sigma_{i}}}{N}}$.
Let $\mu_{i}=\mathbb{E}(f_{s}(G_{i,j}))$ be the expectation of the sample in
the stratum $i$. By definition, we have
$\sigma_{i}=\mathbb{E}(f_{s}(G_{i,j})^{2})-\mu_{i}^{2}=\sum\nolimits_{G_{P}\in\Omega_{i}}{f_{s}(G_{P})^{2}\frac{{\Pr[G_{P}]}}{{\pi_{i}}}}-\mu_{i}^{2}$.
Then, we have
$\begin{array}[]{l}Var(\hat{F}_{BSSI})=\frac{1}{N}\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}(\sum\nolimits_{G_{P}\in\Omega_{i}}{f_{s}(G_{P})^{2}\frac{{\Pr[G_{P}]}}{{\pi_{i}}}}-\mu_{i}^{2})}\\\
\quad\quad\quad\quad\quad\;\;\;=\frac{1}{N}\sum\nolimits_{i=1}^{2^{r}}{(\sum\nolimits_{G_{P}\in\Omega_{i}}{f_{s}(G_{P})^{2}\Pr[G_{P}]}-\pi_{i}\mu_{i}^{2})}\\\
\quad\quad\quad\quad\quad\;\;\;=\frac{1}{N}\sum\nolimits_{G_{P}\in\Omega}{\Pr[G_{P}]f_{s}(G_{P})^{2}}-\frac{1}{N}\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}^{2}}.\\\
\end{array}$
Given this, we can derive the difference between $Var(\hat{F}_{BSSI})$ and
$Var(\hat{F}_{{\sl NMC}})$ (Eq. (5)) as follows:
$\begin{array}[]{l}\quad\quad\;Var(\hat{F}_{{\sl NMC}})-Var(\hat{F}_{BSSI})\\\
\quad\quad\quad\;=\frac{1}{N}(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}^{2}}-(\mathbb{E}[f_{s}(G_{P})])^{2})\\\
\quad\quad\quad\;=\frac{1}{N}(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}^{2}}-(\sum\nolimits_{G_{P}\in\Omega}{\Pr[G_{P}]f_{s}}(G_{P}))^{2})\\\
\quad\quad\quad\;=\frac{1}{N}(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}^{2}}-(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\sum\limits_{G_{P}\in\Omega_{i}}{\frac{{\Pr[G_{P}]}}{{\pi_{i}}}f_{s}}(G_{P})})^{2})\\\
\quad\quad\quad\;=\frac{1}{N}(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}^{2}}-(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}})^{2})\\\
\quad\quad\quad\;=\frac{{Var(\mu_{i})}}{N}\\\ \quad\quad\quad\;\geq 0.\\\
\end{array}$
Note that in the last equality $\mu_{i}$ can be treated as a random variable.
Then, we have
$\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}^{2}}=\mathbb{E}(\mu_{i}^{2})$ and
$(\mathbb{E}(\mu_{i}))^{2}=(\sum\nolimits_{i=1}^{2^{r}}{\pi_{i}\mu_{i}})^{2}$,
thus the last equality holds. This completes the proof.
The BSS-I algorithm: Given the stratification and sample allocation methods,
we present our basic stratified sampling algorithm in Algorithm 3. First,
Algorithm 3 selects $r$ edges to partition the population into $2^{r}$ strata
according to an edge-selection strategy (line 2), either random or BFS edge-
selection. For convenience, we refer to the BSS-I estimator with random edge-
selection and the BSS-I estimator with BFS edge-selection as BSS-I-RM and BSS-
I-BFS estimator, respectively. Second, according to our sample allocation
method, the algorithm draws $\pi_{i}N$ samples from the stratum $i$ (line
8-13). Finally, the algorithm outputs the BSS-I estimator $\hat{F}_{BSSI}$.
Notice that it takes $O(m)$ time for both generating a possible graph $G$ and
performing BFS on $G$. Besides, the algorithm needs to draw $N$ possible
graphs. Hence, the time complexity of Algorithm 3 is $O(mN)$, which has the
same complexity as the NMC estimator. However, our BSS-I estimator
significantly reduces the variance of the NMC estimator. The advantages of the
BSS-I estimator are twofold. On one hand, given the sample size, the BSS-I
estimator is more accurate than the NMC estimator as it has a smaller
variance. On the other hand, to achieve the same variance, the BSS-I estimator
needs a smaller sample size than that of the NMC estimator, thus it reduces
the time complexity of the sampling process.
Input: | Influence network $\mathcal{G}$, sample size $N$, and the seed node $s$.
---|---
Output: The BSS-I estimator $\hat{F}$.
1: $\hat{F}\leftarrow 0$;
2: Choose $r$ edges according to an edge-selection strategy;
3: for $i=1$ to $2^{r}$ do
4: Let $X_{i}$ be the status vector of stratum $i$;
5: Compute $\pi_{i}$ by Eq. (7);
6: $N_{i}\leftarrow[\pi_{i}N]$;
7: $t\leftarrow 0$;
8: for $j=1$ to $N_{i}$ do
9: Flip $m-r$ coins to determine the rest $m-r$ edges;
10: Let $Y_{j}$ be the status vector of the rest $m-r$ edges;
11: Append $X_{i}$ to $Y_{j}$ to generate a possible graph $G_{j}$;
12: Compute $f_{s}(G_{j})$ by the BFS algorithm;
13: $t\leftarrow t+f_{s}(G_{j})$;
14: $t\leftarrow t/N_{i}$;
15: $\hat{F}\leftarrow\hat{F}+\pi_{i}t$;
16: return $\hat{F}$;
ALGORITHM 3 BSS-I ($\mathcal{G}$, $N$, $s$)
### 4.3 Recursive stratified sampling estimator (I)
Recall that the BSS-I estimator splits the entire set of possible graphs into
$2^{r}$ subsets, which corresponds to the top two layers in the enumeration
tree (Fig. 2). Interestingly, we observe that the basic stratified sampling
(BSS-I) can be applied into any internal nodes of the enumeration tree. Based
on this observation, we develop a recursive stratified sampling estimator,
namely RSS-I estimator, which is described in Algorithm 4. The RSS-I estimator
recursively partitions the sample size $N$ to $N_{i}=\pi_{i}N$
($i=1,2,\cdots,2^{r}$) for estimating the influenceability at the stratum $i$
(line 9-19). Note that since the BSS-I estimator is unbiased, the RSS-I
estimator is also unbiased. Moreover, RSS-I reduces the variance at each
partition, thus the variance of RSS-I is significantly smaller than the
variance of BSS-I as stated by the following theorem.
###### Theorem 4.6.
Let $Var(\hat{F}_{RSSI})$ be the variance of RSS-I, then
$Var(\hat{F}_{RSSI})\leq Var(\hat{F}_{BSSI})$.
###### Proof 4.7.
We focus on the case that RSS-I only partitions the population $2^{r}+1$
times. Similar arguments can be used to prove the case of more partitions. At
the first partition, RSS-I splits the population into $2^{r}$ strata, which is
equivalent to BSS-I. In each stratum $i$ ($i=1,\cdots,2^{r}$), RSS-I
recursively partitions it into $2^{r}$ sub-strata. Let $\Omega_{i}$,
$\mu_{i}$, $\sigma_{i}$ and $N_{i}$ be the probability space, the expectation,
the variance, and the sample size of the stratum $i$ at the first partition,
respectively. Let $\pi_{i}=\Pr[G_{P}\in\Omega_{i}]$ be the probability of a
sample in stratum $i$ as defined in Eq. (7). Similarly, for each stratum $i$,
we denote the probability space, the expectation, the variance, and the sample
size of the sub-stratum $k$ ($k=1,\cdots,2^{r}$), as $\Omega_{i,k}$,
$\mu_{i,k}$, $\sigma_{i,k}$, and $N_{i,k}$, respectively. Further, we denote
the probability of a sample in a sub-stratum $k$ as $\pi_{i,k}$, i.e.,
$\pi_{i,k}=\Pr[G_{P}\in\Omega_{i,k}]$. Then, we have
$\pi_{i,k}=\pi_{i}\omega_{k}$, where $\omega_{k}$ denotes the probability of a
sample in sub-stratum $k$ conditioning on it is in stratum $i$, i.e.,
$\omega_{k}=\Pr[G_{P}\in\Omega_{i,k}|G_{P}\in\Omega_{i}]$. The RSS-I estimator
is given by
$\hat{F}_{RSSI}=\sum\nolimits_{i=1}^{2^{r}}{\sum\nolimits_{k=1}^{2^{r}}{\pi_{i,k}\frac{1}{{N_{i,k}}}\sum\nolimits_{j=1}^{N_{i,k}}{f_{s}(G_{i,k,j})}}},$
where $G_{i,k,j}$ ($j=1,\cdots,N_{i,k}$) denotes a possible graph sampled from
the sub-stratum $k$ of the stratum $i$. Then, the variance of RSS-I is
$Var(\hat{F}_{RSSI})=\sum\nolimits_{i=1}^{2^{r}}{\sum\nolimits_{k=1}^{2^{r}}{\frac{{\pi_{i,k}^{2}\sigma_{i,k}}}{{N_{i,k}}}}}.$
By our sample allocation strategy, we have $N_{i,k}=N\pi_{i,k}$, thereby the
variance can be simplified to
$Var(\hat{F}_{RSSI})=\sum\nolimits_{i=1}^{2^{r}}{\frac{1}{N}\sum\nolimits_{k=1}^{2^{r}}{\pi_{i,k}\sigma_{i,k}}}$
Further, by $\pi_{i,k}=\pi_{i}\omega_{k}$, we have
$Var(\hat{F}_{RSSI})=\sum\nolimits_{i=1}^{2^{r}}{\frac{{\pi_{i}}}{N}\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\sigma_{i,k}}}.$
By the proportional sample allocation, we have
$Var(\hat{F}_{BSSI})=\sum\nolimits_{i=1}^{2^{r}}{\frac{{\pi_{i}}}{N}\sigma_{i}}$.
Therefore, the proof is completed followed by
$\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\sigma_{i,k}}\leq\sigma_{i}$. By
definition, we have
$\begin{array}[]{l}\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\sigma_{i,k}}=\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}(\mathbb{E}(f_{s}(G_{i,k,j})^{2})-\mu_{i,k}^{2})}\\\
\quad\quad\quad\quad\;\;\;\;\;\;=\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}(\sum\nolimits_{G_{P}\in\Omega_{i,k}}{\frac{{\Pr[G_{P}]}}{{\pi_{i,k}}}f_{s}(G_{P})^{2}}-\mu_{i,k}^{2})}\\\
\quad\quad\quad\quad\;\;\;\;\;\;=\sum\nolimits_{k=1}^{2^{r}}{\sum\nolimits_{G_{P}\in\Omega_{i,k}}{\frac{{\Pr[G_{P}]}}{{\pi_{i}}}f_{s}(G_{P})^{2}}-\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\mu_{i,k}^{2}}}\\\
\quad\quad\quad\quad\;\;\;\;\;\;=\sum\nolimits_{G_{P}\in\Omega_{i}}{\frac{{\Pr[G_{P}]}}{{\pi_{i}}}f_{s}(G_{P})^{2}}-\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\mu_{i,k}^{2}}.\\\
\end{array}$
Then, we have
$\sigma_{i}-\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\sigma_{i,k}}=\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\mu_{i,k}^{2}}-\mu_{i}^{2}\\\
=\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\mu_{i,k}^{2}}-(\sum\nolimits_{k=1}^{2^{r}}{\omega_{k}\mu_{i,k}})^{2}\geq
0.\\\ $
This completes the proof.
The RSS-I algorithm terminates until the sample size becomes smaller than a
given threshold ($\tau$) or the number of unsampled edges smaller than $r$
(line 2). When the terminative conditions of the RSS-I algorithm satisfy, we
perform a naive Monte-Carlo sampling for estimating the influenceability (line
3-7).
Similar to the BSS-I estimator, the partition approach in RSS-I estimator also
depends on the edge-selection strategy (line 9). Likewise, we have two edge-
selection strategies for the RSS-I estimator, either random edge-selection or
BFS edge-selection. For convenience, we refer to the RSS-I estimator with
random edge-selection and with BFS edge-selection as the RSS-I-RM and RSS-I-
BFS estimator, respectively.
Reconsider the example in Fig. 1(b), the BFS visiting order of the edges is
$\\{v_{5}\to v_{3},v_{5}\to v_{6},v_{3}\to v_{1},v_{3}\to v_{4},v_{6}\to
v_{2},v_{1}\to v_{2},v_{1}\to v_{3},v_{1}\to v_{4},v_{4}\to v_{6},v_{2}\to
v_{6}\\}$. Assume $r=2$, according to the BFS visiting order, then the RSS-I-
BFS first picks edge $v_{5}\to v_{3}$ and $v_{5}\to v_{6}$ for stratification,
and then selects the edges $v_{3}\to v_{1}$ and $v_{3}\to v_{4}$, and so on.
It worth of mentioning that we can invoke the procedure RSS-I
($\mathcal{G},\emptyset,E,\emptyset,N,s$), where $s$ is the seed node, to
calculate the RSS-I estimator.
We analyze the time complexity of Algorithm 4. For sampling a possible graph,
Algorithm 4 needs to traverse the enumeration tree (Fig. 2) from the root node
to the terminative node. Here the terminative node is a node in the
enumeration tree where the terminative conditions of the recursion satisfy at
that node, i.e. $N<\tau$ or $|E_{2}|<r$ holds in Algorithm 4. Let $\bar{d}$ be
the average length of the path from the root node to the terminative node.
Then, by analysis, the time complexity of the algorithm at each internal node
of the path is $O(r)$. Suppose that the total number of such paths is $K$.
Then, the algorithm takes $O(K\bar{d}r)$ time complexity at the internal nodes
of all the paths. Note that $K$ is bounded by the sample size $N$, and
$\bar{d}$ is a very small number w.r.t. $N$. More specifically, we can derive
that $\bar{d}=O(\log_{2^{r}}N)$, which is a very small number. For example,
assume $r=5$ and $N=100,000$, then we can get $\bar{d}\approx 3.3$. For all
the terminative nodes, the time complexity of the algorithm is $O(Nm)$. This
is because the algorithm needs to sample $N$ possible graphs in total over all
the terminative nodes, and for each possible graph the algorithm performs a
BFS to compute the influenceability which takes $O(m)$ time complexity. Since
$O(K\bar{d}r)$ is dominated by $O(Nm)$, the time complexity of Algorithm 4 is
$O(Nm+K\bar{d}r)=O(Nm)$.
Input: | Influence network $\mathcal{G}$, the set of sampled edges $E_{1}$, the set of
---|---
| unsampled edges $E_{2}$, sample size $N$, and the seed node $s$.
Output: The RSS-I estimator $\hat{F}$.
1: $\hat{F}\leftarrow 0$;
2: if $N<\tau$ or $|E_{2}|<r$ then
3: for $j=1$ to $N$ do
4: Flip $|E_{2}|$ coins to generate a possible graph $G_{j}$;
5: Compute $f_{s}(G_{j})$ by the BFS algorithm;
6: $\hat{F}\leftarrow\hat{F}+f_{s}(G_{j})$;
7: return $\hat{F}/N$;
8: else
9: Select $r$ edges from $E_{2}$ according to an edge-selection strategy
{Random or BFS visiting order};
10: Let $T$ be the set of selected edges;
11: for $i=1$ to $2^{r}$ do
12: $Y\leftarrow X$ {Recording the current status vector $X$};
13: Let $X_{i}$ be the status vector of set $T$ in stratum $i$;
14: Append $X_{i}$ to $Y$;
15: Compute $\pi_{i}$ by Eq. (7);
16: $N_{i}\leftarrow\left[{\pi_{i}N}\right]$;
17: $\mu_{i}\leftarrow$ RSS-I ($\mathcal{G}$, $E_{1}\cup T$, $E_{2}\backslash
T$, $Y$, $N_{i}$, $s$);
18: $\hat{F}\leftarrow\hat{F}+\pi_{i}\mu_{i}$;
19: return $\hat{F}$;
ALGORITHM 4 RSS-I ($\mathcal{G}$, $E_{1}$, $E_{2}$, $X$, $N$, $s$)
## 5 New Type-II Estimators
In this section, we propose two new stratified sampling estimators, namely
type-II basic stratified sampling (BSS-II) estimator and type-II recursive
stratified sampling (RSS-II) estimator. The BSS-II and RSS-II are shown to be
unbiased and their variance are significantly smaller than the variance of the
NMC estimator. In the following, we first introduce the BSS-II estimator, and
then present the RSS-II estimator.
### 5.1 Basic stratified sampling estimator (II)
Stratification: We propose a new stratification method for the BSS-II
estimator. This new stratification method splits the entire probability space
$\Omega$ into $r+1$ various subspaces ($\Omega_{0},\cdots,\Omega_{r}$) by
choosing $r$ edges. Specifically, for stratum $0$, we set the statuses of all
the $r$ selected edges to “0”, and for the stratum $i$ ($i\neq 0$), we set the
status of edge $i$ to “1” and the statuses of all the previous $i-1$ edges
(i.e. $e_{1},\cdots,e_{i-1}$) to “0”. Unlike the stratification method of the
BSS-I estimator, this new stratification approach allows us to set $r$ to be a
big number, such as $r=50$. The stratum design method is depicted in Table
5.1.
Stratum design of the BSS-II/RSS-II estimator Edges $e_{1}$ $e_{2}$ $e_{3}$
$\cdots$ $e_{r}$ $e_{r+1}$ $\cdots$ $e_{m}$ Prob. space Stratum 0 0 0 0
$\cdots$ 0 $*$ $\cdots$ $*$ $\Omega_{0}$ Stratum 1 1 $*$ $*$ $\cdots$ $*$ $*$
$\cdots$ $*$ $\Omega_{1}$ Stratum 2 0 1 $*$ $\cdots$ $*$ $*$ $\cdots$ $*$
$\Omega_{2}$ Stratum 3 0 0 1 $\cdots$ $*$ $*$ $\cdots$ $*$ $\Omega_{3}$
$\;\;\;\;\cdots$ $\cdots$ $\cdots$ Stratum $r$ 0 0 0 $\cdots$ 1 $*$ $\cdots$
$*$ $\Omega_{r}$
In Table 5.1, each stratum (Stratum 0, Stratum 1, $\cdots$, Stratum $r$)
corresponds to a subspace ($\Omega_{0},\Omega_{1},\cdots,\Omega_{r}$). For any
$i\neq j$, we have $\Omega_{i}\cap\Omega_{j}=\phi$. Below, we show that
$\bigcup\nolimits_{i=0}^{r}{\Omega_{i}}=\Omega$. Let
$T=(e_{1},e_{2},\cdots,e_{r})$ be the set of $r$ selected edges and $p_{i}$
($i=1$) be the corresponding influence probability, then the probability of a
possible graph in stratum $i$ is given by
$\pi^{\prime}_{i}=\Pr[G_{P}\in\Omega_{i}]=\left\\{\begin{array}[]{l}\prod\limits_{j=1}^{r}{(1-p_{j})},\quad\;\;if\;i=0\\\
p_{i}\prod\limits_{j=1}^{i-1}{(1-p_{j}),\quad otherwise}\\\
\end{array}\right.$ (14)
The following theorem implies $\bigcup\nolimits_{i=0}^{r}{\Omega_{i}}=\Omega$.
###### Theorem 5.1.
$\Pr[G_{P}\in\Omega]=\sum\nolimits_{i=0}^{r}{\Pr[G_{P}\in\Omega_{i}]}=1$.
###### Proof 5.2.
We prove it by the following equalities.
$\begin{array}[]{l}\sum\nolimits_{i=0}^{r}{\Pr[G_{P}\in\Omega_{i}]}\\\
=\prod\nolimits_{j=1}^{r}{(1-p_{j})}+p_{1}+(1-p_{1})p_{2}+\cdots+p_{r}\prod\nolimits_{j=1}^{r-1}{(1-p_{j})}\\\
=\prod\nolimits_{j=1}^{r-1}{(1-p_{j})}+p_{1}+(1-p_{1})p_{2}+\cdots+p_{r-1}\prod\nolimits_{j=1}^{r-2}{(1-p_{j})}\\\
\cdots\\\ =1-p_{1}+p_{1}\\\ =1\\\ \end{array}$
Armed with Theorem 5.1, we conclude that the stratum design approach described
in Table 5.1 is a valid stratification method.
The BSS-II estimator: Similar to the BSS-I estimator, we let $N$ be the total
sample size, and $N_{i}$ be the sample size of the stratum $i$, and $G_{i,j}$
($j=1,2,\cdots,N_{i}$) be a possible graph sampled from the stratum $i$. Then
the BSS-II estimator $\hat{F}_{BSSII}$ is given by
$\hat{F}_{BSSII}=\sum\nolimits_{i=0}^{r}{\pi^{\prime}_{i}\frac{1}{{N_{i}}}}\sum\nolimits_{j=1}^{N_{i}}{f_{s}(G_{i,j})},$
(15)
where $\pi^{\prime}_{i}$ is given in Eq. (14). Similar to Theorem 4.2, the
following theorem shows that the BSS-II estimator is unbiased. The proof is
similar to the proof of Theorem 4.2, thus we omit for brevity.
###### Theorem 5.3.
$F_{s}(\mathcal{G})=\mathbb{E}(\hat{F}_{BSSII})$.
The variance of the BSS-II estimator is given by
$Var(\hat{F}_{BSSII})=\sum\nolimits_{i=0}^{r}{{\pi^{\prime}_{i}}^{2}\frac{{\sigma_{i}}}{{N_{i}}}},$
(16)
where $\sigma_{i}$ denotes the variance of the sample in the stratum $i$.
Sample allocation: Analogous to the BSS-I estimator, for the BSS-II estimator,
we can derive that the optimal sample allocation is given by
$N_{i}=N\pi^{\prime}_{i}\sqrt{\sigma_{i}}/\sum\nolimits_{i=0}^{r}{\pi^{\prime}_{i}\sqrt{\sigma_{i}}}$.
This optimal allocation strategy needs to know the variance of the sample in
each stratum, which is impossible in our problem. Therefore, similar to the
sample allocation approach used in the BSS-I estimator, for the BSS-II
estimator, we set the sample size of the stratum $i$ equals to
$\pi^{\prime}_{i}N$, i.e. $N_{i}=\pi^{\prime}_{i}N$. On the basis of this
sample allocation method, we show that the variance of the BSS-II estimator is
smaller than the variance of the NMC estimator as stated by the following
theorem. The proof of the theorem is similar to theorem 4.4, thus we omitted
for brevity.
###### Theorem 5.4.
If $N_{i}=\pi^{\prime}_{i}N$, $Var(\hat{F}_{BSSII})\leq Var(\hat{F}_{{\sl
NMC}})$.
However, it is very hard to compare the variance of the BSS-II estimator with
the variance of the BSS-I estimator. In our experiments, we find that these
two estimators achieve comparable variance.
The BSS-II algorithm: With the stratification and sample allocation method, we
describe the BSS-II algorithm in Algorithm 5. Algorithm 5 picks $r$ edges to
split the entire population into $r+1$ strata in terms of an edge-selection
strategy (line 2). Any of the two edge-selection strategies (random edge-
selection and BFS edge-selection) used in the BSS-I algorithm can also be used
in the BSS-II algorithm. We refer to the BSS-II estimator with the random
edge-selection and the BSS-II estimator with BFS edge-selection as BSS-II-RM
and BSS-II-BFS estimator, respectively. In terms of the sample allocation
method of the BSS-II estimator, Algorithm 5 picks $N_{i}=\pi^{\prime}_{i}N$
samples from the stratum $i$, for $i=0,1,\cdots,r$, and outputs the BSS-II
estimator $\hat{F}_{BSSII}$. Like the BSS-I estimator, the time complexity of
BSS-II estimator is $O(Nm)$. This is because the BSS-II needs to draw $N$
possible graphs, and both sampling each possible graph $G$ and computing
$F_{s}(G)$ take $O(m)$ time.
Input: | Influence network $\mathcal{G}$, sample size $N$, and the seed node $s$.
---|---
Output: The BSS-II estimator $\hat{F}$.
1: $\hat{F}\leftarrow 0$;
2: Select $r$ edges according to an edge-selection strategy;
3: for $i=0$ to $r$ do
4: Compute $\pi^{\prime}_{i}$ by Eq. (14);
5: $N_{i}\leftarrow[\pi^{\prime}_{i}N]$;
6: $t\leftarrow 0$;
7: if i = 0 then
8: $k\leftarrow r$;
9: else
10: $k\leftarrow i$;
11: Let $E_{i}$ be the set of edges to be determined under stratum $i$;
12: for $j=1$ to $N_{i}$ do
13: Flip $m-k$ coins to determine $E_{i}$, and thus generate a possible graph
$G_{j}$;
14: Compute $f_{s}(G_{j})$ by the BFS algorithm;
15: $t\leftarrow t+f_{s}(G_{j})$;
16: $t\leftarrow t/N_{i}$;
17: $\hat{F}\leftarrow\hat{F}+\pi^{\prime}_{i}t$;
18: return $\hat{F}$;
ALGORITHM 5 BSS-II ($\mathcal{G}$, $N$, $s$)
### 5.2 Recursive stratified sampling estimator (II)
Based on the BSS-II estimator, in this subsection, we develop another new
recursive stratified sampling estimator, namely RSS-II estimator. Similar to
the idea of the RSS-I estimator, the RSS-II estimator makes use of the BSS-II
estimator as the basic component and recursively applies the BSS-II estimator
at each stratum. More specifically, the RSS-II estimator first partitions the
entire probability space $\Omega$ into $r+1$ subspace $\Omega_{i}$
($i=0,1,\cdots,r$) according to the stratification method of the BSS-II
estimator. The same partition procedure is recursively performed in each
subspace $\Omega_{i}$. At each partition, the RSS-II estimator utilizes the
same sample allocation method as the BSS-II estimator to allocate the sample
size. The recursion process of the RSS-II estimator will terminate until the
sample size is smaller than a given threshold ($\tau$) or the number of
unsampled edges is smaller than $r$. Since the BSS-II estimator is unbiased,
the RSS-II estimator is also unbiased. The variance of the RSS-II estimator is
smaller than the variance of the BSS-II estimator, because the RSS-II
estimator recursively reduces variance at each partition while the BSS-II
estimator only reduces variance at one partition. Similar to Theorem 4.6, we
have the following theorem.
###### Theorem 5.5.
Let $Var(\hat{F}_{RSSII})$ be the variance of RSS-I, then
$Var(\hat{F}_{RSSII})\leq Var(\hat{F}_{BSSII})$.
The detail algorithm of the RSS-II estimator is described in Algorithm 6.
Firs, according to an edge-selection strategy, Algorithm 6 selects $r$ edges
from the unsampled edge-set, which is denoted by $E_{2}$, to partition the
population into $r+1$ strata (line 9). Note that the random edge-selection and
BFS edge-selection strategy used in the RSS-I estimator can also be applied in
the RSS-II estimator. We refer to the RSS-II estimator with random edge-
selection and BFS edge-selection as the RSS-II-RM and RSS-II-BFS estimator,
respectively. Second, according to the sample allocation method, the algorithm
recursively invokes the RSS-II algorithm with sample size $N_{i}$ in stratum
$i$, for $i=1,\cdots,r$ (line 11-23). In line 15 and line 20, we let $X_{i}$
be the status vector of the selected edges under the stratum $i$. Unlike the
RSS-I estimator, the status vector of the RSS-II estimator is determined by
the stratification method of the BSS-II estimator (Table 5.1). For example, at
the first partition of the RSS-II estimator, assume
$T=(e_{1},e_{2},\cdots,e_{r})$ is the set of $r$ edges selected, the status
vector of these selected edges under the stratum $0$ is
$X_{0}=(0,0,\cdots,0)$. The status vector under the stratum $i$ is
$X_{i}=(0,\cdots,0,1,*\cdots,*)$, where the statuses of the first $i-1$ edges
are “0”, the status of the $i$-th edge is “1”, and the rest $r-i$ edges are
“$*$”. Finally, the algorithm outputs the RSS-II estimator (line 24).
Input: | Influence network $\mathcal{G}$, the set of sampled edges $E_{1}$,
---|---
| the set of unsampled edges $E_{2}$, sample size $N$,
| and the seed node $s$.
Output: | The RSS-II estimator $\hat{F}$.
1: $\hat{F}\leftarrow 0$;
2: if $N<\tau$ or $|E_{2}|<r$ then
3: for $j=1$ to $N$ do
4: Flip $|E_{2}|$ coins to generate a possible graph $G_{j}$;
5: Compute $f_{s}(G_{j})$ by the BFS algorithm;
6: $\hat{F}\leftarrow\hat{F}+f_{s}(G_{j})$;
7: return $\hat{F}/N$;
8: else
9: Select $r$ edges from $E_{2}$ according to an edge-selection strategy
(random or BFS visiting order);
10: Let $T=(e_{1},e_{2},\cdots,e_{r})$ be the set of selected edges;
11: for $i=0$ to $r$ do
12: Compute $\pi^{\prime}_{i}$ by Eq. (14);
13: $N_{i}\leftarrow\left[{\pi^{\prime}_{i}N}\right]$;
14: if $i=0$ then
15: Let $X_{0}$ be the status vector of set $T$ under stratum $0$;
16: Append $X_{0}$ to $X$;
17: $\mu_{i}\leftarrow$ RSS-II ($\mathcal{G}$, $E_{1}\cup T$, $E_{2}\backslash
T$, $X$, $N_{i}$, $s$);
18: else
19: Let $T_{i}\leftarrow\\{e_{1},\cdots,e_{i}\\}$;
20: Let $X_{i}$ be the status vector of set $T_{i}$ under stratum $i$;
21: Append $X_{i}$ to $X$;
22: $\mu_{i}\leftarrow$ RSS-II ($\mathcal{G}$, $E_{1}\cup T_{i}$,
$E_{2}\backslash T_{i}$, $X$, $N_{i}$, $s$);
23: $\hat{F}\leftarrow\hat{F}+\pi^{\prime}_{i}\mu_{i}$;
24: return $\hat{F}$;
ALGORITHM 6 RSS-II ($\mathcal{G}$, $E_{1}$, $E_{2}$, $X$, $N$, $s$)
Like the RSS-I estimator, to sample a possible graph, the RSS-II algorithm
needs to traverse the recursive tree from the root node to the terminative
node. At all the terminative nodes, the algorithm needs to sample $N$ possible
graphs in total, and for each possible graph it needs to perform a BFS to
compute the influenceability, thus the time complexity is $O(Nm)$. At each
internal node in a path from the root node to the terminative node, the time
complexity is $O(r)$. This is because at each internal node the algorithm only
needs to select $r$ edges and determine their statuses which consume $O(r)$
time complexity. Let $\bar{d}$ be the average length of such path and $K$ be
the total number of paths. Then, for all the internal nodes, the algorithm
takes $O(K\bar{d}r)$ time complexity. According to the terminative condition
given in Algorithm 6, we can derive that
$\bar{d}=\min\\{\log_{r}N,\log_{r}m)$. Since $r$ can be a big number (eg.
$r=50$), $\bar{d}$ is very small. Thus, the time complexity at the internal
nodes $O(K\bar{d}r)$ can be dominated by $O(Nm)$. We conclude that the average
time complexity of Algorithm 6 is $O(Nm)$.
## 6 Experiments
We conduct experimental studies for different estimators over four datasets.
We confirm the efficiency and accuracy of the proposed estimators. In the
following, we first describe the experimental setup, and then report our
results.
### 6.1 Experimental setup
Datasets: We use one synthetic dataset and three real datasets in our
experiments. We apply the same parameters used in [Jin et al. (2011b)] to
generate the synthetic dataset. For the graph topology, we generate an Erdos-
Renyi (ER) random graph with 5,000 vertices and edge density 10. For the
influence probabilities, we generate a probability for each edge according to
a [0,1] uniform distribution.
The three real datasets are given as follows. (1) FacebookLike dataset: this
dataset originates from a Facebook social network for students at University
of California, Irvine. It contains the users who sent or received at least one
message. We collect this dataset from (toreopsahl.com/datasets). The dataset
is a weighted graph, and the weight of each edge denotes the number of
messages passing over the edge. (2) Condmat dataset: this dataset is a
weighted collaboration network, where the weight of an edge represents the
number of co-authored papers between two collaborators. We download this
dataset from (www-personal.umich.edu/~mejn/netdata). (3) DBLP dataset: this
dataset is also a weighted collaboration network, where the weight of the edge
signifies the number of co-authored papers. This dataset is provided by the
authors in [Zhou et al. (2010)]. Table 6.1 summarizes the information for the
four real datasets. To obtain the influence networks, for each real dataset,
we generate the influence probabilities according to the same method used in
[Potamias et al. (2010), Jin et al. (2011b)]. Specifically, to generate the
probability of an edge, we apply an exponential cumulative distribution
function (CDF) with mean 2 to the weight of the edge.
Summary of the datasets Name Nodes Edges Ref. Random graph 5,000 50,616 [Jin
et al. (2011b)] FacebookLike 1,899 20,296 [Opsahl and Panzarasa (2009)]
Condmat 16,264 95,188 [Newman (2001)] DBLP 78,648 376,515 [Zhou et al. (2010)]
Different estimators: In our experiments, we compare 10 estimators. (1) The
NMC estimator, which is the Naive Monte-Carlo estimator. (2) RSS-I-RM ($r=1$),
which is a special RSS-I-RM estimator where the parameter $r=1$, based on work
presented in [Jin et al. (2011b)] for computing distance-constraint
reachability on uncertain graph. We also generalize their estimator to
arbitrary parameter $r$, and apply the generalized estimator for
influenceability evaluation. Recall that beyond the random edge-selection
strategy, we propose a more accurate RSS-I estimator with BFS edge-selection
strategy. (3) BSS-I-RM, which is the BSS-I estimator with the random edge-
selection. (4) BSS-I-BFS, which is the BSS-I estimator with the BFS edge-
selection. (5) RSS-I-RM, which is the RSS-I estimator with the random edge-
selection. (6) RSS-I-BFS, which is the RSS-I estimator with the BFS edge-
selection. (7) BSS-II-RM, which is the BSS-II estimator with the random edge-
selection. (8) BSS-II-BFS, which is the BSS-II estimator with the BFS edge-
selection. (9) RSS-II-RM, which is the RSS-II estimator with the random edge-
selection. (10) RSS-II-BFS, which is the RSS-II estimator with the BFS edge-
selection.
Evaluation metric: Two metrics are used to evaluate the performance of the
estimators: running time and relative variance. The running time evaluates the
efficiency of the estimators. The relative variance is leveraged to evaluate
the accuracy of the estimators. Let $\sigma_{NMC}$ be the variance of the NMC
estimator. We calculate the relative variance of an estimator $\hat{F}$ by
$\sigma_{\hat{F}}/\sigma_{NMC}$. Since computing the exact variance of the
estimators is intractable, we resort to an unbiased estimator of the variance.
Similar evaluation metric has been used in [Jin et al. (2011b)]. Specifically,
for a given seed node $s$ in our experiments, we run all the estimators
$\hat{F}_{s}(\mathcal{G})$ $500$ times, thereby we can obtain $500$ estimating
results:
$\hat{F}^{(1)}_{s}(\mathcal{G}),\hat{F}^{(2)}_{s}(\mathcal{G}),\cdots,\hat{F}^{(500)}_{s}(\mathcal{G})$.
An unbiased variance estimator of $\hat{F}_{s}(\mathcal{G})$ is given by
$\sum\nolimits_{i=1}^{500}{(\hat{F}^{(i)}_{s}(\mathcal{G})-\bar{F}_{s}(\mathcal{G}))^{2}}/499,$
where $\bar{F}_{s}(\mathcal{G}))$ denotes the mean of the $500$ various
estimating results.
Parameter settings and the experimental environment: Without specifically
stated, in all of our experiments, we set the parameters as follows. For all
estimators, we set the sample size $N=1,000$. For the BSS-I and RSS-I
estimators, we set $r=5$, and for the BSS-II and RSS-II estimators, we set
$r=50$. For the threshold parameter $\tau$ in Algorithm 4 and Algorithm 6, we
set $\tau=10$. All the experiments are conducted on the Scientific Linux 6.0
workstation with 2xQuad-Core Intel(R) 2.66 GHz CPU, and 4G memory. All
algorithms are implemented by GCC 4.4.4.
### 6.2 Experimental Results
For all the experiments, we randomly generate 1,000 seed nodes, and the
results are the average result over all the seeds. We report our experimental
results on random graph, FacebookLike, Condmat, and DBLP dataset in Table 6.2,
Table 6.2, Table 6.2, and Table 6.2, respectively.
From Table 6.2, among all the estimators, we can observe that the RSS-I-BFS is
the winner on the random graph dataset, the RSS-I-RM, RSS-II-RM, and RSS-II-
BFS estimators are significantly better than the RSS-I-RM ($r=1$) estimator.
The specific RSS-I-RM ($r=1$) estimator outperforms the BSS estimators, and
all the BSS estimators are better than the NMC estimator. In particular, RSS-
I-BFS reduces the relative variance over the NMC and RSS-I-RM ($r=1$)
estimators by 386% and 227%, respectively. RSS-II-BFS cuts the relative
variance over NMC and RSS-I-RM ($r=1$) by 385% and 226%, respectively. Both
RSS-I-RM and RSS-II-RM estimators cut the relative variance over the NMC and
the RSS-I-RM ($r=1$) estimators more than 185% and 91.4%, respectively. For
the BSS estimators, their performance is worse than the RSS-I-RM ($r=1$)
estimator, but are significantly better than the NMC estimator. In addition,
the running time of all the estimators are comparable. These results consist
with our analysis in Section 4 and Section 5.
From Table 6.2, we can see that RSS-II-BFS achieves the best relative variance
on the FacebookLike dataset, followed by RSS-I-BFS, RSS-II-RM, RSS-I-RM, RSS-
I-RM ($r=1$), the BSS estimators, and the NMC estimator. More specifically,
the RSS-II-BFS estimator reduces the relative variance over the NMC estimator
and the RSS-I-RM ($r=1$) estimators by 317% and 133%, respectively. The RSS-I-
BFS estimator reduces the relative variance over NMC and RSS-I-RM ($r=1$) by
289% and 117%. Both RSS-I-RM and RSS-II-RM estimators cut the relative
variance over NMC and RSS-I-RM ($r=1$) more than 231% and 184%, respectively.
Similar to the result on the random graph dataset, all the BSS estimators are
slightly worse than the RSS-I-RM ($r=1$) estimator but are significantly
better than the NMC estimator. Also, the running time of all the estimators
are comparable because the time complexities of all the estimators are
$O(Nm)$. These results confirm our analysis in the previous sections. Similar
results can be observed in the Condmat (Table 6.2) and DBLP datasets (Table
6.2).
Results on random graph dataset Estimators Relative variance Running time (s)
NMC 1.0000 0.3593 RSS-I-RM ($r=1$) 0.6723 0.3558 BSS-I-RM 0.9429 0.3497 BSS-I-
BFS 0.8938 0.3748 RSS-I-RM 0.3397 0.3373 RSS-I-BFS 0.2056 0.3783 BSS-II-RM
0.9321 0.3633 BSS-II-BFS 0.9042 0.3749 RSS-II-RM 0.3512 0.3716 RSS-II-BFS
0.2063 0.3847
Results on FacebookLike dataset Estimators Relative variance Running time (s)
NMC 1.0000 0.2007 RSS-I-RM ($r=1$) 0.5585 0.2014 BSS-I-RM 0.8898 0.2331 BSS-I-
BFS 0.6819 0.2354 RSS-I-RM 0.3023 0.2002 RSS-I-BFS 0.2570 0.2010 BSS-II-RM
0.6947 0.2250 BSS-II-BFS 0.6672 0.2284 RSS-II-RM 0.2786 0.2027 RSS-II-BFS
0.2397 0.2037
Results on Condmat dataset Estimators Relative variance Running time (s) NMC
1.0000 1.2969 RSS-I-RM ($r=1$) 0.7950 1.2958 BSS-I-RM 0.9068 1.3043 BSS-I-BFS
0.8531 1.3054 RSS-I-RM 0.4883 1.2050 RSS-I-BFS 0.1971 1.2411 BSS-II-RM 0.8553
1.2513 BSS-II-BFS 0.8421 1.3104 RSS-II-RM 0.4891 1.2256 RSS-II-BFS 0.2120
1.2284
Results on DBLP dataset Estimators Relative variance Running time (s) NMC
1.0000 8.5824 RSS-I-RM ($r=1$) 0.5375 8.6536 BSS-I-RM 0.9170 8.6292 BSS-I-BFS
0.8373 8.8173 RSS-I-RM 0.2100 8.3835 RSS-I-BFS 0.1918 8.5933 BSS-II-RM 0.9449
8.8825 BSS-II-BFS 0.7997 9.1305 RSS-II-RM 0.2003 8.6840 RSS-II-BFS 0.1821
8.7052
To summarize, RSS-I-BFS and RSS-II-BFS achieve the best relative variance, and
they reduce the relative variance over the existing estimators several times.
The RSS estimators are better than the BSS estimators. The BSS/RSS estimators
with the BFS edge-selection strategy are better than the BSS/RSS estimators
with the random edge-selection strategy. All of our RSS estimators outperform
the RSS-I-RM ($r=1$) estimator. The proposed BSS estimators are slightly worse
than the RSS-I-RM ($r=1$) estimator, but still significantly outperform the
NMC estimator. The running time of all the estimators are comparable.
Scalability: In order to study the scalability of various estimators, we
generate synthetic probabilistic graphs $\mathcal{G}$ with nodes ranging from
200,000 (200k) to 800,000 and the edges ranging from 800,000 to 3,200,000
(3.2m) according to the ER random graph model. And the probability of each
edge is randomly generated according to a [0, 1] uniform distribution. Also,
for each estimator, we set the sample size $N$ to 1,000. Table 6.2 shows the
running time of different estimators on four large synthetic probabilistic
graphs. As can be seen in Table 6.2, the running time increases as the size of
the graph increases. In general, all the estimators achieve comparable running
time, and they have linear growth w.r.t. the graph size. These results consist
with the complexities of our estimators, i.e. $O(Nm)$.
Scalability: Running time on synthetic graphs. Here the two numbers in the
2nd-5th columns (eg. 200k/800k) indicate the numbers of nodes and edges
respectively Time (s) 200k/800k 400k/1.3m 600k/1.6m 800k/3.2m NMC 26.0820
156.9600 289.7720 365.0280 BSS-I-RM 25.2090 159.1990 281.6810 343.0350 BSS-I-
BFS 27.2120 169.6120 286.2180 368.0910 RSS-I-RM 23.3430 143.6700 264.9790
342.3920 RSS-I-BFS 25.2090 169.6120 286.2180 344.0180 BSS-II-RM 26.1450
161.4100 287.1500 371.4770 BSS-II-BFS 29.5760 162.3930 290.9340 374.6830 RSS-
II-RM 26.4440 156.8120 270.6670 363.1590 RSS-II-BFS 26.4990 162.7940 271.1370
365.9630
Effect of parameter $r$: We study the effectiveness of the parameter $r$ in
our proposed estimators on Condmat dataset. Similar results can be observed
from other datasets. Fig. 3 and Fig. 4 show the relative variance of our
type-I and type-II estimators w.r.t. various $r$. As can be seen in Fig. 3,
the BSS-I estimators exhibit similar relative variance over different $r$
values. However, the relative variance of the RSS-I-RM estimator decreases as
the $r$ increases when $r\leq 5$, and otherwise it increases as the $r$
increases. For the RSS-I-BFS estimator, the relative variance decreases as $r$
increases, and when $r\geq 5$ the descent rate is very small, and the curve
tends to be smooth. Based on this observation, $r=5$ is the best choice, which
is used in the previous experiments. For the type-II estimators, we test the
parameter $r$ from 10 to 70, and the results (Fig. 4) show that all of our
type-II estimators except RSS-I-BFS are not very sensitive w.r.t. the
parameter $r$. As an exception, the relative variance of the RSS-I-BFS
estimator decreases as the $r$ increases when $r\leq 50$, and when $r\geq 50$
the the curve tends to be smooth. Therefore, $r=50$ is a good choice. In our
previous experiments, we set $r$ to 50\. Table 6.2 and Table 6.2 report the
running time of type-I estimators and type-II estimators under different $r$
values. We can see that the running time of both type-I estimators and type-II
estimators are comparable.
Figure 3: Effect of $r$ of BSS-I/RSS-I estimators. Figure 4: Effect of $r$ of
BSS-II/RSS-II estimators.
BSS-I/RSS-I estimators: Running time vs $r$ Time (s) $r=1$ $r=2$ $r=3$ $r=5$
$r=10$ BSS-I-RM 1.2642 1.2705 1.2770 1.3043 1.2986 BSS-I-BFS 1.2731 1.2791
1.2798 1.3054 1.3686 RSS-I-RM 1.2082 1.1993 1.1810 1.2050 1.1172 RSS-I-BFS
1.2158 1.2140 1.2189 1.2411 1.1833
BSS-II/RSS-II estimators: Running time vs $r$ Time (s) $r=10$ $r=20$ $r=30$
$r=50$ $r=70$ BSS-II-RM 1.2515 1.2502 1.2511 1.2513 1.2524 BSS-II-BFS 1.2579
1.2719 1.2836 1.3104 1.3447 RSS-II-RM 1.2279 1.2246 1.2162 1.2256 1.2092 RSS-
II-BFS 1.2358 1.2278 1.2258 1.2284 1.2295
Effect of sample size: As shown in the previous experiments, the RSS-I-BFS and
the RSS-II-BFS estimators are the best two estimators. Here we study how
sample size affects the estimating accuracy of these two estimators on the
Condmat dataset. Similar results can be observed on the other dataset. Fig. 5
shows the relative variance of the estimators under various sample size. As
can be observed in Fig. 5, the curves of RSS-I-BFS and RSS-II-BFS estimators
are very smooth, which indicate that the relative variance of both RSS-I-BFS
and RSS-II-BFS estimators are robust w.r.t. the sample size.
Figure 5: Relative variance vs sample size.
## 7 Related work
After the seminal work by Kempe, et al. [Kempe et al. (2003)], influence
maximization in social networks has recently attracted much attention in data
mining and social network analysis research communities [Kempe et al. (2005),
Leskovec et al. (2007), Chen et al. (2009), Chen et al. (2010), Goyal et al.
(2010), Chen et al. (2011), Goyal et al. (2011)]. A crucial subroutine in
influence maximization is the influence function evaluation to which the
influenceability estimation problem presented in this paper is closely
related. In the following, we first review some notable work on influence
maximization problem, and then discuss the existing work on influence function
evaluation. In [Leskovec et al. (2007)], the authors study the influence
maximization problem under the context of water distribution and blogosphere
monitoring. They propose a so-called CELF framework for optimizing the
influence maximization algorithms. To further accelerate the influence
maximization algorithms, Chen, et al. in [Chen et al. (2009)] propose a
scalable algorithm by sampling $N$ possible graphs and estimating the
influence spread of all vertices on each possible graph at one time.
Subsequently, the same authors propose a series of scalable algorithms in
[Chen et al. (2010)] and [Chen et al. (2011)] for influence maximization by
developing the heuristic vertices-selection strategies on unsigned and signed
networks, respectively. Recently, Goyal, et al. in [Goyal et al. (2010), Goyal
et al. (2011)] consider the problem of learning the influence probabilities,
and study the influence maximization from a data-driven perspective. Note that
all the mentioned methods focus on the influence maximization problem. For the
influence function evaluation problem, Kempe, et al. firstly pose it as an
open problem in [Kempe et al. (2005)]. Then, Chen, et al. in [Chen et al.
(2010)] show that the influence function evaluation problem is #P complete.
Given the hardness of the problem, most of the existing work for this problem,
such as [Kempe et al. (2003), Leskovec et al. (2007), Chen et al. (2009), Chen
et al. (2010)], are based on the Naive Monte-Carlo (NMC) sampling. In this
paper, we study the influenceability evaluation problem and develop more
accurate RSS estimators for estimating the influenceability, and our
algorithms can also be used for influence function evaluation.
Our work is also related to the uncertain graph mining. Recently, uncertain
graphs mining have been attracted increased interest because of the increasing
applications in biological database [Sevon et al. (2006)], network routing
[Ghosh et al. (2007)], and influence networks [Goyal et al. (2011)]. There are
a large body of works have been proposed in the literature. Notable work
includes finding the reliable subgraph in a large uncertain graph [Hintsanen
and Toivonen (2008), Jin et al. (2011a)], frequent subgraph mining in
uncertain graph database [Zou et al. (2010a), Zou et al. (2010b)], subgraph
search in large uncertain graph [Yuan et al. (2011)], K-nearest neighbor
search in uncertain graph [Potamias et al. (2010)], and distance constraint
reachability computation in uncertain graph [Jin et al. (2011b)]. In general,
all the mentioned uncertain graph mining problems are shown to be #P-complete,
and thereby finding the exact solution is intractable in large uncertain
graphs. Consequently, most existing work, such as [Potamias et al. (2010)] and
[Jin et al. (2011a)], are based on NMC sampling. Basically, the NMC sampling
based methods lead to a large variance, thus reduce the performance of the
algorithms. Recently, Jin, et al. in [Jin et al. (2011b)] propose a recursive
stratified sampling method for distance-constraint reachability computation on
uncertain graph, although they do not claim their method is a stratified
sampling. It is important to note that their method is a very special case of
our RSS-I algorithm. In their method, they select only one edge for
stratification at a time, and then recursively perform this procedure. Unlike
their algorithm, first, we develop a generalized algorithm (RSS-I) that
selects $r$ edges for stratification. Second, unlike their reachability
problem, here we study the influenceability evaluation problem using the RSS-I
sampling. Moreover, in our work, we also develop another RSS estimator, i.e.
RSS-II estimator. Note that all of our RSS estimators can also be applied into
the distance-constraint reachability computation problem.
In addition, our work is related to the network reliability estimation
problem, where a network is modeled as an uncertain graph and the goal is to
estimate some reliability metrics of the network [Fishman (1986a), Rubino
(1999)]. There are many work on this topic in the last five decades. Surveys
can be found in [Colbourn (1987), Rubino (1999)].
Below, we review the Monte-Carlo algorithms for network reliability
estimation. Kumamoto, et al. [Kumamoto et al. (1977)] propose an efficient
Monte-Carlo algorithm by exploiting the bound of the reliability metric.
Fishman [Fishman (1986b)] proposes a more generalized Monte-Carlo algorithm
based on such bound techniques for reliability estimation. Subsequently,
Fishman [Fishman (1986a)] compares four Monte-Carlo algorithms for network
reliability estimation problem. Cancela, et al. in [Cancela and Khadiri
(2003)] propose a recursive variance-reduction algorithm for network
reliability estimation. Note that all the mentioned Monte-Carlo algorithms are
tailored for the network reliability estimation problem, and the reliability
measure is typically a Boolean metric thus they cannot be used in our problem.
## 8 Conclusions
In this paper, we focus on the influenceability evaluation problem, which is a
fundamental issue for influence analysis in social network. This problem is
known to be #P-complete, and the only existing algorithm is based on the Naive
Monte-Carlo (NMC) sampling. To reduce the variance of the NMC estimator, we
propose two basic stratified sampling (BSS) estimators. Furthermore, based on
our BSS estimators, we present two recursive stratified sampling (RSS)
estimators. We conduct comprehensive experiments on one synthetic and three
real datasets, and the results confirm that our RSS estimators reduce the
variance of the NMC estimator by several times. There are several future
directions that deserve further investigation. First, most of our estimators
except the RSS estimators with BFS edge selection do not take the graph
structural information into account. In our experiments, the RSS estimators
with BFS edge selection are shown much better performance than the RSS
estimators with random edge selection. A promising direction is to exploit the
graph structural information to develop more efficient and more accurate
estimators for influenceability evaluation. Second, our estimation techniques
are quite general. For many uncertain graph mining problems, such as shortest
path [Potamias et al. (2010)], reachability [Jin et al. (2011b)], and reliable
subgraph discovery [Jin et al. (2011a)], our estimators can be directly used.
For these problems, we only need to replace the $\phi_{s}(G_{P})$ to other
quantities, such as the length of the shortest path, the reachability function
between two nodes, and the reliable subgraph metric. Most of these uncertain
graph mining problems are based on NMC. Another promising future direction is
to apply our estimation techniques to these problems.
## References
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|
arxiv-papers
| 2012-07-04T06:49:22 |
2024-09-04T02:49:32.620971
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rong-Hua Li, Jeffrey Xu Yu, Zechao Shang",
"submitter": "Rong-Hua Li",
"url": "https://arxiv.org/abs/1207.0913"
}
|
1207.0938
|
# Symbol Error Rate of Space-Time Network Coding in Nakagami-$m$ Fading
Ang Yang, Zesong Fei, Nan Yang, Chengwen Xing, and Jingming Kuang A. Yang, Z.
Fei, C. Xing, and J. Kuang are with the School of Information and Electronics,
Beijing Institute of Technology, Beijing, China (email:taylorkingyang@163.com,
feizesong@bit.edu.cn, chengwenxing@ieee.org, JMKuang@bit.edu.cn).N. Yang is
with the Wireless and Networking Technologies Laboratory, CSIRO ICT Centre,
Marsfield, NSW 2122, Australia (email: jonas.yang@csiro.au).
###### Abstract
In this paper, we analyze the symbol error rate (SER) of space-time network
coding (STNC) in a distributed cooperative network over independent but not
necessarily identically distributed (i.n.i.d.) Nakagami-$m$ fading channels.
In this network, multiple sources communicate with a single destination with
the assistance of multiple decode-and-forward (DF) relays. We first derive new
exact closed-form expressions for the SER with $M$-ary phase shift-keying
modulation ($M$-PSK) and $M$-ary quadrature amplitude modulation ($M$-QAM). We
then derive new compact expressions for the asymptotic SER to offer valuable
insights into the network behavior in the high signal-to-noise ratio (SNR)
regime. Importantly, we demonstrate that STNC guarantees full diversity order,
which is determined by the Nakagami-$m$ fading parameters of all the channels
but independent of the number of sources. Based on the new expressions, we
examine the impact of the number of relays, relay location, Nakagami-$m$
fading parameters, power allocation, and nonorthogonal codes on the SER.
###### Index Terms:
Space-time network coding, symbol error rate, Nakagami-$m$ fading.
## I Introduction
Cooperative communication has been recognized as a promising low-cost solution
to combat fading and to extend coverage in wireless networks [1, 2]. The key
idea behind this solution is to employ relays to receive and transmit the
source information to the destination, which generates a virtual multiple-
input and multiple-output (MIMO) system to provide spatial cooperative
diversity [3, 4]. Apart from diversity, throughput enhancement is another
critical challenge for wireless networks. Against this background, network
coding (NC) is proposed as a potentially powerful tool to enable efficient
information transmission, where data flows coming from multiple sources are
combined to increase throughput [5, 6, 7].
Recently, the joint exploitation of cooperative diversity and NC has become a
primary design concern in distributed networks with multiple users and
multiple relays [8, 9, 10, 11]. Motivated by this, [8] investigated various
sink network decoding approaches for the network with intermediate node
encoding. In [9], linear network coding (LNC) was applied in distributed
uplink networks to facilitate the transmission of independent information from
multiple users to a common base station. In [10], low-density parity-check
(LDPC) code and NC was jointly designed for a multi-source single-relay FDMA
system over uniform phase-fading Gaussian channels. In [11], cooperative
network coding strategies were proposed for a relay-aided two-source two-
destination wireless network with a backhaul connection between the sources.
In order to increase the capacity and transmission reliability of wireless
cooperative networks, multiple antennas are deployed to gain the merits of
MIMO processing techniques [12, 13, 14, 15, 16, 17]. In this strategy,
distributed space-time coding (DSTC) was proposed to further boost network
performance, where the antennas at the distributed relays are utilized as
transmit antennas to generate a space-time code for the receiver [18, 19]. A
differential DSTC was proposed in [20] to eliminate the requirement of channel
information at the relays and the receiver. The combined benefits of maximum-
ratio combining (MRC) and DSTC were investigated in [21]. In [22], the impact
of DSTC in two-way amplify-and-forward relay channels was characterized.
One of the principal challenges in distributed cooperative networks is to
leverage the benefits from both NC and DSTC. A promising solution that
addresses this challenge is space-time network coding (STNC), which was
proposed in [23]. Fundamentally, STNC combines the information from different
sources at a relay, which involves the concept of NC. Moreover, STNC transmits
the combined signals in several time slots using a set of relays, which
involves the concept of DSTC. Based on these, STNC achieves spatial diversity
with low transmission delay under the impact of imperfect frequency and timing
synchronization. We note that in [23], the performance of STNC was evaluated
for Rayleigh fading channels, where no closed-form expression was presented.
In this paper, we consider a distributed cooperative network using STNC over
independent but not necessarily identically distributed (i.n.i.d.)
Nakagami-$m$ fading channels, which generalizes the result in [23]. In this
network, multiple sources communicate with a single destination with the
assistance of multiple relays. Here, we focus on decode-and-forward (DF)
protocol at the relays, which arises from the fact that this protocol has been
successfully deployed in practical wireless standards, e.g., 3GPP Long Term
Evolution (LTE) and IEEE 802.16j WiMAX [24]. Different from [23], we examine
two fundamental questions as follows: “ _1) What is the impact of STNC on the
symbol error rate (SER) in general Nakagami- $m$ fading channels?_” and “ _2)
Can we provide closed-form expressions for the SER in Nakagami- $m$ fading to
alleviate the burden of Monte Carlo simulations?_” The rationale behind these
questions is that Nakagami-$m$ fading covers a wide range of typical fading
scenarios in realistic wireless applications via the $m$ parameter. Notably,
Nakagami-$m$ fading encompasses Rayleigh fading ($m=1$) as a special case
[25]. To tackle these questions, we first derive new closed-form expressions
for the exact SER, which are valid for multiple phase shift-keying modulation
($M$-PSK) and $M$-ary quadrature amplitude modulation ($M$-QAM). To further
provide valuable insights at high signal-to-noise ratios (SNRs), we derive new
compact expressions for the asymptotic SER, from which the diversity gain is
obtained. Specifically, it is demonstrated that the diversity order is
determined by the Nakagami-$m$ fading parameters of all the channels, but
independent of the number of sources. Various numerical results are utilized
to examine the impact of the number of relays, relay location, Nakagami-$m$
fading parameters, power allocation, and nonorthogonal codes on the SER.
Importantly, it is shown that nonorthogonal codes provide higher throughput
than orthogonal codes, while guaranteeing full diversity over Nakagami-$m$
fading channels. Our analytical expressions are substantiated via Monte Carlo
simulations.
## II System Model
Fig. 1 depicts a distributed cooperative network where $L$ sources,
$U_{1},U_{2},\ldots,U_{L}$, transmit their own information to a common
destination $D$ with the aid of $Q$ relays, $R_{1},R_{2},\ldots,R_{Q}$. In
this network, each node is equipped with a single antenna. We consider a
practical and versatile operating scenario where the source-relay, the relay-
destination, and the source-destination channels experience i.n.i.d.
Nakagami-$m$ fading. As such, we denote the Nakagami-$m$ fading parameter
between $U_{l}$ and $R_{q}$ as $m_{lq}$, the Nakagami-$m$ fading parameter
between $R_{q}$ and $D$ as $m_{qd}$, and the Nakagami-$m$ fading parameter
between $U_{l}$ and $D$ as $m_{ld}$. We further denote the channel coefficient
between $U_{l}$ and $R_{q}$ as $h_{lq}$, the channel coefficient between
$R_{q}$ and $D$ as $h_{qd}$, and the channel coefficient between $U_{l}$ and
$D$ as $h_{ld}$, where $1\leq{}l\leq{}L$ and $1\leq{}q\leq{}Q$. Throughout
this paper, we define the variances of these channel coefficients as
$h_{\phi\varphi}\sim\mathcal{CN}\left(0,d_{\phi\varphi}^{-\alpha}\right)$,
where $\phi\in\left\\{l,q\right\\}$, $\varphi\in\left\\{q,d\right\\}$, and
$\phi\neq\varphi$. Here, we incorporate the path loss in the signal
propagation such that $d_{\phi\varphi}$ denotes the distance between $\phi$
and $\varphi$ and $\alpha$ denotes the path loss exponent.
In this network, the STNC transmission between the sources and the destination
is divided into two consecutive phases: 1) source transmission phase and 2)
relay transmission phase. In the source transmission phase, the sources
transmit their symbols in the designated time slots. In this phase, the relays
receive a set of overheard symbols from the sources. In the relay transmission
phase, each relay encodes the set of overheard symbols to a single signal and
then transmits it to the destination in its designated time slot. As
illustrated in Fig. 2, $(L+Q)$ time slots are required to complete the STNC
transmission to eliminate the detrimental effects of imperfect synchronization
on any point-to-point transmission in this network at any time slot.
We proceed to detail the transmission in the two phases, as follows:
In the source transmission phase, the signals received at the destination from
$U_{l}$ in the time slot $l$ is given by
$y_{ld}\left(t\right)=h_{ld}\sqrt{P_{l}}x_{l}s_{l}\left(t\right)+w_{ld}\left(t\right),$
(1)
where $P_{l}$ denotes the transmit power at $U_{l}$, $x_{l}$ denotes the
symbol transmitted by $U_{l}$, $s_{l}\left(t\right)$ denotes the spreading
code of $x_{l}$, and $w_{ld}$ is the additive white Gaussian noise (AWGN) with
zero mean and the variance of $N_{0}$. The cross correlation between
$s_{p}\left(t\right)$ and $s_{q}\left(t\right)$ are expressed as
$\rho_{pq}=\left\langle{s_{p}\left(t\right),s_{q}\left(t\right)}\right\rangle$,
where
$\left\langle{f\left(t\right),g\left(t\right)}\right\rangle\triangleq\frac{1}{T}\int_{0}^{T}{f\left(t\right)g^{\ast}\left(t\right)dt}$
is the inner product between $f\left(t\right)$ and $g\left(t\right)$ during
the symbol interval $T$. Moreover, we assume that
$\rho_{ll}=\left\|{s_{l}\left(t\right)}\right\|^{2}=1$. The signals received
at $R_{q}$ from $U_{l}$ is given by
$y_{lq}\left(t\right)=h_{lq}\sqrt{P_{l}}x_{l}s_{l}\left(t\right)+w_{lq}\left(t\right),$
(2)
where $w_{lq}\left(t\right)$ is AWGN with zero mean and the variance of
$N_{0}$.
In the relay transmission phase, the signal received at the destination from
$R_{q}$ is given by
$y_{qd}\left(t\right)={h_{qd}}\underbrace{\sum_{l=1}^{L}{{\beta_{ql}}\sqrt{{P_{ql}}}{x_{l}}{s_{l}}\left(t\right)}}_{{f_{q}}\left(x\right)}+{w_{qd}}\left(t\right),$
(3)
where $P_{ql}$ denotes the transmit power at $R_{q}$ and $w_{qd}$ is AWGN with
zero mean and the variance of $N_{0}$. In (3), the scalar $\beta_{ql}$ denotes
the state whether $R_{q}$ decodes $x_{l}$ correctly. Specifically,
$\beta_{ql}$ is equal to $1$ if $R_{q}$ decodes $x_{l}$ correctly, but $0$
otherwise.
For the detection of the received signals at the destination, we assume that
the full knowledge of the channel state information are available at the
receivers with the aid of a preamble in the transmitted signal. We also assume
that the destination has the detection states at the relays, which can be
obtained via an indicator in the relaying signal. At the destination, the
spreading codes $s_{l}$ is employed such that the information symbols $x_{l}$
is separated from $y_{ld}$ and $y_{qd}$, where $l\in\\{1,\cdots,L\\}$. For any
desired symbol $x_{l}$, the destination combines the information of $x_{l}$
from $U_{l}$ and the $Q$ relays using maximum ratio combining (MRC).
Therefore, the instantaneous signal-to-noise ratio (SNR) of $x_{l}$ is
expressed as [23]
$\gamma_{l}=\frac{{P_{l}\left|{h_{ld}}\right|^{2}}}{{N_{0}}}+\sum_{q=1}^{Q}{\frac{{\beta_{ql}P_{lq}\left|{h_{qd}}\right|^{2}}}{{N_{0}\varepsilon_{l}}}},$
(4)
where $\varepsilon_{l}$ is the $l$th diagonal element of matrix
${\bf{R}}^{-1}$ associated with symbol $x_{l}$ and $\bf{R}$ is given by
${\bf{R}}=\left[{\begin{array}[]{*{20}c}1&{\rho_{21}}&\cdots&{\rho_{Q1}}\\\
{\rho_{12}}&1&\cdots&{\rho_{Q2}}\\\ \vdots&\vdots&\ddots&\vdots\\\
{\rho_{1Q}}&{\rho_{2Q}}&\cdots&1\\\ \end{array}}\right].$ (5)
To facilitate the performance analysis in the following section, we re-express
(4) as a unitary expression given by
$\gamma_{l}=c_{0}\left|{h_{0}}\right|^{2}+\sum_{q=1}^{Q}\beta_{ql}c_{q}\left|{h_{q}}\right|^{2},$
(6)
where $c_{0}=P_{l}d_{dl}^{-\alpha}/N_{0}$ denotes the equivalent SNR at $D$
received from $U_{l}$, $h_{0}$ denotes the unitary Nakagami-$m$ fading
coefficient between $U_{l}$ and $D$ with variance one,
$c_{q}=P_{l}d_{ql}^{-\alpha}/N_{0}\varepsilon_{l}$ denotes the $q$th
equivalent SNR received at $D$ from $R_{q}$, and $h_{q}$ denotes the unitary
Nakagami-$m$ fading coefficient between $R_{q}$ and $D$ with variance one.
## III SER Analysis over Nakagami-$m$ Fading Channel
In this section, we first derive new closed-form expressions for the exact SER
with $M$-PSK and $M$-QAM. We then derive new compact expressions for the
asymptotic SER, which will allow us to examine the network behavior in the
high SNR regime.
### III-A Exact SER
In DF protocol, $\beta_{ql}$ denotes the decoding state at $R_{q}$ associated
with $x_{l}$. Based on the values of all $\beta_{ql}$’s, we define a decimal
number as
$S_{l}=\left[\beta_{1l}~{}\beta_{2l}~{}\cdots~{}\beta_{Ql}\right]_{2}$ to
represent one of $2^{Q}$ network decoding states at $Q$ relays associated with
$x_{l}$. Since all the channels in this network are mutually independent, the
events that whether $R_{q}$ correctly decodes the received signal are
independent. It follows that $\beta_{ql}$’s are independent Bernoulli random
variables, the distribution of which is written as
$G\left({\beta_{ql}}\right)=\left\\{{\begin{array}[]{ll}{1-{\rm{SER}}_{ql},}&{{\rm{if}}~{}{\beta_{ql}=1}}\\\
{{\rm{SER}}_{ql},}&{{\rm{otherwise,}}}\\\ \end{array}}\right.$ (7)
where ${\rm{SER}}_{ql}$ denotes the SER of detecting $x_{l}$ at $R_{q}$.
Therefore, the joint probability of a particular combination of $x_{l}$ in
$S_{l}$ is written as
${\Pr}\left(S_{l}\right)=\prod_{q=1}^{Q}{G\left(\beta_{ql}\right)}.$ (8)
Applying Bayesian rule, the SER of detecting $x_{l}$ at $D$ is derived as
$\displaystyle{\rm{SER}}_{l}=\sum_{S_{l}=0}^{2^{Q}-1}{{\rm{SER}}_{\gamma_{l|S_{l}}}\Pr\left({S_{l}}\right)}=\sum_{S_{l}=0}^{2^{Q}-1}{{\rm{SER}}_{\gamma_{l|S_{l}}}\prod_{q=1}^{Q}{G\left({\beta_{ql}}\right)}},$
(9)
where ${{\rm{SER}}_{\gamma_{l|S_{l}}}}$ denotes the SER of detecting $x_{l}$
at $D$ conditioned on $S_{l}$. To facilitate the calculation of (9), we
present the exact closed-form results for ${{\rm{SER}}_{\gamma_{l|S_{l}}}}$
and ${G\left({\beta_{ql}}\right)}$, as follows.
#### III-A1 Exact Results for ${\rm{SER}_{\gamma_{l|S_{l}}}}$
We commence the derivation of ${\rm{SER}_{\gamma_{l|S_{l}}}}$ by presenting
the PDF of $\gamma_{l|S_{l}}$, $f_{\gamma_{l|S_{l}}}\left(v\right)$. If there
are $N$ “$1$” elements in one set $S_{l}$, let
$a_{1},a_{2},\cdots,a_{N}\in\left\\{c_{1},c_{2},\cdots,c_{Q}\right\\}$ denote
the equivalent SNRs of the $N$ relays which decode $x_{l}$ successfully, where
$N\leq Q$. We then define $a_{0}=c_{0}$ to make the source equivalent to the
zeroth relay. As such, the SNR at $D$ to one desired symbol $x_{l}$ can be
rewritten as
$\gamma_{l|S_{l}}=a_{0}\left|{h_{0}}\right|^{2}+\sum_{n=1}^{N}{a_{n}\left|{h_{n}}\right|^{2}}=\sum_{n=0}^{N}{\underbrace{a_{n}\left|{h_{n}}\right|^{2}}_{Y_{n}}}.$
(10)
According to [25], the PDF of $Y_{n}$ in Nakagami-$m$ fading is given by
$f_{Y_{n}}\left(y\right)=\frac{{{m_{n}}^{m_{n}}y^{m_{n}-1}}}{{{a_{n}}^{m_{n}}\Gamma\left(m_{n}\right)}}\exp\left({-\frac{{m_{n}y}}{{a_{n}}}}\right),$
(11)
where $m_{n}$ is the Nakagami-$m$ fading parameter between the $n$th
successful relay and the destination. For example, if the first and the second
relays out of three relays successfully decode the information from $U_{l}$,
we have $m_{1}=m_{1d}$ and $m_{2}=_{2d}$. Using Fourier transform together
with [26, eq. (3.351.3)], the characteristic function (CF) of $Y_{n}$ is
calculated as
$\begin{split}C_{Y_{n}}\left(u\right)&=\int_{-\infty}^{\infty}{f_{Y_{n}}\left(y\right)e^{juy}dy}=\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}.\end{split}$
(12)
Given that $\gamma_{l|S_{l}}$ is the sum of $Y_{n}$’s, the CF of
$\gamma_{l|S_{l}}$ is obtained as
$\begin{split}C_{\gamma_{l|S_{l}}}\left(u\right)&=\prod\limits_{n=0}^{N}{C_{Y_{n}}\left(u\right)}=\prod\limits_{n=0}^{N}{\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}}.\end{split}$
(13)
Applying inverse Fourier transform, the PDF of $\gamma_{l|S_{l}}$ is derived
as
$\begin{split}f_{\gamma_{l|S_{l}}}\left(v\right)&=\frac{1}{{2\pi}}\int_{-\infty}^{\infty}{C_{\gamma_{l|S_{l}}}\left(u\right)e^{-juv}du}\\\
&=\frac{1}{{2\pi}}\int_{-\infty}^{\infty}{\underbrace{\left({\prod\limits_{n=0}^{N}{\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}}}\right)e^{-juv}}_{g\left(u\right)}du}.\end{split}$
(14)
We next seek the solution for $g\left(u\right)$. Due to the randomicity of the
wireless channels, we note that $g(u)$ has $N+1$ different poles
$z_{0}=-jm_{0}/a_{0}$, $z_{1}=-jm_{1}/a_{1}$, $\cdots$, $z_{N}=-jm_{N}/a_{N}$
in complex field. As such, based on residue theorem [27], the residue of
$k^{\rm{th}}$ pole of $g(u)$ in complex field can be expressed as
$\displaystyle{\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]=$
$\displaystyle\frac{1}{{\left({m_{k}-1}\right)!}}\mathop{\lim}\limits_{u\to
z_{k}}\frac{{d^{m_{k}-1}}}{{du^{m_{k}-1}}}\left[{\left({u-z_{k}}\right)^{m_{k}}g\left(u\right)}\right]$
$\displaystyle=$
$\displaystyle\frac{{{m_{k}}^{m_{k}}}}{{(-j)^{m_{k}}{a_{k}}^{m_{k}}\left({m_{k}-1}\right)!}}\mathop{\lim}\limits_{u\to
z_{k}}\frac{{d^{m_{k}-1}}}{{du^{m_{k}-1}}}\left[{e^{-juv}\left({\prod\limits_{n=0,n\neq
k}^{N}{\left({1-\frac{{jua_{n}}}{m_{n}}}\right)^{-m_{n}}}}\right)}\right].$
(15)
As per the general Leibniz’s rule, we derive
${\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]$ in (III-A1) as
$\displaystyle{\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]$
$\displaystyle=$
$\displaystyle\frac{{{m_{k}}^{m_{k}}}}{{(-j)^{m_{k}}a_{k}^{m_{k}}\left({m_{k}-1}\right)!}}\mathop{\lim}\limits_{u\to
z_{k}}\sum_{i=0}^{m_{k}-1}{\left[{\left({-jv}\right)^{m_{k}-1-i}e^{-juv}}\right]}\sum_{i_{0}=0}^{i}{\sum_{i_{1}=0}^{i_{0}}{\cdots\sum_{i_{k-1}=0}^{i_{k-2}}{\sum_{i_{k+1}=0}^{i_{k-1}}{\cdots\sum_{i_{N-1}=0}^{i_{N-2}}}}}}$
$\displaystyle\times{{m_{k}-1}\choose i}{i\choose i_{0}}{i_{0}\choose
i_{1}}\cdots{i_{k-2}\choose i_{k-1}}{i_{k-1}\choose
i_{k+1}}\cdots{i_{N-2}\choose
i_{N-1}}\left[{\frac{{d^{i_{N-1}}}}{{du^{i_{N-1}}}}\left({1-\frac{{jua_{N}}}{m_{N}}}\right)^{-m_{N}}}\right]$
$\displaystyle\times\left[{\frac{{d^{i_{N-2}-i_{N-1}}}}{{du^{i_{N-2}-i_{N-1}}}}\left({1-\frac{{jua_{N-1}}}{m_{N-1}}}\right)^{-m_{N-1}}}\right]\cdots\left[{\frac{{d^{i_{k-1}-i_{k+1}}}}{{du^{i_{k-1}-i_{k+1}}}}\left({1-\frac{{jua_{k+1}}}{m_{k+1}}}\right)^{-m_{k+1}}}\right]$
$\displaystyle\times\left[{\frac{{d^{i_{k-2}-i_{k-1}}}}{{du^{i_{k-2}-i_{k-1}}}}\left({1-\frac{{jua_{k-1}}}{m_{k-1}}}\right)^{-m_{k-1}}}\right]\cdots\left[{\frac{{d^{i_{0}-i_{1}}}}{{du^{i_{0}-i_{1}}}}\left({1-\frac{{jua_{1}}}{m_{1}}}\right)^{-m_{1}}}\right]\left[{\frac{{d^{i-i_{0}}}}{{du^{i-i_{0}}}}\left({1-\frac{{jua_{0}}}{m_{0}}}\right)^{-m_{0}}}\right].$
(16)
Upon close observation, we simplify (III-A1) as
${\rm{Res}}\left[{g\left({z_{k}}\right),z_{k}}\right]=\sum_{i=0}^{m_{k}-1}{jB_{N,k,i}}v^{m_{k}-1-i}\exp\left({-\frac{{m_{k}v}}{{a_{k}}}}\right),$
(17)
where $B_{N,k,i}$ is defined as
$\displaystyle B_{N,k,i}$ (18) $\displaystyle=$
$\displaystyle\frac{{{m_{k}}^{m_{k}}}{(-1)}^{-i}}{{a_{k}^{m_{k}}\left({{m_{k}}-1}\right)!}}\sum_{i_{0}=0}^{i}{\sum_{i_{1}=0}^{i_{0}}{\cdots\sum_{i_{k-1}=0}^{i_{k-2}}{\sum_{i_{k+1}=0}^{i_{k-1}}{\cdots\sum_{i_{N-1}=0}^{i_{N-2}}}}}}{{m_{k}-1}\choose
i}{i\choose i_{0}}{i_{0}\choose i_{1}}\cdots{i_{k-2}\choose
i_{k-1}}{i_{k-1}\choose i_{k+1}}\cdots{i_{N-2}\choose i_{N-1}}$
$\displaystyle\times\left({\frac{{a_{N}}}{{m_{N}}}}\right)^{i_{N-1}}\left({\frac{{a_{N-1}}}{{m_{N-1}}}}\right)^{i_{N-2}-i_{N-1}}\cdots\left({\frac{{a_{k+1}}}{{m_{k+1}}}}\right)^{i_{k-1}-i_{k+1}}\left({\frac{{a_{k-1}}}{{m_{k-1}}}}\right)^{i_{k-2}-i_{k+1}}\cdots\left({\frac{{a_{1}}}{{m_{1}}}}\right)^{i_{0}-i_{1}}\left({\frac{{a_{0}}}{{m_{0}}}}\right)^{i-i_{0}}$
$\displaystyle\times\left(m_{N}\right)_{i_{N-1}}\left(m_{N-1}\right)_{i_{N-2}-i_{N-1}}\cdots\left(m_{k+1}\right)_{i_{k-1}-i_{k+1}}\left(m_{k-1}\right)_{i_{k-2}-i_{k-1}}\cdots\left(m_{1}\right)_{i_{0}-i_{1}}\left(m_{0}\right)_{i-i_{0}}$
$\displaystyle\times\left({1-\frac{{m_{k}a_{N}}}{{m_{N}a_{k}}}}\right)^{-m_{N}-i_{N-1}}\left({1-\frac{{m_{k}a_{N-1}}}{{m_{N-1}a_{k}}}}\right)^{-m_{N-1}-i_{N-2}+i_{N-1}}\cdots\left({1-\frac{{m_{k}a_{k+1}}}{{m_{k+1}a_{k}}}}\right)^{-m_{k+1}-i_{k-1}+i_{k+1}}$
$\displaystyle\times\left({1-\frac{{m_{k}a_{k-1}}}{{m_{k-1}a_{k}}}}\right)^{-m_{k-1}-i_{k-2}+i_{k-1}}\cdots\left({1-\frac{{m_{k}a_{1}}}{{m_{1}a_{k}}}}\right)^{-m_{1}-i_{0}+i_{1}}\left({1-\frac{{m_{k}a_{0}}}{{m_{0}a_{k}}}}\right)^{-m_{0}-i+i_{0}},$
(19)
and
$\left(m_{n}\right)_{i}=\Gamma\left({m_{n}+i}\right)/\Gamma\left(m_{n}\right)$
is the Pochmann symbol. Based on the residues of the poles, we confirm that
$g(u)$ is available for the residue theorem. Specifically, using the residue
of $k^{\rm{th}}$ pole of $g(u)$ in complex field, we obtain
$\int_{-\infty}^{\infty}{g\left(u\right)}du=-2\pi j\sum_{k=0}^{N}{{\mathop{\rm
Res}\nolimits}\left[{g\left({z_{k}}\right),z_{k}}\right]}.$ (20)
The proof of (20) is shown in Appendix A. Based on (17) and (20), the PDF of
$\gamma_{l|S_{l}}$ in (14) is derived as
$f_{\gamma_{l|S_{l}}}\left(v\right)=\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}v^{m_{k}-1-i}\exp\left({-\frac{{m_{k}v}}{{a_{k}}}}\right)}.$
(21)
With the aid of $f_{\gamma_{l|S_{l}}}\left(v\right)$ in (21), we are capable
to derive ${\rm{SER}}_{\gamma_{l|S_{l}}}$ for $M$-PSK and $M$-QAM. First,
${\rm{SER}}_{\gamma_{l|S_{l}}}$ for $M$-PSK is derived as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\int_{0}^{\infty}{\alpha\exp\left({-\frac{{b\gamma}}{{\sin^{2}\theta}}}\right)f_{\gamma_{l|S_{l}}}\left(\gamma\right)}d\gamma
d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}}\int_{0}^{\left({M-1}\right)\pi/M}{\int_{0}^{\infty}{\gamma^{m_{k}-1-i}\exp\left({\left({-\frac{b}{{\sin^{2}\theta}}-\frac{m_{k}}{{a_{k}}}}\right)\gamma}\right)}d\gamma
d\theta},$ (22)
where $\alpha$ and $b$ are modulation specific constants. For $M$-PSK,
$\alpha=1$ and $b=\sin^{2}(\pi/M)$. With the aid of [26, eq. (3.351.3)] and
[28, eq. (5A.17)], (III-A1) is derived as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{\pi}\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}}\Gamma\left({m_{k}-i}\right)\int_{0}^{\left({M-1}\right)\pi/M}{\left({\frac{b}{{\sin^{2}\theta}}+\frac{m_{k}}{{a_{k}}}}\right)^{-m_{k}+i}d\theta}$
$\displaystyle=$
$\displaystyle\alpha\sum_{k=0}^{N}{\sum_{i=0}^{m_{k}-1}{B_{N,k,i}}}\Gamma\left({m_{k}-i}\right)\left({\frac{{a_{k}}}{m_{k}}}\right)^{m_{k}-i}\left[\frac{{M-1}}{M}-\frac{1}{\pi}\sqrt{\frac{{a_{k}b}}{{a_{k}b+m_{k}}}}\left(\left({\frac{\pi}{2}+\tan^{-1}\omega}\right)\sum_{p=0}^{m_{k}-i-1}{2p\choose
p}\right.\right.$
$\displaystyle\left.\left.\times\left(4\left({1+\frac{{a_{k}b}}{m_{k}}}\right)\right)^{-p}+\sin\left({\tan^{-1}\omega}\right)\sum_{p=1}^{m_{k}-i-1}\sum_{t=1}^{p}T_{p,t}\left(1+\frac{a_{k}b}{m_{k}}\right)^{-p}\left({\cos\left({\tan^{-1}\omega}\right)}\right)^{2\left({p-t}\right)+1}\right)\right],$
(23)
where $\omega=\sqrt{\frac{{a_{k}b}}{{a_{k}b+m_{k}}}}\cot\frac{\pi}{M}$ and
$T_{p,t}=\frac{{2p\choose
p}}{{2(p-t)\choose{p-t}}4^{t}\left[{2\left(p-t\right)+1}\right]}$.
We then derive ${\rm{SER}}_{\gamma_{l|S_{l}}}$ for $M$-QAM as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}=$
$\displaystyle\frac{4}{\pi}\int_{0}^{\pi/2}{\int_{0}^{\infty}{\alpha\exp\left({-\frac{{b\gamma}}{{{{\sin}^{2}}\theta}}}\right){f_{{\gamma_{l|{S_{l}}}}}}\left(\gamma\right)}d\gamma
d\theta}$
$\displaystyle-\frac{4}{\pi}\int_{0}^{\pi/4}{\int_{0}^{\infty}{\alpha^{2}\exp\left({-\frac{{b\gamma}}{{{{\sin}^{2}}\theta}}}\right){f_{{\gamma_{l|{S_{l}}}}}}\left(\gamma\right)}d\gamma
d\theta},$ (24)
where $\alpha=4\left({1-1/\sqrt{M}}\right)$ and
$b=3/\left(2\left({M-1}\right)\right)$. Calculating the integrals in (III-A1),
${\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}$ is derived as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}$ $\displaystyle=$
$\displaystyle
4\alpha\sum\limits_{k=0}^{N}{\sum\limits_{i=0}^{{m_{{}_{k}}}-1}{{B_{k,i}}}}\Gamma\left({{m_{k}}-i}\right){\left({\frac{{{a_{k}}}}{{{m_{k}}}}}\right)^{{m_{i}}-i}}\left[\frac{1}{2}-\frac{1}{\pi}\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\left(\left({\frac{\pi}{2}+{{\tan}^{-1}}{\omega_{1}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}\right.\right.$
$\displaystyle\times\left.\left.{\left({4\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}+\sin\left({{{\tan}^{-1}}{\omega_{1}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\omega_{1}}}\right)}\right)^{2\left({p-q}\right)+1}}\right)\right]$
$\displaystyle-4\alpha^{2}\sum\limits_{k=0}^{N}{\sum\limits_{i=0}^{{m_{{}_{k}}}-1}{{B_{k,i}}}}\Gamma\left({{m_{k}}-i}\right){\left({\frac{{{a_{k}}}}{{{m_{k}}}}}\right)^{{m_{i}}-i}}\left[\frac{1}{4}-\frac{1}{\pi}\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\left(\left({\frac{\pi}{2}-{{\tan}^{-1}}{\omega_{2}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}\right.\right.$
$\displaystyle\times\left.\left.{\left({4\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}-\sin\left({{{\tan}^{-1}}{\omega_{2}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{a_{k}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\omega_{2}}}\right)}\right)^{2\left({p-q}\right)+1}}\right)\right],$
(25)
where
${\omega_{1}}=\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\cot\frac{\pi}{2}$
and
${\omega_{2}}=\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\cot\frac{\pi}{4}$.
#### III-A2 Exact Results for $G\left(\beta_{ql}\right)$
We now analyze $G\left(\beta_{ql}\right)$ for $M$-PSK and $M$-QAM,
respectively. According to (7), it is equivalent to analyze ${\rm{SER}}_{ql}$.
Using (2), the received SNR at $R_{q}$ is written as
$\displaystyle\gamma_{ql}=\frac{{P_{l}\left|{h_{ql}}\right|^{2}}}{{N_{0}}}=\frac{{P_{l}d_{ql}^{-\alpha}}}{{N_{0}}}\left|h\right|^{2}=c_{ql}\left|h\right|^{2},$
(26)
where $c_{ql}=P_{l}d_{ql}^{-\alpha}/N_{0}$ denotes the equivalent SNR at
$R_{q}$ received from $U_{l}$, and $h$ denotes the unitary Nakagami-$m$ fading
coefficients between $U_{l}$ and $R_{q}$ with variance one.
We first derive ${\rm{SER}}_{ql}$ for $M$-PSK as
$\displaystyle{\rm{SER}}_{ql,{\textrm{MPSK}}}=$
$\displaystyle\frac{{\alpha(M-1)}}{M}-\frac{\alpha}{\pi}\sqrt{\frac{{c_{ql}b}}{{c_{ql}b+m_{lq}}}}\left[\left({\frac{\pi}{2}+\tan^{-1}\varpi}\right)\sum_{p=0}^{m_{lq}-1}{2p\choose
p}\left({4\left({1+\frac{{c_{ql}b}}{m_{lq}}}\right)}\right)^{-p}\right.$
$\displaystyle+\left.\sin\left({\tan^{-1}\varpi}\right)\sum_{p=1}^{m_{lq}-1}{\sum_{t=1}^{p}{T_{p,t}}}\left({1+\frac{{c_{ql}b}}{m_{lq}}}\right)^{-p}\left({\cos\left({\tan^{-1}\varpi}\right)}\right)^{2\left({p-t}\right)+1}\right],$
(27)
where $\varpi=\sqrt{\frac{{c_{ql}b}}{{c_{ql}b+m_{lq}}}}\cot\frac{\pi}{M}$.
We then derive ${\rm{SER}}_{ql}$ for $M$-QAM as
$\displaystyle{\rm{SER}}_{ql,{\textrm{MQAM}}}=$ $\displaystyle
2\alpha-\frac{4\alpha}{\pi}\sqrt{\frac{{{c_{ql}}b}}{{{c_{ql}}b+{m_{k}}}}}\left[\left({\frac{\pi}{2}+{{\tan}^{-1}}{\varpi_{1}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}{\left({4\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}\right.$
$\displaystyle+\left.\sin\left({{{\tan}^{-1}}{\varpi_{1}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\varpi_{1}}}\right)}\right)^{2\left({p-q}\right)+1}}\right]$
$\displaystyle-\alpha^{2}+\frac{{{4\alpha^{2}}}}{\pi}\sqrt{\frac{{{a_{k}}b}}{{{a_{k}}b+{m_{k}}}}}\left[\left({\frac{\pi}{2}-{{\tan}^{-1}}{\varpi_{2}}}\right)\sum\limits_{p=0}^{{m_{k}}-i-1}{2p\choose
p}{\left({4\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)}\right)^{-p}}\right.$
$\displaystyle\left.-\sin\left({{{\tan}^{-1}}{\varpi_{2}}}\right)\sum\limits_{p=1}^{m-i-1}{\sum\limits_{q=1}^{p}{{T_{p,q}}}}{\left({1+\frac{{{c_{ql}}b}}{{{m_{k}}}}}\right)^{-p}}{\left({\cos\left({{{\tan}^{-1}}{\varpi_{2}}}\right)}\right)^{2\left({p-q}\right)+1}}\right],$
(28)
where
${\varpi_{1}}=\sqrt{\frac{{{c_{ql}}b}}{{{a_{k}}b+{m_{lq}}}}}\cot\frac{\pi}{2}$
and
${\varpi_{2}}=\sqrt{\frac{{{c_{ql}}b}}{{{a_{k}}b+{m_{lq}}}}}\cot\frac{\pi}{4}$.
Substituting (III-A2) and (III-A2) into (7), we obtain
${G\left({\beta_{ql}}\right)}$ for $M$-PSK and $M$-QAM, respectively.
Therefore, we insert (7) and (III-A1) into (9), which yields the exact closed-
form SER for $M$-PSK, and substitute (7) and (III-A1) into (9), which gives
the exact SER for $M$-QAM. Observing (III-A1), (III-A1), (III-A2), and
(III-A2), we see that the exact SER expressions for $M$-PSK and $M$-QAM are
given in closed-form and are valid to arbitrary numbers of sources and relays.
### III-B Asymptotic SER
We now provide useful insights into the network behavior in the high SNR
regime. In doing so, new compact closed-form expressions are presented for the
asymptotic SER.
We first focus on $M$-PSK. Based on (10), ${\rm{SER}}_{\gamma_{l|S_{l}}}$ for
$M$-PSK can be alternatively written as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\int_{0}^{\infty}{\exp\left({-\frac{{b\gamma_{l|S_{l}}}}{{\sin^{2}\theta}}}\right)f_{{}_{\gamma_{l|S_{l}}}}\left({\gamma_{l|S_{l}}}\right)}d\gamma_{l|S_{l}}d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\left({\int_{0}^{\infty}{\int_{0}^{\infty}{\cdots\int_{0}^{\infty}{\exp\left({-\frac{{b\sum_{i=0}^{N}{y_{i}}}}{{\sin^{2}\theta}}}\right)\left({\prod\limits_{i=0}^{N}{f_{Y_{i}}\left({y_{i}}\right)}}\right)\prod\limits_{i=0}^{N}{dy_{i}}}}}}\right)d\theta}.$
(29)
Substituting (11) into (III-B) and using [26, eq. (3.351.3)], we obtain
${\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$ as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\prod\limits_{i=0}^{N}{\left({\int_{0}^{\infty}{\frac{{{m_{i}}^{m_{i}}y_{i}^{{m_{i}}-1}}}{{{a_{i}}^{m_{i}}\Gamma\left({m_{i}}\right)}}\exp\left({-\left({\frac{b}{{\sin^{2}\theta}}+\frac{{m_{i}}}{{a_{i}}}}\right)y_{i}}\right)dy_{i}}}\right)}d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}^{m_{i}}}}{{a_{i}^{m_{i}}}}\left({\frac{b}{{\sin^{2}\theta}}+\frac{{m_{i}}}{{a_{i}}}}\right)^{-{m_{i}}}}\right)}d\theta}.$
(30)
We next use [26, eq. (2.513.1)] to develop an asymptotic expression for
${\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}$ as
$\displaystyle{\rm{SER}}_{{\gamma_{l|{S_{l}}}},{\rm{MPSK}}}^{\infty}$
$\displaystyle\leq\frac{\alpha}{\pi}\int_{0}^{\left({M-1}\right)\pi/M}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}^{{m_{i}}}}}{{a_{i}^{{m_{i}}}}}{{\left({\frac{b}{{{{\sin}^{2}}\theta}}}\right)}^{-{m_{i}}}}}\right)}d\theta}$
$\displaystyle=\frac{\alpha}{\pi}\left({\prod\limits_{i=0}^{N}{{{\left({\frac{{{m_{i}}}}{{{a_{i}}b}}}\right)}^{{m_{i}}}}}}\right)\int_{0}^{\left({M-1}\right)\pi/M}{{{\sin}^{2\sum\nolimits_{i=0}^{N}{{m_{i}}}}}\theta
d\theta}$
$\displaystyle=\frac{\alpha}{{\pi{b^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}}{A_{M,N}}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}}}{{{a_{i}}}}}\right)}^{{m_{i}}}}$
(31)
where
$\displaystyle A_{M,N}=$
$\displaystyle\frac{1}{{2^{2\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}{{2\sum\nolimits_{i=0}^{N}{{m_{i}}}}\choose{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}\frac{\left({M-1}\right)\pi}{M}+\frac{{\left({-1}\right)^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}{{2^{2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-1}}}$
$\displaystyle\times\sum_{k=0}^{{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-1}{\left({-1}\right)^{k}{2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}\choose
k}\frac{{\sin\left({\left({2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-2k}\right)\frac{\left({M-1}\right)\pi}{M}}\right)}}{{2{\sum\nolimits_{i=0}^{N}{{m_{i}}}}-2k}}}.$
(32)
Similarly, an asymptotic expression for ${\rm{SER}}_{ql,{\textrm{MPSK}}}$ is
obtained as
${\rm{SER}}_{ql,{\textrm{MPSK}}}^{\infty}\leq\frac{\alpha}{\pi}\left({\frac{{{m_{lq}}}}{{b}}}\right)^{{m_{lq}}}\frac{A_{M,{m_{lq}},0}}{{c_{ql}^{m_{lq}}}}.$
(33)
Correspondingly, the asymptotic $G\left({\beta_{ql}}\right)$ for $M$-PSK is
given as
$G\left({\beta_{ql}}\right)_{\textrm{MPSK}}^{\infty}=\left\\{{\begin{array}[]{ll}{1,}&{{\rm{if}}~{}{\beta_{ql}=1}}\\\
{{\rm{SER}}_{ql,{\textrm{MPSK}}}^{\infty},}&{{\rm{otherwise.}}}\\\
\end{array}}\right.$ (34)
Based on (III-B), (33), and (34), the asymptotic SER for $M$-PSK is derived as
${\rm{SER}}_{l,{\textrm{MPSK}}}^{\infty}=\sum_{S_{l}=0}^{2^{Q}-1}{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MPSK}}}^{\infty}\prod_{q=1}^{Q}G\left({\beta_{ql}}\right)_{\textrm{MPSK}}^{\infty}.$
(35)
Following the same procedure outlined for $M$-PSK, we derive the asymptotic
SER for $M$-QAM as
${\rm{SER}}_{l,{\textrm{MQAM}}}^{\infty}=\sum_{S_{l}=0}^{2^{Q}-1}{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}^{\infty}\prod_{q=1}^{Q}G\left({\beta_{ql}}\right)_{\textrm{MQAM}}^{\infty},$
(36)
where the asymptotic ${\rm{SER}}_{\gamma_{l|S_{l}},}$ for $M$-QAM is derived
as
$\displaystyle{\rm{SER}}_{\gamma_{l|S_{l}},{\textrm{MQAM}}}^{\infty}\leq\frac{4\alpha}{{\pi{b^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}}{A_{2,N}}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}}}{{{a_{i}}}}}\right)}^{{m_{i}}}}-\frac{{4{\alpha^{2}}}}{{\pi{b^{\sum\nolimits_{i=0}^{N}{{m_{i}}}}}}}{A_{4/3,N}}{\prod\limits_{i=0}^{N}{\left({\frac{{{m_{i}}}}{{{a_{i}}}}}\right)}^{{m_{i}}}},$
(37)
with ${A_{2,{m_{i}},N}}$ and ${A_{4/3,{m_{i}},N}}$ being defined in (III-B),
and the asymptotic $G\left({\beta_{ql}}\right)$ for $M$-QAM is derived as
$G\left({\beta_{ql}}\right)_{\textrm{MQAM}}^{\infty}=\left\\{{\begin{array}[]{ll}{1,}&{{\rm{if}}~{}{\beta_{ql}=1}}\\\
{{\rm{SER}}_{ql,{\textrm{MQAM}}}^{\infty},}&{{\rm{otherwise,}}}\\\
\end{array}}\right.$ (38)
with
${\rm{SER}}_{ql,{\textrm{MQAM}}}^{\infty}\leq\frac{{4\alpha}}{\pi}{\left({\frac{{{m_{lq}}}}{b}}\right)^{{m_{lq}}}}\frac{{{A_{2,{m_{lq}},0}}}}{{c_{ql}^{{m_{lq}}}}}-\frac{{4{\alpha^{2}}}}{\pi}{\left({\frac{{{m_{lq}}}}{b}}\right)^{{m_{lq}}}}\frac{{{A_{4/3,{m_{lq}},0}}}}{{c_{ql}^{{m_{lq}}}}}.$
(39)
Based on (35) and (36), we next examine the diversity order of the network,
which represents the slope of the SER against average SNR in a log-log scale.
According to (35) and (36), the asymptotic SER of $U_{l}$ can rewritten as
${\rm{SER}}_{l}^{\infty}=\sum\limits_{{S_{l}}=0}^{{2^{Q}}-1}{{\Theta_{{S_{l}}}}\left[{\prod\limits_{i=0}^{N}{\frac{1}{{a_{i}^{{m_{i}}}}}}}\right]}\prod\limits_{q=1}^{Q}{\frac{1}{{c_{ql}^{{m_{lq}}}}}},$
(40)
where $\Theta_{{S_{l}}}$ denotes the coefficient independent of $a_{i}$ and
$c_{ql}$.in which the diversity order can be confirmed as
${\rm{div}}={m_{ld}}+\sum\nolimits_{q=1}^{Q}{\min\left({{m_{qd}},{m_{lq}}}\right)}.$
(41)
It is evident from (41) that the full diversity order is achieved, which is
determined by the Nakagami-$m$ fading parameters of all the channels. Notably,
the diversity order is independent of the number of sources. In particular,
this full diversity order is preserved even non-orthogonal STNC codes are
employed.
## IV Simulation and Numerical Results
In this section, simulation and numerical results are presented to examine the
impact of network parameters with STNC (e.g., the number of relays, the relay
location, Nakagami-$m$ fading parameters, and power allocation) on the SER of
$U_{l}$. In the figures, we consider a practical scenario where the relays are
placed at different distances from $D$ and $U_{l}$ with $c_{j}\neq c_{i}$ and
$c_{jl}\neq c_{il}$ for $j\neq i$. We set the distance between $U_{l}$ and $D$
as $d_{ld}=1$. The cross correlations between different spread codes, defined
in (1), are set to be zero. We also assume equal transmit power at each node.
Further, our results concentrate on the practical example of a highly shadowed
area with the path loss exponent as $\alpha=3.5$ [29]. In the figures, the
exact SER for $M$-PSK is evaluated by substituting (7), (III-A1), and (III-A2)
into (9), and the exact SER for $M$-QAM is evaluated by substituting (7),
(III-A1), and (III-A2) into (9). The asymptotic SER for $M$-PSK and $M$-QAM is
calculated from (35) and (36), respectively.
### IV-A Impact of Number of Relays and Equal Nakagami-$m$ Fading Parameters
In this subsection, we focus on equal Nakagami-$m$ fading parameters with
$m_{i}=m$. The average received SNRs are set as
$c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$ and
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Fig. 3 plots the exact and asymptotic SER
with 4QAM. Fig. 4 plots the exact and asymptotic SER with 8PSK. From Figs. 3
and 4, we see that the asymptotic SER curves accurately predict the exact ones
in the high SNR regime. By observing these asymptotic curves, it is evident
that the diversity order increases with $Q$, which indicates that increasing
the number of relays brings an improved performance. It is also seen that the
diversity order increases with $m$, which indicates that the improvement in
fading channels leads to a reduction in the SER. Moreover, we see that the
simulation points are in precise agreement with our exact analytical curves,
which demonstrates the correctness of our analysis in Section III. Comparing
the SER in Fig. 3 with that in Fig. 4, we further see a poorer network
performance is achieved by higher order modulation schemes.
### IV-B Impact of Relay Location
In this subsection, we consider ${d_{lq}}\neq{d_{qd}}$, which leads to
${c_{q}}\neq{c_{qd}}$, and consider equal Nakagami-$m$ fading parameters with
$m_{i}=m=2$. We further normalize ${d_{ld}}$ to unity with ${d_{ld}}=1$. Fig.
5 plots the exact SER with BPSK for $Q=2$. In this figure, _Cases 1_ , _2_ ,
_3_ represent the scenario where the relays are located close to the source,
while _Cases 4_ , _5_ , _6_ represent the scenario where the relays are
located close to the destination.
We first consider _Cases 1_ , _2_ , and _3_. We see that _Case 1_ offers a
prominent SNR advantage relative to _Case 2_. This indicates that the
reduction in the distance between the relay and the destination brings a
substantial SER improvement. We also see that _Case 1_ and _Case 3_ achieve
almost the same SER across the entire SNR range. This indicates that the SER
improvement from the reduced distance between the source and the relay is
negligible. These observations are due to the fact that the network
performance is dominant by the relay-destination link when the relays are
close to the source. As such, the quality improvement of the relay-destination
link has a higher positive impact on the SER than that of the source-relay
link.
We next consider _Cases 4_ , _5_ , and _6_. It is seen that _Case 4_ provides
a substantial SNR advantage compared to _Case 5_. It is also seen that _Case
4_ achieves a slight SNR advantage compared to _Case 6_. These observations
are explained by the fact that the network performance is dominant by the
source-relay link when the relays are close to the destination.
### IV-C Impact of Unequal Nakagami-$m$ Fading Parameters
We concentrate on unequal Nakagami-$m$ fading parameters and set the average
received SNRs as $c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$
and $c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Fig. 6 plots the exact SER with 4QAM
for $Q=2$. This figure clearly shows that the diversity order in (41) is
accurate. For example, it is evident that the asymptotic SER curves of _Cases
1_ , _2_ , and _3_ are in parallel, which indicates that they achieve the same
diversity order. As indicated in (41), _Cases 1_ , _2_ , and _3_ achieve
identical diversity order of $3$. Moreover, we see that the diversity order of
_Case 4_ increases to $4$, and the diversity order of _Case 5_ increases to
$6$. This is predicted by (41), which shows that the diversity order is
determined by the Nakagami-$m$ fading parameters of all the channels.
### IV-D Impact of Power Allocation
We now focus on arbitrary transmit power at each node. We consider equal
Nakagami-$m$ fading parameters with $m_{i}=m=2$, set the relay location as
${d_{l1}}=0.8$, ${d_{l2}}=1$, ${d_{1d}}=0.9$, and ${d_{2d}}=0.7$, and
normalize ${d_{ld}}$ as ${d_{ld}}=1$. We denote the transmit powers at
$U_{l}$, $R_{1}$, and $R_{2}$ as $P_{0}$, $P_{1}$, and $P_{2}$, respectively.
Under the total power constraint, we have $P_{0}+P_{1}+P_{2}=3P$. Fig. 7 plots
the exact SER versus $\xi=P_{1}/(P_{1}+P_{2})$ with 4QAM for $Q=2$ and
$P/N_{0}=12~{}{\rm{dB}}$. We see that the optimal value of $P_{1}/P_{2}$
depends on $P_{0}$. For example, the optimal power allocation is at
$P_{1}=1.75P$ and $P_{2}=0.75P$ when $P_{0}=0.5P$. Moreover, the optimal power
allocation is at $P_{1}=P_{2}=0.5P$ when $P_{0}=2P$. Using our SER expressions
with different relay locations, iterative search method can be used to find
the optimal power allocation that minimizes the SER.
### IV-E Impact of Nonorthogonal Codes
We now turn our attention to the impact of nonorthogonal codes. We consider
$\rho_{pq}=\rho\neq 0$ for all $p,q$ and equal Nakagami-$m$ fading parameters
with $m_{i}=2$. Fig. 8 plots the exact SER with 4QAM for $Q=2$ and $N=3$. The
case of $\rho=0$ represents orthogonal codes. We see a reduction in the SER as
$\rho$ increases. We also see that the diversity order is not affected by
cross correlation. As such, the nonorthogonal codes which permit broader
applications can be used for higher throughput without sacrificing the error
rate significantly.
## V Conclusions
In this paper, we analyzed the SER of STNC in a distributed cooperative
network where $L$ sources communicate with a single destination with the
assistance of $Q$ relays. For $M$-PSK and $M$-QAM modulation, new exact
closed-form expressions of SER over independent but not necessarily
identically distributed Nakagami-$m$ fading channels were derived. Moreover,
the asymptotic SER was derived to reveal the network performance in the high
SNR regime. Specifically, the asymptotic SER reveals that the diversity order
of STNC was determined by the Nakagami-$m$ fading parameters of all the
channels. Simulation results were used to validate our analytical expressions
and to examine the impact of Nakagami-$m$ fading parameters, relay location,
power allocation, and nonorthogonal codes on the SER.
## Appendix A Proof of (20)
According to Fig. 9, all poles are located in a closed curve. According to the
residue theorem, the integral over a closed curve can be expressed as the
linear combination of the residues of the poles in the curve. Mathematically,
we have
$\int_{R}^{-R}{g\left(u\right)}du+\int_{C_{R}}{g\left(z\right)dz}=2\pi
j\sum_{k=0}^{N}{\rm Res}\left[{g\left({z_{k}}\right),z_{k}}\right],$ (42)
where $C_{R}$ is the counterclockwise semicircular curve in Fig. 3. The
absolute value of $\int_{C_{R}}{g\left(z\right)dz}$ in (42) can be upper
bounded as
$\displaystyle\left|{\int_{C_{R}}{g\left(z\right)dz}}\right|$
$\displaystyle\leq\int_{C_{R}}{\left|{g\left(z\right)}\right|dz}$
$\displaystyle=\int_{C_{R}}{\left|{\left({\prod\limits_{n=0}^{N}{\left({1-\frac{{jza_{n}}}{m_{n}}}\right)^{-m_{n}}}}\right)e^{-jzv}}\right|dz}$
$\displaystyle\leq\int_{C_{R}}{\left|{\prod\limits_{n=0}^{N}{\frac{1}{{\left({1-ja_{n}z}\right)^{m_{n}}}}}}\right|\left|{e^{-jzv}}\right|dz}$
$\displaystyle=\int_{C_{R}}{\prod\limits_{n=0}^{N}{\left|{\frac{1}{{\left({1-ja_{n}z}\right)^{m_{n}}}}}\right|}dz}.$
(43)
When $R\to\infty$, which indicates that the integral range of
$g\left({u}\right)$ is from $\infty$ to $-\infty$, we find the property of (A)
as
$\displaystyle\left|{\int_{C_{R}}{g\left(z\right)dz}}\right|$
$\displaystyle=\int_{C_{R}}{\prod\limits_{n=0}^{N}{\frac{1}{{{a_{n}}^{m_{n}}\left|z\right|^{m_{n}}+O\left({z^{m_{n}}}\right)}}}dz}$
$\displaystyle=\left({\prod\limits_{n=0}^{N}{\frac{1}{{{a_{n}}^{m_{n}}}}}}\right)\int_{C_{R}}{\frac{1}{{\left|z\right|^{\sum\nolimits_{n=0}^{N}{{m_{n}}}}}}dz}$
$\displaystyle=\left({\prod\limits_{n=0}^{N}{\frac{1}{{{a_{n}}^{m_{n}}}}}}\right)\frac{\pi}{{R^{\sum\nolimits_{n=0}^{N}{{m_{n}}}-1}}}$
$\displaystyle\to 0.$ (44)
Substituting (A) into (42), we obtain (20) and thus complete this proof.
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Figure 1: System model. Figure 2: A framework of space-time network coding.
Figure 3: Exact and asymptotic SER with 4QAM for $Q=1,2,3$,
$c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$, and
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Figure 4: Exact and asymptotic SER with
8PSK for $Q=1,2,3$,
$c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$, and
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$. Figure 5: Exact SER with BPSK for $Q=2$,
$m=2$ and 6 cases: _Case 1_ : $d_{l1}=0.2,d_{l2}=0.3,d_{1d}=1.1,d_{2d}=1.2$;
_Case 2_ : $d_{l1}=0.2,d_{l2}=0.3,d_{1d}=2,d_{2d}=2.2$; _Case 3_ :
$d_{l1}=0.8,d_{l2}=0.9,d_{1d}=1.1,d_{2d}=1.2$; _Case 4_ :
$d_{l1}=1.1,d_{l2}=1.2,d_{1d}=0.2,d_{2d}=0.3$; _Case 5_ :
$d_{l1}=2,d_{l2}=2.2,d_{1d}=0.2,d_{2d}=0.3$; and _Case 6_ :
$d_{l1}=1.1,d_{l2}=1.2,d_{1d}=0.8,d_{2d}=0.9$. Figure 6: Exact SER with 4QAM
for $Q=2$, $c_{\left({i+1}\right)l}=c_{i+1}=c_{i}+0.1\gamma_{\Delta}$,
$c_{0}=c_{0l}=0.6\gamma_{\Delta}$ and 6 cases: _Case 1_ : $m_{i}=m=1$; _Case
2_ : $m_{l1}=1,m_{l2}=1,m_{ld}=1,m_{1d}=2,m_{2d}=2$; _Case 3_ :
$m_{l1}=2,m_{l2}=2,m_{ld}=1,m_{1d}=1,m_{2d}=1$; _Case 4_ :
$m_{l1}=1,m_{l2}=2,m_{ld}=1,m_{1d}=1,m_{2d}=2$; _Case 5_ : $m_{i}=m=2$. Figure
7: Exact SER with 4QAM for $Q=2$, $m=2$ with ${d_{l1}}=0.8$, ${d_{l2}}=1$,
${d_{1d}}=0.9$, ${d_{2d}}=0.7$, and different power allocation $P_{1}$,
$P_{2}$, $P_{3}$. Figure 8: Exact SER with 4QAM for $Q=2$, $N=3$, $m_{i}=2$
and different cross correlation $\rho$ with ${d_{l1}}=0.6$, ${d_{l2}}=0.8$,
${d_{1d}}=0.6$ and ${d_{2d}}=0.8$, where ${d_{ld}}$ is normalized as
${d_{ld}}=1$. Figure 9: The distribution of the poles.
|
arxiv-papers
| 2012-07-04T10:23:03 |
2024-09-04T02:49:32.633516
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ang Yang, Zesong Fei, Nan Yang, Chengwen Xing, and Jingming Kuang",
"submitter": "Ang Yang",
"url": "https://arxiv.org/abs/1207.0938"
}
|
1207.1057
|
# Coarsening rates for the dynamics of slipping droplets
Georgy Kitavtsev111Max Planck Institute for Mathematics in the Sciences,
Inselstr. 22, D–04103 Leipzig, Germany. E-mail: Georgy.Kitavtsev@mis.mpg.de
###### Abstract
We derive reduced finite dimensional ODE models starting from one dimensional
lubrication equations describing coarsening dynamics of droplets in nanometric
polymer film interacting on a hydrophobically coated solid substrate in the
presence of large slippage at the liquid/solid interface. In the limiting case
of infinite slip length corresponding in applications to free films a
collision/absorption model then arises and is solved explicitly. The exact
coarsening law is derived for it analytically and confirmed numerically.
Existence of a threshold for the decay of initial distributions of droplet
distances at infinity at which the coarsening rates switch from algebraic to
exponential ones is shown.
## 1 Introduction.
Dewetting processes of a liquid polymer film of nanometer thickness
interacting on a hydrophobically coated solid substrate attracted an intensive
research during last several decades, see. e.g. a review in [1]. In general,
such processes can be divided into three stages. During the first stage a
liquid polymer film is susceptible to instability due to small perturbations
of the film profile. Typically such films rupture, thereby initiating a
complex dewetting process, see e.g. [2, 3, 4]. The influence of intermolecular
forces play an important part in the rupture and subsequent dewetting process,
see e.g. [5, 6] and references therein. Typically the competition between the
long-range attractive van der Waals and short-range Born repulsive
intermolecular forces reduces the unstable film to _an ultra-thin layer_ that
connects the evolving patterns and is given by the minimum of the
corresponding intermolecular potential, i.e. the film settles into an
energetically more favorable state, see [7, 8]. The second stage is associated
with the formation of regions of this minimal thickness, bounded by moving
rims that connect to the undisturbed film, see e.g. [10, 11, 9].
In this study we are interested in the third and the last stage of the
dewetting process, namely the long-time coarsening process that originates in
the breaking up of the evolving patterns into small droplets and is
characterized by its subsequent slow-time coarsening dynamics, which has been
observed and investigated experimentally by Limary and Green [12, 13]. They
show that during the coarsening the average size of droplets increases and the
number of droplets decreases. The coarsening mechanisms that were observed in
such films are typically subsequent collapses of smaller droplets and
collisions of neighboring ones. During collapse the size of a droplet shrinks
in time and its mass is distributed in the ultra-thin layer. In turn,
collisions among droplets occur due to the mass transfer through the ultra-
thin layer between them that causes a translation movement of them, _droplet
migration_ , eventually leading to the formation of new droplets. A numerical
example of the coarsening dynamics in two-dimensional films is shown in Fig.
2.
Figure 1: Plots of intermolecular pressure $\Pi_{\varepsilon}(h)$ (blue) and
potential function $U_{\varepsilon}(h)$ (green) for $\varepsilon=0.1$ Figure
2: Numerical solution to (1.2a)–(1.2b) with
$\varepsilon=0.1,\,\beta=2.5,\,\mathrm{Re}=1$ showing an example of a
coarsening process (collapse of the 4th small droplet and collision of 2nd and
3rd ones) in the array of five quasiequilibrium droplets.
Besides intermolecular forces and surface tension at the free surface of the
film the dewetting of polymer films on hydrophobic substrates also involves
such boundary effect as slippage on a solid substrate [14]. Recently in Münch
et al. [15] closed-form one-dimensional lubrication equations over a wide
range of slip lengths were derived from the underlying equations for
conservation of mass and momentum, together with boundary conditions for the
tangential and normal stresses, as well as the kinematic condition at the free
boundary, impermeability and Navier-slip condition at the liquid-solid
interface. Asymptotic arguments, based on the magnitude of the slip length
show that within a lubrication scaling there are two _distinguished regimes_ ,
see [15].
These are the well-known _weak-slip_ model
$\partial_{t}h=-\partial_{x}\Big{(}M(h)\partial_{x}\left(\sigma\partial_{xx}h-\Pi_{\varepsilon}(h)\right)\Big{)}$
(1.1)
with $M(h):=h^{3}+b\,h^{2}$ and $b$ denoting the slip length parameter; and
the _strong-slip_ model
$\displaystyle\mathrm{Re}\,(\varepsilon\partial_{t}(hu)+\partial_{x}(hu^{2}))$
$\displaystyle=\nu\partial_{x}(h\partial_{x}u)+h\partial_{x}(\sigma\partial_{xx}h-\Pi_{\varepsilon}(h))-\frac{u}{\beta}$
(1.2a) $\displaystyle\varepsilon\partial_{t}h$
$\displaystyle=-\,\partial_{x}\left(hu\right),$ (1.2b)
respectively. Here, $u(x,t)$ and $h(x,t)$ denote the average velocity in the
lateral direction and the height profile for the free surface, respectively.
The positive slip length parameters $b$ and $\beta$ are related by orders of
magnitude via $b\sim\eta^{2}\beta$, where the (small) parameter $\eta$,
$0<\eta\ll 1$, refers to the vertical to horizontal scale separation of the
thin film.
The high order of the lubrication equations (1.1) and (1.2a)–(1.2b) is a
result of the contribution from surface tension at the free boundary,
reflected by the linearized curvature term $\sigma\partial_{xx}h$ with
parameter $\sigma\geq 0$. A further contribution to the pressure is denoted by
$\Pi_{\varepsilon}(h)$ and represents that of the intermolecular forces,
namely long-range attractive van der Waals and short-range Born repulsive
intermolecular forces. A commonly used expression for it [8, 18] is given by
$\Pi_{\varepsilon}(h)=\frac{\varepsilon^{2}}{h^{3}}-\frac{\varepsilon^{2}}{h^{4}}\
\ \text{with}\ \ 0\leq\varepsilon\ll 1.$ (1.3)
It can be written as a derivative of the potential function
$U_{\varepsilon}(h)=\mathcal{U}(h/\varepsilon)$ (see Fig. 1) where
$\mathcal{U}(H)=-\frac{1}{2\,H^{2}}+\frac{1}{3\,H^{3}},$ (1.4)
The parameter $0<\varepsilon\ll 1$ is the global minimum of
$U_{\varepsilon}(h)$ and gives to the leading order thickness of the ultra-
thin layer. Below we often use notation for the pressure and flux functions
function
$p(h):=\sigma\partial_{xx}h-\Pi_{\varepsilon}(h),\quad j(h)=hu$ (1.5)
$\mathrm{Re}\,(\partial_{t}(hu)+\partial_{x}(hu^{2}))$ and
$\nu\partial_{x}(h\partial_{x}u)$ in (1.2a)–(1.2b), with
$\mathrm{Re},\,\nu\geq 0$ denoting the Reynolds number and viscosity
parameter, represent inertial and Trouton viscosity terms, respectively.
Additionally, the weak-slip and the strong-slip models contain as limiting
cases three further lubrication models. One of them is the _no-slip model_ ,
which is obtained setting $b=0$ in the weak-slip model:
$\partial_{t}h=-\partial_{x}\Big{(}h^{3}\partial_{x}\left(\partial_{xx}h-\Pi_{\varepsilon}(h)\right)\Big{)}\,.$
(1.6)
The second one is obtained from the strong-slip model in the limit
$\beta\rightarrow\infty$ and describes the dynamics of suspended or falling
free films:
$\displaystyle\mathrm{Re}\,(\varepsilon\partial_{t}(hu)+\partial_{x}(hu^{2}))$
$\displaystyle=\nu\partial_{x}(h\partial_{x}u)+h\partial_{x}(\sigma\partial_{xx}h-\Pi_{\varepsilon}(h))$
(1.7a) $\displaystyle\varepsilon\partial_{t}h$
$\displaystyle=-\,\partial_{x}\left(hu\right),$ (1.7b)
For the third limiting case the slip-length parameter $\beta_{I}$ is of order
of magnitude lying in between those that lead to the weak and the strong-slip
model, i.e. $b\ll\beta_{I}\ll\beta$. The corresponding _intermediate-slip_
model is given by
$\partial_{t}h=-\partial_{x}\Big{(}h^{2}\partial_{x}\left(\partial_{xx}h-\Pi_{\varepsilon}(h)\right)\Big{)}\,.$
(1.8)
It can be obtained by rescaling time in (1.1) by $b$ and letting
$b\rightarrow\infty$ or by rescaling time and the horizontal velocity by
$\beta$ in (1.2a)–(1.2b) and taking the limit $\beta\rightarrow 0$. Existence
of weak solutions to (1.2a)–(1.2b) and (1.7a)–(1.7b) and rigorous convergence
of the former ones to the classical solutions of (1.8) as $\beta\rightarrow 0$
was shown recently in [17].
As in [17] we consider systems (1.2a)–(1.2b) and (1.7a)–(1.7b) on a bounded
interval $(0,L)$ with the boundary conditions
$u=0,\quad\mbox{and}\quad\partial_{x}h=0\quad\mbox{at}\quad x=0,\,L,$ (1.9)
whereas equations (1.1),(1.6) and (1.8) with
$\partial_{xxx}h=0,\quad\mbox{and}\quad\partial_{x}h=0\quad\mbox{at}\quad
x=\pm L.$ (1.10)
Both (1.9) and (1.10) incorporate zero flux at the boundary and as a
consequence imply the conservation of mass law
$\frac{1}{L}\int_{0}^{L}h(x,t)\,dx=\mathrm{const},\ \forall t>0.$
Within the context of thin liquid films one of the first studies of the
coarsening dynamics can be found in Glasner and Witelski [18, 19]. The authors
considered the one-dimensional no-slip lubrication model (1.6) with (1.10).
They confirmed numerically existence of the two coarsening driven mechanisms,
namely collision and collapse. One of the typical problems considered in [18,
19] was the calculation of the coarsening rates, i.e. how fast the number of
droplets decreases due to coarsening in time. Often in order to identify the
characteristic dependence for coarsening rates one needs to model very large
arrays of droplets (around $10^{4}$). But due to the presence of the
ultrathin-layer of order $\varepsilon$ between droplets the problem of
numerical solution for any lubrication equation becomes very stiff in time and
demands high space resolution as the number of droplets increases. Therefore,
there exists a need for further reduction of lubrication models to more
simple, possibly finite-dimensional ones.
Basing on the observation that solutions of lubrication equations describing
coarsening dynamics stay in time very close to a perturbed finite combination
of quasistationary droplets and can be therefore parameterized by a finite
number of parameters, namely positions and pressures of drops, in [18, 19] for
the first time a reduced ODE model describing evolution of the latter ones on
the slow time scale was derived from the lubrication equation (1.6). Using
this reduced model the authors derived also the corresponding coarsening law
in the form
$n(t)\sim t^{-2/5},$ (1.11)
where $n(t)$ denotes the number of droplets remaining at time $t$. Later,
analogous reduced ODE models from lubrication equation (1.1) with a general
mobility $M(h)=h^{q}$, $q>0$ in one and two dimensional case were derived and
analyzed in [21, 20]. An step to a rigorous justification of these models
basing on a center manifold approach was made recently in [22]. For the case
$M(h)=h$ the coarsening law (1.11) was justified rigorously in [23] using the
gradient flow structure of the corresponding lubrication equation. The work of
[21, 20] concerns migration of droplet. There it was shown that the direction
of the migration of droplets governed by (1.1) with a general mobility
$M(h)=h^{q}$, $q>0$ is opposite to the mass flux applied to them. Moreover,
for $g\leq 2$ the driving coarsening mechanism is collapse of droplets that is
due to the mass diffusion in the ultra-thin layer between droplets and similar
to Ostwald ripening in binary alloys, see [24, 25, 26]. Note, that also in the
no-slip case $q=3$, i.e. one described by (1.6), as was shown in [19] the
coarsening rates even for the systems coarsening solely due collisions obey
the law (1.11).
Recently, in Kitavtsev and Wagner [27] it was shown that the coarsening
dynamics of quasistationary droplets governed by (1.2a)–(1.2b) with
sufficiently small $\mathrm{Re}$ number is driven also by collapse and
collision. There reduced ODE models analogous to that one of [18, 19] were
derived for system (1.2a)–(1.2b) and its limiting case (1.7a)–(1.7b) as well.
In contrast, to the case of (1.1) it was found there that the coefficients of
the strong-slip reduced ODE model depend explicitly on the slip length
$\beta$. In particular, there exists a critical length
$\beta_{cr}=O(\varepsilon)$ such that the migration of droplets proceeds in
the direction of the applied mass flux for $\beta>\beta_{cr}$ and opposite to
it for $\beta<\beta_{cr}$. Moreover, it was shown that for moderate and large
$\beta$ the driving coarsening mechanism switches from collapse to collision
of droplets. Basing, on these observations it was conjectured and shown
numerically in [27] that the coarsening rates for systems (1.2a)–(1.2b) and
(1.7a)–(1.7b) can be remarkably different from ones for (1.11).
In this study we continue the research initiated in [27]. Our aim here is to
derive explicit coarsening laws for the dynamics of droplets in the strong-
slip and free film regimes, i.e. governed by lubrication system (1.2a)–(1.2b)
and its limiting case (1.7a)–(1.7b). The missing point in [27] was a
derivation of flux representation between interacting droplets for moderate
and large slip lengths $\beta$ which was important for closure of the derived
there reduced ODE models. Therefore, inspired with the matched asymptotics
technique applied in Glasner [20] to the lubrication equation (1.1), we
present in section 2 a new closed form derivation of reduced ODE models for
(1.2a)–(1.2b) and (1.7a)–(1.7b) that incorporate the explicit flux
representation for all $0<\beta\leq\infty$.
In section 3 we concentrate on the reduced ODE model corresponding to
(1.7a)–(1.7b), i.e. on the regime of free films characterized by the infinite
slip length $\beta=\infty$. In this case migration and subsequent collisions
of droplets dominate completely collapse component of the coarsening dynamics.
Therefore, we look only at the migration subsystem of the derived reduced ODE
model such that droplet pressures are kept constant during evolution of
droplets and updated only after each subsequent collision event. We observe
further that for a special initial data this migration subsystem can be solved
explicitly while its solution represents subsequent collisions of $N-1$
droplets with the largest last one. Therefore, we call it as an exactly
solvable collision/absorption model. It turns out that the coarsening law for
this model depends only on the initial distribution of the distances between
droplets and can be derived analytically. Finally, we derive the continuous
counterpart of the coarsening law proceeding to the limit
$N\rightarrow\infty$.
In section 4 we consider several examples of initial distributions of
distances between droplets and show that the corresponding coarsening rates
depend only on the distribution decay at infinity. Moreover, for an explicit
family of distributions decaying as $1/x^{1+\alpha}$ with $\alpha>0$ we show
existence of a threshold at $\alpha=1$ at which the coarsening rates switch
from algebraic to exponential ones.
In section 5.1 we justify the derived hierarchy of the reduced models by
numerical comparison of their solutions to ones of the initial PDE system
(1.2a)–(1.2b) and its limiting cases (1.8) and (1.2a)–(1.2b). We observe that
the deviation between them stays $O(\varepsilon)$ uniformly in time. Besides
we compare solutions of the collision/absorption model from section 3 with
those of the full reduced ODE system for the case $\beta=\infty$. Finally, in
section 5.2 we check numerically the derived coarsening law for the
collision/absorption model in the case of finite $N$ and its continuous
counterpart.
## 2 Derivation of reduced ODE models.
We consider a solution to (1.2a)–(1.2b) which stays close in time to a union
of $N+1$ droplets, which precise characterization to be described below.
Similar to the derivation of reduced coarsening models for the classical thin
film equation in [20] we distinguish three regions in our matched asymptotic
analysis.
* •
Droplet core (DC) region: This region corresponds to droplets and is composed
of the union of disjoint intervals $(X_{i}(t)-R_{i}(t),X_{i}(t)+R_{i}(t))$, so
that $X_{i}(t)$ and $R_{i}(t)$ are the center and the radius of the $i$-th
droplet, $i=0,...,N$. The dynamical points $X_{i}(t)\pm R_{i}(t)$ are called
contact line points and are defined through the relation
$h(X_{i}\pm R_{i})=\varepsilon H^{*},$ (2.1)
where $H^{*}$ is the global maximum of $U^{\prime}(H)$ with function $U(H)$
defined in (1.4). We expand
$R_{i}=R_{i,0}+\varepsilon R_{i,1}+...,X_{i}=X_{i,0}+\varepsilon X_{i,1}+...$
(2.2)
and denote
$\dot{R}=\frac{dR}{dt},\ \ \dot{X}=\frac{dX}{dt}.$
* •
Contact line (CL) region is a microscopic internal layer around the contact
line points where $h$ and $x$ scale like $\varepsilon$. Here we employ the
moving rescaled spatial coordinate
$z=\frac{R(t)-|x-X(t)|}{\varepsilon},\ \
\partial_{x}=\frac{1}{\varepsilon}\partial_{z},\ \
\partial_{t}=\partial_{\tau}-\frac{1}{\varepsilon}\partial_{z}(\dot{R}_{0}\mp\dot{X}_{0}),$
(2.3)
where in this section the sign $\mp$ corresponds to two CL regions around the
points $X\mp R$, respectively. Accordingly, by definition (2.1) we have
$h(z=0)=\varepsilon H^{*}$.
* •
Precursor layer (PL) region is the complement
$(-L,L)\setminus\cup_{i}(X_{i}-R_{i},X_{i}+R_{i})$. In this region $h$ scales
like $\varepsilon$.
The main goal is to determine the evolution of $R_{i}(t)$ and $X_{i}(t)$. To
do so as in [20] we propose self-consistent asymptotic expansions in each of
three regions and connect them via matching conditions. Corrections to the
leading order base solutions solve linear equations, and Fredholm-type
solvability conditions will yield information about the dynamics.
Let us first consider the motion of the $i$-th droplet with $i\in 1,...,N-1$.
For a time being we skip below the subscript $i$. Let us start with the CL
region. Here the solution to (1.2a)–(1.2b) is expanded as
$h=\varepsilon H_{1}+\varepsilon^{2}H_{2}+...,\ \ u=\varepsilon
U_{1}+\varepsilon^{2}U_{2}+...$
We will also use the induced expansions
$P=P_{0}+\varepsilon P_{1}+...,\ \
J=\varepsilon^{2}J_{2}+\varepsilon^{3}J_{3}+...$
for the pressure and flux functions defined in (1.5). The corresponding
leading order system in $\varepsilon$ in this region is given by
$\displaystyle\partial_{z}(\sigma\partial_{zz}H_{1}-U^{\prime}(H_{1}))=0,$
$\displaystyle\partial_{z}H_{1}(\dot{R}_{0}\mp\dot{X}_{0})=\mp\partial_{z}(H_{1}U_{1}).$
Integrating the last system and using matching conditions to the DC and PL
regions
$\displaystyle\partial_{z}H_{1}\rightarrow 0,\ \partial_{zz}H_{1}\rightarrow
0,H_{1}\rightarrow 1\ \text{as}\ \ z\rightarrow-\infty$
$\displaystyle\partial_{zz}H_{1}\rightarrow 0,\ H_{1}\rightarrow+\infty\
\text{as}\ \ z\rightarrow+\infty$ (2.4)
one obtains
$\displaystyle\frac{\sigma}{2}(\partial_{z}H_{1})^{2}$ $\displaystyle=$
$\displaystyle U(H_{1})-U(1),$ (2.5a) $\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle-(1-\frac{1}{H_{1}})(\dot{R}_{0}\mp\dot{X}_{0})\mp\frac{J_{2}(-\infty)}{H_{1}}$
(2.5b)
In particular,
$\displaystyle\lim_{z\rightarrow+\infty}U_{1}=-(\dot{R}_{0}\mp\dot{X}_{0}),\ \
\lim_{z\rightarrow-\infty}U_{1}=\mp J_{2}(-\infty).$ (2.6)
Next, in the DC region we expand the solution as
$h=h_{0}+\varepsilon h_{1}+\varepsilon^{2}h_{2}+...,\ \ u=u_{0}+\varepsilon
u_{1}+\varepsilon^{2}u_{2}+...$
and correspondingly pressure as
$p=p_{0}+\varepsilon p_{1}+...$
In turn, the leading order system in this region is given by
$\displaystyle\sigma
h_{0}\partial_{x}(\partial_{xx}h_{0})-\frac{u_{0}}{\beta}=0,$
$\displaystyle-\partial_{x}(h_{0}u_{0})=0$
Integrating the system and using the matching condition
$h_{0}(X\mp R)=0$ (2.7)
one obtains its solution in the form
$h_{0}=\frac{1}{R\sqrt{12\sigma}}(R^{2}-(x-X(t))^{2}),\ \ u_{0}\equiv 0.$
(2.8)
Correspondingly, the leading order pressure is given by
$p_{0}\equiv\frac{1}{R\sqrt{3\sigma}}.$ (2.9)
In the PL region we expand the solution as
$h=\varepsilon h_{1}+\varepsilon^{2}h_{2}+...,\ \ u=\varepsilon
u_{1}+\varepsilon^{2}u_{2}+...$
and correspondingly pressure and flux as
$p=p_{0}+\varepsilon p_{1}+...,\ \
j=\varepsilon^{2}j_{2}+\varepsilon^{3}j_{3}+...$
The leading order system in this region is given by
$\displaystyle h_{1}\partial_{x}(U^{\prime}(h_{1}))=0,$
$\displaystyle\partial_{t}h_{1}=-\partial_{x}(h_{1}u_{1})$
Integrating the system and using the matching condition $h_{1}(X\mp R)=1$ one
obtains
$h_{1}\equiv 1,\ \ u_{1}=j_{2}\equiv\mathrm{const}$ (2.10)
For the next order corrections $h_{2},u_{2}$ in PL region one has the system
$\displaystyle
h_{1}\partial_{x}(U^{\prime\prime}(h_{1})h_{2})=-\frac{u_{1}}{\beta},$
$\displaystyle\partial_{t}h_{2}=-\partial_{x}(h_{1}u_{2}+h_{2}u_{1})$
From the first equation and (2.10) one obtains
$\partial_{xx}h_{2}=\partial_{xx}p_{0}=0\ \ \text{and}\ \
j_{2}=-\beta\partial_{x}p_{0}.$ (2.11)
Proceeding further in the expansion in the CL region for the second order
corrections $H_{2},U_{2}$ one obtains the system
$\displaystyle-\nu\partial_{z}(H_{1}\partial_{z}U_{1})$ $\displaystyle=$
$\displaystyle H_{1}(\sigma\partial_{zz}H_{2}-U^{\prime\prime}(H_{1})H_{2}),$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{z}H_{2}(\dot{R}_{0}-\dot{X}_{0})-\partial_{z}(H_{1}U_{2}+H_{2}U_{1})$
(2.12)
Let us introduce a linear operator
$\boldsymbol{\mathcal{L}}\left[\begin{array}[]{ccc}H\\\
U\end{array}\right]=\left[\begin{array}[]{ccc}H_{1}\partial_{z}(\sigma\partial_{zz}H_{2}-U^{\prime\prime}(H_{1})H_{2}))\\\
\partial_{z}(H_{2}(\dot{R}_{0}-\dot{X}_{0}-U_{1})-H_{1}U_{2})\end{array}\right]\,.$
Formal adjoint operator to $\boldsymbol{\mathcal{L}}$ is given by
$\boldsymbol{\mathcal{L}}^{*}\left[\begin{array}[]{ccc}g\\\
v\end{array}\right]=\left[\begin{array}[]{ccc}-\sigma\partial_{zzz}(H_{1}g)+U^{\prime\prime}(H_{1})\partial_{z}(H_{1}g)-\partial_{z}V(\dot{R}_{0}-\dot{X}_{0}-U_{1})\\\
\partial_{z}vH_{1}\end{array}\right].$
The kernel of it contains two linear independent functions
$\left[\begin{array}[]{ccc}g_{1}\\\
v_{1}\end{array}\right]:=\left[\begin{array}[]{ccc}1\\\ 0\end{array}\right],\
\ \left[\begin{array}[]{ccc}g_{2}\\\
v_{2}\end{array}\right]:=\left[\begin{array}[]{ccc}1/H_{1}\\\
0\end{array}\right].$ (2.13)
To derive necessary Fredholm-type solvability conditions for the system (2.12)
we multiply the first equation in (2.12) by $g_{2}$ and integrate it on
$(-\infty,+\infty)$ to obtain
$P_{0}(+\infty)-P_{0}(-\infty)=\int_{-\infty}^{+\infty}\frac{\nu}{H_{1}}\partial_{z}(H_{1}\partial_{z}U_{1}),$
where we have used that in the CL region
$P_{0}=U^{\prime\prime}(H_{1})H_{2}-\sigma\partial_{zz}H_{2}.$ (2.14)
Substituting in the previous expression (2.5b) one obtains
$P_{0}(+\infty)-P_{0}(-\infty)=-\nu I(\mp
J_{2}(-\infty)+\dot{R}_{0}\mp\dot{X}_{0}),$ (2.15)
where a constant integral $I$ is given by
$I=\int_{-\infty}^{+\infty}\frac{1}{H_{1}}\partial_{z}\left(\frac{\partial_{z}H_{1}}{H_{1}}\right)\,dz=\frac{1}{35(3+\sqrt{3})}.$
(2.16)
and can be effectively calculated from (2.5a) (see Appendix A). Formula (2.15)
is an analog of Gibbs-Thomson boundary condition and shows that the pressure
experiences a jump at the CL region. Note, that this is a first considerable
difference between the coarsening dynamics driven by (1.2a)–(1.2b) and (1.1).
In contrast to (2.15) as was shown in [18] the pressure is constant through
the CL region in the case of (1.1).
Next, multiplying the first equation in (2.12) by $g_{3}$ and integrating it
on $(-\infty,+\infty)$ one obtains
$0=\nu
H_{1}\partial_{z}U_{1}\Big{|}_{-\infty}^{+\infty}+\int_{-\infty}^{+\infty}H_{1}\partial_{z}(\sigma\partial_{zz}H_{2}-U^{\prime\prime}(H_{1})H_{2}).$
Integrating further three times by parts and using (2.5b), (2.14) one arrives
at
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\nu\frac{\partial_{z}H_{1}}{H_{1}}(\mp
J_{2}(-\infty)+\dot{R}_{0}\mp\dot{X}_{0})\Big{|}_{-\infty}^{+\infty}-H_{1}P_{0}\Big{|}_{-\infty}^{+\infty}$
$\displaystyle-$
$\displaystyle\sigma\partial_{z}H_{1}\partial_{z}H_{2}\Big{|}_{-\infty}^{+\infty}+\sigma\partial_{zz}H_{1}H_{2}\Big{|}_{-\infty}^{+\infty},$
Using the matching condition (2.4) and additionally
$\displaystyle\partial_{z}H_{2}\rightarrow\mathrm{const}\ \text{as}\ \
z\rightarrow-\infty,$
$\displaystyle\partial_{z}H_{1}\rightarrow\partial_{x}h_{0},\
\partial_{z}H_{2}\sim\partial_{x}h_{1}+\partial_{xx}h_{0}z,\ H_{1}\sim
h_{1}+\partial_{x}h_{0}z\ \text{as}\ \ z\rightarrow+\infty$ (2.17)
one arrives at
$(H_{1}P_{0})\Big{|}_{-\infty}^{+\infty}=\sigma\partial_{z}H_{1}(+\infty)\partial_{z}H_{2}(+\infty).$
The last expression again using (2.4) and (2.17) implies
$\displaystyle\sigma\partial_{xx}h_{0}$ $\displaystyle=$
$\displaystyle-P(+\infty),$
$\displaystyle\sigma(\partial_{x}h_{0}\partial_{x}h_{1})\Big{|}_{X\mp R}$
$\displaystyle=$ $\displaystyle P(-\infty)-P(+\infty)h_{1}(X\mp R).$ (2.18)
Note, that the first relation in (2.18) is consistent with already derived
(2.8)–(2.9), whereas the second one is new.
Finally, let us consider the system for the first order corrections
$h_{1},u_{1}$ in the DC region which has the form
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nu\partial_{x}(h_{0}\partial_{x}u_{1})+\sigma
h_{0}\partial_{xxx}h_{1}-u_{1}/\beta,$ (2.19a) $\displaystyle\frac{\partial
h_{0}}{\partial R}\dot{R}_{0}-\frac{\partial h_{0}}{\partial x}\dot{X}_{0}$
$\displaystyle=$ $\displaystyle-\partial_{x}(h_{0}u_{1})$ (2.19b)
Let us introduce a linear operator
$\boldsymbol{\mathcal{L}}\left[\begin{array}[]{ccc}h\\\
u\end{array}\right]=\left[\begin{array}[]{ccc}\nu\partial_{x}(h_{0}\partial_{x}u_{1})+\sigma
h_{0}\partial_{xxx}h_{1}-u_{1}/\beta\\\
-\partial_{x}(h_{0}u_{1})\end{array}\right]\,.$
Formal adjoint operator to $\boldsymbol{\mathcal{L}}$ is given by
$\boldsymbol{\mathcal{L}}^{*}\left[\begin{array}[]{ccc}g\\\
v\end{array}\right]=\left[\begin{array}[]{ccc}\nu\partial_{x}(h_{0}\partial_{x}g)-\frac{g}{\beta}+h_{0}\partial_{x}v\\\
-\sigma\partial_{xxx}(h_{0}g)\end{array}\right].$
The kernel of it contains two linear independent functions
$\left[\begin{array}[]{ccc}g_{1}\\\
v_{1}\end{array}\right]:=\left[\begin{array}[]{ccc}0\\\ 1\end{array}\right],\
\ \left[\begin{array}[]{ccc}g_{2}\\\
v_{2}\end{array}\right]:=\left[\begin{array}[]{ccc}1\\\
\int_{X}^{x}\frac{d\tau}{\beta h_{0}}\end{array}\right].$ (2.20)
To derive necessary Fredholm-type solvability conditions for the system
(2.19a)–(2.19b) we multiply (2.19b) by $v_{1}$, integrate it and using the
matching condition (2.7) obtain
$\dot{R}_{0}=0.$ (2.21)
In turn, multiplying the right hand side of the second equation in (2.12) by
$v_{2}$ and integrating it on $(X-R,\,X+R)$ one obtains
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\dot{X}_{0}\int_{X-R}^{X+R}\frac{\partial h_{0}}{\partial
x}v_{2}\,dx+\int_{X-R}^{X+R}\partial_{x}(h_{0}u_{1})v_{2}\,dx=h_{0}u_{1}v_{2}\Big{|}_{X-R}^{X+R}-\int_{X-R}^{X+R}h_{0}u_{1}\partial_{x}v_{2}\,dx-$
$\displaystyle-$ $\displaystyle\dot{X}_{0}\int_{X-R}^{X+R}\frac{\partial
h_{0}}{\partial
x}v_{2}\,dx=\dot{X}_{0}(h_{0}v_{2})\Big{|}_{X-R}^{X+R}-\frac{2\dot{X}_{0}R}{\beta}-\frac{1}{\beta}\int_{X-R}^{X+R}u_{1}\,dx=$
$\displaystyle=$
$\displaystyle-\frac{2\dot{X}_{0}R}{\beta}-\frac{1}{\beta}\int_{X-R}^{X+R}u_{1}\,dx.$
In the last equality we used that $h_{0}\sim O(R-|x-X|)$ and
$v_{2}\sim\log(R-|x-X|)$ as $x\rightarrow X\mp R$. Next, using (2.19a) and
integrating three times by parts one arrives at
$\displaystyle\frac{2\dot{X}_{0}R}{\beta}$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\int_{X-R}^{X+R}u_{1}\,dx=\int_{X-R}^{X+R}\nu\partial_{x}(h_{0}\partial_{x}u_{1})+\sigma
h_{0}\partial_{xxx}h_{1}\,dx=$ (2.22) $\displaystyle=$ $\displaystyle\left[\nu
h_{0}\partial_{x}u_{1}+\sigma
h_{0}\partial_{xx}h_{1}-\sigma\partial_{x}h_{0}\partial_{x}h_{1}+\sigma\partial_{xx}h_{0}h_{1}\right]\Big{|}_{X-R}^{X+R}$
Let us note that from (2.19a)–(2.19b) and (2.21), (2.7) it follows that
$\displaystyle u_{1}\equiv\dot{X}_{0},$ (2.23a)
$\displaystyle\partial_{xxx}h_{1}=\frac{\dot{X}_{0}}{\beta\sigma h_{0}}$
(2.23b)
Hence, by (2.23a) the first term in the square brackets in (2.22) vanishes. By
(2.23b) one has $\partial_{xx}h_{1}\sim\log(R-|x-X|)$ as $x\rightarrow X\pm
R$. Due to this and (2.7)–(2.8) the second term in the square brackets in
(2.22) also vanishes. In turn, due to the matching condition
$h_{1}(X\mp R)=H_{1}(0)$ (2.24)
and (2.8) the fourth fourth term in the square brackets in (2.22) vanishes.
Therefore, relation (2.22) reduces to
$\frac{2\dot{X}_{0}R}{\beta}=-\left[\sigma\partial_{x}h_{0}\partial_{x}h_{1}\right]\Big{|}_{X-R}^{X+R}.$
(2.25)
At this moment let us introduce back the droplet subscript $i=1,...,N-1$ and
denote by $J_{i}$ the flux $j_{2}$ in the PL region between $i-1$-th and
$i$-th droplets. Combining (2.11) with (2.15) and (2.21) one obtains that
$J_{i}$ is constant and satisfies
$J_{i}=\beta\frac{P_{i}-P_{i-1}-2\nu J_{i}I-\nu
I(\dot{X}_{i,0}+\dot{X}_{i-1,0})}{d_{i}}.$ (2.26)
In the last formula we introduced two more notations: the constant leading
order pressure inside $i$-th droplet
$P_{i}=\frac{1}{R_{i}\sqrt{3\sigma}}.$ (2.27)
according to (2.9) and the distance between the neighboring DC regions
$d_{i}=X_{i}-X_{i-1}-R_{i}-R_{i-1}.$
From (2.26) one obtains an explicit expression for $J_{i}$:
$J_{i}=\beta\frac{P_{i}-P_{i-1}-\nu
I(\dot{X}_{i,0}+\dot{X}_{i-1,0})}{d_{i}+2\nu I\beta},\ \ i=1,...,N.$ (2.28)
Next, from (2.25), the matching conditions (2.18), (2.24) and equations(2.15),
(2.21) one obtains the leading order equation for $i$-th droplet position
evolution
$\dot{X}_{i,0}=-\frac{I\beta\nu}{2R_{i}+2I\beta\nu}(J_{i+1}+J_{i}).$
Substituting in the last expression the flux representation (2.28), denoting
$\widetilde{d}_{i}=\frac{d_{i}}{I\nu\beta},$ (2.29)
and using (2.27) one obtains
$\displaystyle\dot{X}_{i,0}$ $\displaystyle=$
$\displaystyle-\frac{P_{i}}{2/(\sqrt{3\sigma}\beta)+2I\nu
P_{i}}\left(\frac{(P_{i+1}-P_{i})-I\nu(\dot{X}_{i+1}+\dot{X}_{i})}{\widetilde{d}_{i+1}+2}+\frac{(P_{i}-P_{i-1})-I\nu(\dot{X}_{i}+\dot{X}_{i-1})}{\widetilde{d}_{i}+2}\right),$
(2.30) $\displaystyle\text{for}\ \ i=1,...,N-1.$
In turn by (2.21) and definition (2.27) one has
$\dot{P}_{i,0}=0,\ \ \text{for}\ \ i=0,...,N.$
The derived ODE system describing the leading order in $\varepsilon$ evolution
of pressures and positions of $N+1$ droplets will be closed if we additionally
prescribe that the first and the last droplet do not move, i.e
$\dot{X}_{0,0}=\dot{X}_{N,0}=0,\ X_{0}=0,\,X_{N}=L.$ (2.31)
The condition (2.31) corresponds to the situation when one extends the array
on $N+1$ droplets from interval $(0,L)$ to an infinite array on the whole real
line $\mathbb{R}\,$ by reflection around the points $x=0$ and $x=L$. It stays
also in agreement with boundary conditions (1.9) and (1.10)
Let us point out that the evolutions of pressures is slower than one of
positions and proceeds on the order $\varepsilon$. One can potentially obtain
it by going further in the expansion of the solution to (1.2a)–(1.2b), while
an easier way is to derive it from the conservation of droplet volume as was
done in [18, 20] for the case of equation (1.1). Namely, the volume of the
$i$-th droplet $V_{i}$ is changing due to the difference of the fluxes in the
surrounding it PL regions. Using (2.8) and (2.27) one obtains
$\dot{V}_{i,0}=\varepsilon\frac{4A^{3}}{3P^{3}}\dot{P}_{i,1}=\varepsilon^{2}(J_{i+1}-J_{i}),$
Substituting in the last expression the flux representation (2.28) and
denoting
$C_{i}=\varepsilon\frac{3P^{3}}{4A^{3}}$
one obtains
$\varepsilon\dot{P}_{i,1}=\frac{C_{i}}{I\nu}\left(\frac{P_{i+1}-P_{i}}{\widetilde{d}_{i+1}+2}-\frac{P_{i}-P_{i-1}}{\widetilde{d}_{i}+2}\right)-C_{i}\left(\frac{\dot{X}_{i+1}-\dot{X}_{i}}{\widetilde{d}_{i+1}+2}-\frac{\dot{X}_{i}-\dot{X}_{i-1}}{\widetilde{d}_{i}+2}\right),\
i=1,...,N-1.$
Finally, combining the last expression with (2.30) and (2.31) the closed ODE
system for the leading order evolution of positions and pressures in the array
of $N+1$ droplets takes the following form:
$\displaystyle\dot{X}_{i}$ $\displaystyle=$
$\displaystyle-\frac{P_{i}}{2/(\sqrt{3\sigma}\beta)+2I\nu
P_{i}}\left(\frac{(P_{i+1}-P_{i})-I\nu(\dot{X}_{i+1}+\dot{X}_{i})}{\widetilde{d}_{i+1}+2}+\frac{(P_{i}-P_{i-1})-I\nu(\dot{X}_{i}+\dot{X}_{i-1})}{\widetilde{d}_{i}+2}\right),$
$\displaystyle\dot{P}_{i}$ $\displaystyle=$
$\displaystyle\frac{C_{i}}{I\nu}\left(\frac{P_{i+1}-P_{i}}{\widetilde{d}_{i+1}+2}-\frac{P_{i}-P_{i-1}}{\widetilde{d}_{i}+2}\right)-C_{i}\left(\frac{\dot{X}_{i+1}-\dot{X}_{i}}{\widetilde{d}_{i+1}+2}-\frac{\dot{X}_{i}-\dot{X}_{i-1}}{\widetilde{d}_{i}+2}\right),\
i=1,...,N-1;$ (2.32)
and
$\displaystyle\dot{P}_{1}+2C_{1}\frac{\dot{X}_{2}}{\widetilde{d}_{1}+2}$
$\displaystyle=$ $\displaystyle
2\frac{C_{1}}{I\nu}\frac{P_{2}-P_{1}}{\widetilde{d}_{1}+2},\ \ \dot{X}_{1}=0,$
$\displaystyle\dot{P}_{N}-2C_{N}\frac{\dot{X}_{2}}{\widetilde{d}_{N-1}+2}$
$\displaystyle=$
$\displaystyle-2\frac{C_{N}}{I\nu}\frac{P_{N}-P_{N-1}}{\widetilde{d}_{N-1}+2},\
\ \dot{X}_{N}=0.$ (2.33)
Let us consider certain limiting cases. In the case $\beta\rightarrow\infty$
the limiting system for evolution of pressures and positions has the form
$\displaystyle\dot{P}_{i}+C_{i}(\dot{X}_{i+1}-\dot{X}_{i-1})$ $\displaystyle=$
$\displaystyle\frac{C_{i}}{I\nu}(P_{i+1}-2P_{i}+P_{i-1}),$
$\displaystyle\dot{X}_{i+1}-2\dot{X}_{i}+\dot{X}_{i-1}$ $\displaystyle=$
$\displaystyle\frac{P_{i+1}-P_{i-1}}{\nu I},\ \ \text{for}\ \ i=1,...,N-1;$
(2.34)
and
$\displaystyle\dot{P}_{1}+C_{1}\dot{X}_{2}$ $\displaystyle=$
$\displaystyle\frac{C_{1}}{I\nu}(P_{2}-P_{1}),\ \ \dot{X}_{1}=0,$
$\displaystyle\dot{P}_{N}-C_{N}\dot{X}_{N-1}$ $\displaystyle=$
$\displaystyle-\frac{C_{N}}{I\nu}(P_{N}-P_{N-1}),\ \ \dot{X}_{N}=0.$ (2.35)
Next, rescaling the time by $\beta\nu$ and proceeding to the limit
$\beta\rightarrow 0$ the limiting system for evolution of pressures and
positions takes the form
$\displaystyle\dot{P}_{i}$ $\displaystyle=$ $\displaystyle
C_{i}\left(\frac{P_{i+1}-P_{i}}{d_{i+1}}-\frac{P_{i}-P_{i-1}}{d_{i}}\right),$
$\displaystyle\dot{X}_{i}$ $\displaystyle=$
$\displaystyle-\frac{P_{i}\sqrt{3\sigma}I}{2}\left(\frac{P_{i+1}-P_{i}}{d_{i+1}}+\frac{P_{i}-P_{i-1}}{d_{i}}\right),\
\ \text{for}\ \ i=1,...,N-1;$ (2.36)
and
$\displaystyle\dot{P}_{1}$ $\displaystyle=$ $\displaystyle
2C_{1}\frac{P_{2}-P_{1}}{d_{1}},\ \ \dot{X}_{1}=0,$ $\displaystyle\dot{P}_{N}$
$\displaystyle=$ $\displaystyle-2C_{N}\frac{P_{N}-P_{N-1}}{d_{N}},\ \
\dot{X}_{N}=0.$ (2.37)
Note, that the last system coincides with one derived in [20] for the
intermediate-slip equation (1.8) in the one-dimensional case. This stays in
agreement with the fact that (1.8) is the limiting case of (1.2a)–(1.2b) as
$\beta\rightarrow 0$ as was shown in [15, 17]. Finally, note that after time
rescaling by $\beta\nu$ taking limits $\nu\rightarrow\infty$ or
$\nu\rightarrow 0$ results again in (2.34)–(2.35) and (2.36)–(2.37),
respectively. This is also natural, because (1.7a)–(1.7b) and (1.8) are the
limiting cases of (1.2a)–(1.2b) as well as $\nu\rightarrow\infty$ or
$\nu\rightarrow 0$, respectively.
Let us summarize the algorithm for simulation of coarsening dynamics in large
arrays of droplets using the derived reduced ODE models. Starting with an
array of $N+1$ droplets after each subsequent coarsening event (i.e a collapse
of one droplet or collision of two droplets) one can model the coarsening
process further by reducing the dimension of the model by two and solving the
reduced ODE model with the updated initial data. Practically, as in [19] we
say that a collapse event occurs at a moment when pressure of one droplet
increases a certain threshold, namely when
$P>0.5P_{max}(\varepsilon),\ \ \text{with}\ \
P_{max}(\varepsilon):=\frac{27}{256\varepsilon}.$ (2.38)
Then we take the final pressures and positions for remaining droplets from the
previous run of the reduced ODE model as initial conditions for the next one.
In the case of collision in Glasner and Witelski [19] was suggested that
coarsening event occurs when the distance between two colliding $i$-th and
$i+1$-th droplets becomes smaller then a certain threshold
$\delta=O(\varepsilon)$, i.e. when
$d_{i}\leq\delta,$ (2.39)
After the collision we calculate the position and the pressure for the new
formed droplet by formulas
$\displaystyle X_{i,new}$ $\displaystyle=1/2(X_{i+1}-R_{i+1}+X_{i}-R_{i}),$
$\displaystyle P_{i,new}$
$\displaystyle=\left(\frac{1}{P_{i}^{2}}+\frac{1}{P_{i+1}^{2}}\right)^{-1/2}.$
(2.40)
The last formula for $P_{i,new}$ is based on the observation that mass of the
new droplet is to the leading order in $\varepsilon$ given by the sum of the
masses of the collided droplets (see [19]). In section 5.1 we compare
solutions of the derived reduced ODE model (2.32)–(2.33) with those of the
initial PDE system (1.2a)–(1.2b) and show that the former ones provide high
accuracy $O(\varepsilon)$ also after subsequent coarsening events.
## 3 An exactly solvable collisions/absorption model
Let us consider the limiting case of infinite slip length $\beta=\infty$,
namely the ODE system (2.34)–(2.35) describing coarsening in free films. As
pressure evolution proceeds on a slower time scale then that one of positions
as $\varepsilon\rightarrow 0$ let us consider only migration of droplets.
Namely, we investigate the zero order system
$\displaystyle\dot{X}_{0}$ $\displaystyle=$
$\displaystyle\dot{X}_{N}=\dot{P}_{i}=0,\ \ \text{for}\ \ i=0,...,N,\ \;$
$\displaystyle\dot{X}_{i+1}-2\dot{X}_{i}+\dot{X}_{i-1}$ $\displaystyle=$
$\displaystyle\frac{P_{i+1}-P_{i-1}}{\nu I},\ \ \text{for}\ \ i=2,...,N-1;$
(3.1)
As will be justified numerically in section 5.1 for given $\varepsilon,\,T>0$
one can choose initial data with sufficiently small $P_{i}(0)\ll 1,\
i=0,1,...,N$ such that the difference between solutions to (3.1) and
(2.34)–(2.35) stays uniformly $O(\varepsilon)$ for all times $t\in(0,T]$. Note
that for such initial data there is no other constraint on the location of
$X_{i}(0)$ other than that $d_{i}(0)$ should be larger then the collision
threshold introduced in (2.39).
Moreover, for certain initial data one can solve (3.1) explicitly. Indeed, if
$P_{i}(0)=p,\ \text{for}\ i=0,1,...,N-1\ \text{and}\ P_{N}(0)=\bar{p}\
\text{with}\ 1\gg p>\bar{p}$ (3.2)
then the solution to (3.1) is given by
$X_{i}(t)=X_{i}(0)+\frac{Bi}{N}t,\ \text{for}\ i=1,...,N-1;\
X_{0}=0,\,X_{N}=L,\ \text{where}\ B=\frac{p-\bar{p}}{\nu I}.$ (3.3)
In this and the next sessions for convenience reasons we redefine the notation
for the distances between droplets from the previous section as
$d_{i}(t)=X_{i}(t)-X_{i-1}(t)\ \text{for}\ i=1,...,N-1\ \text{and}\
d_{N}(t)=L-X_{N-1}(t)-R_{N}(t)-R_{N-1}(t)$
and call usually $d_{i}(t)$ as the distance of the $i$-th droplet at time $t$.
Using this notation one can rewrite the solution (3.3) in the following form
$\displaystyle d_{i}(t)$ $\displaystyle=$ $\displaystyle
d_{i}(0)+\frac{B}{N}t,\ \ \text{for}\ \ i=1,...,N-1$ $\displaystyle d_{N}(t)$
$\displaystyle=$ $\displaystyle d_{N}(0)-\frac{B(N-1)}{N}t\ \text{for}\
t\in(0,T_{c}),\ \text{where}\ T_{c}=\frac{d_{N}N}{(N-1)B}.$ (3.4)
Note, that $T_{c}$ denotes the time proceeding until $(N-1)$-th droplet
collides with the largest last one. Iterating (3.4) one observes that first
$N-1$ droplets collide one after another with the last one like rings in the
famous rubber band toy. Due to (3.3) all droplets except the first and the
last ones move to the right. The last droplet consequently absorbs the
neighbor droplet, while the distance between them is uniformly distributed
between the remaining droplets. Therefore, the distances of the remaining
droplets at the collision time $T_{c}$ are given by
$d_{i}(T_{c})=d_{i}(0)+d_{N}(0)/(N-1),\ \ i=1,...,N-1.$ (3.5)
Writing the solution to (3.1) in the form (3.4) is convenient because one can
substitute $d_{i}(T_{c})$ as the initial distances for the modelling of the
next collision event.
Note, that due to (3.5) distance monotonicity is preserved in time for
solutions (3.4), i.e. if $d_{l}(0)>d_{m}(0)$ for some $l,m\in 1,...,N$ then
$d_{l}(t)>d_{m}(t)$ for all times $t>0$. This property allows us basing only
on a given initial distribution of the distances in the array of droplets to
derive the coarsening laws analytically for solutions to (3.1) considered with
(3.2) and additional assumption
$1>>p>>P_{N}.$ (3.6)
This assumption prescribes that the last droplet is much larger then others
and allows us to simplify further the dynamics by assuming that its pressure
$P_{N}$ remains constant in time. In turn, this implies that the coarsening
dynamics in this case depends solely on the evolution of droplet positions
without change of their pressures after subsequent collisions.
Indeed, let initial distances be prescribed by $k\in\mathbb{N}\,$ families
such that there are $i_{m}$ distances in $m$-th family ($1\leq m\leq k$), all
of them are equal to $d_{m}$ and
$d_{1}\geq d_{2}\geq...\geq d_{k},\quad i_{1}+i_{2}+...+i_{k}=N$ (3.7)
holds. Additionally, let us allocate these $k$ families in the initial
configuration so that the distances between droplets non increase coming from
the first to the last droplet. Then due to the distance monotonicity property
this ordering will be preserved in time, i.e. first the members of the family
$k$ will be absorbed by the last droplet, then those of $k-1$-th one and etc.
Moreover, the distances in each family will stay equal for all $t>0$. This
implies that for initial data satisfying (3.2), (3.6) and (3.7) all collision
times are uniquely determined having given $k$ and the set
$\\{d_{m},i_{m}\\},\ m=1,...,k$. Therefore, using the explicit solution (3.4)
holding between subsequent collision events the coarsening law can be derived
analytically by a recursive procedure.
Indeed, let us fix an index $1\leq m\leq k$ and look at the moment when all
families with the indexes $m+1,...,k$ and also $l-1$ members of the $m$-th
family have been absorbed for some given $1\leq l\leq i_{m}$. Let us calculate
the time $t(n)$ needed for the absorption of the $l$-th member with $n$
denoting the remaining number of droplets after the latter event. Using (3.4)
one can easily calculate by recursion that
$\displaystyle
t(n)=\frac{n+1}{nB}\left(\widetilde{d}_{m}+\sum_{r=1}^{l-1}\frac{t(n+r)B}{n+r+1}\right)=\frac{(n+l)\widetilde{d}_{m}}{nB},$
(3.8)
where by $\widetilde{d}_{m}$ we denote the distance in the $m$-th family at
the time when $(m+1)$-th family has been absorbed. From (3.8) one can obtain
the total time needed for the $m$-th family to be absorbed $T_{m}$ in the form
$\displaystyle
T_{m}=\frac{\widetilde{d}_{m}}{B}\left(N-\sum_{p=m+1}^{k}i_{p}\right)\sum_{r=1}^{i_{p}}\frac{1}{N-\sum_{p=m+1}^{k}i_{p}-r}.$
(3.9)
In turn, using again (3.4) one recursively finds
$\displaystyle\widetilde{d}_{m}=\sum_{p=m+1}^{k}\frac{\widetilde{d}_{p}i_{p}}{N-\sum_{p^{\prime}=p}^{k}i_{p}}=\frac{\sum_{p=m+1}^{k}d_{p}i_{p}}{N-\sum_{p^{\prime}=p}^{k}i_{p}}+d_{m}$
Substituting the last expression in (3.9) one obtains
$\displaystyle
T_{m}=\frac{1}{B}\left(Nd_{m}+\sum_{p=m}^{k}(d_{p}-d_{m})i_{p}\right)\sum_{r=1}^{i_{m}}\frac{1}{N-\sum_{p=m+1}^{k}i_{p}-r}$
Therefore, the total time needed for all families up to $m$-th to be absorbed
is given by
$\displaystyle
T(d_{m})=\sum_{p=m}^{k}\frac{1}{B}\left(Nd_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})i_{p}^{\prime}\right)\sum_{r=1}^{i_{p}}\frac{1}{N-\sum_{p^{\prime}=p+1}^{k}i_{p}^{\prime}-r}$
(3.10)
Let us now derive the continuum version for the discrete coarsening law in
(3.10) proceeding to the limit $N\rightarrow\infty$ and $k\rightarrow\infty$.
Suppose we are given a probability density function $f(d)$ on $(0,+\infty)$,
i.e.
$\int_{0}^{+\infty}f(x)\,dx=1,\ \ f(d)\geq 0\ \text{and}\ f(d)=0\ \text{if}\
d\leq 0.$
Defining $d_{i}=i\Delta d$ for $i\in\mathbb{N}\,\cup\\{0\\}$ and a fixed
$\Delta d\ll 1$ we approximate $f(d)$ by a piece-wise constant function
$f_{a}(d)$ as follows.
$f_{a}(d)=f(d_{i+1})\ \text{for}\ d\in[d_{i},\,d_{i+1})\ \text{and}\
f_{a}(d)=0\ \text{if}\ d\leq 0.$
Accordingly to this approximation suppose we are given an array of $N+1$
droplets with $N\gg 1$ such that the number of droplets with the distances
lying in the interval $[d_{i},\,d_{i+1})$ is equal to $[Nf(d_{i+1})\Delta d]$.
As before we suppose that droplets are allocated so that the distances between
them are non increasing and (3.2), (3.6) hold. Then using (3.10) one obtains
$\displaystyle T_{\Delta d,N}(d_{m})$ $\displaystyle=$
$\displaystyle\sum_{p=0}^{m}\frac{1}{B}\left(Nd_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})Nf(d_{p}^{\prime})\Delta
d\right)\sum_{r=1}^{Nf(d_{p})\Delta
d}\frac{1}{N-\sum_{p^{\prime}=p+1}^{k}Nf(d_{p}^{\prime})\Delta d-r}+O(\Delta
d,1/N)$ $\displaystyle=$
$\displaystyle\sum_{p=0}^{m}\frac{1}{B}\left(d_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})f(d_{p}^{\prime})\Delta
d\right)\sum_{r=1}^{Nf(d_{p})\Delta
d}\frac{1}{1-\sum_{p^{\prime}=p+1}^{k}f(d_{p}^{\prime})\Delta d-r/N}+O(\Delta
d,1/N)$ $\displaystyle=$
$\displaystyle\sum_{p=0}^{m}\frac{N}{B}\left(d_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})f(d_{p}^{\prime})\Delta
d\right)\sum_{s=1/N}^{f(d_{p})\Delta d}\frac{\Delta
s}{1-\sum_{p^{\prime}=p+1}^{k}f(d_{p}^{\prime})\Delta d-s}+O(\Delta d,1/N)$
Taking the limit $N\rightarrow+\infty$ in the last expression and introducing
$\displaystyle T_{\Delta d}(d)=\lim_{N\rightarrow+\infty}\frac{T(d)_{\Delta
d,N}}{N}$
one obtains
$T_{\Delta
d}(d)=\sum_{p=0}^{m}\frac{N}{B}\left(d_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})f(d_{p}^{\prime})\Delta
d\right)\int_{s=0}^{f(d_{p})\Delta
d}\frac{ds}{1-\sum_{p^{\prime}=p+1}^{k}f(d_{p}^{\prime})\Delta d-s}+O(\Delta
d).$ (3.11)
Applying the Taylor expansion to the last integral in (3.11) one finds
$\int_{s=0}^{f(d_{p})\Delta
d}\frac{ds}{1-\sum_{p^{\prime}=p+1}^{k}f(d_{p}^{\prime})\Delta
d-s}=\frac{f(d_{p})\Delta d}{1-\sum_{p^{\prime}=p}^{k}f(d_{p}^{\prime})\Delta
d}+O(\Delta d^{2})$
Inserting this into (3.11) one obtains
$T_{\Delta
d}(d)=\frac{1}{B}\sum_{p=0}^{m}\left(d_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})f(d_{p}^{\prime})\Delta
d\right)\frac{f(d_{p})\Delta
d}{1-\sum_{p^{\prime}=p}^{k}f(d_{p}^{\prime})\Delta d}+O(\Delta d).$
Finally, taking the limit $\Delta d\rightarrow 0$ and introducing
$\displaystyle T(d)=\lim_{\Delta d\rightarrow 0}\frac{T_{\Delta d}(d)}{N}$
one arrives at
$T(d)=\frac{1}{B}\int_{0}^{d}\left(x+\int_{0}^{x}(y-x)f(y)\,dy\right)\frac{f(x)}{1-\int_{0}^{x}f(y)\,dy}\,dx$
(3.12)
Introducing function $n(d)$ as the relative number of droplets with initial
distances larger or equal $d$, i.e. as
$n(d)=1-\int_{0}^{d}f(x)\,dx$ (3.13)
one obtains from (3.12) that
$\displaystyle T(d)$ $\displaystyle=$
$\displaystyle\frac{1}{B}\int_{0}^{d}\left(-x+\int_{0}^{x}(y-x)n^{\prime}(y)\,dy\right)\frac{n^{\prime}(x)}{1-\int_{0}^{x}n(y)\,dy}\,dx=$
(3.14) $\displaystyle=$
$\displaystyle\frac{1}{B}\int_{0}^{d}n(x)\ln\left[\frac{n(x)}{n(d)}\right]\,dx.$
The last expression provides an exact coarsening law, i.e. it tells what time
$T(d)$ is needed until all droplets having initially distances smaller then
$d$ are absorbed by the last large droplet.
In appendix B, we show that the discrete coarsening law (3.10) can be
recovered back from (3.14) if the initial distribution $f(x)$ has the form
$f(d)=\sum_{m=1}^{k}i_{m}^{\prime}\delta(d-d_{m}),$ (3.15)
i.e. if initial distance distribution is represented by $k\in\mathbb{N}\,$
families as in (3.7) while the number of droplets $N\rightarrow\infty$.
Moreover, we justify the connection between (3.14), (3.10) and the starting
ODE system (3.1) as well numerically in the section 5.2.
## 4 Examples of coarsening rates
a) We consider an explicit family of initial distributions $f(x)$ and show
that depending on their decay as $x\rightarrow+\infty$ the coarsening rates
reproduce all possible algebraic decays. Moreover, there is a certain
threshold after which the decay becomes exponential. Namely, let us consider a
family
$f(x)=\frac{1}{x^{1+\alpha}}\Big{/}\int_{A}^{+\infty}\frac{dx}{x^{1+\alpha}}=\frac{A^{\alpha}}{x^{1+\alpha}}\
\ \text{with}\ \ \alpha,\,A>0.$ (4.1)
From (3.13) it follows that
$n(x)=\left(\frac{A}{x}\right)^{\alpha}.$ (4.2)
Substituting this in (3.14) one obtains
$\displaystyle T(d)$ $\displaystyle=$ $\displaystyle\frac{\alpha
A}{B(\alpha-1)}\left(\frac{1}{\alpha-1}\left[\left(\frac{d}{A}\right)^{1-\alpha}-1\right]+\alpha\ln\left[\frac{d}{A}\right]\right)\
\text{if}\ \alpha\neq 1,$ (4.3a) $\displaystyle T(d)$ $\displaystyle=$
$\displaystyle\frac{A}{B}\left(\left[\ln\left(\frac{d}{A}\right)\right]^{2}/2+\ln\left(\frac{d}{A}\right)\right)\
\text{if}\ \alpha=1.$ (4.3b)
Combining (4.3b) and (4.2) one obtains the exact coarsening law for the case
$\alpha=1$
$n(t)=\exp\left[1-\sqrt{1+2Bt/A}\right]$ (4.4)
In the case $\alpha\neq 1$ one obtains from (4.3a) and (4.2)
$T(n)=\frac{\alpha
A}{B(\alpha-1)}\left(\frac{1}{1-\alpha}\left[n^{\frac{\alpha-1}{\alpha}}-1\right]+\ln(n)\right).$
For the latter exact law one obtains the following asymptotics
$n(t)\sim\left\\{\begin{aligned}
\displaystyle&\left(\frac{tB(\alpha-1)^{2}}{\alpha
A}\right)^{\frac{\alpha}{\alpha-1}},\hskip 56.9055pt\text{if}\ \
\alpha<1\\\\[8.61108pt] \displaystyle&\exp\left\\{-\frac{tB(\alpha-1)}{\alpha
A}\right\\},\hskip 56.9055pt\text{if}\ \ \alpha>1\end{aligned}\right.\ \
\text{as}\ \ t\rightarrow\infty.$ (4.5)
Therefore, from (4.4)–(4.5) one finds out that for $0<\alpha<1$ the coarsening
rates are algebraic at least for large times, while at $\alpha=1$ they become
exponential and stay so for $\alpha\in(1,\,+\infty)$.
b) Consider $f(x)=\exp(-x)$. Substituting it in (3.13) and (3.14),
consequently, one obtains the exact law
$T(n)=\frac{1}{B}(n-1-\ln(n)).$
Thus, in this case the following asymptotics holds
$n(t)\sim\exp(-Bt)\ \ \text{as}\ \ t\rightarrow\infty.$ (4.6)
c) Consider a Gaussian distribution $f(x)=2/\sqrt{\pi}\exp(-x^{2})$. In this
case by (3.13) one has $n(x)=\mathrm{erfc}(x)$. Substituting it in (3.14) one
obtains
$\displaystyle T(d)$ $\displaystyle=$
$\displaystyle\frac{1}{B}\int_{0}^{d}\left[\int_{0}^{x}n(y)\,dy\right]\frac{n^{\prime}(x)}{n(x)}\,dx=$
$\displaystyle=$
$\displaystyle\frac{1}{B\sqrt{\pi}}\int_{0}^{d}\left(\frac{(1-\exp(-x^{2}))\exp(-x^{2})}{\int_{x}^{+\infty}\exp(-x^{2})\,dt}-2x\exp(-x^{2})\right)\,dx$
$\displaystyle=$
$\displaystyle\frac{1}{B\sqrt{\pi}}\left(-C+O\left(\exp(-d^{2})\right)+\int_{0}^{d}\frac{\exp(-x^{2})}{\int_{x}^{+\infty}\exp(-x^{2})\,dt}\,dx\right)$
$\displaystyle=$ $\displaystyle\frac{1}{B\sqrt{\pi}}(-C-\ln
n(d))+O(\exp(-d^{2})),$
where constant $C\approx 0.74$. Therefore the following asymptotics holds
$n(t)\sim\exp\\{-C-B\sqrt{\pi}t\\}\ \ \text{as}\ \ t\rightarrow\infty.$ (4.7)
and the coarsening rates show an exponential decay as in the example b).
Figure 3: Initial distributions in the example d) (left) and the example e)
with $\alpha=2$ (right)
d) Finally let us show that the coarsening rates for large times depend only
on how fast the initial distribution $f(x)$ decays as $x\rightarrow+\infty$
and not its behavior for moderate $x$. In this and the next example we
consider non-monotone distributions having a local maximum at $x>0$. Consider
$f(x)=(1-x)^{2}\exp(-x)$ (see Fig. 3) with $n(x)=(1+x^{2})\exp(-x)$,
correspondingly. By (3.14) one obtains
$\displaystyle BT(d)$ $\displaystyle=$
$\displaystyle\int_{0}^{d}(1+x^{2})\exp(-x)\ln(1+x^{2})\,dx-\int_{0}^{d}x(1+x^{2})\exp(-x)\,dx$
(4.8) $\displaystyle-$
$\displaystyle(\ln(1+d^{2})-d)\int_{0}^{d}(1+x^{2})\exp(-x)\,dx$
$\displaystyle=$
$\displaystyle\int_{0}^{d}(1+x^{2})\exp(-x)\ln(1+x^{2})\,dx-7+3(d-\ln(1+d^{2}))+O(\exp(-d)).$
The first integral in the last expression can be estimated as follows.
$\displaystyle\int_{0}^{d}(1+x^{2})\exp(-x)\ln(1+x^{2})\,dx$
$\displaystyle\leq$
$\displaystyle\ln(1+d^{2})\int_{0}^{d}(1+x^{2})\exp(-x)\,dx$ $\displaystyle=$
$\displaystyle\ln(1+d^{2})(3+O(\exp-d))$
Combining this with (4.8) one obtains
$T(d)=\frac{3d}{B}+o(d)$
and hence the following asymptotics holds
$n(t)\sim\left(1+\frac{9}{B^{2}}t^{2}\right)\exp(-Bt)\ \ \text{as}\ \
t\rightarrow\infty.$
Therefore, the coarsening rates show an exponential decay as in the example
b).
e) Consider distributions
$\frac{\alpha}{\alpha+1}\left[(1-x)^{2}\exp(-x)+1/(1+x)^{1+\alpha}\right]\ \
\text{with}\ \ \alpha>0,\,\alpha\neq 1.$ (4.9)
They have a local maximum at $x>0$ and a decay $\sim 1/x^{1+\alpha}$ as
$x\rightarrow\infty$ (see Fig. 3). Correspondingly, one has
$n(x)=\frac{1}{1+\alpha}\left[(1+x)^{-\alpha}+\alpha\exp-x(1+x^{2})\right].$
Substituting it in (3.14) one obtains
$\displaystyle BT(d)$ $\displaystyle=$
$\displaystyle\int_{0}^{d}\left[\int_{0}^{x}n(y)\,dy\right]\frac{n^{\prime}(x)}{n(x)}\,dx=$
$\displaystyle=$
$\displaystyle\int_{0}^{d}\left[\frac{1}{1-\alpha}\left((1+x)^{1-\alpha}-1\right)+\alpha\left(3-\exp(-d)(3+d(2+d))\right)\right]\times$
$\displaystyle\times$
$\displaystyle\frac{(1-x)^{2}\exp(-x)+(1+x)^{-1-\alpha}}{(1+x)^{-\alpha}+\alpha\exp(-x)(1+x^{2})}\,dx.$
The last integral can be bounded from below and from above by integrals of the
following type
$I^{*}=\int_{0}^{d}\left[\frac{1}{1-\alpha}\left((1+x)^{1-\alpha}-1\right)+C_{1}\right]\frac{C_{2}\exp(-x/2)+(1+x)^{-1}}{1+C_{3}}$
with some nonnegative constants $C_{i},\ i=1,2,3$. Integrals in (4) have the
following asymptotics:
$I^{*}=C_{3}\left(\frac{1}{1-\alpha}\left[(d+1)^{1-\alpha}-1\right]-C_{4}\ln(d+1)\right)+O(1)\
\ \text{as}\ \ d\rightarrow\infty$
with some positive constants $C_{3},\,C_{4}$. Therefore, using the asymptotics
$n(d)\sim\frac{1}{1+\alpha}(1+d)^{-\alpha}\ \ \text{as}\ \ d\rightarrow\infty$
(4.10)
one obtains that the asymptotics of the coarsening law for (4.9) coincides up
to multiplicative constants with (4.3a) already obtained in the example a) for
the monotone distributions. Consequently, we conclude that as in the example
a) the coarsening rates are algebraic with power $\alpha/(\alpha-1)$ for
$\alpha<1$ and exponential for $\alpha>1$.
A more simple but rather formal proof of this fact is as follows. Let us fix a
large number $A$ such that asymptotics (4.10) holds for all $d>A$ with a good
precision. The one has
$\displaystyle BT(d)$ $\displaystyle=$
$\displaystyle\int_{0}^{A}n(x)\ln\left[\frac{n(x)}{n(d)}\right]\,dx+\int_{A}^{d}n(x)\ln\left[\frac{n(x)}{n(d)}\right]\,dx$
$\displaystyle\sim$ $\displaystyle O(1)+\alpha\ln(d+1)\times
O(1)+\int_{A}^{d}n(x)\ln\left[\frac{n(x)}{n(d)}\right]\,dx.$
The last integral in view of (4.10) is of the type considered already in
example a). Therefore, the term $O(1)+\alpha\ln(d+1)\times O(1)$ produces no
change in the asymptotics of $T(d)$ and, consequently, the coarsening law
coincides up to multiplicative constants with (4.3a).
## 5 Numerics
In the last two sections starting from the system (1.2a)-(1.2b) we first
derived a closed ODE model (2.32)–(2.33) describing coarsening dynamics in an
array of initially $N+1$ metastable droplets. Then we looked at its limiting
case $\beta\rightarrow\infty$ described by (2.34)–(2.35) and more precisely on
its leading order version as $\varepsilon\rightarrow 0$ given by (3.1). Next,
we found out that for a special initial data satisfying (3.2) one can obtain
the explicit solution to (3.1) given by (3.4). Assuming additionally (3.6) and
that the distances in the array are ordered decreasingly we derived an
explicit coarsening law (3.10). Finally, we obtained its continuous
counterpart (3.14). In this section we systematically compare numerically the
solutions of subsequent models in the derived model hierarchy and check
coarsening laws (3.10), (3.14).
### 5.1 Comparison between models
Here we compare solutions to the full ODE system (2.32)–(2.33) with those of
the strong-slip system (1.2a)-(1.2b) and its limiting cases (1.8) and
(1.7a)-(1.7b) as $\beta\rightarrow 0$ and $\beta\rightarrow+\infty$,
respectively. For the solution of PDE systems we used a fully implicit finite
difference scheme derived and applied already to (1.2a)-(1.2b) and its
limiting cases in [28, 15, 29, 30]. The numerical solutions for (2.32)–(2.33)
were obtained applying a fourth-order adaptive time step Runge-Kutta method
and using updating rules (2.38)–(2.40) after each subsequent coarsening event.
In the case of PDE system the corresponding pressure evolution was calculated
using finite-difference discretization of the term
$\Pi_{\varepsilon}(h)-\partial_{xx}h$.
Figure 4: Comparison of droplet position evolution obtained from the ODE model
(dots) and the lubrication model (solid line) in the strong-slip case with
$\varepsilon=0.025$, $Re=0$, $\beta=10$. Initial profile of four droplets
presented on the top-left plot of Fig. 5 was used. Subsequent collisions of
the second (left) and the third (right) droplets with the first one are shown.
In Fig. 4 starting from an array of four droplets we compare evolution of
positions resulting from PDE and ODE models for two subsequent collisions.
Fig. 4 shows that the absolute deviation between results stays uniformly
$O(\varepsilon)$ also after subsequent collision events.
In Fig. 5 starting from the same array of four droplets we compare solutions
for different slip lengths $\beta$. In the cases $\beta=0$ and $\beta=\infty$
we compared solutions to (1.8) and (1.7a)-(1.7b) and those of (2.36)–(2.37)
and (2.34)–(2.35), respectively. Again for all $\beta$ the absolute deviation
between PDE and ODE results is $O(\varepsilon)$.
Figure 5: Comparison of droplet evolution obtained from the ODE model (dots)
and the lubrication model (solid line) in the strong-slip case with
$\varepsilon=0.025$, $Re=0$, and different $\beta$. Upper row: Initial profile
of four droplets (left) and their pressure evolution in the intermediate-slip
case until collapse of the second droplet (right). Lower row: Starting from
the same initial profile position evolution until collision of the second
droplet with the first one for $\beta=5$ (left) and $\beta=\infty$ (right) is
shown.
Note, Fig. 5 demonstrates the fact pointed already in [21, 27] that in the
intermediate-slip case the coarsening dynamics is governed mostly by collapse
mechanism while in the strong-slip case with moderate and large $\beta$ by
collisions.
Finally, Fig. 6 shows that for arrays being taken initially with sufficiently
small pressures the migration subsystem (3.1) approximates with a high
accuracy (at least $O(\varepsilon)$) the full ODE system (2.34)–(2.35)
describing the case $\beta=\infty$. Note, that the migration path of the
fourth droplet Fig. 6 is considerably large $\approx 50$. Hence, the reduction
from (2.34)–(2.35) to (3.1) does not constrains the initial distances to be
small.
Figure 6: Comparison of droplet position evolution obtained from the ODE model
(2.34)–(2.35) (dots)with $\varepsilon=0.025$ and its leading order subsystem
(3.1) (solid line). Left plot: The initial profile of five droplets. Pressure
of the first four and the last droplet are $0.01$ and $0.001$, respectively.
Right plot: Collision of the fourth droplet with the largest last one.
### 5.2 Coarsening rates
Here using the explicit solution (3.4) for the system (3.1) we check
numerically the discrete and continuous coarsening laws (3.10) and (3.14). In
Fig. 7 we take $k=20$ families of initial distances as prescribed in (3.7)
with the corresponding pressures satisfying (3.2), (3.6) and order them non
increasingly in space. Next, we compare the subsequent times for each $m$-th
family ($1\leq m\leq k$) to be absorbed given on one hand by the analytical
law (3.10) and by iterative calculation using (3.4),(3.5) between each
subsequent collision on the other. Naturally, one finds out the exact
coincidence between them. Note, as each collision is comprised of an
absorption by the largest droplet of a smaller one one do not need to update
the position and pressure of the former one. This is because its position is
fixed due to (2.31) to $x=L$ and the pressure to the leading order does not
change due to (3.6).
Figure 7: Comparison of the subsequent collision/absorption times for each
$m$-th family ($1\leq m\leq 20$) given by the discrete coarsening law (3.10)
(dots) and iterative calculation using the solution (3.4) (solid line). Left
plot: A part of a typical initial profile under consideration. Right plot:
plot of the absorption times versus the family number.
In Fig. 8 we show numerical results for continuous coarsening rates for three
initial distributions taken from the family (4.1) with different $\alpha$ and
one Gaussian distribution considered in examples a) and c) of the previous
section, respectively. To obtain the coarsening rates numerically we first
sampled $N\gg 1$ distances according to the given initial distribution. After
ordering them non increasingly in the initial configuration we substitute them
as initial data into (3.1) and solve the latter one iteratively using (3.4).
Note, that due to an extremal simplicity of (3.4) one can effectively model
numerically a huge number of droplets $N\approx 10^{7}$ just using
capabilities of a personal computer.
Fig. 8 shows that thus obtained numerical coarsening rates coincide for large
times very well with the analytical ones prescribed by the law (3.14) and
found out in examples a) and c) of the section 4. In the case of (4.1) with
$\alpha=1$ one has the exact coarsening law (4.4) and therefore a good
coincidence for all times. In the case of (4.1) with $\alpha\neq 1$ we
compared our numerical results with the asymptotic law (4.5), while for the
Gaussian initial distribution with (4.7). Note that a certain deviation
between numerics and analytical laws starting at the very end of the
considered time interval is caused by a numerical error increase in sampling
of large distances according to initial probability distributions.
Figure 8: Comparison of numerical coarsening rates using sampling of the
initial data with $N\gg 1$ and subsequent iterative calculation using (3.4)
(solid line) with those provided by the analytical law (3.14) (dots). Upper
row: loglog and semilog plots for the initial distribution (4.1) with
$\alpha=1/2$ (left) and $\alpha=1$ (right), respectively. Lower row: semilog
plots for (4.1) with $\alpha=20$ (left) and for the Gaussian initial
distribution (right). According to (4.5) and (4.7) in the chosen axe scales
the dots reproduce the linear functions except for the upper-right plot where
they represent the law (4.4)
## 6 Conclusions and discussion
In this paper we started from the lubrication equations (1.2a)–(1.2b)
describing dewetting process in nanometric polymer film interacting on a
hydrophobically coated solid substrate in the presence of large slippage at
the liquid/solid interface. This model describes a distinguished and important
regime within a lubrication scaling. In particular, it incorporates as a
limiting case of the infinite slip length the well-known model of free films
(1.7a)–(1.7b) studied intensively in applications [16, 31]. Note, that a
similar to (1.7a)–(1.7b) system appears in the study of viscoelastic threads
for which coarsening dynamics of interacting droplets was observed also at the
experiments, see e.g. [32].
Motivated by this we derived the reduced ODE models (2.32)-(2.37) describing
coarsening dynamics of droplets governed by (1.2a)–(1.2b) and its limiting
cases. In the limiting case $\beta=\infty$ we observed that the migration
subsystem (3.1) can be solved explicitly for special initial data satisfying
(3.2), (3.6). By (3.4)-(3.5) the dynamics of droplets consists of sequential
collisions of smaller ones with the largest one while their distance is
distributed uniformly between the remaining drops.
Similar models were suggested for collapse/collision dynamics of breath
figures by Derrida et al. [33] basing on heuristic arguments. There authors
considered the ’cut-in-two’ and ’past-all’ models where the distance of the
smallest droplet was divided between two neighbors or pasted as a whole to one
of them, respectively. The breath figures of [33] found later interesting
analogs in the reduced coarsening models arising from the Allen-Cahn and
Ginsburg-Landau equations,see e.g. a review article [34]. A recent
generalization of these models and rigorous analysis of their self-similar
solutions can be found in [35]. Note that (3.1) can be classified due to (3.5)
as a ’cut uniformly between all’ type model. We are not aware if any heuristic
or rigorous analog of it was considered so far in the literature.
Remarkably, our model (3.1) appears as a reduction of a complicated dynamics
governed by a high-order lubrication system (1.7a)–(1.7b). Moreover, in
contrast to stochastic in nature models of [33] if the initial distances in
(3.1) are ordered non increasingly then they coarse in a deterministic and
exactly solvable scenario. For the latter case we derived the coarsening laws
(3.10) and (3.14) analytically and confirmed them numerically. Interestingly,
the derived law (3.14) have a form similar to the Shannon’s entropy with
respect to a certain normalization of the droplet number function $n(x)$. The
explanation of this fact from the statistical point of view will be presented
elsewhere.
Surprisingly, in contrast to the coarsening dynamics governed by reduced ODE
models arising from (1.1) which obey always the law (1.11) our simple model
(3.1) can reproduce any algebraic coarsening rates between zero and infinity
as well as exponential ones. Moreover, for a family of initial distributions
(4.1) we showed existence of a threshold for their decay at infinity at which
the corresponding coarsening rates switch from algebraic to exponential ones.
Note, that a similar situation was accounted recently for self-similar
solutions to Smoluchowski coagulation equation with certain kernels, see [36,
37].
In view of the above observations it would be natural to extend our
deterministic collision/absorption model to its stochastic variant withdrawing
the non increasing order of the distances and thus allowing collisions of
random droplets with the largest one. As in [33] one could probably look for
self-similar solutions of the mean-field approximations for thus arising
stochastic collision models. A further generalization of the model could be a
withdrawal of the constraints on the initial data (3.2), (3.6) and thus
allowing droplets to collide and collapse inside of the domain. Note, that an
additional difficulty to handle the pressure and position update according to
the coarsening rules (2.38)–(2.40) would appear then.
Finally, it could be possible to derive two-dimensional analogs of the reduced
ODE models (2.32)-(2.37) describing physically coarsening of three dimensional
droplets on a plane substrate. The two dimensional reduced ODE models arising
from (1.1) were derived in [20, 21]. In [20] a mean-filed approximation for
the fluxes between droplets was suggested under an assumption of well
separation of droplets that is unfortunately not suitable for the modelling of
droplet collisions because the distance between them tends to zero then. In
this case one should face a problem of solving a Laplace equation (counterpart
to equation (2.11) in two dimensions) in a complex domain between droplets
occupied by the PL region. We expect the same problem to appear for the two-
dimensional reduced ODE models corresponding to (1.2a)–(1.2b).
## 7 Acknowledgments
The author acknowledges the postdoctoral scholarship at the Max-Planck-
Institute for Mathematics in the Natural Sciences, Leipzig and thanks Gennady
Chuev, Christian Seis and Andre Schlichting for fruitful discussion.
## Appendix A Integral I
Here we show that integral $I$ defined in (2.16) converges and integrate it
explicitly. Changing variable in (2.16) according to the explicit solution
(2.5a) in the CL region, and using matching conditions (2.4) one obtains
$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{1}{H_{1}}\partial_{z}\left(\frac{\partial_{z}H_{1}}{H_{1}}\right)\,dz=\int_{1}^{+\infty}\left[\frac{U^{\prime}(H)}{H^{2}}-\frac{2(U(H)-U(1))}{H^{3}}\right]\frac{dH}{\sqrt{2(U(H)-U(1))}}$
$\displaystyle=$
$\displaystyle\int_{1}^{+\infty}\frac{-5/3+2H-1/3H^{3}}{\sqrt{2/3-H+H^{3}/3}\,H^{9/2}}\,dH.$
Let us make a further change of variables $t=1/H$ and integrate $I$ explicitly
as follows.
$I=\int_{0}^{1}\frac{-5/3t^{4}+2t^{3}-1/3}{\sqrt{2/3t^{3}-t^{2}+1/3}}\,dt=\int_{0}^{1}\frac{5t^{3}-t^{2}-t}{\sqrt{6t+3}}\,dt=\frac{1}{35(3+\sqrt{3})}.$
## Appendix B Connection between discrete and continuous
coarsening laws
Here we show that the discrete coarsening law (3.10) can be recovered back
from (3.14) if the initial distribution $f(x)$ has the form (3.15) i.e. if it
is represented by $k\in\mathbb{N}\,$ families as in (3.7) where we denote
$\displaystyle i_{m}^{\prime}=\lim_{N\rightarrow\infty}\frac{i_{m}}{N}.$
In this case (3.13) implies
$n(d)=1-\sum_{p=m}^{k}i_{m}^{\prime}\ \ \text{if}\ \ d\in[d_{m},d_{m-1}).$
Substituting the last expression in (3.14) implies
$\displaystyle BT(d_{m})$ $\displaystyle=$ $\displaystyle
d_{k}\ln\left[\frac{1}{1-\sum_{p=m}^{k}i_{m}^{\prime}}\right]+(d_{k-1}-d_{k})\ln\left[\frac{1-i^{\prime}_{k}}{1-\sum_{p=m}^{k}i_{m}^{\prime}}\right]+...+$
$\displaystyle+$
$\displaystyle(d_{m+1}-d_{m})\ln\left[\frac{1-\sum_{p=m+1}^{k}i_{m}^{\prime}}{1-\sum_{p=m}^{k}i_{m}^{\prime}}\right]=BT(h_{m+1})+$
$\displaystyle+$
$\displaystyle\left(Nd_{m}+\sum_{p=m}^{k}(d_{p}-d_{m})i_{p}\right)\ln\left[\frac{1-\sum_{p=m+1}^{k}i_{m}^{\prime}}{1-\sum_{p=m}^{k}i_{m}^{\prime}}\right]$
Therefore,one obtains recursively that
$T(d_{m})=\sum_{p=m}^{k}\frac{1}{B}\left(Nd_{p}+\sum_{p^{\prime}=p}^{k}(d_{p}^{\prime}-d_{p})i_{p}^{\prime}\right)\ln\left[\frac{1-\sum_{p=m+1}^{k}i_{m}^{\prime}}{1-\sum_{p=m}^{k}i_{m}^{\prime}}\right]$
(B.1)
On the other hand dividing (3.10) by $N$ and proceeding to the limit
$N\rightarrow\infty$ with $k$ fixed one obtains exactly (B.1).
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|
arxiv-papers
| 2012-07-04T17:00:35 |
2024-09-04T02:49:32.645738
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Georgy Kitavtsev",
"submitter": "Georgy Kitavtsev",
"url": "https://arxiv.org/abs/1207.1057"
}
|
1207.1126
|
# Two New Zeta Constants: Fractal String, Continued Fraction, and
Hypergeometric Aspects of the Riemann Zeta Function
Stephen Crowley
###### Abstract.
The Riemann zeta function at integer arguments can be written as an infinite
sum of certain hypergeometric functions and more generally the same can be
done with polylogarithms, for which several zeta functions are a special case.
An analytic continuation formula for these hypergeometric functions exists and
is used to derive some infinite sums which allow the zeta function at integer
arguments $n$ to be written as a weighted infinite sum of hypergeometric
functions at $n-1$. The form might be considered to be a shift operator for
the Riemann zeta function which leads to the curious values
$\zeta^{F}(0)=I_{0}(2)-1$ and $\zeta^{F}(1)=\operatorname{Ei}(1)-\gamma$ which
involve a Bessel function of the first kind and an exponential integral
respectively and differ from the values $\zeta\left(0\right)=-\frac{1}{2}$ and
$\zeta\left(1\right)=\infty$ given by the usual method of continuation.
Interpreting these “hypergeometrically continued” values of the zeta constants
in terms of reciprocal common factor probability we have
$\zeta^{F}\left(0\right)^{-1}\cong 78.15\%$ and
$\zeta^{F}\left(1\right)^{-1}\cong 75.88\%$ which contrasts with the standard
known values for sensible cases like
$\zeta\left(2\right)^{-1}=\frac{6}{\pi^{2}}\cong 60.79\%$ and
$\zeta\left(3\right)^{-1}\cong 83.19\%$. The combinatorial definitions of the
Stirling numbers of the second kind, and the $2$-restricted Stirling numbers
of the second kind are recalled because they appear in the differential
equation satisfied by the hypergeometric representation of the polylogarithm.
The notion of fractal strings is related to the (chaotic) Gauss map of the
unit interval which arises in the study of continued fractions, and another
chaotic map is also introduced called the “Harmonic sawtooth” whose Mellin
transform is the (appropritately scaled) Riemann zeta function. These maps are
within the family of what might be called “deterministic chaos”. Some number
theoretic definitions are also recalled.
Email: stephen.crowley@mavs.uta.edu
###### Contents
1. 1 The Zeta Function
1. 1.1 Riemann’s $\zeta(t)$ Function
1. 1.1.1 The Generalized Hurwitz Zeta Function $\zeta(t,a)$
2. 1.1.2 Hypergeometric Representations of the Lerch Transcendent: $\Phi(z,t,v)$
3. 1.1.3 The Hypergeometric Polylogarithm
4. 1.1.4 The Differential Equation Solved by $\operatorname{Li}^{F}_{n}(t)$ and Some Combinatorics
5. 1.1.5 The “Hypergeometric Form” of the Zeta Function
2. 2 Number Theory, Continued Fractions, and Fractal Strings
1. 2.1 Fractal Strings and Dynamical Zeta Functions
2. 2.2 The Gauss Map $h(x)$
1. 2.2.1 The Frobenius-Perron Transfer Operator
2. 2.2.2 Piecewsise Integration of $h(x)$
3. 2.2.3 The Mellin Transform
3. 2.3 The Harmonic Sawtooth w(x)
4. 2.4 The Prime Numbers
1. 2.4.1 The Prime Counting Function: $\pi(x)$
2. 2.4.2 von Mangoldt and Chebyshev’s Functions: $\Lambda(x),\theta(x),\text{$\psi(x)$}$
3. 3 Analytic Continuation
1. 3.1 Continuation of ${}_{n+1}F_{n}$ Near Unit Argument
2. 3.2 The Continuation of $\operatorname{Li}_{n}^{F}(t)\operatorname{and}\zeta^{F}(n)$ via Contiguous Functions
1. 3.2.1 $\operatorname{Li}_{1}^{F}(t)\rightarrow\operatorname{Li}_{2}^{F}(t)$ and $\zeta_{1}^{F}(t)\rightarrow\zeta_{2}^{F}(t)$
2. 3.2.2 $\zeta^{F}(2)\rightarrow\zeta^{F}(3)$
3. 3.2.3 $\zeta^{F}(3)\rightarrow\zeta^{F}(4)$
4. 4 Appendix
1. 4.1 $\operatorname{The}\operatorname{Generalized}\operatorname{Hypergeometric}\operatorname{Function}:_{p}F_{q}$
1. 4.1.1 The Differential Equation and Convergence
2. 4.1.2 Contiguous Functions and Linear Relations
2. 4.2 Other Special Functions
1. 4.2.1 Polygamma $\Psi^{(n)}(x)$
3. 4.3 Notation
## 1\. The Zeta Function
### 1.1. Riemann’s $\zeta(t)$ Function
Riemann’s zeta function, named after Bernhard Riemann(1826-1866), is defined
by
(1)
$\begin{array}[]{lll}\zeta(t)&=\sum_{n=1}^{\infty}n^{-t}&\forall\\{t\in\mathbbm{C}:\mathcal{R}(t)>1\\}\\\
&=\frac{1}{1-2^{-t}}\sum_{n=0}^{\infty}(2n+1)^{-t}&\forall\\{t\in\mathbbm{C}:\mathcal{R}(t)>1\\}\\\
&=\left(1-2^{-t}\right)\eta\left(t\right)&\forall\\{t\in\mathbbm{C}:\mathcal{R}(t)>0\\}\end{array}$
where $\mathcal{R}(t)$ and $\mathcal{I}(t)$ denote real and imaginary parts of
$t$ respectively and $\eta\left(t\right)$ is the Dirichlet eta function, also
known as the alternating zeta function, named after Johann
Dirichlet(1805-1859)
(2)
$\begin{array}[]{lll}\eta\left(t\right)&=\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n^{t}}&\forall\\{t\in\mathbbm{C}:\mathcal{R}(t)>0\\}\\\
&=\frac{1}{\Gamma\left(s\right)}\int
x^{s-1}\frac{1}{e^{x}+1}\mathrm{d}x&\forall\\{t\in\mathbbm{C}:\mathcal{R}(t)>0\\}\end{array}$
where the integral is a Mellin transform of $\left(e^{x}+1\right)^{-1}$. The
function $\zeta(t)$ is analytic and uniformly convergent when
$\mathfrak{R}(t)>1$ or $\mathfrak{R}(t)>0$ when using the eta function form.
The only singularity of $\zeta(t)$ is at $t=1$ where it becomes the divergent
harmonic series. The reflection functional equation [48, 13.151] which relates
$\zeta(t)$ to $\zeta(1-t)$ is given by
(3)
$\zeta(t)\pi^{-t}2^{1-t}\Gamma(t)\cos\left(\frac{t\pi}{2}\right)=\zeta(1-t)$
The interpretation of zeta in terms of frequentist probability is that given
$n$ integers chosen at random, the probability that no common factor will
divide them all is $\zeta\left(n\right)^{-1}$. In other words, given an array
$i$ of $n$ random intgers, $\zeta\left(n\right)^{-1}$ is the probabability
that $\gcd\left(i_{1},i_{2},\ldots,i_{n}\right)=1$ where $\gcd$is the greatest
common denominator function. So for example, the probability that a pair of
randomly chosen integers is coprime is
$\zeta\left(2\right)^{-1}=\frac{6}{\pi^{2}}\cong 60.79\%$, and the probability
that a triplet of randomly chosen integers is relatively prime is
$\zeta\left(3\right)^{-1}\cong 83.19\%$. [37][48, 13.1][7, 1.4]
#### 1.1.1. The Generalized Hurwitz Zeta Function $\zeta(t,a)$
A more general function which includes Riemann’s Zeta function was defined by
A. Hurwitz.
(4) $\begin{array}[]{ll}\zeta(t,a)&=\end{array}\sum_{n=0}^{\infty}(n+a)^{-t}$
Notice that the summation starts at $n=0$ whereas Riemann’s starts at $n=1$.
It is apparent that $\zeta(t)$ is a special case of $\zeta(t,a)$ where
(5)
$\zeta(t)=\sum_{n=1}^{\infty}n^{-t}=\zeta(t,1)=\sum_{n=0}^{\infty}(n+1)^{-t}$
[14][48, 13.11]
#### 1.1.2. Hypergeometric Representations of the Lerch Transcendent:
$\Phi(z,t,v)$
The Lerch transcendent $\Phi(z,t,v)$ [10, 1.11] is a further generalization of
the Hurtwitz zeta function
(6)
$\begin{array}[]{ll}\Phi(z,t,v)&=\sum_{n=0}^{\infty}\frac{z^{n}}{(v+n)^{t}}\end{array}$
valid $\forall|z|<1$ or $\\{|z|=1:\mathcal{R}(t)>1\\}$ which is related to
$\zeta(t,v)$ and $\zeta(t)$ by
(7) $\begin{array}[]{lll}\Phi(1,t,v)&=\zeta(t,v)&\\\ \Phi(1,t,1)&=\zeta(t)&\\\
\Phi(1,t,1/2)&=\zeta(t,1/2)&=(2^{t}-1)\zeta(t)\end{array}$
When $t=1$ the Lerch transcendent reduces to
(8)
$\begin{array}[]{ll}\Phi(z,1,v)&=\frac{{}_{2}F_{1}\left(\begin{array}[]{ll}1&v\\\
1+v&\end{array}|z\right)}{v}\end{array}$
and when $n\in\\{0,1,2,\ldots.\\}$, $\Phi(z,n,v)$ has the hypergeometric
representation [19]
(9)
$\begin{array}[]{ll}\Phi(z,n,v)&=\end{array}v^{-n}_{n+1}F_{n}\left(\begin{array}[]{ll}1&\vec{v}_{n}\\\
&\overrightarrow{1+v}_{n}\end{array}|z\right)$
yielding
(10)
$\begin{array}[]{ll}\zeta(n,v)&=v^{-n}_{n+1}F_{n}\left(\begin{array}[]{ll}1&\vec{v}_{n}\\\
&\overrightarrow{1+v}_{n}\end{array}\right)\end{array}$
and
(11)
$\begin{array}[]{ll}\zeta(n)&=\left(\frac{2^{n}}{2^{n}-1}\right)_{n+1}F_{n}\left(\begin{array}[]{ll}1&\overrightarrow{1/2}_{n}\\\
&\overrightarrow{3/2}_{n}\end{array}\right)\\\
&=_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}\right)\end{array}$
and thus due to (1) and (7) we have the hypergeometric transformation
(12) ${}_{n+1}F_{n}\left(\begin{array}[]{ll}1&\overrightarrow{1/2}_{n}\\\
&\overrightarrow{3/2}_{n}\end{array}\right)=(1-2^{-n})_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}\right)$
where the argument absent in ${}_{p}F_{q}$ is assumed to be 1 and the symbol
$\vec{c}_{n}$ denotes a parameter vector of length $n$ where each element is
equal to $c$ (e.g. $\vec{5}_{3}=[5,5,5]$).
#### 1.1.3. The Hypergeometric Polylogarithm
The polylogarithm, also known as Jonquière’s function, is defined
$\forall|t|\leqslant 1,n\in\\{0,1,2,\ldots\\}$ by
(13)
$\begin{array}[]{lll}\operatorname{Li}_{n}(t)&=\sum_{k=1}^{\infty}\frac{t^{k}}{k^{n}}&\\\
&=_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}|t\right)t&\\\
&=t\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\frac{\prod_{i=1}^{n+1}(1)_{k}}{\prod_{j=1}^{n}(2)_{k}}&\\\
&=t\sum_{k=0}^{\infty}t^{k}\frac{(1)_{k}^{n+1}}{(2)_{k}^{n}}&\\\
&=t\sum_{k=0}^{\infty}t^{k}\frac{\Gamma(k+1)^{n}}{\Gamma(k+2)^{n}}&\\\
&=t\sum_{k=0}^{\infty}\frac{t^{k}}{(1+k)^{n}}&\end{array}$
The hypergeometric representation (116) of
$\operatorname{Li}_{n}(t)=_{n+1}F_{n}\left(\begin{array}[]{l}a_{1}\ldots
a_{n+1}\\\ b_{1}\ldots
b_{n}\end{array}|t\right)t=\operatorname{Li}_{n}^{F}(t$) where $a_{1}\ldots
a_{n+1}=\vec{1}_{n+1}$ and $b_{1}\ldots b_{n}=\vec{2}_{n}$ is nearly-poised of
the first kind [41, 2.1.1] since $a_{1}+b_{1}=3=\ldots=a_{n}+b_{n}=3$. The
notation $\operatorname{Li}_{n}^{F}(t)$ refers specifically the hypergeometric
form of $\operatorname{Li}_{n}(t$). The derivatives and integrals of
$\operatorname{Li}_{n}(t$) satisfy the recurrence relations
(14)
$\begin{array}[]{ll}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Li}_{n}(t)&=\frac{\operatorname{Li}_{n-1}(t)}{t}\\\
\frac{\mathrm{d}}{\mathrm{d}t}_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}|t\right)t&=_{n}F_{n-1}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}|t\right)\end{array}$ (15)
$\begin{array}[]{ll}\int_{0}^{t}\frac{\operatorname{Li}_{n}(s)}{s}\mathrm{d}s&=\operatorname{Li}_{n+1}(t)\\\
\int_{0}^{t}{{}_{n+1}F_{n}}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}|s\right)\mathrm{d}s&={{}_{n+2}F_{n+1}}\left(\begin{array}[]{l}\vec{1}_{n+2}\\\
\vec{2}_{n+1}\end{array}|t\right)t\end{array}$
and the reflection functional equation for $\operatorname{Li}_{n}(1)=\zeta(n)$
is
(16)
$\begin{array}[]{ll}\operatorname{Li}_{n}(1)&=\frac{\operatorname{Li}_{n}(-1)}{(2^{1-n}-1)}\end{array}$
$\operatorname{Li}_{n}^{F}(t)$ is seen to be ($n-1$)-balanced (117) with the
trivial calculation
(17) $\sum_{k=1}^{n}2-\sum_{k=1}^{n+1}1=2n-(n+1)=n-1$
The usual defintion of $\operatorname{Li}_{n}(t)$ requires analytic
continuation at $t=1$ but this is not necessary because the hypergeometric
function converges absolutely on the unit circle when it is at least
$1$-balanced (117) which is true $\forall n\geqslant 2$. The only
Saalschützian polylogarithm is $\operatorname{Li}_{2}(t$) [32, Eq3.8] [20,
25:12][26, 1.4.2]
#### 1.1.4. The Differential Equation Solved by
$\operatorname{Li}^{F}_{n}(t)$ and Some Combinatorics
Some combinatorial functions need to be defined before writing the
differential equation solved by $\operatorname{Li}_{n}^{F}(t)$. Let a
partition be an arrangement of the set of elements $1,\ldots,k$ into $n$
subsets where each element is placed into exactly one set. The number of
partitions of the set $1,\ldots,k$ into $n$ subsets is given by the Stirling
numbers of the second kind [2, 1.1.3][38, 2.7] defined by
(18) $\begin{array}[]{ll}\left\\{\begin{array}[]{l}k\\\
n\end{array}\right\\}&=\sum_{j=0}^{n}\frac{j^{k}}{n!(-1)^{j-n}}\left(\begin{array}[]{l}n\\\
j\end{array}\right)\\\
&=\sum_{j=0}^{n}\frac{j^{k}(-1)^{n-j}}{\Gamma(j+1)\Gamma(n-j+1)}\\\
&=\frac{(-1)^{n+1}}{\Gamma(n)}_{k}F_{k-1}\left(\begin{array}[]{l}1-n,\vec{2}_{k-1}\\\
\vec{1}_{k-1}\end{array}\right)\end{array}$
The ${}_{k}F_{k-1}$ representation of $\left\\{\begin{array}[]{l}k\\\
n\end{array}\right\\}$ is ($n-k$)-balanced (117) since
$(k-1)-((1-n)+2(k-1))=n-k$. The $r$-restricted Stirling numbers of the second
kind $\left\\{\begin{array}[]{l}k\\\ n\end{array}\right\\}_{r}$, or simply the
$r$-Stirling numbers, counts the number of partitions of the set 1,…,n into
$k$ subsets with the restriction that the numbers $1,\ldots,r$ belong to
distinct subsets. [29] The recursion satisfied by
$\left\\{\begin{array}[]{l}k\\\ n\end{array}\right\\}_{r}$ is given by
(19) $\begin{array}[]{ll}\left\\{\begin{array}[]{l}k\\\
n\end{array}\right\\}_{r}&=\left\\{\begin{array}[]{ll}0&k<r\\\
\delta_{n,r}&k=r\\\ n\left\\{\begin{array}[]{l}k-1\\\
n\end{array}\right\\}_{r}+\left\\{\begin{array}[]{l}k-1\\\
n-1\end{array}\right\\}_{r}&n>r\end{array}\right.\end{array}$
where $\delta_{n,m}=\left\\{\begin{array}[]{ll}1&n=m\\\ 0&n\neq
m\end{array}\right.$ is the Kronecker delta. Specifically, the $2$-restricted
Stirling numbers[15, A143494] appearing in the differential equation for
$\operatorname{Li}_{n}^{F}(t)$ are given by
(20) $\begin{array}[]{lll}\left\\{\begin{array}[]{l}k\\\
n\end{array}\right\\}_{2}&&=\left\\{\begin{array}[]{l}k\\\
n\end{array}\right\\}-\left\\{\begin{array}[]{l}k-1\\\
n\end{array}\right\\}\\\
&&=\frac{1}{(k-2)!}\sum_{j=0}^{k-2}(-1)^{j-k}\left(\begin{array}[]{c}k-2\\\
j\end{array}\right)(j+2)^{n-2}\\\
&&=(-1)^{k}\sum_{j=0}^{k-2}\frac{(j+2)^{n-2}(-1)^{j}}{j!(k-2-j)!}\end{array}$
The ($n+1$)-th order hypergeometric differential equation (119) satisfied by
f(t)=$\operatorname{Li}^{F}_{n}(t)$ (13)
(21)
$0=\left\\{\begin{array}[]{ll}f(t)+\frac{\mathrm{d}}{\mathrm{d}t}f(t)(t^{2}-t)&n=0\\\
\frac{\mathrm{d}}{\mathrm{d}t}f(t)+\sum_{m=2}^{n+1}\left(\frac{\mathrm{d}}{\mathrm{d}t^{m}}f(t)\right)\left(t^{m-1}\left\\{\begin{array}[]{l}n+1\\\
m\end{array}\right\\}-t^{m-2}\left\\{\begin{array}[]{l}n+1\\\
m\end{array}\right\\}_{2}\right)&n\geqslant 1\end{array}\right.$
has a most general solution of the form
(22)
$\begin{array}[]{l}\begin{array}[]{l}f\left(t\right)=x+yG_{n}(t)+\sum_{m=1}^{n-1}z_{m}\ln(t)^{m}\end{array}\end{array}$
where $x,y,z_{1},\ldots,z_{n-1}$ are arbitrary parameters and $G_{n}(t)$
satifies the recursion
(23)
$\begin{array}[]{ll}G_{n}(t)&=\left\\{\begin{array}[]{ll}\frac{t}{1-t}&n=0\\\
\ln(t-1)&n=1\\\ \operatorname{Li}_{2}(1-t)+\ln(t-1)\ln(t)&n=2\\\
\int\frac{G_{n-1}(t)}{t}\mathrm{d}t&n\geqslant 3\end{array}\right.\end{array}$
which has the explicit solution
(24)
$\begin{array}[]{ll}G_{n}(t)&=\left\\{\begin{array}[]{ll}\frac{t}{1-t}&n=0\\\
\ln(t-1)&n=1\\\ \operatorname{Li}_{2}(1-t)+\ln(t-1)\ln(t)&n=2\\\
\frac{(\ln(t-1)-\ln(1-t))\ln(t)^{n-1}}{\Gamma(n)}+\frac{\pi^{2}}{6}\frac{\ln(t)^{n-2}}{\Gamma(n-1)}-\operatorname{Li}_{n}(t)&n>2\end{array}\right.\end{array}$
The indicial equation of (21) at the $t=1$ is
(25)
$\begin{array}[]{l}\operatorname{ind}(\operatorname{Li}_{n}^{F}(t))=-\frac{t(-1)^{n-1}\Gamma(n-1-t)(t-n+1)^{2}}{\Gamma(1-t)}\end{array}$
The ($n+1$) roots of $\operatorname{ind}(\operatorname{Li}_{n}^{F}(t))$ are
the exponents of (21) which are simply
(26)
$\left\\{t:\operatorname{ind}(\operatorname{Li}_{n}^{F}(t))=0\\}=0,1,\ldots,n-1,n-1\right.$
where the last root $n-1$ of
$\operatorname{ind}(\operatorname{Li}_{n}^{F}(t))$ is the balance of
$\operatorname{Li}_{n}^{F}(t)$ (17) having multiplicity 2 thus inducing the
logarithmic terms of (22). [16, 15.31 and 16.33] These equations were derived
by writing the differential equation for increasing values of $n$ and then
noticing that the developing pattern of coefficients were combinatorial. After
deriving the general combinatorial differential equation, it was solved for
increasing values of $n$ which resulted in nested integrals of prior solutions
and then the general solution was derived from that pattern.
#### 1.1.5. The “Hypergeometric Form” of the Zeta Function
The main focus will be on the special case $\operatorname{Li}^{F}_{n}(t)$ at
unit argument where it coincides with the Riemann Zeta function at the
integers. As with $\operatorname{Li}^{F}_{n}(t)$, the symbol $\zeta^{F}(n)$
refers specifically to the hypergeometric representation of $\zeta(n)$ at non-
negative integer values of $n$. Using (5) and (13), it can easily be seen that
$\zeta(n$) can be expressed as a generalized hypergeometric function (116)
with
(27) $\begin{array}[]{ll}\zeta^{F}(n)&=\operatorname{Li}^{F}_{n}(1)\\\
&={{}_{n+1}F_{n}}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}\right)\\\
&=\sum_{k=0}^{\infty}\frac{1}{k!}\frac{\prod_{i=1}^{n+1}(1)_{k}}{\prod_{j=1}^{n}(2)_{k}}\\\
&=\sum_{k=0}^{\infty}\frac{(1)^{n+1}_{k}}{k!(2)^{n}_{k}}\\\
&=\sum_{k=0}^{\infty}\frac{\Gamma(k+1)^{n}}{\Gamma(k+2)^{n}}\\\
&=\sum_{k=0}^{\infty}(k+1)^{-n}\\\ &=\zeta(n,1)\\\ &=\zeta(n)\end{array}$
The value $\zeta^{F}(0)=_{1}F_{0}\left(\begin{array}[]{l}1\\\
\end{array}|1\right)$ is singular and so must be calculated with the
reflection equation (16) to get
$\operatorname{Li}^{F}_{0}(-1)=_{1}F_{0}\left(\begin{array}[]{l}1\\\
\end{array}|-1\right)=-\frac{1}{2}=\zeta(0)$ which agrees with the integral
form of $\zeta(t)\forall t\neq 1$
(28)
$\begin{array}[]{lllll}\left.\zeta(t)\right|_{t=0}&\left.=\left(\frac{1}{2}+\frac{1}{t-1}+2\int_{0}^{\infty}\frac{\sin(s\arctan(s))(1+s^{2})^{-\frac{s}{2}}}{e^{2\pi
s}-1}\mathrm{d}s\right)\right|_{t=0}&=&\operatorname{Li}_{0}(-1)=-\frac{1}{2}&\end{array}$
## 2\. Number Theory, Continued Fractions, and Fractal Strings
### 2.1. Fractal Strings and Dynamical Zeta Functions
A fractal string is defined as a nonempty open subset of the real line
$\Omega\subseteq\mathbbm{R}$ which can be expressed as a disjoint union of
open intervals $I_{j}$ being the connected components of $\Omega$. [25,
3.1][30][23][22][11][24]
(29) $\begin{array}[]{ll}\Omega&=\bigcup_{j=1}^{\infty}I_{j}\end{array}$
The length of the $j$-th interval $I_{j}$ will be denoted $\ell_{j}$. It will
be assumed that $\Omega$ is standard, meaning that its length is finite, and
that $\ell_{j}$ is a nonnegative monotically decreasing sequence.
(30)
$\begin{array}[]{l}|\Omega|_{d}=\sum_{j=1}^{\infty}(\ell_{j})^{d}<\infty\exists
d>0\\\
\ell_{1}\geqslant\ell_{2}\geqslant\ldots\geqslant\text{$\ell_{j}\geqslant\ell_{j+1}\geqslant\cdots\geqslant
0$}\end{array}$
where $\exists d>0$ means there is at least one value of $d$ for which the
statement is true. It can be the case that $\ell_{j}=0$ for some $j$ in which
case $\ell_{j}$ is a finite sequence. The sequence of lengths of the
components of the fractal string is denoted by
(31)
$\begin{array}[]{ll}\mathcal{L}&=\\{\ell_{j}\\}_{j=1}^{\infty}\end{array}$
The boundary of $\Omega$ in $\mathbbm{R}$ will be denoted by
$\partial\Omega=K\subset\Omega$ which will also denote the boundary of
$\mathcal{L}$. Any totally disconnected bounded perfect subset
$K\subset\mathbbm{R}$, or generally, any compact subset $K\subset\mathbbm{R}$,
can be represented as a string of finite length $|\Omega|_{1}$. A subset $K$
of a topological space $\Omega$ is said to be perfect if it is closed and each
of its points is a limit point. Since here $\Omega$ is a metric space and
$K\subset\Omega$ is closed, the Cantor-Bendixon lemma states that there exists
a perfect set $P\subset K$ such that $K-P$ is a most countable. [35, 2.2 Ex17]
As such, $\Omega$ can be defined as the complenent of $K$ in its closed convex
hull, that is, $\Omega=\Omega(K)$ is the smallest compact interval $[a,b]$
containing $K$. The connected components of the bounded open set
$\Omega=(a,b)\backslash K$ are the intervals $I_{j}$ of the fractal string
$\mathcal{L}$ associated with $K$.
### 2.2. The Gauss Map $h(x)$
Let $\Omega_{h}=(0,1)\backslash\partial\Omega_{h}$ where
$\partial\Omega_{h}=\left\\{\pm\infty,0,\frac{1}{n}:n\in\mathbbm{Z}\right\\}$
is the set of discontinous boundary points of the Gauss map
$h(x)\in\Omega_{h}\forall x\not\in\partial\Omega_{h}$, also known as the Gauss
function or Gauss transformation, which maps unit intervals onto unit
intervals and by iteration gives the continued fraction expansion of a real
number
(32)
$\begin{array}[]{ll}h(x)&=\frac{1}{x}-\left\lfloor\frac{1}{x}\right\rfloor\\\
&=-\frac{\left\lfloor\frac{1}{x}\right\rfloor x-1}{x}\\\ &=\\{x^{-1}\\}\\\
&=\frac{1}{x}\operatorname{mod}1\end{array}$
where $\left\lfloor x\right\rfloor$ is the floor function, the greatest
integer$\leqslant x$ and $\\{x\\}=x-\left\lfloor x\right\rfloor$ is the
fractional part of $x$. [13, 2.1,3.9.1,9.1,9.3,9.7.1][42, A.1.7] Clearly
$h(x)$ is also defined outside of $\Omega$
(33) $h(x)=\left\\{\begin{array}[]{ll}\frac{1}{x}&x>1\\\ h(x)&-1\leqslant
x\leqslant 1\\\ \frac{1}{x}+1&x<-1\end{array}\right.$
since
(34)
$\begin{array}[]{l}\end{array}\left\lfloor\frac{1}{x}\right\rfloor=\left\\{\begin{array}[]{ll}0&x>1\\\
\left\lfloor\frac{1}{x}\right\rfloor&-1\leqslant x\leqslant 1\\\
-1&x<-1\end{array}\right.$
As can be seen in Figure 1, $h(x)$ is discontinuous at a countably infinite
set of points of Lebesgue measure zero on its boundary $\partial\Omega_{h}$
(35) $\text{$\left\\{y:\lim_{x\rightarrow y^{-}}h(x)\neq\lim_{x\rightarrow
y^{+}}h(x)\right\\}=$}\partial\Omega_{h}=\left\\{\pm\infty,0,\frac{1}{n}:n\in\mathbbm{Z}\right\\}$
The left and right limits of $h(x)$ when $x$ approaches an element on the
boundary $\partial\Omega_{h}$ is given by
(36) $\begin{array}[]{lll}\lim_{x\rightarrow\partial\Omega^{-}}h(x)&=0&\\\
\lim_{x\rightarrow\partial\Omega^{+}}h(x)&=1&\end{array}$
Figure 1. The Gauss Map
#### 2.2.1. The Frobenius-Perron Transfer Operator
The Frobenius-Perron transfer operator [42, 2.4.4][31, 2.3.3][13,
1.3.1,8.2][40, 1.8,2.4] of a unit interval mapping $f(y)$ describes how a
probablility density $\rho(y)$ transforms under the action of the map.
(37)
$\begin{array}[]{ll}{}[U_{f}\rho](x)&=\int\delta(x-f(y))\rho(y)\mathrm{d}y\end{array}$
where $\delta$ is the Dirac delta function and $\theta$ is the Heaviside step
function.
(38) $\begin{array}[]{ll}\text{$\int\delta(x)\mathrm{d}x$}&=\theta(x)\\\
\theta(x)&=\left\\{\begin{array}[]{ll}0&x<0\\\ 1&x\geqslant
0\end{array}\right.\end{array}$
The function $f(y)$ is the map being iterated and $\rho(y)$ is some density on
which the transfer operator $U$ acts. Essentially, iteration of the map
transforms points to points and iteration of the transfer operator maps point
densities to point densities. The Gauss-Kuzmin-Wirsing(GKW) operator is
obtained by applying the transfer operator to the Guass map. [44, 2] [50] [46]
(39)
$\begin{array}[]{ll}{}[U_{h}\rho](x)&=\sum_{n=1}^{\infty}\frac{\rho\left(\frac{1}{n+x}\right)}{(n+x)^{2}}\end{array}$
By changing the variables and order of integration in (65) an operator
equation for $\zeta(s$) is obtained.
(40)
$\begin{array}[]{ll}\zeta(s)&=\frac{s}{s-1}-s\int_{0}^{1}x[U_{h}x^{s-1}]\mathrm{d}x\\\
&=\frac{s}{s-1}-s\int_{0}^{1}x\int\delta(x-h(y))y^{s-1}\mathrm{d}y\mathrm{d}x\\\
&=\frac{s}{s-1}-s\int_{0}^{1}x\int\delta(x-\left(y^{-1}-\left\lfloor
y^{-1}\right\rfloor\right))y^{s-1}\mathrm{d}y\mathrm{d}x\\\
&=\frac{s}{s-1}-s\int_{0}^{1}x\sum_{n=1}^{\infty}\frac{\left(\frac{1}{n+x}\right)^{s-1}}{(n+x)^{2}}\mathrm{d}x\end{array}$
An operator similiar to (39) is
(41)
$\begin{array}[]{ll}{}[S_{h}\rho](x)&=\sum_{n=1}^{\infty}\rho\left(\frac{1}{n}\right)-\rho\left(\frac{1}{n+x}\right)\end{array}$
The action of $[U_{h}\rho](x)$ on the identity function $x\rightarrow x$ is
given by
(42)
$\begin{array}[]{ll}{}[U_{h}x](x)&=\sum_{n=1}^{\infty}\frac{\frac{1}{n+x}}{(n+x)^{2}}\\\
&=\sum_{n=1}^{\infty}\frac{1}{(n+x)^{3}}\\\ &=-\frac{\Psi^{(2)}(x+1)}{2}\\\
&=\frac{{}_{4}F_{3}\left(\begin{array}[]{llll}1&x+1&x+1&x+1\\\
&x+2&x+2&x+2\end{array}\right)}{(x+1)^{3}}\end{array}$
where $\Psi^{(n)}(x)$ is the polygamma function (122). The area under the
curve of $[U_{h}x](x)$ is
(43)
$\begin{array}[]{l}\int_{0}^{1}\end{array}[U_{h}x](x)\mathrm{d}x=\begin{array}[]{l}\int_{0}^{1}\end{array}-\frac{\Psi^{(2)}(x+1)}{2}\mathrm{d}x=\frac{1}{2}$
The identity action of $[S_{h}\rho](x)$ is
(44)
$\begin{array}[]{ll}{}[S_{h}x](x)&=\sum_{n=1}^{\infty}\frac{1}{n}-\frac{1}{n+x}\\\
&=\gamma+\Psi(x+1)\end{array}$
where $\gamma$ is Euler’s constant
(45)
$\begin{array}[]{ll}\gamma&=\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{1}{k}-\ln(n)\\\
&=\lim_{s\rightarrow 1}\zeta(s)-\frac{1}{s-1}\\\ &=\lim_{s\rightarrow
1}\zeta(s)+\int_{1}^{\infty}h(x)x^{s-1}\mathrm{d}x\\\ &=\lim_{s\rightarrow
1}\frac{1}{s-1}-s\int_{0}^{1}h(x)x^{s-1}\mathrm{d}x+\int_{1}^{\infty}h(x)x^{s-1}\mathrm{d}x\\\
&=1-\int_{0}^{1}h(x)\mathrm{d}x\\\ &\approx
0.57721566490153286060651209\end{array}$
and the area under its curve is given by
(46)
$\begin{array}[]{ll}\int_{0}^{1}[S_{h}x](x)\mathrm{d}x&=\int_{0}^{1}\gamma+\Psi(x+1)\mathrm{d}x=1-\gamma\end{array}$
#### 2.2.2. Piecewsise Integration of $h(x)$
The Guass map $h(x)\in\Omega_{h}$ is piecewise monotone [40, 2.1] between the
points of $\partial\Omega_{h}$, and thus partitions the unit interval infinite
covering set of decreasing open intervals seperated by $\partial\Omega_{h}$.
[13, 5.7.1] Let $I_{n}$ be an infinite set of open intervals
(47) $\begin{array}[]{ll}I_{n}&=\left\\{\begin{array}[]{ll}(1,\infty)&n=0\\\
\left(\frac{1}{n+1},\frac{1}{n}\right)&0<n<\infty\\\
(0,0)=\emptyset&n=\infty\end{array}\right.\end{array}$
It is easy to see that
(48)
$\begin{array}[]{lll}\Omega_{h}\cup\partial\Omega_{h}=&[0,1]&=\text{$\bigcup_{n=1}^{\infty}I_{n}$}\\\
&[0,\infty]&=\text{$\bigcup_{n=0}^{\infty}I_{n}$}\end{array}$
Define the Gauss map partition $h_{n}(x)$ where $\\{h_{n}(x)\neq 0:x\in
I_{n}\\}$ as a piecewise step function
(49)
$\begin{array}[]{ll}h_{n}(x)&=\left\\{\begin{array}[]{ll}\frac{1-xn}{x}&\frac{1}{n+1}<x<\frac{1}{n}\\\
0&\operatorname{otherwise}\end{array}\right.\\\
&=\left(\frac{1-xn}{x}\right)\left(\theta\left(\frac{xn+x-1}{n+1}\right)-\theta\left(\frac{xn-1}{n}\right)\right)\end{array}$
where $\theta(t)$ is the Heaviside step function (38). We can reassemble all
of the $\\{h_{n}(x)\\}_{n=1}^{\infty}$ to recover $h(x$)
(50) $\begin{array}[]{ll}h(x)&=\sum_{n=1}^{\infty}h_{n}(x)\\\
&=\sum_{n=1}^{\infty}\left(\frac{1-xn}{x}\right)\left(\theta\left(\frac{xn+x-1}{n+1}\right)-\theta\left(\frac{xn-1}{n}\right)\right)\end{array}$
where only one of the $h_{n}(x)$ is $\operatorname{nonzero}$ for each $x$. By
setting $n=\left\lfloor\frac{1}{x}\right\rfloor$ in (49) we get
(51)
$\begin{array}[]{ll}h(x)&=\left\\{\begin{array}[]{ll}\frac{1-x\left\lfloor\frac{1}{x}\right\rfloor}{x}&\frac{1}{\left\lfloor\frac{1}{x}\right\rfloor+1}<x<\frac{1}{\left\lfloor\frac{1}{x}\right\rfloor}\\\
0&\operatorname{otherwise}\end{array}\right.\\\
&=\left(\frac{1-x\left\lfloor\frac{1}{x}\right\rfloor}{x}\right)\left(\theta\left(\frac{x\left\lfloor\frac{1}{x}\right\rfloor+x-1}{\left\lfloor\frac{1}{x}\right\rfloor+1}\right)-\theta\left(\frac{x\left\lfloor\frac{1}{x}\right\rfloor-1}{\left\lfloor\frac{1}{x}\right\rfloor}\right)\right)\end{array}$
Define the partitioned integral operator $\left[Pf(x);x\right](n)$ by
(52)
$\begin{array}[]{lllll}\text{$\left[Pf(x);x\right](n)$}&=\text{$\left[Pf(x)\right](n)$}&=\left[Pf\right](n)&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}f(x)\mathrm{d}x&=\int_{I_{n}}f(x)\mathrm{d}x\end{array}$
where by convention we have
(53) $\begin{array}[]{llll}\left[Pf(x)\right](0)&=\lim_{n\rightarrow
0^{+}}\int_{I_{n}}f(x)\mathrm{d}x&=\int_{1}^{\infty}f(x)\mathrm{d}x&\\\
\left[Pf(x)\right](\infty)&=\lim_{n\rightarrow\infty}\int_{I_{n}}f(x)\mathrm{d}x&=\int_{0}^{0}f(x)\mathrm{d}x&=0\end{array}$
Thus
(54)
$\begin{array}[]{lll}\int_{0}^{1}f(x)\mathrm{d}x&=\sum_{n=1}^{\infty}\left[Pf(x)\right](n)&=\sum_{n=1}^{\infty}\int_{I_{n}}f(x)\mathrm{d}x\\\
\int_{0}^{\infty}f(x)\mathrm{d}x&=\sum_{n=0}^{\infty}\left[Pf(x)\right](n)&=\sum_{n=0}^{\infty}\int_{I_{n}}f(x)\mathrm{d}x\end{array}$
Each interval $I_{n}$ has the length
(55) $\begin{array}[]{ll}\ell I_{n}&=\left[P1\right](n)\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}1\mathrm{d}x\\\
&=\frac{1}{n}-\frac{1}{n+1}\\\ &=\frac{1}{n(n+1)}\end{array}$
The elements $n(n+1)$ are known as the oblong numbers [15, A002378]. It is
seen, together with (44), that
(56) $\begin{array}[]{ll}|\Omega_{h}|&=\int_{0}^{1}1\mathrm{d}x\\\
&=\sum_{n=1}^{\infty}\ell I_{n}\\\ &=\sum_{n=1}^{\infty}\frac{1}{n(n+1)}\\\
&=[S_{h}x](1)\\\ &=\gamma+\Psi(2)\\\ &=1\end{array}$
The piecewise integral operator $\left[Pf(x);x\right](n)$ can be used to
calculate the area under the curve of $h(x)$ which is also equal to the area
under the curve of $[S_{h}x](x)$. Let the length of the $n$-th component
$h_{n}(x)$ be denoted by
(57) $\begin{array}[]{ll}\ell h_{n}&=\left[Ph(x);x\right](n)\\\
&=\int_{I_{n}}h(x)\mathrm{d}x\\\ &=\int_{I_{n}}h_{n}(x)\mathrm{d}x\\\
&=\int_{0}^{1}h_{n}(x)\mathrm{d}x\\\
&=\frac{\ln(n+1)n+\ln(n+1)-\ln(n)n-\ln(n)-1}{n+1}\end{array}$
Regarding $h(x)$ as a fractal string
$\mathcal{L}_{h}=\\{h_{n}(x)\\}_{n=1}^{\infty}$ its length $|\mathcal{L}_{h}|$
is given by
(58)
$\begin{array}[]{ll}\text{$|\mathcal{L}_{h}|$}&=\int_{0}^{1}h(x)\mathrm{d}x\\\
&=\sum_{n=1}^{\infty}\ell h_{n}\\\
&=\sum_{n=1}^{\infty}\frac{\ln(n+1)n+\ln(n+1)-\ln(n)n-\ln(n)-1}{n+1}\\\
&=\int_{0}^{1}[S_{h}x](x)\mathrm{d}x\\\
&=\int_{0}^{1}\gamma+\Psi(x+1)\mathrm{d}x\\\ &=1-\gamma\end{array}$
If $n=0$ in (48) we get the interval
$\left.I_{0}=\left(\frac{1}{1},\frac{1}{0}\right)=(1,\infty\right)$ and
(59) $\begin{array}[]{ll}\ell
h_{0}=&\int_{I_{0}}h(x)\mathrm{d}x=\int_{1}^{\infty}\frac{1}{x}\mathrm{d}x=\infty\end{array}$
but if we choose a finite cutoff then
(60)
$\begin{array}[]{ll}\int_{1}^{y}h(x)\mathrm{d}x&=\int_{1}^{y}\frac{1}{x}\mathrm{d}x\\\
&=\ln(y)\end{array}$
and
(61)
$\begin{array}[]{l}\int_{1}^{\infty}\end{array}\frac{\ln(y)}{y^{n}}\mathrm{d}y=\frac{1}{(n-1)^{2}}$
thus
(62)
$\begin{array}[]{ll}\sum_{n=2}^{\infty}\begin{array}[]{l}\int_{1}^{\infty}\end{array}\frac{\int_{1}^{y}h(x)\mathrm{d}x}{y^{n}}\mathrm{d}y&=\sum_{n=2}^{\infty}\begin{array}[]{l}\int_{1}^{\infty}\end{array}\frac{\int_{1}^{y}\frac{1}{x}\mathrm{d}x}{y^{n}}\mathrm{d}y\\\
&=\sum_{n=2}^{\infty}\begin{array}[]{l}\int_{1}^{\infty}\end{array}\frac{\ln(y)}{y^{n}}\mathrm{d}y\\\
&=\sum_{n=2}^{\infty}\frac{1}{(n-1)^{2}}\\\ &=\zeta(2)\\\
&=\frac{\pi^{2}}{6}\end{array}$
#### 2.2.3. The Mellin Transform
The Mellin transform [36, II.10.8][3, 3.6] is defined as
(63) $\begin{array}[]{ll}M^{(a,b)}_{x\rightarrow
s}f(x)&=\int_{a}^{b}f(x)x^{s-1}\mathrm{d}x\end{array}$
where the usual definition of the Mellin transform is
$M^{(0,\infty)}_{x\rightarrow s}f(x)$. Somewhat incredibly, by taking the
Mellin transformation of $h(x)$ over the unit interval, we get an analytic
continuation of $\zeta(s)$ which is convergent when $s$ is not equal to a
negative integer, $0$, or $1$. When $s$ is a negative integer or 0 the limit
or analytic continuation must be taken since the series is formally divergent
at these points, and of course the series $s=1$ diverges. [45] [44] [43]
(64) $\begin{array}[]{ll}M_{x\rightarrow
s}^{I_{n}}h(x)&=\left[Ph(x)x^{s-1};x\right](n)\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}\left(\frac{1}{x}-\left\lfloor\frac{1}{x}\right\rfloor\right)x^{s-1}\mathrm{d}x\\\
&=-\frac{n(n+1)^{-s}+s(n+1)^{-s}-n^{1-s}}{s(s-1)}\end{array}$ (65)
$\begin{array}[]{ll}\zeta(s)&=\frac{1}{s-1}-sM_{x\rightarrow s}^{(0,1)}h(x)\\\
&=\frac{1}{s-1}-s\int_{0}^{1}h(x)x^{s-1}\mathrm{d}x\\\
&=\frac{1}{s-1}-s\int_{0}^{1}\left(\frac{1}{x}-\left\lfloor\frac{1}{x}\right\rfloor\right)x^{s-1}\mathrm{d}x\\\
&=\frac{s}{s-1}-s\sum_{n=1}^{\infty}M_{x\rightarrow s}^{I_{n}}h(x)\\\
&=\frac{s}{s-1}-s\sum_{n=1}^{\infty}-\frac{n(n+1)^{-s}+s(n+1)^{-s}-n^{1-s}}{s(s-1)}\end{array}$
The term $\frac{1}{s-1}$ changes to
$\frac{s}{s-1}=\text{$\frac{1}{s-1}-(-1)$}$ by subtracting the residue [47,
10.41][48, 6.1] of
(66) $M_{x\rightarrow
s}^{I_{0}}h(x)=\int_{I_{0}}h(x)x^{s-1}\mathrm{d}x=\int_{1}^{\infty}\frac{x^{s-1}}{x}\mathrm{d}x=-\frac{1}{s-1}$
at the singular point $s=1$, which happens to coincide with
$\sum_{s=2}^{\infty}\frac{-\frac{1}{s-1}}{s}$
(67)
$\begin{array}[]{ll}\operatorname{Res}\left(\int_{1}^{\infty}\frac{x^{s-1}}{x}\mathrm{d}x;1\right)&=\operatorname{Res}\left(-\frac{1}{s-1};1\right)\\\
&=\sum_{s=2}^{\infty}-\frac{\frac{1}{s-1}}{s}\\\ &=-1\end{array}$
### 2.3. The Harmonic Sawtooth w(x)
Define the harmonic sawtooth map
$w(x)\in\Omega_{h}\backslash\partial\Omega_{h}$ which shares the same domain
and boundary as the Gauss map $h(x)$ to which it is similiar, and also has the
property that its Mellin transform is the (appropriately scaled) zeta
function. The $n$-th component $w_{n}(x)$ is defined over the $n$-th interval
$I_{n}$
(68)
$\begin{array}[]{ll}w_{n}(x)&=\left\\{\begin{array}[]{ll}n(xn+x-1)&\frac{1}{n+1}<x<\frac{1}{n}\\\
0&\operatorname{otherwise}\end{array}\right.\\\
&=n(xn+x-1)\left(\theta\left(\frac{xn+x-1}{n+1}\right)-\theta\left(\frac{xn-1}{n}\right)\right)\end{array}$
and by the substitution $n\rightarrow\left\lfloor\frac{1}{x}\right\rfloor$ we
have
(69) $\begin{array}[]{ll}w(x)&=\sum_{n=1}^{\infty}w_{n}(x)\\\
&=\sum_{n=1}^{\infty}n(xn+x-1)\left(\theta\left(\frac{xn+x-1}{n+1}\right)-\theta\left(\frac{xn-1}{n}\right)\right)\\\
&=\left\lfloor\frac{1}{x}\right\rfloor\left(x\left\lfloor\frac{1}{x}\right\rfloor+x-1\right)\end{array}$
Unlike $h(x)$ which is nonzero outside of $|x|>1$, the (harmonic) sawtooth map
has $w(x)=0\forall|x|>1$.
Figure 2. The Harmonic Sawtooth
The length of each component of $w(x)$ is
(70) $\begin{array}[]{ll}\ell w_{n}&=\left[Pw(x);x\right](n)\\\
&=\int_{I_{n}}w(x)\mathrm{d}x\\\ &=\frac{1}{2(n+1)n}\end{array}$
So that the total length of the harmonic sawtooth string $\mathcal{L}_{w}$ is
(71) $\begin{array}[]{ll}|\mathcal{L}_{w}|&=\int_{0}^{1}w(x)\mathrm{d}x\\\
&=\sum_{n=1}^{\infty}\ell w_{n}\\\ &=\sum_{n=1}^{\infty}\frac{1}{2(n+1)n}\\\
&=\frac{1}{2}\end{array}$
The infinite set of Mellin transforms of $w_{n}(x)$
(72) $\begin{array}[]{ll}M_{x\rightarrow s}^{I_{n}}w(x)&=M_{x\rightarrow
s}^{(0,1)}w_{n}(x)\\\ &=\left[Pw(x)x^{s-1};x\right](n)\\\
&=\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)x^{s-1}\mathrm{d}x\\\
&=\int_{0}^{1}n(xn+x-1)\left(\theta\left(\frac{xn+x-1}{n+1}\right)-\theta\left(\frac{xn-1}{n}\right)\right)x^{s-1}\mathrm{d}x\\\
&=-\frac{n(n+1)^{-s}+s(n+1)^{-s}-n^{1-s}}{s(s-1)}\end{array}$
are summed to get an expression for
$\\{\zeta(s):\mathfrak{R}(s)\not\in\mathbbm{N}_{0^{-}}\\}$
(73)
$\begin{array}[]{ll}\zeta(s)&=s\frac{s+1}{s-1}\int_{0}^{1}\left\lfloor\frac{1}{x}\right\rfloor\left(x\left\lfloor\frac{1}{x}\right\rfloor+x-1\right)x^{s-1}\mathrm{d}x\\\
&=\sum_{n=1}^{\infty}s\frac{s+1}{s-1}M_{x\rightarrow s}^{I_{n}}w(x)\\\
&=\sum_{n=1}^{\infty}s\frac{s+1}{s-1}\int_{\frac{1}{n+1}}^{\frac{1}{n}}n(xn+x-1)x^{s-1}\mathrm{d}x\\\
&=\sum_{n=1}^{\infty}s\frac{s+1}{s-1}\left(-\frac{n(n+1)^{-s}+s(n+1)^{-s}-n^{1-s}}{s(s-1)}\right)\\\
&=\sum_{n=1}^{\infty}\frac{n(n+1)^{-s}-n^{1-s}+sn^{-s}}{s-1}\end{array}$
### 2.4. The Prime Numbers
Let $\mathbbm{P}=\\{2,3,5,7,11,13,17,19,23,29,\ldots\\}$ denote the set of
prime numbers and $\mathbbm{N}_{1}=\\{1,2,\ldots\\}$ and
$\mathbbm{N}_{0}=\\{0,1,2,\ldots\\},\mathbbm{N}=\\{\ldots,-2,-1,0,1,2,\ldots\\}$
be the set of positive, non-negative, and signed integers.
#### 2.4.1. The Prime Counting Function: $\pi(x)$
The prime counting function $\pi(x)$ counts the number of primes less than a
given number. It can written as
(74) $\begin{array}[]{ll}\pi(x)&=\sum_{p<x}^{p\in\mathbbm{P}}1\end{array}$
which is essentially a step function which increases by 1 for each prime. [9,
15.11]
#### 2.4.2. von Mangoldt and Chebyshev’s Functions:
$\Lambda(x),\theta(x),\text{$\psi(x)$}$
Chebyshev’s function of the first kind $\theta(x)$ is the sum of the logarithm
of all primes $\leqslant x$
(75) $\begin{array}[]{ll}\theta(x)&=\sum_{k=1}^{\pi(x)}\ln(p_{k})\\\
&=\ln\left(\sum_{k=1}^{\pi(x)}p_{k}\right)\end{array}$
where $p_{k}\in\mathbbm{P}$ is the $k$-th prime. [7, 4.4] The generalization
of $\pi(x)$ is the Chebyshev function of the second kind
(76) $\begin{array}[]{ll}\psi(x)&=\sum_{p^{r}\leqslant
x}^{\\{p\in\mathbbm{P},r\in\mathbbm{N}_{1}\\}}\ln(p)\\\
&=\sum_{k=1}^{\left\lfloor\log_{2}(x)\right\rfloor}\theta(x^{\frac{1}{k}})\\\
&=\ln(\operatorname{lcm}(1,2,3,\ldots,\left\lfloor x\right\rfloor))\\\
&=\sum^{n\leqslant x}_{n}\Lambda(n)\\\
&=x-\frac{\ln(1-x^{-2})}{2}-\ln(2\pi)-\sum^{\zeta(\rho)=0}_{\rho}\frac{x^{\rho}}{\rho}\forall\mathcal{I}(\rho)\neq
0\end{array}$
where the first sum ranges over the primes $p\in\mathbbm{P}$ and positive
integers $r$ and the sum over $\rho$ is von Mangoldt’s formula where $\rho$
ranges over the non-trivial roots of $\zeta(s)$ in increasing order. The
function $\operatorname{lcm}(\ldots.)$ is the least common multiple, and
$\Lambda(x)$ is the von Mangoldt function.
(77)
$\begin{array}[]{ll}\Lambda(x)&=\left\\{\begin{array}[]{ll}\ln(p)&\\{n=p^{k}:p\in\mathbbm{P},k\in\mathbbm{N}_{1}\\}\\\
0&\operatorname{otherwise}\end{array}\right.\\\
&=\ln\left(\frac{\operatorname{lcm}(1,2,\ldots,n)}{\operatorname{lcm}(1,2,\ldots,n-1)}\right)\end{array}$
$\Lambda(s)$ is related to $\zeta(s)$ by
(78)
$-\frac{\frac{\mathrm{d}}{\mathrm{d}s}\zeta(s)}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n^{s}}\forall\mathcal{R}(s)>1$
Chebyshev proved that $\pi(x),\theta(x),\operatorname{and}\text{$\psi(x)$}$
have the same scaled asymptotic limit.
(79)
$\lim_{x\rightarrow\infty}\frac{\pi(x)}{\left(\frac{x}{\ln(x)}\right)}=\lim_{x\rightarrow\infty}\frac{\psi(x)}{x}=\lim_{x\rightarrow\infty}\frac{\theta(x)}{x}=1$
[18, 1.3] [17, I.4] [7, 4.3-4.4&3.1-3.2] [8] [9, 15.11]. Note that [9]
incorrectly defines $\psi(x)$ as $\ln(\gcd(\ldots)).$
## 3\. Analytic Continuation
### 3.1. Continuation of ${}_{n+1}F_{n}$ Near Unit Argument
The continuation formula for Gauss’s hypergeometric function ${}_{2}F_{1}$
near unit argument is well known
(80)
$\begin{array}[]{ll}\frac{\Gamma(a_{1})\Gamma(a_{2})}{\Gamma(b_{1})}_{2}F_{1}\left(\begin{array}[]{ll}a_{1}&a_{2}\\\
b_{1}&\end{array}|z\right)&=\sum_{n=0}^{\infty}\frac{(-1)^{n}(1-z)^{n}}{n!}\frac{\Gamma(a_{1}+n)\Gamma(a_{2}+n)\Gamma(s_{1}-n)}{\Gamma(a_{1}+s_{1})\Gamma(a_{2}+s_{1})}\\\
&+(1-z)^{s_{1}}\sum_{n=0}^{\infty}\frac{(-1)^{n}(1-z)^{n}}{n!}\frac{\Gamma(a_{1}+s_{1}+n)\Gamma(a_{2}+s_{1}+n)\Gamma(-s_{1}-n)}{\Gamma(a_{1}+s_{1})\Gamma(a_{2}+s_{1})}\end{array}$
where $s_{1}=b_{1}-a_{1}-a_{2}$ is the balance (117) of ${}_{2}F_{1}$ which
must not be equal to an integer, that is, ${}_{2}F_{1}$ cannot be
$s_{1}$-balanced. A function is said to be $k$-balanced only when $k$ is an
integer. When $\mathcal{R}(s_{1})>0$ the value at $z=1$ is finite and given by
the Gaussian summation formula
(81)
$\begin{array}[]{ll}\frac{{}_{2}F_{1}\left(\begin{array}[]{ll}a_{1}&a_{2}\\\
b_{1}&\end{array}\right)}{\Gamma(b_{1})}&=\frac{\Gamma(b_{1}-a_{1}-a_{2})}{\Gamma(b_{1}-a_{1})\Gamma(b_{1}-a_{2})}\\\
&=\frac{\Gamma(s_{1})}{\Gamma(a_{1}+s_{1})\Gamma(a_{2}+s_{1})}\end{array}$
It is obvious that $\lim_{t\rightarrow
1}\operatorname{Li}_{1}^{F}(t)=\lim_{t\rightarrow
1}\text{${}_{2}F_{1}\left(\begin{array}[]{ll}1&1\\\
2&\end{array}|t\right)=\zeta^{F}(1)$}=\infty$ is $0$-balanced and of course
equal to the divergent harmonic series so the continuation formula does not
apply. However, Bühring and Srivastava [6][5] generalized this relation to all
${}_{n+1}F_{n}$ by expanding (81) as a series then interchanging the order of
summations to derive a recurrence with respect to $n$
(82)
$\begin{array}[]{ll}{}_{n+1}F_{n}\left(\begin{array}[]{l}a_{1},\ldots,a_{n+1}\\\
b_{1},\ldots,b_{n}\end{array}|t\right)&=\frac{\Gamma(b_{n})\Gamma(b_{n-1})}{\Gamma(a_{n+1})\Gamma(b_{n}+b_{n-1}-a_{n+1})}\\\
&\cdot\sum_{m=0}^{\infty}\frac{(b_{n}-a_{n+1})_{m}(b_{n-1}-a_{n-1})_{m}}{(b_{n}+b_{n-1}-a_{n+1})_{m}m!}_{n}F_{n-1}\left(\begin{array}[]{l}a_{1},\ldots,a_{n}\\\
b_{1},\ldots,b_{n-2},b_{n-1}+b_{n}-a_{n+1}+m\end{array}|t\right)\end{array}$
which is valid $\forall\\{\mathcal{R}(a_{i})>0:1\leqslant i\leqslant n+1\\}$.
The $m$-th term of the summand in (82) is contiguous (4.1.2) to the ($m-1$)-th
and ($m+1$)-th terms and thus a linear relationship can always be found
between neighboring terms.
### 3.2. The Continuation of
$\operatorname{Li}_{n}^{F}(t)\operatorname{and}\zeta^{F}(n)$ via Contiguous
Functions
There are 4 functions contiguous (4.1.2) to $\operatorname{Li}_{n}^{F}(t)$,
only 3 of them are unique, and just 1 of them is interesting. The functions
are obtained by shifting one of the numerator parameters $a_{i}\pm 1$ or
shifting one of the denominator parameters $b_{i}\pm 1$. Shifting any of the
$a$ parameters or any of the $b$ parameters will suffice since they are all
equal and ${}_{p}F_{q}$ is invariant with respect to the ordering of
parameters. Let $\vec{c}^{+}_{n}$ and $\vec{c}^{-}_{n}$ denote the parameter
vector $\vec{c}_{n}$ where one element is shifted up or down by $1$.
(83)
$\begin{array}[]{ll}\vec{c}^{+}_{n}&=\vec{c}_{n-1},c+1=\underbrace{c,\ldots,c}_{n-1},c+1\\\
\vec{c}^{-}_{n}&=\vec{c}_{n-1},c-1=\underbrace{c,\ldots,c}_{n-1},c-1\end{array}$
For example, $\vec{4}_{3}^{+}=4,4,5$. Two of the four functions contiguous to
$\operatorname{Li}_{n}^{F}(t)$ are identical
(84)
$\begin{array}[]{ll}{}_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}^{+}_{n+1}\\\
\vec{2}_{n}\end{array}|t\right)t&=_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}^{-}_{n}\end{array}|t\right)t=\operatorname{Li}_{n-1}^{F}(t)\end{array}$
Shifting any $a_{i}$ up is equivalent to shifting any $b_{i}$ down, both
operations take $\operatorname{Li}_{n}^{F}(t)$ to
$\operatorname{Li}_{n-1}^{F}(t)$. Shifting any $a_{i}$ down results in the
identity function since it puts a $0$ in the numerator.
(85) ${}_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}^{-}_{n+1}\\\
\vec{2}_{n}\end{array}|t\right)t=t$
Thus, the only interesting function continguous to
$\operatorname{Li}_{n}^{F}(t)$ is obtained by shifting one of the denominator
parameters up. Let this function be denoted by
$\operatorname{Li}_{n}^{F+1}(t)$
(86)
$\begin{array}[]{ll}\text{$\operatorname{Li}_{n}^{F+1}(t)$}=_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}^{+}_{n}\end{array}|t\right)&=\left\\{\begin{array}[]{ll}I_{0}\left(2\sqrt{t}\right)-\frac{1}{\sqrt{t}}I_{1}\left(2\sqrt{t}\right)&n=0\\\
\frac{e^{t}}{t}-\frac{1}{t}-1&n=1\\\
(-1)^{n}\left(1-\frac{\operatorname{Li}_{1}(t)}{t}+\sum_{k=1}^{n-1}(-1)^{k+1}\operatorname{Li}_{k}(t)\right)&n\geqslant
2\end{array}\right.\end{array}$
where $I_{n}(x)$ is a modified Bessel function of the first kind [34, 65] [10,
6.9.1]
(87)
$\begin{array}[]{ll}I_{n}(x)&=\frac{x^{n}}{\Gamma(n+1)2^{n}}_{0}F_{1}\left(\begin{array}[]{l}\\\
n+1\end{array}|\frac{x^{2}}{4}\right)\end{array}$
Before applying (82), the notation will be simplified by extending (83) so
that repeated shifts can be written more easily
(88)
$\begin{array}[]{ll}\vec{c}^{+j}_{n}&=\vec{c}_{n-1},c+j=\underbrace{c,\ldots,c}_{n-1},c+j\\\
\vec{c}^{-j}_{n}&=\vec{c}_{n-1},c-j=\underbrace{c,\ldots,c}_{n-1},c-j\end{array}$
where clearly $\vec{c}^{+}_{n}=\vec{c}^{+1}_{n}$ and
$\vec{c}^{-}_{n}=\vec{c}^{-1}_{n}$. The goal is to extend
$\operatorname{Li}_{n}^{F+1}(t)$ to all $\operatorname{Li}_{n}^{F+m}(t)$ by
repeated application of $\vec{c}^{+1}_{n}$. Applying (82) to (13) gives the
continuation of
$\operatorname{Li}_{n}^{F}(t)\rightarrow\operatorname{Li}_{n+1}^{F}(t)\forall
n\geqslant 1$ by setting $a_{1\ldots n+1}=\vec{1}_{n+1}$ and $b_{1\ldots
n}=\vec{2}_{n}$ which results in
(89)
$\begin{array}[]{ll}\operatorname{Li}_{n}^{F}(t)&={{}_{n+1}F_{n}}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}|t\right)t\forall n\geqslant 0\\\
&=t\sum_{m=0}^{\infty}\left(\frac{{}_{n}F_{n-1}\left(\begin{array}[]{l}\vec{1}_{n}\\\
\vec{2}_{n-2},3+m\end{array}|t\right)}{(m+1)(m+2)}\right)\forall n\geqslant
2\\\
&=t\sum_{m=0}^{\infty}\left(\frac{{}_{n}F_{n-1}\left(\begin{array}[]{l}\vec{1}_{n}\\\
\vec{2}^{+m+1}_{n-1}\end{array}|t\right)}{(m+1)(m+2)}\right)\forall n\geqslant
2\end{array}$
since
(90)
$\frac{\Gamma(b_{n})\Gamma(b_{n-1})}{\Gamma(a_{n+1})\Gamma(b_{n}+b_{n-1}-a_{n+1})}=\frac{\Gamma(2)\Gamma(2)}{\Gamma(1)\Gamma(2+2-1)}=\frac{1}{2}$
and
(91)
$\frac{(b_{n}-a_{n+1})_{m}(b_{n-1}-a_{n-1})_{m}}{(b_{n}+b_{n-1}-a_{n+1})m!}=\frac{(1)_{m}(1)_{m}}{(2+2-1)_{m}m!}=\frac{2}{(m+1)(m+2)}$
The denominator parameters $\vec{2}^{+1+m}_{n-1}$ in (89) are simply
(92)
$\begin{array}[]{lll}\vec{2}^{+1+m}_{n-1}&=\vec{2}_{n-2},3+m&=\underbrace{2,\ldots,2}_{n-2},3+m\end{array}$
The numbers ($m+1)(m+2$) are known as the oblong numbers, [15, A002378]. By
simply setting $t=1$ in (89) we get the continuation from
$\zeta^{F}(n)\rightarrow\zeta^{F}(n+1)\forall n\geqslant 1$
(93)
$\begin{array}[]{ll}\zeta^{F}(n)=_{n+1}F_{n}\left(\begin{array}[]{l}\vec{1}_{n+1}\\\
\vec{2}_{n}\end{array}\right)&=\sum_{m=0}^{\infty}\left(\frac{{}_{n}F_{n-1}\left(\begin{array}[]{l}\vec{1}_{n}\\\
\vec{2}^{+m+1}_{n-1}\end{array}\right)}{(m+1)(m+2)}\right)\forall n\geqslant
2\end{array}$
The justification in saying that $\operatorname{Li}_{n}^{F}(t)$ and
$\zeta^{F}(n)$ are continued to $\operatorname{Li}_{n+1}^{F}(t)$ and
$\zeta^{F}(n+1)$ comes from the fact that the first term in the summand of the
continuation (89) from
$\operatorname{Li}_{n-1}^{F}(t)\rightarrow\operatorname{Li}_{n}^{F}(t$) is
contiguous to $\operatorname{Li}_{n-1}^{F}(t)$, that is,
${}_{n}F_{n-1}\left(\begin{array}[]{l}\vec{1}_{n}\\\
\vec{2}^{+1}_{n-1}\end{array}|t\right)$ is contiguous to
$\operatorname{Li}_{n-1}^{F}(t)=\text{${}_{n}F_{n-1}\left(\begin{array}[]{l}\vec{1}_{n}\\\
\vec{2}_{n-1}\end{array}|t\right)$}$. The continuation formula (89) gives
interesting answers for $n=0$ and $n=1$ which suggest an alternative to “the
analytic continuation” of $\zeta\left(t\right)$ which is different from the
usual $\frac{1}{1-2^{-t}}\sum_{n=0}^{\infty}(2n+1)^{-t}$. We have
(94)
$\begin{array}[]{llll}\zeta^{F}(0)&=\sum_{m=0}^{\infty}\left(\frac{{}_{0}F_{1}\left(\begin{array}[]{l}\\\
m+3\end{array}\right)}{(m+1)(m+2)}\right)&&\\\
&=\sum_{m=0}^{\infty}\frac{-\left(I_{m+1}\left(2\right)m+I_{m+1}\left(2\right)-I_{m}\left(2\right)\right)\Gamma\left(m+3\right)}{(m+1)(m+2)}&&\\\
&=I_{0}(2)-1&&\\\ &\approx 1.2795853023360&&\\\
\zeta^{F}(1)&=\sum_{m=0}^{\infty}\left(\frac{{}_{1}F_{1}\left(\begin{array}[]{l}1\\\
m+3\end{array}\right)}{(m+1)(m+2)}\right)&&\\\
&=\sum_{m=0}^{\infty}\frac{e\left(\Gamma\left(m+3\right)-\Gamma\left(m+2,1\right)m-2\Gamma\left(m+2,1\right)\right)}{(m+1)(m+2)}&&\\\
&=\operatorname{Ei}(1)-\gamma&&\\\ &\approx 1.3179021514544&&\end{array}$
where $\operatorname{Ei}(x)$ is the exponential integral [10, 6.9.2]
(95)
$\begin{array}[]{ll}\operatorname{Ei}(x)&=\gamma-\frac{\ln(x^{-1})}{2}+\frac{\ln(x)}{2}+\sum_{k=1}^{\infty}\frac{x^{k}}{k\Gamma(k+1)}\\\
&=\gamma-\frac{\ln(x^{-1})}{2}+\frac{\ln(x)}{2}+x_{2}F_{2}\left(\begin{array}[]{ll}1&1\\\
2&2\end{array}|x\right)\end{array}$
and $\Gamma\left(a,z\right)$ is the incomplete Gamma function
(96)
$\Gamma\left(a,z\right)=\Gamma\left(z\right)-\frac{z^{a}_{1}F_{1}\left(\begin{array}[]{l}a\\\
a+1\end{array}|-z\right)}{a}$
So we have the “hypergeometrically continued” values $\zeta^{F}(0)=I_{0}(2)-1$
and $\zeta^{F}(1)=\operatorname{Ei}(1)-\gamma$ whereas the “real” values are
$\zeta\left(0\right)=-\frac{1}{2}$ and $\zeta\left(1\right)=\infty$. In terms
of reciprocal probability we have
(97) $\begin{array}[]{ll}\zeta^{F}\left(0\right)^{-1}&\cong 78.15\%\\\
\zeta^{F}\left(1\right)^{-1}&\cong 75.88\%\end{array}$
#### 3.2.1.
$\operatorname{Li}_{1}^{F}(t)\rightarrow\operatorname{Li}_{2}^{F}(t)$ and
$\zeta_{1}^{F}(t)\rightarrow\zeta_{2}^{F}(t)$
The continuation $\zeta_{n}^{F}(t)$ from $n=1\rightarrow 2$ via (93) is
straightforward
(98)
$\begin{array}[]{ll}\text{$\zeta^{F}(2)$}&=_{3}F_{2}\left(\begin{array}[]{lll}1&1&1\\\
2&2&\end{array}\right)\\\
&=\sum_{m=0}^{\infty}\left(\frac{{}_{2}F_{1}\left(\begin{array}[]{ll}1&1\\\
&3+m\end{array}\right)}{(m+1)(m+2)}\right)\\\
&=\sum_{m=0}^{\infty}\left(\frac{\sum_{k=0}^{\infty}\frac{\Gamma(m+3)\Gamma(k+1)}{\Gamma(m+k+3)}}{(m+1)(m+2)}\right)\\\
&=\sum_{m=0}^{\infty}\frac{1}{(m+1)(m+2)}\frac{(m+2)}{(m+1)}\\\
&=\sum_{m=0}^{\infty}\frac{1}{(m+1)^{2}}\\\ &=\frac{\pi^{2}}{6}\end{array}$
The continuation of $\operatorname{Li}^{F}_{1}(t)$ to
$\operatorname{Li}^{F}_{2}(t)$ is a bit more complicated
(99)
$\begin{array}[]{ll}\operatorname{Li}_{2}^{F}\left(t\right)&=_{3}F_{2}\left(\begin{array}[]{lll}1&1&1\\\
&2&2\end{array}|t\right)\\\
&=\sum_{m=0}^{\infty}\left(\frac{{}_{2}F_{1}\left(\begin{array}[]{ll}1&1\\\
&m+3\end{array}|t\right)}{(m+1)(m+2)}\right)\\\
&=\sum_{m=0}^{\infty}r_{2}(m,t)\end{array}$
then $r_{2}(m,t)$ is given by
(100)
$\begin{array}[]{ll}r_{2}(m,t)&=\frac{\sum_{n=0}^{m}\frac{(-1)^{n+1}\Gamma(m+2)(\Psi(m-n+1)-\Psi(m+2))(-1)^{m}e^{\psi(m+2)}t^{n}}{\Gamma(n+2)\Gamma(m-n+1)}}{(m+1)e^{\psi(m+2)}t^{m+1}}\\\
&-\frac{(t-1)^{m+1}t^{-2-m}\ln(1-t)}{m+1}\end{array}$
so $\operatorname{Li}_{2}^{F}(t)$ is equal to
(101)
$\text{$\operatorname{Li}_{2}^{F}(t)$=}\sum_{m=0}^{\infty}\frac{\sum_{n=0}^{m}\frac{(-1)^{n+1}\Gamma(m+2)(\Psi(m-n+1)-\Psi(m+2))(-1)^{m}e^{\psi(m+2)}t^{n}}{\Gamma(n+2)\Gamma(m-n+1)}}{(m+1)e^{\psi(m+2)}t^{m+1}}-\frac{(t-1)^{m+1}t^{-2-m}\ln(1-t)}{m+1}$
where $\psi(m)=\ln(\operatorname{lcm}(1,2,3,\ldots,m))$ is Chebyshev’s
function of the 2nd kind (76) and $\Psi(m$) is the digamma function
(102)
$\Psi(x)=\frac{\mathrm{d}}{\mathrm{d}x}\ln(\Gamma(x))=\frac{\frac{\mathrm{d}}{\mathrm{d}x}\Gamma(x)}{\Gamma(x)}$
#### 3.2.2. $\zeta^{F}(2)\rightarrow\zeta^{F}(3)$
The continuation from $\zeta^{F}(2)$ to $\zeta^{F}(3)$ via (93) is carried out
like so
(103)
$\begin{array}[]{ll}\text{$\zeta^{F}(3)$}&=_{4}F_{3}\left(\begin{array}[]{llll}1&1&1&1\\\
&2&2&2\end{array}\right)\\\
&=\sum_{m=0}^{\infty}\left(\frac{{}_{3}F_{2}\left(\begin{array}[]{lll}1&1&1\\\
&2&3+m\end{array}\right)}{(m+1)(m+2)}\right)\\\
&=\sum_{m=0}^{\infty}r_{3}(m)\end{array}$
Each term in the summand $r_{3}(m)$ has the form
$\frac{\zeta(2)}{m+1}+q_{3}(m)$ where of course $\zeta(2)=\frac{\pi^{2}}{6}$
and $q_{3}(m)$ is a rational function of $m$ which follows a 3rd order linear
recurrence equation[28, 8.2] given by
(104) $\begin{array}[]{ll}q_{3}(m)&=q_{3}\left(m+1\right)\left(m^{3}+8\hskip
2.5ptm^{2}+21\hskip 2.5ptm+18\right)\\\ &+q_{3}\left(m+2\right)\left(-2\hskip
2.5ptm^{3}-20\hskip 2.5ptm^{2}-67\hskip 2.5ptm-75\right)\\\
&+q_{3}\left(m+3\right)\left(m^{3}+12\hskip 2.5ptm^{2}+48\hskip
2.5ptm+64\right)\end{array}$ (105)
$q_{3}(m)=\left\\{\begin{array}[]{ll}-1&m=0\\\ -\frac{5}{8}&m=1\\\
-\frac{49}{108}&m=2\end{array}\right.$
The solution to which is given by
(106)
$\begin{array}[]{ll}q_{3}(m)&=\frac{\Psi^{(1)}(m+2)-\zeta\left(2\right)}{m+1}\end{array}$
so the summand $r_{3}(m)$ is
(107)
$\begin{array}[]{ll}r_{3}(m)&=\frac{\zeta(2)}{m+1}+q_{3}(m)=\frac{\Psi^{(1)}(2+m)}{m+1}\end{array}$
Thus (111) is also equal to
(108)
$\begin{array}[]{ll}\zeta^{F}(3)&=\sum_{m=0}^{\infty}\frac{\Psi^{(1)}(m+2)}{m+1}\end{array}$
Thus
(109) $\begin{array}[]{ll}r_{3}(m)&=\frac{\Psi^{(1)}(m+2)}{m+1}\\\
&=\frac{\zeta(2,m+2)}{m+1}\\\
&=\frac{\pi^{2}}{6}-\sum_{k=1}^{m-1}\frac{1}{k^{2}}\\\
&=\frac{\sum_{k=1}^{\infty}\frac{1}{(k+m-1)^{2}}}{m+1}\\\
&=\frac{{}_{3}F_{2}\left(\begin{array}[]{lll}1&m+2&m+2\\\
&m+3&m+3\end{array}\right)}{(m+1)(m+2)^{2}}\\\
&=\frac{{}_{3}F_{2}\left(\begin{array}[]{lll}1&1&1\\\
&2&3+m\end{array}\right)}{(m+1)(m+2)}\end{array}$
The first 10 terms of $\\{q_{3}(m):m=0\ldots 9\\}$ are
(110)
$\left[-1,-\frac{5}{8},-\frac{49}{108},-\frac{205}{576},-\frac{5269}{18000},-\frac{5369}{21600},-\frac{266681}{1234800},-\frac{1077749}{5644800},-\frac{9778141}{57153600},-\frac{1968329}{12700800}\right]$
The denominator of (110) appears to be [15, A119936], the least common
multiple of denominators of the rows of a certain triangle of rationals and
the numerators are [15, A007406], the numerator of
$\sum_{k=1}^{n}\frac{1}{k^{2}}$ from (123) which, according to a theorem
Wolstenholme, $p$ divides $\operatorname{numer}(q_{3}(p-1$)) where
$p\in\mathbbm{P}$ is prime. [12] [4] [1]
#### 3.2.3. $\zeta^{F}(3)\rightarrow\zeta^{F}(4)$
The continuation from $\zeta^{F}(3)$ to $\zeta^{F}(4)$ via (93) is given by
(111)
$\begin{array}[]{ll}\text{$\zeta^{F}(4)$}&=_{5}F_{4}\left(\begin{array}[]{lllll}1&1&1&1&1\\\
2&2&2&2&\end{array}\right)\\\
&=\sum_{m=0}^{\infty}\left(\frac{{}_{4}F_{3}\left(\begin{array}[]{llll}1&1&1&1\\\
&2&2&3+m\end{array}\right)}{(m+1)(m+2)}\right)\\\
&=\sum_{m=0}^{\infty}r_{4}(m)\end{array}$
The summand $r_{4}(m)$ has the form
(112)
$\begin{array}[]{ll}r_{4}(m)&=a(t,m)-b(t,m)-\frac{H(m+1)\operatorname{Li}_{2}(t)}{(m+1)t}+\frac{\operatorname{Li}_{3}(t)}{(m+1)t}\end{array}$
where $a(t,m)$ is an $(m+1)$-th degree polynomial and $b(t,m)$ is a $(m+2)$-th
degree polynomial(the determination of which is left to an excercise for the
reader or the topic of another article, but is readily obtained with the help
of Maple[27]), and $H(n)$ is the $n$-th Harmonic number
(113) $\begin{array}[]{ll}H(n)&=\sum_{i=1}^{n}\frac{1}{n}\\\
&=\Psi(n+1)+\gamma\\\ &=\sum_{k=1}^{\infty}\frac{n}{k^{2}+kn}\\\
&=\frac{n}{n+1}_{3}F_{2}\left(\begin{array}[]{lll}1&1&n+1\\\
&2&n+2\end{array}\right)\end{array}$
The polynomial $b(t,m)$ vanishes when $t=1$. An interesting set of formulas
for $\zeta(4)$ is
(114)
$\begin{array}[]{ll}\zeta(4)&=\sum_{n=1}^{\infty}\frac{\Psi^{(2)}(n+1)+2\zeta(3)}{2n\left(n+1\right)}\\\
&=\sum_{n=1}^{\infty}\frac{\Psi^{(2)}(n+1)+2\sum_{m=0}^{\infty}\frac{\Psi^{(1)}(m+2)}{m+1}}{2n\left(n+1\right)}\\\
&=\frac{\pi^{4}}{90}\end{array}$
## 4\. Appendix
### 4.1.
$\operatorname{The}\operatorname{Generalized}\operatorname{Hypergeometric}\operatorname{Function}:_{p}F_{q}$
The Pochhammer symbol is defined according to
(115) $\begin{array}[]{ll}(n)_{k}&=\frac{\Gamma(n+k)}{\Gamma(n)}\end{array}$
The generalized hypergeometric function [39][48, 4.1] is defined as an
infinite sum of quotients of finite products of Pochhammer symbols
(116)
$\begin{array}[]{ll}\begin{array}[]{l}{}_{p}F_{q}\left(\begin{array}[]{l}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{array}|t\right)\end{array}&=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\frac{\prod_{i=1}^{p}(a_{i})_{k}}{\prod_{j=1}^{q}(b_{j})_{k}}\end{array}$
The function ${}_{p}F_{q}$ is said to be $k$-balanced [5] if the sum of the
denominator parameters $b_{1}\ldots b_{p}$ minus the sum of the numerator
parameters $a_{1}\ldots a_{p+1}$ is an integer.
(117)
$k=\operatorname{bal}(_{p}F_{q})=\sum_{n=1}^{q}b_{n}-\sum_{n=1}^{p}a_{n}$
The value $k$ is the characteristic exponent of the hypergeometric
differential equation at unit argument which is equal to the maximum root of
the corresponding indicial equation and so determines the behaviour of the
function near this point. A $1$-balanced function is said to be Saalschützian.
[41, 2.1.1]
#### 4.1.1. The Differential Equation and Convergence
The function ${}_{p}F_{q}$ converges when
(118) $\begin{array}[]{l}\left\\{\begin{array}[]{lll}p\leqslant
q&\forall|t|\neq\infty&\\\ p=q+1&\forall|t|<1&\\\
\\{p=q+1:\operatorname{bal}(_{p}F_{q})\geqslant 1\\}&\forall|t|=1&\\\
p>q+1&\forall t=0&\end{array}\right.\end{array}$
where $\operatorname{bal}(_{p}F_{q})=\sum_{n=1}^{q}b_{n}-\sum_{n=1}^{p}a_{n}$
is the balance of the parameters (117). The differential equation solved by
${}_{p}F_{q}\operatorname{is}\operatorname{of}\operatorname{order}\text{max(p,q+1)}$
(119)
$\left(\theta_{t}\prod_{j=1}^{q}(\theta_{t}+b_{j}-1)-t\prod_{i=1}^{p}(\theta_{t}+a_{i})\right)f(t)=0$
where $f(t)=_{p}F_{q}\left(\begin{array}[]{l}a_{1},\ldots,a_{p}\\\
b_{1},\ldots,b_{q}\end{array}|t\right)$ and
$\theta_{t}=t\frac{\mathrm{d}}{\mathrm{d}t}$ is the differential operator.
When $p=q+1$ (119) has the form
(120)
$a_{0}f(t)+t^{q}\frac{\mathrm{d}}{\mathrm{d}t^{q+1}}f(t)+\sum_{n=1}^{q}t^{n-1}(ta_{n}-b_{n})\frac{\mathrm{d}}{\mathrm{d}t^{n}}f(t)=0$
[48, 4.2][21, Ch3][34, 44-46] [41, 2.1.2]
#### 4.1.2. Contiguous Functions and Linear Relations
Any two hypergeometric functions ${{}_{p}F_{q}}(a_{\ldots},b_{\ldots};z)$ and
${}_{p}F_{q}(c_{\ldots},d_{\ldots};z)$ are said to be contiguous if all $p+q$
pairs of parameters
$(a_{1},c_{1}),\ldots,(a_{p},c_{p}),(b_{1},d_{1}),\ldots,(b_{q},d_{q})$ are
equal except for one pair which differs only by 1. There are $2p+q$ linearly
independent relations between the $(2p+2q)$ functions contiguous to
${}_{p}F_{q}(a_{\ldots},b_{\ldots};z)$ where the relations are linear
functions of $z$ and polynomial functions of the parameters
$a_{\ldots},b_{\ldots}$. When any $\left\\{a_{i}=a_{j}:i\neq j\\}\right.$ or
$\left\\{b_{i}=b_{j}:i\neq j\\}\right.$ in
${}_{p}F_{q}(a_{\ldots},b_{\ldots};z)$ there will fewer unique contiguous
functions than if all the parameters were unique since the hypergeometric
function is invariant with respect to the ordering of parameters. [41, 2.2.1]
[34, 48] [39] [10, 4.3] [49] [33]
### 4.2. Other Special Functions
#### 4.2.1. Polygamma $\Psi^{(n)}(x)$
The polygamma function is the $n$-th derivative of the digamma (102) function
(121) $\Psi^{(n)}(x)=\frac{\mathrm{d}}{\mathrm{d}x^{n}}\Psi(x)$
and is defined as an infinite sum, a Hurwitz zeta function (4), or a
hypergeometric function when $x$ is positive integer
(122)
$\begin{array}[]{ll}\Psi^{(n)}(x)&=\left\\{\begin{array}[]{ll}\left(\sum_{k=1}^{\infty}\frac{1}{k}-\frac{1}{k+x-1}\right)-\gamma&n=0\\\
\sum_{k=0}^{\infty}-\frac{n!(-1)^{n}}{(k+x)^{n+1}}&n\geqslant
1\end{array}\right.\\\
&=\left\\{\begin{array}[]{ll}\left(\frac{x-1}{x}_{3}F_{2}\left(\begin{array}[]{lll}1&1&x\\\
&2&x+1\end{array}\right)\right)-\gamma&n=0\\\
\frac{n!(-1)^{n+1}}{x^{n+1}}_{n+2}F_{n+1}\left(\begin{array}[]{ll}1&\vec{x}_{n+1}\\\
\overrightarrow{(1+x)}_{n+1}&\end{array}\right)&n\geqslant
1\end{array}\right.\\\ &=\left\\{\begin{array}[]{ll}\Psi(x)&n=0\\\
(-1)^{n+1}n!\zeta(x,n+1)&n\geqslant 1\end{array}\right.\end{array}$
or as a finite sum when $x$ is a positive integer and $n=1$ [10, 1.16]
(123)
$\begin{array}[]{ll}\Psi^{(1)}(x)&=\frac{\pi^{2}}{6}-\sum_{k=1}^{x-1}\frac{1}{k^{2}}\end{array}$
### 4.3. Notation
(124) $\begin{array}[]{ll}\mathbbm{Z}&\\{\ldots,-2,-1,0,1,2,\ldots\\}\\\
\mathbbm{N}&\\{1,2,3,\ldots.\\}\\\
\mathbbm{N}_{1^{-}}&\\{\ldots,-3,-2,-1\\}\\\
\mathbbm{N}_{0}&\\{0,1,2,3,\ldots.\\}\\\
\mathbbm{N}_{0^{-}}&\\{\ldots,-3,-2,-1,0\\}\end{array}$
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|
arxiv-papers
| 2012-07-04T21:05:25 |
2024-09-04T02:49:32.658974
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stephen Crowley",
"submitter": "Stephen Crowley",
"url": "https://arxiv.org/abs/1207.1126"
}
|
1207.1218
|
Approximate bound state solutions of the deformed Woods-Saxon potential using
asymptotic iteration method
Babatunde J. Falaye111E-mail: fbjames11@physicist.net
Theoretical Physics Section, Department of Physics
University of Ilorin, P. M. B. 1515, Ilorin, Nigeria.
Majid Hamzavi222E-mail: majid.hamzavi@gmail.com
Department of Basic Sciences, Shahrood Branch,
Islamic Azad University, Shahrood, Iran
Sameer M. Ikhdair333E-mail: sikhdair@neu.edu.tr
Physics Department, Near East University,
922022 Nicosia, North Cyprus, Mersin 10, Turkey
###### Abstract
By using the Pekeris approximation, the Schr$\ddot{o}$dinger equation is
approximately solved for the nuclear deformed Woods-Saxon potential within the
framework of the asymptotic iteration method. The energy levels are worked out
and the corresponding normalized eigenfunctions are obtained in terms of
hypergeometric function.
Keywords: Schr$\ddot{o}$dinger equation; nuclear deformed Woods-Saxon
potential; asymptotic
iteration method.
PACs No. 03.65.Pm; 03.65.Ge; 03.65.-w; 03.65.Fd; 02.30.Gp
## 1 Introduction
The deformed Woods-Saxon (dWS) potential is a short range potential and widely
used in nuclear, particle, atomic, condensed matter and chemical physics
[1-7]. This potential is reasonable for nuclear shell models and used to
represent the distribution of nuclear densities. The dWS and spin-orbit
interaction are important and applicable to deformed nuclei [8] and to
strongly deformed nuclides [9]. The dWS potential parameterization at large
deformations for plutonium ${}^{237,239,241}Pu$ odd isotopes was analyzed
[10]. The structure of single-particle states in the second minima of
${}^{237,239,241}Pu$ has been calculated with an exactly dWS potential. The
Nuclear shape was parameterized.
The parameterization of the spin-orbit part of the potential was obtained in
the region corresponding to large deformations (second minima) depending only
on the nuclear surface area. The spin-orbit interaction of a particle in a
non-central self consistent field of the WS type potential was investigated
for light nuclei and the scheme of single-particle states has been found for
mass number $A_{o}=10$ and $25$ [8]. Two parameters of the spin-orbit part of
the dWS potential, namely the strength parameter and radius parameter were
adjusted to reproduce the spins for the values of the nuclear deformation
parameters [11].
Badalov et al. investigated Woods-Saxon potential in the framework of
Schr$\ddot{o}$dinger and Klein-Gordon equations by means of Nikiforov-Uvarov
method [12, 13]. In our recent works [14, 15], we have studied the
relativistic Duffin-Kemmer-Petiau and the Dirac equation for dWS potential. In
addition, we have also obtained the bound state solutions of the $PT-/non-
PT$-symmetric and non-Hermitian modified Woods-Saxon potential with the real
and complex-valued energy levels [16].
In this paper we will study Schr$\ddot{o}$dinger equation with dWS potential
for any arbitrary orbital quantum number $\ell$. We obtain the analytical
expressions for the energy levels and wave functions in closed form.
Therefore, this work is arranged as follows: In section $2$, a brief
introduction to asymptotic iteration method (AIM) is given. In section $3$,
the Schr$\ddot{o}$dinger equation with dWS potential is solved. Finally,
results and conclusions are presented in section $4$.
## 2 The Asymptotic Iteration Method
We briefly outline the AIM here; the details can be found in references [17,
18].
### 2.1 Energy eigenvalues
AIM is proposed to solve the homogenous linear second-order differential
equation of the form
$y_{n}^{\prime\prime}(x)=\lambda_{o}(x)y_{n}^{\prime}(x)+s_{o}(x)y_{n}(x),$
(1)
where $\lambda_{o}(x)\neq 0$ and the prime denotes the derivative with respect
to $x$, the extral parameter $n$ is thought as a radial quantum number. The
variables, $s_{o}(x)$ and $\lambda_{o}(x)$ are sufficiently differentiable. To
find a general solution to this equation, we differentiate equation (1) with
respect to $x$ as
$y_{n}^{\prime\prime\prime}(x)=\lambda_{1}(x)y_{n}^{\prime}(x)+s_{1}(x)y_{n}(x),$
(2)
where
$\displaystyle\lambda_{1}(x)$ $\displaystyle=$
$\displaystyle\lambda_{o}^{\prime}(x)+s_{o}(x)+\lambda_{o}^{2}(x),$
$\displaystyle s_{1}(x)$ $\displaystyle=$ $\displaystyle
s_{o}^{\prime}(x)+s_{o}(x)\lambda_{o}(x),$ (3)
and the second derivative of equation (1) is obtained as
$y_{n}^{\prime\prime\prime\prime}(x)=\lambda_{2}(x)y_{n}^{\prime}(x)+s_{2}(x)y_{n}(x),$
(4)
where
$\displaystyle\lambda_{2}(x)$ $\displaystyle=$
$\displaystyle\lambda_{1}^{\prime}(x)+s_{1}(x)+\lambda_{o}(x)\lambda_{1}(x),$
$\displaystyle s_{2}(x)$ $\displaystyle=$ $\displaystyle
s_{1}^{\prime}(x)+s_{o}(x)\lambda_{1}(x).$ (5)
Equation (1) can be iterated up to $(k+1)th$ and $(k+2)th$ derivatives,
$k=1,2,3...$ Therefore we have
$\displaystyle y_{n}^{(k+1)}(x)$ $\displaystyle=$
$\displaystyle\lambda_{k-1}(x)y_{n}^{\prime}(x)+s_{k-1}(x)y_{n}(x),$
$\displaystyle y_{n}^{(k+2)}(x)$ $\displaystyle=$
$\displaystyle\lambda_{k}(x)y_{n}^{\prime}(x)+s_{k}(x)y_{n}(x),$ (6)
where
$\displaystyle\lambda_{k}(x)$ $\displaystyle=$
$\displaystyle\lambda_{k-1}^{\prime}(x)+s_{k-1}(x)+\lambda_{o}(x)\lambda_{k-1}(x),$
$\displaystyle s_{k}(x)$ $\displaystyle=$ $\displaystyle
s_{k-1}^{\prime}(x)+s_{o}(x)\lambda_{k-1}(x).$ (7)
From the ratio of the (k+2)th and (k+1)th derivatives, we obtain
$\frac{d}{dx}ln\left[y_{n}^{k+1}(x)\right]=\frac{y_{n}^{(k+2)}(x)}{y_{n}^{(k+1)}(x)}=\frac{\lambda_{k}(x)\left[y_{n}^{\prime}(x)+\frac{s_{k}(x)}{\lambda_{k}(x)}y_{n}(x)\right]}{\lambda_{k-1}(x)\left[y_{n}^{\prime}(x)+\frac{s_{k-1}(x)}{\lambda_{k-1}(x)}y_{n}(x)\right]},$
(8)
if $k>0$, for sufficiently large $k$, we obtain $\alpha$ values from
$\frac{s_{k}(x)}{\lambda_{k}(x)}=\frac{s_{k-1}(x)}{\lambda_{k-1}(x)}=\alpha(x),$
(9)
with quantization condition
$\delta_{k}(x)=\left|\begin{array}[]{lr}\lambda_{k}(x)&s_{k}(x)\\\
\lambda_{k-1}(x)&s_{k-1}(x)\end{array}\right|=0\ \ ,\ \ \ k=1,2,3....$ (10)
Then equation (8) reduces to
$\frac{d}{dx}ln\left[y_{n}^{(k+1)}(x)\right]=\frac{\lambda_{k}(x)}{\lambda_{k-1}(x)},$
(11)
which yields the general solution of Eq. (1)
$y_{n}(x)=\exp\left(-\int^{x}\alpha(x^{\prime})dx^{\prime}\right)\left[C_{2}+C_{1}\int^{x}\exp\left(\int^{x^{\prime}}\left[\lambda_{o}(x^{\prime\prime})+2\alpha(x^{\prime\prime})\right]dx^{\prime\prime}\right)dx^{\prime}\right].$
(12)
For a given potential, the idea is to convert the radial Schr$\ddot{o}$dinger
equation to the form of equation (1). Then $\lambda_{o}(x)$ and $s_{o}(x)$ are
determine and $s_{k}(x)$ and $\lambda_{k}(x)$ parameters are calculated by the
recurrence relations given by equation (7). The energy eigenvalues are then
obtained by the condition given by equation (10) if the problem is exactly
solvable.
### 2.2 Energy eigenfunction
Suppose we wish to solve the radial Schr$\ddot{o}$dinger equation for which
the homogenous linear second-order differential equation takes the following
general form
$y^{\prime\prime}(x)=2\left(\frac{tx^{N+1}}{1-bx^{N+2}}-\frac{m+1}{x}\right)y^{\prime}(x)-\frac{Wx^{N}}{1-bx^{N+2}}.$
(13)
The exact solution $y_{n}(x)$ can be expressed as [18]
$y_{n}(x)=(-1)^{n}C_{2}(N+2)^{n}(\sigma)_{{}_{n}}{{}_{2}F_{1}(-n,\rho+n;\sigma;bx^{N+2})},$
(14)
where the following notations has been used
$(\sigma)_{{}_{n}}=\frac{\Gamma{(\sigma+n)}}{\Gamma{(\sigma)}}\ \ ,\ \
\sigma=\frac{2m+N+3}{N+2}\ \ and\ \ \rho=\frac{(2m+1)b+2t}{(N+2)b}.$ (15)
## 3 Any $\ell$-state solutions
The deformed Woods-Saxon potential we investigate in this study is defined as
[14, 15, 19]
$V(r)=-\frac{V_{o}}{1+q\exp{(\frac{r-R}{a})}},\ \ \ \ R=r_{o}A_{o}^{1/3},\ \ \
\ V_{o}=(40.5+0.13A_{o})MeV,\ \ \ \ R>>a,\ \ \ \ q>0,$ (16)
where $V_{o}$ is the depth of potential, $q$ is a real parameter which
determines the shape (deformation) of the potential, $a$ is the diffuseness of
the nuclear surface, $R$ is the width of the potential, $A_{o}$ is the atomic
mass number of target nucleus and $r_{o}$ is radius parameter. By inserting
this potential into the Schr$\ddot{o}$dinger equation [17, 18] as
$\left(-\frac{\hbar^{2}}{2\mu}\left[\frac{1}{r^{2}}\frac{\partial}{\partial
r}r^{2}\frac{\partial}{\partial
r}+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}\right]+V(r)\right)\Psi_{n\ell
m}(r)=E\Psi_{n\ell m}(r),$ (17)
and setting the wave functions $\Psi_{n\ell m}(r)=R_{n\ell}(r)Y_{\ell
m}(\theta,\phi)r^{-1}$, we obtain the radial part of the equation as
$\left[\frac{d^{2}}{dr^{2}}+\frac{2\mu}{\hbar^{2}}\left(E_{n\ell}+\frac{V_{o}}{1+q\exp{(\frac{r-R}{a})}}\right)-\frac{\ell(\ell+1)}{r^{2}}\right]R_{n\ell}(r)=0.$
(18)
Because of the total angular momentum centrifugal term, equation (18) cannot
be solved analytically for $\ell\neq 0$. Therefore, we shall use the Pekeris
approximation in order to deal with this centrifugal term and we may express
it as follows[20, 21]
$U_{cent.}(r)=\frac{1}{r^{2}}=\frac{1}{R^{2}\left(1+\frac{x}{R}\right)^{2}}\cong\frac{1}{R^{2}}\left(1-2\left(\frac{x}{R}\right)+3\left(\frac{x}{R}\right)^{2}+\cdots\right),$
(19)
with $x=r-R$. In addition, we may also approximately express it in the
following way
$\tilde{U}=\frac{1}{r^{2}}\cong\frac{1}{R^{2}}\left[D_{o}+\frac{D_{1}}{1+q\exp{(\nu
x)}}+\frac{D_{2}}{\left(1+q\exp{(\nu x)}\right)^{2}}\right],$ (20)
where $\nu=1/a$. After expanding (20) in terms of $x$, $x^{2}$,
$x^{3}$,$\cdots$ and next, comparing with equation (19), we obtain expansion
coefficients $D_{o}$, $D_{1}$ and $D_{2}$ as follows:
$\displaystyle D_{o}$ $\displaystyle=$ $\displaystyle 1-\frac{(1+q)^{2}}{\nu
Rq^{2}}\left(1-\frac{3}{\nu R}\right),$ $\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle\frac{(1+q)^{2}}{\nu Rq^{2}}\left(-1+3q-\frac{6(1+q)}{\nu
R}\right),$ (21) $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle\frac{(1+q)^{3}}{\nu
Rq^{2}}\left(\frac{1-q}{2}+\frac{3(1+q)}{\nu R}\right).$
Now, inserting the approximation expression (20) into equation (18) and
changing the variables $r\rightarrow z$ through the mapping function
$z=(1+q\exp{(\nu x)})^{-1}$, equation (18) turns to
$\displaystyle\frac{d^{2}R_{n\ell}(z)}{dz^{2}}+\frac{1-2z}{z(1-z)}\frac{dR_{n\ell}(z)}{dz}+\frac{1}{\left[\nu
z(1-z)\right]^{2}}$ $\displaystyle\times\left[\left(\frac{2\mu
E_{n\ell}}{\hbar^{2}}-\frac{D_{o}\ell(\ell+1)}{R^{2}}\right)-z\left(-\frac{2\mu
V_{o}}{\hbar^{2}}+\frac{D_{1}\ell(\ell+1)}{R^{2}}\right)-\frac{D_{2}z^{2}\ell(\ell+1)}{R^{2}}\right]R_{n\ell}(z)=0.$
(22)
Before applying the AIM to this problem, we have to obtain the asymptotic wave
functions and then transform equation (22) into a suitable form of the AIM.
This can be achieved by the analysis of the asymptotic behaviours at the
origin and at infinity. As a result the boundary conditions of the wave
functions $R_{n\ell}(z)$ are taken as follows:
$\displaystyle R_{n\ell}(z)$ $\displaystyle\rightarrow$ $\displaystyle 0\ \ \
\ when\ \ \ \ z\rightarrow 1,$ $\displaystyle R_{n\ell}(z)$
$\displaystyle\rightarrow$ $\displaystyle 0\ \ \ \ when\ \ \ \ z\rightarrow
0,$ (23)
thus, one can write the wave functions for this problem as
$R_{n\ell}(z)=z^{\alpha}(1-z)^{\gamma}F_{n\ell}(z),$ (24)
where we have introduced parameters $\alpha$ and $\gamma$ defined by
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{1}{\nu}\left[\frac{D_{o}\ell(\ell+1)}{R^{2}}-\frac{2\mu
E_{n\ell}}{\hbar^{2}}\right]^{\frac{1}{2}},$ $\displaystyle\gamma$
$\displaystyle=$
$\displaystyle\frac{1}{\nu}\left[-\frac{2\mu}{\hbar^{2}}(V_{o}+E_{n\ell})+\frac{\ell(\ell+1)}{R^{2}}(D_{o}+D_{1}+D_{2})\right]^{\frac{1}{2}},$
(25)
for simplicity. By substituting equation (24) into equation (22), we have the
second-order homogeneous differential equation of the form:
$F_{n\ell}^{\prime\prime}(z)+\left[\frac{(2\alpha+1)-2z(\alpha+\gamma+1)}{z(1-z)}\right]F_{n\ell}^{\prime}(z)-\left[\frac{\alpha^{2}+\gamma^{2}+\alpha+\gamma+2\alpha\gamma-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}{z(1-z)}\right]F_{n\ell}(z)=0,$
(26)
which is now suitable to an AIM solutions. By comparing this equation with
equation (1), we can write the $\lambda_{o}(z)$ and $s_{o}(z)$ values and
consequently; by means of equation (7), we may derive the $\lambda_{k}(z)$ and
$s_{k}(z)$ as follows:
$\displaystyle\lambda_{o}(z)$ $\displaystyle=$
$\displaystyle\frac{2z(\alpha+\gamma+1)-(2\alpha+1)}{z(1-z)},$ $\displaystyle
s_{o}(z)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}+\gamma^{2}+\alpha+\gamma+2\alpha\gamma-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}{z(1-z)},$
$\displaystyle\lambda_{1}(z)$ $\displaystyle=$
$\displaystyle\frac{2+z\left(2-3\gamma+\gamma^{2}-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right)-z^{2}\left(7\gamma-3\gamma^{2}-2-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right)}{z^{2}(1-z)^{2}}$
$\displaystyle\frac{[a(z-1)(z(6\gamma-7)-6)]+a^{2}(4-7z+3z^{2})}{z^{2}(1-z)^{2}}$
$\displaystyle s_{1}(z)$ $\displaystyle=$
$\displaystyle\frac{2\left[(\alpha+\gamma)^{2}+\alpha+\gamma-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right]\left[\gamma
z-(1+\alpha(1-z))\right]}{z^{2}(1-z)^{2}}$ $\displaystyle\ldots etc.$
The substitution of the above equations into equation (10), we obtain the
first $\delta$ values as
$\delta_{o}(z)=\frac{\left[(\alpha+\gamma)^{2}+(\alpha+\gamma)-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right]\left[\alpha(3+2\gamma)+\gamma(3+\gamma)-2+\alpha^{2}-\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}\right]}{z^{2}(1-z)^{2}}.$
(28)
From the root of equation (28), we obtain the first relation between $\alpha$
and $\gamma$ as
$\gamma_{o}+\alpha_{o}=-\frac{1}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}$.
In a similar fashion, we can obtain other $\delta$ values and consequently
establish a relationship between $\alpha_{n}$ and $\gamma_{n}$, $n=1,2,3,$
$\cdots$ as
$\displaystyle\delta_{1}(z)=\left|\begin{array}[]{lr}\lambda_{2}(z)&s_{2}(z)\\\
\lambda_{1}(z)&s_{1}(z)\end{array}\right|=0\ \ \ \ \Rightarrow\ \ \ \
\gamma_{1}+\alpha_{1}=-\frac{3}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}$
(31)
$\displaystyle\delta_{2}(z)=\left|\begin{array}[]{lr}\lambda_{3}(z)&s_{3}(z)\\\
\lambda_{2}(z)&s_{2}(z)\end{array}\right|=0\ \ \ \ \Rightarrow\ \ \ \
\gamma_{2}+\alpha_{2}=-\frac{5}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}$
(34) $\displaystyle\ldots etc.$ (35)
The nth term of the above arithmetic progression is found to be
$\alpha_{n}+\gamma_{n}=-\frac{2n+1}{2}-\frac{1}{2}\sqrt{1+4\frac{D_{2}\ell(\ell+1)}{\nu^{2}R^{2}}}.$
(36)
By substituting for $\alpha$ and $\gamma$, we obtain a more explicit
expression for the eigenvalues energy as
$E_{n\ell}=\frac{\hbar^{2}D_{o}\ell(\ell+1)}{2\mu R^{2}}-\frac{\hbar^{2}}{8\mu
a^{2}}\left[\frac{\left(\frac{1}{2}\sqrt{1+4\frac{D_{2}a^{2}\ell(\ell+1)}{R^{2}}}+\frac{2n+1}{2}\right)^{2}+a^{2}\left(\frac{2\mu
V_{o}}{\hbar^{2}}-\frac{\ell(\ell+1)}{R^{2}}(D_{1}+D_{2})\right)}{\frac{1}{2}\sqrt{1+4\frac{D_{2}a^{2}\ell(\ell+1)}{R^{2}}}+\frac{2n+1}{2}}\right]^{2}.$
(37)
Let us now turn to the calculation of the normalized wave functions. By
comparing equation (26) with equation (13) we have the following:
$t=\frac{2\gamma+1}{2},\ \ \ \ b=1,\ \ \ \ N=-1,\ \ \ \
m=\frac{2\alpha-1}{2},\ \ \ \ \sigma=2\alpha+1,\ \ \ \
\rho=2(\alpha-\gamma)+1.$ (38)
Having determined these parameters, we can easily find the wave functions as
$F_{n\ell}(z)=(-1)^{n}C_{2}\frac{\Gamma(2\alpha+n+1)}{\Gamma{(2\alpha+1)}}\
_{2}F_{1}\left(-n,2(\alpha-\gamma)+1+n;2\alpha+1;z\right),$ (39)
where $\Gamma$ and ${}_{2}F_{1}$ are the Gamma function and hypergeometric
function respectively. By using equations (24) and (39), the total radial wave
function can be written as follows:
$R_{n\ell}(r)=(-1)^{n}N_{n\ell}\frac{\left[1+q^{-1}\exp{\left(\frac{R-r}{a}\right)}\right]^{\gamma}}{\left[1+q\exp{\left(\frac{r-R}{a}\right)}\right]^{\alpha}}\
{{}_{2}F_{1}\left(-n,2(\alpha-\gamma)+1+n;2\alpha+1;\left(1+q\exp{\left(\frac{r-R}{a}\right)}\right)^{-1}\right)},$
(40)
where $N_{n\ell}$ is the normalization constant
Figure 1: The variation of energy spectrum (37) as a function of the
deformation parameter. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $a=0.65fm$. The radial quantum number is fixed
to $n=0$ Figure 2: The variation of energy spectrum (37) as a function of the
deformation parameter. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $a=0.65fm$. The radial quantum number is fixed
to $n=1$.
## 4 Results and Conclusion
In Figures (1) and (2) we plot the energy levels $E_{n\ell}$ versus
deformation constant $q$ for ground $n=0$ and first exited $n=1$ states,
respectively. In Figure (1), it is seen that when $q\approx 0.6$, the orbital
quantum numbers $\ell=0,1,2,$ will have the same energy eigenvalues, i.e.,
$E\approx-0.165fm^{-1}$. For the case when $q<0.6$, we noticed that
$E_{00}>E_{01}>E_{02}$, whereas when $q>0.6$, then $E_{02}>E_{01}>E_{00}$
(less negative or attractive). In Figure (2), taking same parameter set, we
noticed that when $q\approx 1.0$, the three states have energy;
$E\approx-1.04fm^{-1}$. In the case when $q<1.0$, $E_{10}>E_{11}>E_{12}$ but
when $q>1.0$, $E_{12}>E_{11}>E_{10}$. The system becomes weakly attractive. We
have seen that the energy states are sensitive to the deformation constant
$q$. There are no physical energy states for $\ell=0,1,2$ when $q\approx 0.6$
for $n=0$ and when $q<1.0$ for $n=1$
Figure 3: The variation of energy spectrum (37) as a function of the
diffuseness of the nuclear surface. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $q=1.5fm$. The radial quantum number is fixed
to $n=0$. Figure 4: The variation of energy spectrum (37) as a function of the
diffuseness of the nuclear surface. For example we select $\mu=1fm^{-1}$,
$V_{o}=0.3fm^{-1}$, $R=7fm$ and $q=1.5fm$. The radial quantum number is fixed
to $n=1$.
In Figures (3) and (4), we plot the energy levels $E_{n\ell}$ versus the
diffuseness of nuclear surface $a$ for $n=0$ and $n=1$ states, respectively.
For $n=0$ case, the energy eigenvalues increasing as $a$ increases, i.e.,
becoming weakly attractive ($E_{02}>E_{01}>E_{00}$). However, for $n=1$, the
case is same as former (i.e. $n=1$) but the three curves overlap each other.
Figure 5: The variation of energy spectrum (37) as a function of the potential
depth. For example we select $\mu=1fm^{-1}$, $a=0.65fm$, $R=7fm$ and
$q=1.5fm$. The radial quantum number is fixed to $n=0$. Figure 6: The
variation of energy spectrum (37) as a function of the potential depth. For
example we select $\mu=1fm^{-1}$, $a=0.65fm$ ,$R=0.7fm$ and $q=1.5fm$. The
radial quantum number is fixed to $n=1$.
In Figures (5) and (6), we plot the energy levels $E_{n\ell}$ versus the
potential depth potential depth $V_{o}$ for $n=0$ and $n=1$ states,
respectively. For $n=0$ case, we found that there is a substantial change in
the energy eigenvalues $E_{02}>E_{01}>E_{00}$, they increase with increasing
$V_{o}$ and becoming less attractive. However, the same behaviour is seen with
slow change in three curves when $n=1$.
Figure 7: The variation of energy spectrum (37) as a function of the potential
width. For example we select $\mu=1fm^{-1}$, $V_{o}=0.3fm^{-1}$, $a=0.65fm$
and $q=1.5$. The radial quantum number is fixed to $n=0$. Figure 8: The
variation of energy spectrum (37) as a function of the potential width. For
example we select $\mu=1fm^{-1}$, $V_{o}=0.3fm^{-1}$, $a=0.65fm$ and $q=1.5$.
The radial quantum number is fixed to $n=1$.
In Figures (7) and (8) we plot the energy levels $E_{n\ell}$ versus the
potential width $R$ for the two radial quantum numbers. At $R\approx 5.1fm$,
the energy is same (coincides). For $R<5.1fm$, $E_{00}>E_{01}>E_{02}$ (less
attractive) but when $R>5.1fm$, $E_{02}>E_{01}>E_{00}$. It is obvious that the
energy levels are sensitive to potential width $R$. There are no physical
energies when $R<5.1fm$ for $n=0$, and $R<6.2fm$ for $n=1$ . For $E\approx 0$,
the particles are no longer attracted. However, for $E_{01}$ and
$E_{00}\approx-0.2$. When $n=1$, $R\approx 6.2fm$, $E_{02}=E_{01}=E_{00}$. For
different values of $R$, the behavior is seen as former for $n=0$.
Figure 9: The variation of energy spectrum (37) as a function of the particle
mass. For example we select $V_{o}=0.3fm^{-1}$, $R=7fm$, $a=0.65fm$ and
$q=1.5$. The radial quantum number is fixed to $n=0$. Figure 10: The variation
of energy spectrum (37) as a function of the particle mass. For example we
select $V_{o}=0.3fm^{-1}$, $R=7fm$, $a=0.65fm$ and $q=1.5$. The radial quantum
number is fixed to $n=1$.
In Figures (9) and (10) we plot the energy levels $E_{n\ell}$ versus the
reduced mass $\mu$ for the two radial quantum numbers. In the ground level the
energy becomes repulsive (positive) as $\mu$ increases. For example, $E_{02}$
becomes positive for $\mu>2.2fm^{-1}$, $E_{01}>0$ for $\mu>2.5fm^{-1}$ and
$E_{00}>0$ for $\mu>3.0fm^{-1}$. This means that the mass has limit. When mass
increases, then we have no attractive particles and hence no energy spectrum.
However, in the excited level, the energies of the three states are close to
each other and the system remains attractive when the reduced mass increases.
We have seen that the approximately analytical bound states solutions of the
$\ell-$wave Schr$\ddot{o}$dinger equation for the nuclear deformed Woods-Saxon
potential can be solved by proper approximation to the centrifugal term within
the framework of the AIM. By using the AIM, closed analytical forms for the
energy eigenvalues are obtained and the corresponding wave functions have been
presented in terms of hypergeometric functions.
The method presented in this paper is an elegant and powerful technique. If
there are analytically solvable potentials, it provides the closed forms for
the eigenvalues and the corresponding eigenfunctions. However, the case if the
solution is not available, the eigenvalues are obtained by using an iterative
approach [23, 24, 25].
## References
* [1] R. D. Woods and D. S. Saxon, Phys. Rev. 95 (1954) 577.
* [2] W. S. C. Williams, Nuclear and Particle Physics, Clarendon, Oxford, 1996.
* [3] F. Garcia et al., Eur. Phys. J. A 6 (1999) 49.
* [4] V. Goldberg et al., Phys. Rev. C 69 (2004) 031302.
* [5] A. Syntfelt et al., Eur. Phys. J. A 20 (2004) 359.
* [6] A. Diaz-Torres and W. Scheid, Nucl. Phys. A 757 (2005) 373.
* [7] J. Y. Guo and Q. Sheng, Phys. Lett. A 338 (2005) 90.
* [8] V. A.Chepurnov and P. E. Nemirovsky, Nucl. Phys. 49 (1963) 90.
* [9] R. R. Chasman and B. D. Wilkins, Phys. Lett. B 149 (1984) 433.
* [10] J. Dudek and T. Wemer, J. Phys. G: nuclear Phys. 4 (1978) 1543.
* [11] S. Fl$\ddot{u}$gge, Practical Quantum Mechanics, Springer-Verlag, Berlin, 1974.
* [12] V. H. Badalov, H. I. Ahmadov, S. V. Badalov, Int. J. Mod. Phys. E 18 (2010) 1463.
* [13] V. H. Badalov, H. I. Ahmadov, S. V. Badalov, Int. J. Mod. Phys. E 18 (2009) 631.
* [14] S. M. Ikhdair and R. Sever, Cent. Eur. J. Phys. 8 (2010) 652; Int. J. Mod. Phys. A 25 (2010) 3941.
* [15] M. Hamzavi and S. M. Ikhdair, Few-Body Syst, DOI 10.1007/s00601-012-0452-9 (2012).
* [16] S.M. Ikhdair and R. Sever, Int. J. Theor. Phys. 46 (2007) 1643.
* [17] H. Ciftci, R. L. Hall and N. Saad, J. Phys. A: Math Gen. 36 (2003) 11807.
* [18] H. Ciftci, R. L. Hall and N. Saad, Phys. Lett. A: 340 (2005) 388.
* [19] S.M. Ikhdair and R. Sever, Ann. Phys. (Leipzig) 16 (2007) 218.
* [20] L. I. Schiff, Quantum Mechanics 3rd edn. (McGraw-Hill Book Co., New York, 1968).
* [21] L. D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-relativistic Theory, 3rd edn. (Pergamon, New York, 1977).
* [22] V. H. Badalov, H. I. Ahmadov, and S. V. Badalov, arXiv:0912.3890v2 math-ph.
* [23] T. Barakat, J. Phys. A: Math. Gen. 36 (2006) 823.
* [24] F. M. Fernandez, J. Phys. A: Math. Gen. 37 (2004) 6173.
* [25] T. Barakat, Phys. Lett. A 344 (2005) 411.
|
arxiv-papers
| 2012-07-05T10:53:10 |
2024-09-04T02:49:32.671986
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Babatunde J. Falaye, Majid Hamzavi and Sameer M. Ikhdair",
"submitter": "Sameer Ikhdair",
"url": "https://arxiv.org/abs/1207.1218"
}
|
1207.1219
|
# Relating the Kick Velocities of Young Pulsars with Magnetic Field Growth
Timescales Inferred From Braking Indices
A. Güneydaş†∗ and K. Y. Ekşi†
İstanbul Technical University, Faculty of Science and Letters, Department of
Physics, Istanbul 34469, Turkey
† E-mails: guneydas@gmail.com, eksi@itu.edu.tr
∗ Present address: Sabancı University, İstanbul, Turkey
###### Abstract
A nascent neutron star may be exposed to fallback accretion soon after the
proto-neutron star stage. This high accretion episode can submerge the
magnetic field deep in the crust. The diffusion of the magnetic field back to
the surface will take hundreds to millions of years depending on the amount of
mass accreted and the consequent depth the field is buried. Neutron stars with
large kick velocities will accrete less amount of fallback material leading to
shallower submergence of their fields and shorter time-scales for the growth
of their fields. We obtain the relation $\tau_{\rm Ohm}\propto v^{-1}$ between
the space velocity of the neutron star and Ohmic time-scale for the growth of
the magnetic field. We compare this with the relation between the measured
transverse velocities, $v_{\perp}$ and the field growth time-scales,
$\mu/\dot{\mu}$, inferred from the measured braking indices. We find that the
observational data is consistent with the theoretical prediction though the
small number of data precludes a strong conclusion. Measurement of the
transverse velocities of pulsars B1509$-$58, J1846$-$0258, J1119$-$6127 and
J1734$-$3333 would increase the number of the data and strongly contribute to
understanding whether pulsar fields grow following fallback accretion.
††pagerange: Relating the Kick Velocities of Young Pulsars with Magnetic Field
Growth Timescales Inferred From Braking Indices–Relating the Kick Velocities
of Young Pulsars with Magnetic Field Growth Timescales Inferred From Braking
Indices††pubyear: 2012
## 1 INTRODUCTION
Soon after the discovery of radio pulsars (Hewish et al., 1968) their nature
as rapidly rotating highly magnetized neutron stars (NSs) radiating at the
expense of their rotational energy was established (Gold, 1968). A
dimensionless parameter related to the spin-down torques on these rotationally
powered pulsars (RPPs) is the braking index defined operationally as
$n\equiv\nu\ddot{\nu}/\dot{\nu}^{2}$, where $\nu$ is the spin frequency and
dots represent time derivatives. The value of $n$ should be 3 if RPPs are
spinning down with magnetic dipole radiation $\dot{\nu}\propto-\mu^{2}\nu^{3}$
where $\mu$ is the magnetic dipole moment. Most of the measured pulsar braking
indices (Lyne et al., 1993; Livingstone et al., 2007; Weltevrede et al., 2011;
Roy et al., 2012) are close to 3 but slightly less as shown in Table 1 which
indicate that another process also contributes to the MDR in braking these
objects. The recently measured braking index of PSR J1734$-$3333 as $n=0.9\pm
0.2$ (Espinoza et al., 2011) and that of J0537$-$6910 as $n=-1.5$ (Middleditch
et al., 2006) together with the earlier measurement as $n=1.4\pm 0.2$ of the
Vela pulsar (Lyne et al., 1996) imply that this process might severely alter
the spin history of young neutron stars.
Table 1: Pulsars with accurately measured braking indices.
Pulsar | $\nu$ | $\tau_{c}$ | $n$ | $\tau_{\mu}$ | $v_{\perp}$ | References
---|---|---|---|---|---|---
| (Hz) | (kyr) | | (kyr) | (km s-1) |
B0531$+$21(Crab) | 30.225 | 1.2399 | 2.51(1) | 10.1(2) | 140$\pm$8 | Lyne et al. (1993); Ng & Romani (2006)
B0833$-$45(Vela) | 11.2(5) | 11.303 | 1.4(2) | 28(4) | 62$\pm$2 | Lyne et al. (1996); Ng & Romani (2007)
J1833$-$1034 | 16.159 | 4.8535 | 1.8569(6) | 16.984(9) | 125$\pm$30 | Roy et al. (2012); Ng & Romani (2007)
B0540$-$69 | 19.738 | 1.6763 | 2.087(7) | 7.34(6) | 1300$\pm$612 | Gradari et al. (2011); Ng & Romani (2007)
J0537$-$6910 | 62.038 | 4.9743 | -1.5 | 4.422 | 634$\pm$50 | Middleditch et al. (2006); Ng & Romani (2007)
B1509$-$58 | 6.6115 | 1.5648 | 2.832(3) | 37.3(7) | $*$ | Livingstone & Kaspi (2011)
J1846$-$0258 | 3.0743 | 0.72736 | 2.65(1) | 8.3(2) | $*$ | Livingstone et al. (2007)
J1119$-$6127 | 2.4512 | 1.6078 | 2.684(2) | 20.4(1) | $*$ | Weltevrede et al. (2011)
J1734$-$3333 | 0.85518 | 81.280 | 0.9(2) | 0.16(2) | $*$ | Espinoza et al. (2011)
Numbers in parenthesis are last digit errors.
Different mechanisms have been invoked (e.g. Melatos, 1997; Menou et al.,
2001; Wu et al., 2003; Ho & Andersson, 2012) for addressing what makes the
braking indices of RPPs less than 3. An early suggestion (Blandford & Romani,
1988) which has been recently advocated (Bernal & Page, 2011; Espinoza et al.,
2011) is that the braking indices of RPPs are less than 3 because their
magnetic fields are growing in time. If the magnetic dipole moment of a pulsar
is changing in time, the braking index becomes
$n=3+2\frac{\dot{\mu}}{\mu}\frac{\nu}{\dot{\nu}}\ \ ,$ (1)
(e.g. Chanmugam & Sang, 1989) where $\mu$ is the magnetic dipole moment of the
NS. The characteristic time-scale for the growth of the magnetic dipole
moment, $\tau_{\mu}\equiv\mu/\dot{\mu}$, is then
$\tau_{\mu}=\tau_{c}\frac{4}{3-n}\ \ ,$ (2)
where $\tau_{c}\equiv-\nu/2\dot{\nu}$ is the characteristic spin-down age. We
have assumed here that the field growth is the dominant process modifying the
braking index from its value 3 and the effect of other processes, e.g. those
of magnetospheric currents (Contopoulos & Spitkovsky, 2006; Spitkovsky, 2006)
is small. The field growth timescales inferred from Equation (2) are shown in
Table 1. The wide range of $\tau_{\mu}$ values, from hundreds to millions of
years, indicate that explaining the observed values of braking indices by
growing magnetic dipole fields requires a process which could accomodate such
different timescales.
The magnetic dipole moment of a RPP could be growing because it was submerged
(Muslimov & Page, 1995; Young & Chanmugam, 1995; Geppert et al., 1999) in the
crust due to fallback accretion (Colgate, 1971; Zel’dovich et al., 1972;
Chevalier, 1989) following the supernova explosion. In this work we predict
from this hypothesis that pulsars which get large kick velocities during their
birth should accrete less amount of matter, have shallower field burial and so
shorter field growth time-scales. We plot field growth time-scales
$\tau_{\mu}$ as inferred from the measured braking indices via Equation (2)
against the measured transverse velocities. We derive a relation between the
Ohmic timescale for field evolution in the crust and the mass of the accreted
matter finally relating the latter to the space velocity of pulsars. The
observational data, low in number, marginally support the theoretical
prediction favoring the idea that the magnetic field of young pulsars could be
growing as a consequence of field diffusion to the surface following fallback
accretion.
## 2 A RELATION BETWEEN SPACE VELOCITY AND FIELD GROWTH TIMESCALE
A common ingredient of supernova models is the fallback of some material that
can not reach the escape velocity (Colgate, 1971; Zel’dovich et al., 1972;
Chevalier, 1989). According to Chevalier (1989) the fallback due to the
reversed shock in SN87A reached to the surface of the neutron star 2 hours
after bounce. The initial accretion rate of fallback matter can be very large.
For SN87A the initial accretion rate is estimated (Chevalier, 1989) to be
$350\,M_{\odot}\mathrm{yr}^{-1}$. The amount of matter that reaches the
surface and the time required for this to happen varies depending on many
details like the initial density of the surrounding medium, relative velocity
of the NS with respect to the medium, the magnetic field and spin frequency of
the NS before fallback (Colpi et al., 1996).
The rapid accretion of fallback matter can bury the preexisting magnetic field
that was formed in the proto-NS stage (Muslimov & Page, 1995; Young &
Chanmugam, 1995; Muslimov & Page, 1996; Geppert et al., 1999; Ho, 2011; Viganò
& Pons, 2012; Bernal et al., 2012). The field would then diffuse back to the
surface in an Ohmic time-scale $\tau_{\rm Ohm}=(\Delta R)^{2}/\eta$ where
$\Delta R\sim 0.1$ km is the depth of submergence and $\eta\equiv
c^{2}/4\pi\sigma$ is the magnetic diffusivity. The ohmic time-scale varies
within a large range, hundreds to millions of years, depending on the
conductivity, $\sigma$ of the crust and $\Delta R$ which, in turn, depends on
the amount of fallback, $\Delta M$. If the initial accretion rate is very
large the magnetic field is submerged rapidly, and the amount of accreted
matter is determined by the density of the medium and the velocity of the NS.
Old NSs in low mass X-ray binaries have magnetic fields $\sim 10^{9}$ G which
is three orders of magnitude smaller than fields inferred for young NSs in
high mass X-ray binaries and young isolated radio pulsars. The magnetic
moments of NSs in binary systems is inversely correlated with accretion
history (Taam & van den Heuvel, 1986) supporting the recycling scenario
(Bisnovatyi-Kogan & Komberg, 1974, 1976; Alpar et al., 1982; Radhakrishnan &
Srinivasan, 1982) which suggests that the millisecond radio pulsars are spun
up in x-ray binaries. Three distinct mechanisms have been suggested for the
reduction of the magnetic field: (i) accretion induced heating decreases the
conductivity of the star thus leading to the accelerated ohmic decay (Konar &
Bhattacharya, 1997; Geppert & Urpin, 1994; Urpin & Konenkov, 1997); (ii)
vortex-fluxoid interactions in the superconducting core (Muslimov & Tsygan,
1985; Srinivasan et al., 1990); and (iii) magnetic screening or burial
(Bisnovatyi-Kogan & Komberg, 1974; Cheng & Zhang, 1998; Konar & Bhattacharya,
1997; Payne & Melatos, 2004; Wang et al., 2012). A well known problem with the
field burial scenario is that the buried magnetic field will be prone to
instabilities that rapidly will overturn the field (Litwin et al., 2001;
Vigelius & Melatos, 2008; Mukherjee & Bhattacharya, 2012). This problem, a
solution of which is beyond the scope of this _letter_ , is also relevant for
fields buried by fallback accretion. We simply assume that whatever mechanism
suppresses the instabilities in the case of binary accretion could also work
in the case of fallback accretion.
Figure 1: The relation between the measured transverse velocities $v_{\perp}$
and characteristic magnetic field growth time-scale
$\tau_{\mu}\equiv\mu/\dot{\mu}$ inferred from measured braking indices via
Equation (2) for the 5 pulsars, B0531$+$21(Crab), B0833$-$45(Vela),
J1833$-$1034, B0540$-$69 and J0537$-$6910\. These 5 objects form the subset of
pulsars with accurately measured braking indices and measured transverse
velocities. Also shown on the plot is the estimate of the Ohmic time-scale
given in Eqn.(9) depending on the space velocity $v$: Dotted line stands for
the case $\rho_{0}\ t_{0}^{3}={\rm 10^{14}\ gr\ cm^{-3}\ s^{3}}$, dashed line
stands for the case $\rho_{0}\ t_{0}^{3}={\rm 10^{15}\ gr\ cm^{-3}\ s^{3}}$
and dashed-dotted line stands for the case $\rho_{0}\ t_{0}^{3}={\rm 10^{16}\
gr\ cm^{-3}\ s^{3}}$.
In binary accretion the burial of the magnetic field will proceed by the
formation of a magnetically confined mountain (Woosley & Wallace, 1982;
Hameury et al., 1983; Mukherjee & Bhattacharya, 2012) of accreted plasma on
the polar caps which then spreads laterally thus transporting the magnetic
flux towards the equator (Cheng & Zhang, 1998). The mountain then relaxes
(Vigelius & Melatos, 2009) due to finite resistivity and sinks while the
accreted plasma becomes a part of the crust (Payne & Melatos, 2004, 2007;
Choudhuri & Konar, 2002; Vigelius & Melatos, 2008; Wette et al., 2010). Simply
put, the burial of the field in a rather stable configuration requires the
break down of spherical symmetry which is achieved by the modulation of the
flow by the magnetosphere. As we show in the following, for the supercritical
accretion rates expected to prevail at the initial stages of fallback
accretion the magnetosphere, overwhelmed by the accretion flow, can not
channel the inflowing matter to the polar caps. In this case the spherical
symmetry is broken down by the asymmetry due to the space velocity of the
pulsar i.e. the face of the NS in the direction of motion is subject to higher
accretion rate than the opposite face. We expect in this case that the spin of
the NS will convert the magnetic flux in the pre-existing dipole field to the
toroidal component in the equator. To our knowledge burial of the magnetic
field subject to Bondi-Hoyle accretion has not been explored numerically yet.
For a depth at which the density is greater than $10^{6}$ g cm-3 the electrons
in the crust are ultra-relativistic and the relativity parameter $x_{\rm
F}\equiv p_{\rm F}/m_{\rm e}c\gg 1$ where $p_{\rm F}$ is the Fermi momentum
and $m_{\rm e}$ is the electron mass. At the age of $\sim 10^{4}$ years which
is the order of magnitude age of most of the considered pulsars with measured
braking indices, $T\gtrsim 10^{8}$ K at the center of the star and $T\gtrsim
10^{6}$ K at the surface. In such regimes the conductivity is determined by
the electron-ion scattering in the melted layer at the surface and electron-
phonon scattering in the solid crust
$\sigma_{\rm e-ph}=1.21\times 10^{22}x_{\rm F}^{2}T_{6}^{-1}$ (3)
(Yakovlev & Urpin, 1980) where $T_{6}\equiv T/10^{6}$ K and we assumed $x_{\rm
F}\gg 1$ (Urpin et al., 1994). At lower temperatures conductivity will depend
on the impurity scattering (Flowers & Itoh, 1976) which is independent of
temperature.
To a good approximation the mass $\Delta M$ and thickness $\Delta R$ of the
accreted layer are small fractions of the total mass, $M$ and radius $R$ of
the star, respectively. Within this approximation the relativity parameter
depends on $z\equiv\Delta R/H_{\rm R}$ as $x_{\rm F}=\sqrt{z(z+2)}$ (Urpin &
Yakovlev, 1979) where $H_{\rm R}=m_{\rm e}c^{2}/g\mu_{\rm e}m_{\rm p}$ is the
scale-height. Here $g=GMR^{-2}(1-2GM/Rc^{2})^{-1}$, $\mu_{\rm e}=A/Z$ ($A$ is
the atomic weight) and $m_{\rm p}$ is the proton mass. For $M=1.4M_{\odot}$,
$R=10$ km and $\mu_{\rm e}=2$ one obtains $H_{\rm R}=7.25$ m. The mass of the
crust region of thickness $\Delta R$ is $\Delta M\simeq 4\pi
R^{2}\int_{R-\Delta R}^{R}\rho(r)dr$ where $\rho$ is the density. By using the
hydrostatic equilibrium equation (Oppenheimer & Volkoff, 1939) this can be
written as $\Delta M=(4\pi R^{2}/g)P_{\rm B}$ where $P_{\rm B}$ is the
pressure at the bottom of the layer. For ultra-relativistic electrons
$P=P_{0}x_{\rm F}^{4}$ where $P_{0}=(2\pi/3)m_{\rm e}c^{2}(m_{\rm
e}c/h)^{3}=1.2\times 10^{23}$ dyne cm-2. If one approximates
$\sqrt{z(z+2)}\sim z$ as appropriate for $x_{\rm F}\gg 1$, then
$\Delta R=H_{\rm R}\left(\frac{g\Delta M}{4\pi R^{2}P_{0}}\right)^{1/4}\simeq
77.5\,{\rm m}\,\left(\frac{\Delta M}{10^{-6}M_{\odot}}\right)^{1/4}.$ (4)
relates the accreted mass to the thickness of the resulting layer. Assuming
the conductivity due to electron-phonon scattering dominates we find the Ohmic
time scale to be
$\tau_{\rm Ohm}=36.7\,{\rm kyr}\,\left(\frac{\Delta
M}{10^{-6}M_{\odot}}\right)T_{6}^{-1}.$ (5)
As the mass of the NS increases by accretion its radius becomes smaller and
hence $g$ increases. The relation $\Delta M=(4\pi R^{2}/g)P_{\rm B}$ implies
that the mass of the crust will decrease meaning that the difference is to be
assimilated to the core (Konar & Bhattacharya, 1997). The movement of the
current-carrying parts to larger depths where conductivity is larger will
increase the Ohmic timescale and finally will lead to the freezing of the
field if superconducting core is reached. We ignore this effect here as we
infer that the amount of fallback mass accreted by any of the objects
considered in this work is sufficiently small. Yet we think the mechanism
could be important for central compact objects like Cas A which appear to have
very small magnetic fields and could have relevance to the hidden magnetic
field scenario (Viganò & Pons, 2012; Ho, 2011; Shabaltas & Lai, 2012).
The newborn neutron star will be accreting from a uniformly expanding medium
where density decreases as $\rho\propto t^{-3}$ (Chevalier, 1989). The neutron
star moving with velocity $v$ in this medium is subject to a mass inflow rate
of $\dot{M}_{\rm in}=4\pi(GM)^{2}\rho v^{-3}$ (Hoyle & Lyttleton, 1939). We
combine these together as
$\dot{M}_{\rm
in}=\dot{M}_{0}\left(1+\frac{t}{t_{0}}\right)^{-3},\qquad\dot{M}_{0}=\frac{4\pi
G^{2}M^{2}\rho_{0}}{v^{3}}$ (6)
where $\rho_{0}$ is the initial density of the medium and $t_{0}$ is the
dynamical time-scale at which the accretion rate declines. We use the symmetry
of the fluid dynamic equations under translations in time to get rid of the
singularity at $t=0$ (Ertan et al., 2009). Initially, the mass inflow rate can
be very high and the amount of matter accreting onto the neutron star will be
Eddington limited ($\dot{M}_{\ast}=\dot{M}_{\rm E}$) until
$t_{1}=t_{0}[(\dot{M}_{0}/\dot{M}_{\rm E})^{1/3}-1]$ at which $\dot{M}_{\rm
in}$ drops down to $\dot{M}_{\rm E}\equiv L_{\rm E}R/GM$ where $L_{\rm E}=4\pi
GMm_{\rm p}c/\sigma_{\rm T}$ is the Eddington luminosity.
The magnetospheric radius is determined by the Alfvén radius
$R_{\mathrm{A}}=\left(\frac{\mu^{2}}{\sqrt{2GM}\dot{M}_{\mathrm{in}}}\right)^{2/7}$
(7)
though this is very likely modified by the radiation pressure due to super-
Eddington mass infall rates at the initial stage. Initial value of the Alfvén
radius, referring Equation (6), is
$R_{\mathrm{A0}}=1\times 10^{6}\,{\rm
cm}\,\,\left(\frac{\rho_{0}}{10^{-7}\,{\rm
g\,cm^{-3}}}\right)^{-2/7}\mu_{30}^{4/7}v_{100}^{6/7}$ (8)
where $v_{100}\equiv v/(\rm 100\ km\ s^{-1})$ and we assumed $M=1.4M_{\odot}$.
For $\rho_{0}>10^{-7}\,{\rm g\,cm^{-3}}$, the initial value of the Alfvén
radius is smaller than the radius of the star suggesting that the
magnetospheric radius is dynamically not important at the initial rapid
accretion stage. In fact the initial accretion rate of fallback may be
hypercritical i.e. photons are trapped in the accretion flow and the energy
released in the accretion process is lost by neutrino emission (see e.g.
Chevalier, 1989; Bernal & Page, 2011; Bernal et al., 2012) as signaled by the
divergence of $\dot{M}_{\rm in}$ for $v=0$. We do not assume this to be the
case for the pulsars with large initial kicks that we consider in this work.
Even with the more modest initial accretion rates we consider, the NS starts
its life in the accretion mode and the propeller mechanism does not work at
the initial stage at least not before most of the fallback matter accretes.
Relying on this we assume that the spin and magnetic field of the NS will not
change the amount of accreted matter significantly unless they have magnetar-
like initial values, $\mu\sim 10^{33}$ G and $P_{0}\sim 1$ ms.
The accretion rate onto the neutron star, for $t>t_{1}\gg t_{0}$ is
$\dot{M}_{\ast}=\dot{M}_{\rm E}(t/t_{1})^{-3}$. This implies that the total
mass accreted onto the neutron star will be $\Delta M=\frac{3}{2}\dot{M}_{\rm
E}t_{1}$. Assuming $\dot{M}_{0}\gg\dot{M}_{\rm E}$ we find $\Delta
M=\frac{3}{2}\dot{M}_{\rm E}t_{0}(\dot{M}_{0}/\dot{M}_{\rm E})^{1/3}\propto
v^{-1}$ and this leads to
$\tau_{\rm Ohm}=202\,{\rm kyr}\left(\frac{\rho_{0}t_{0}^{3}}{\rm 10^{15}\ g\
cm^{-3}\ s^{3}}\right)^{1/3}T_{6}^{-1}v_{100}^{-1}$ (9)
where we scale the poorly known quantities $\rho_{0}$ and $t_{0}$ together. As
the fallback is expected to occur a few hours after the supernova explosion
(Chevalier, 1989; Bernal et al., 2012) typically $t_{0}\sim 10^{4}$ s which
means $\rho_{0}\sim 10^{3}$ g cm-3 would provide the observed time-scale for
the field evolution.
As a test of this relation $\tau_{\rm Ohm}\propto v^{-1}$ we check the
relation between $\tau_{\mu}$ inferred from braking indices via Equation (2)
and measured transverse velocities $v_{\perp}$ expecting
$\tau_{\mu}\sim\tau_{\rm Ohm}$ and $v_{\perp}\sim v$. Only 9 RPPs have
accurately measured braking indices that would allow for inferring the time-
scale $\tau_{\mu}$ for the growth of their magnetic moment via Equation (2).
Of these RPPs, there are only 5 with measured transverse velocities as shown
in Table 1. In Figure 1 we plot $\tau_{\mu}$ versus $v_{\perp}$ of these
objects together with the estimate of the Ohmic time-scale in Eqn.(9)
depending on the space velocity $v$. Although the number of data is small, an
inverse relation between the field growth time-scale and the transverse
velocities can still be noticed as a result of the strong dependence of
accretion rate on $v$. The scattering in the data is most likely the result of
differences in the temperatures of the pulsars, the initial density $\rho_{0}$
of the surrounding media, and possibly the weaker factors like the initial
spin and magnetic field of the neutron star. Note also that $\tau_{\mu}$ does
not necessarily coincide with $\tau_{\rm Ohm}$ if the field growth is not
exponential. Add to this that $v_{\perp}$ may not be a good estimate of the
space velocity $v$ if the radial component of the velocity is not negligible
compared to $v_{\perp}$. Given these deteriorating factors the relation
obtained with 5 data can be considered promising.
## 3 DISCUSSION
As a prediction of the fallback submergence and post-fallback growth of
magnetic fields, we have derived the relation $\tau_{\rm Ohm}\propto v^{-1}$
between the space velocity $v$ and Ohmic time-scale for the diffusion of a
buried magnetic field, $\tau_{\rm Ohm}$. We looked for a similar relation
between the measured transverse velocities $v_{\perp}$ and characteristic
field growth time-scales $\tau_{\mu}\equiv\mu/\dot{\mu}$ inferred from the
measured braking indices. Such an inverse relation between the kick velocity
and the field-growth time scale of pulsars is a prediction of only post-
fallback magnetic field growth model and not of any other model.
As the number of pulsars with both measured braking indices and transverse
velocities is small (only 5), it is not possible to claim for a strong
relation. Yet we found the result promising given there are many factors that
could deteriorate the predicted simple relation. It appears from this analysis
that measuring the transverse velocities of pulsars B1509$-$58, J1846$-$0258,
J1119$-$6127 and J1734$-$3333 would almost double the number of data and allow
for a stronger conclusions for the model.
Our estimations indicate that the depth of field submergence for the
considered population of young neutron stars is very shallow, possibly leading
to the selection of these objects as rotationally powered pulsars. Smaller
space velocities result with larger amount of accretion and very long
diffusion time-scales of buried magnetic fields leading to young neutron stars
not appearing as rotationally powered pulsars. This hidden magnetic field
scenario is recently studied by Viganò & Pons (2012); Ho (2011); Shabaltas &
Lai (2012). This, as well as a more accurate numerical analysis of the field
growth of pulsars, is the subject of a following work.
## Acknowledgments
K. Y. E. acknowledges support from the Faculty of Science and Letters of
İstanbul Technical University. K. Y. E. thanks A. Patruno for hospitality
during his visit to Amsterdam and M. A. Alpar for his support and careful
reading of an early version of the manuscript. We thank T. Tauris and W. Ho
for helpful comments. We thank the anonymous referee for comments that helped
to improve the paper.
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|
arxiv-papers
| 2012-07-05T11:00:22 |
2024-09-04T02:49:32.678686
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. G\\\"uneyda\\c{s}, K. Y. Ek\\c{s}i",
"submitter": "Abdullah G\\\"uneyda\\c{s}",
"url": "https://arxiv.org/abs/1207.1219"
}
|
1207.1298
|
# Witnessed entanglement and the geometric measure of quantum discord
Tiago Debarba debarba@fisica.ufmg.br Thiago O. Maciel Reinaldo O. Vianna
reinaldo@fisica.ufmg.br Departamento de Física - ICEx - Universidade Federal
de Minas Gerais, Av. Pres. Antônio Carlos 6627 - Belo Horizonte - MG - Brazil
- 31270-901.
###### Abstract
We establish relations between geometric quantum discord and entanglement
quantifiers obtained by means of optimal witness operators. In particular, we
prove a relation between negativity and geometric discord in the Hilbert-
Schmidt norm, which is slightly different from a previous conjectured one [1].
We also show that, redefining the geometric discord with the trace norm,
better bounds can be obtained. We illustrate our results numerically.
###### pacs:
03.67.Mn, 03.65.Aa
## I Introduction
Entanglement has been widely investigated in the last years RMP-Horodecki ,
and is a resource that allows for tasks that cannot be performed classically,
as teleportation b93 , quantum key distribution ekert , superdense coding bw92
, and speed-up of some algorithms speedup , just to cite a few examples.
Therefore entanglement is an indisputable signature of the non-classicallity
of a state. Nevertheless, some authors have argued that there is more to the
quantumness of a state than just its entanglement hv2001 ; oz2001 ; discordia-
review . This notion of quantumness beyond entanglement is captured by the
quantum discord, which is defined as all the correlations contained in a state
but the classical ones hv2001 , or as a measure of disturbance of a state
after local measurements oz2001 , both definitions being compatible with a
class of separable states with non-null quantum discord.
Recent investigations suggest that quantum discord can be considered a
resource that gives a quantum advantage discordia-recurso . In order to deepen
our understanding of both the usefulness of such a resource and how quantum it
really is, it is important to devise operational means to quantify it, and
also to relate it to quantum entanglement. In this respect, the geometric
discord dvb2010 defined as the distance between the state of interest and a
properly defined classical state is an invaluable tool. Unhappily the
classical states do not form a convex set facca2010 , and therefore one cannot
use the well known separating hyperplane theorem to characterize discord as is
done with the witness operators in the case of the entanglement problem RMP-
Horodecki .
Interesting investigations relating entanglement and discord have been done
recently ga2011 ; f2011 . In f2011 , entanglement of formation is related to
discord in a conservation equation, and in ga2011 geometric discord is
conjectured to be bounded by the negativity. While entanglement of formation
is not computable in general, many other interesting entanglement quantifiers
can be expressed in terms of optimal entanglement witnesses b2005 which, by
its turn, can be calculated numerically by means of efficient semidefinite
programs b2004 ; b2006 . In this work we will explore bounds for geometric
discord by means of optimal entanglement witnesses. In particular, we will
prove that negativity bounds the geometric discord.
This paper is organized as follows. In Sec.II we briefly revise quantum
discord, and propose a redefinition of geometric discord using the Schatten
$p$-norm. In Sec.III, we recall the witnessed entanglement, with special
attention to both negativity and robustness. In Sec.IV, we derive bounds for
geometric discord using witnessed entanglement. In Sec.V, we illustrate our
results for Werner states and some families of bound entangled states. We
conclude in Sec.VI.
## II Quantum Discord
The total amount of correlations of a bipartite system $AB$ is quantified by
the well known mutual information, which in the classical case can be written
in two equivalent forms linked by Bayes’ rule, namely:
$I(A:B)=H(A)+H(B)-H(AB)=H(A)-H(A|B)$, being $H(X)$ the Shanon entropy of $X$
and $H(X|Y)$ the conditional entropy of $X$ given $Y$. For a quantum system
$\rho_{AB}$, the mutual information is defined in terms of the von Neuman
entropy $S(\rho)=-Tr(\rho\log\rho)$, and reads:
$I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (1)
with $\rho_{A}$ and $\rho_{B}$ being the marginals of $\rho_{AB}$. However,
the definition of a quantum conditional entropy is dependent on the choice of
a given POVM $\\{\Pi_{k}\\}$ to be measured on party B, and the two
expressions for the mutual information are no longer equivalent. While Eq.1
still quantifies the total amount of correlations in the quantum state, the
other expression involving the conditional entropy needs some attention. After
the measurement of $\Pi_{k}$ on $B$, party $A$ is left in the state
$\rho_{A|k}=Tr_{B}(\mathbb{I}\otimes\Pi_{k}\rho_{AB})/p_{k}$, with probability
$p_{k}=Tr(\mathbb{I}\otimes\Pi_{k}\rho_{AB})$. Now we can write the
conditional entropy associated to the POVM $\\{\Pi_{k}\\}$ as:
$S(\rho_{A|B})=\sum_{k}p_{k}S(\rho_{A|k}).$ (2)
$J(\rho_{AB},\\{\Pi_{k}\\})\equiv S(\rho_{A})-S(\rho_{A|B})$ quantifies the
classical correlations contained in $\rho_{AB}$ under measurements in the
given POVM. Therefore, maximizing $J(\rho_{AB},\\{\Pi_{k}\\})$ over all POVMs
quantifies the classical correlations in the quantum state, namely hv2001 ;
discordia-review :
$J_{AB}(\rho)=S(\rho_{A})-\min_{\Pi_{k}}\sum_{k}p_{k}S(\rho_{A|k}),$ (3)
where the POVMs can be chosen to be rank-one hkz2004 . Finally, the quantum
discord is the disagreement between the nonequivalent expressions of mutual
information in the quantum case, namely oz2001 ; discordia-review :
$D(\rho_{AB})=I(\rho_{AB})-J(\rho_{AB}).$ (4)
Note that $D(\rho_{AB})$ is non-negative and asymmetric with respect to
$A\leftrightarrow B$.
As discord is supposed to measure the quantumness of a state, it is no wonder
that the maximally entangled states Eq.5 are the most discordant, while states
which are a mere encoding of classical probability distributions Eq.6 are
concordant (i.e. $I(\rho_{AB})=J(\rho_{AB})$) o3h2002 . Bipartite maximally
entangled states in $\mathcal{B}(\mathbb{C}^{d=d_{A}\times d_{B}})$, with
$d_{A}=d_{B}$, have the form:
$\phi=\frac{1}{d_{A}}\sum_{i,j=1}^{d_{A}}|ii\rangle\langle jj|,$ (5)
while classical states can be written as:
$\xi=\sum_{i,j=1}^{d_{A}}p_{ij}|e_{i}\rangle\langle
e_{i}|\otimes|f_{j}\rangle\langle f_{j}|,$ (6)
where $|e_{i}\rangle$ and $|f_{j}\rangle$ are two orthonormal bases. Note
however, that as discord is asymmetric, if the measurements are to be done in
subsystem $B$, the following class of states are also concordant or classical:
$\xi=\sum_{j=1}^{d_{A}}p_{j}\rho_{j}\otimes|f_{j}\rangle\langle f_{j}|.$ (7)
To distinguish these two classes of classical states, sometimes the former is
referred to as classical-classical, while the later is quantum-classical.
An alternative definition for quantum discord is based on the distance between
the given quantum state and the closest classical state dvb2010 ; mpsvw2010 ;
bggfcz2011 ; lf2010 ; bm2011 . Adopting the Hilbert-Schmidt norm $\|\|_{(2)}$,
we can write dvb2010 :
$D_{(2)}(\rho_{AB})=\min_{\xi\in\Omega}\|\rho-\xi\|^{2}_{(2)},$ (8)
where $\Omega$ is the set of zero-discord states. This measure can be
interpreted as the minimal disturbance after local measurements on subsystem
$B$. In this case $\Omega$ contains the states in Eq.7, and $D_{(2)}$ can be
calculated analytically for some states lf2010 .
Consider the Schatten $p$-norm for some matrix $A$ and positive integer $p$:
$\|A\|_{(p)}=\\{Tr[(A^{{\dagger}}A)^{p/2}]\\}^{1/p},$ (9)
which, for $p=2$ is the Hilbert-Schmidt norm. In finite Hilbert spaces, these
norms induce the same ordering zyc . Therefore we propose to extend the
geometric discord for any $p$-norm, namely:
$D_{(p)}(\rho_{AB})=\min_{\xi\in\Omega}\|\rho_{AB}-\xi\|^{p}_{(p)}.$ (10)
Note that for $p\geq q$, we have $\|A\|_{p}\leq\|A\|_{q}$. It follows that the
1-norm is the most distinguishable distance in Hilbert space. Therefore, we
shall investigate the geometric discord in the 1-norm ($D_{(1)}$), besides the
usual $D_{(2)}$. As we shall see, it is easy to bound these geometric discords
by entanglement witnesses.
## III Witnessed Entanglement
Entanglement witnesses are Hermitian operators (observables - $W$) whose
expectation values contain information about the entanglement of quantum
states. The operator $W$ is an entanglement witness for a given entangled
quantum state $\rho$ if the following conditions are satisfied hhh1996 : its
expectation value is negative for the particular entangled quantum state
($Tr(W\rho)<0$), while it is non-negative on the set of separable states ($S$)
($\forall\sigma\in\mathcal{S},\,\,\,Tr(W\sigma)\geq 0$). We are particularly
interested in optimal entanglement witnesses. $W_{opt}$ is the OEW for the
state $\rho$ if
$Tr(W_{opt}\rho)=\min\limits_{W\in\mathcal{M}}\,Tr(W\rho),$ (11)
where $\mathcal{M}$ represents a compact subset of the set of entanglement
witnesses $\mathcal{W}$ b2005 .
OEWs can be used to quantify entanglement. Such quantification is related to
the choice of the set $\mathcal{M}$, where different sets will determine
different quantifiers b2005 . We can define these quantifiers by:
$E_{w}(\rho)=\max{(0,-\min\limits_{W\in\mathcal{M}}\,Tr(W\rho))}.$ (12)
An example of a quantifier that can be calculated using OEWs is the
Generalized Robustness of entanglement vt1999 ( $\mathcal{R}_{g}(\rho)$),
which is defined as the minimum required mixture such that a separable state
is obtained. Precisely, it is the minimum value of $s$ such that
$\sigma=\frac{\rho+s\varphi}{1+s}$ (13)
be a separable state, where $\varphi$ can be any state. We know that the
Generalized Robustness can be calculated from Eq.12, using
$\mathcal{M}=\\{W\in\mathcal{W}\,|\,W\leq\mathbb{I}\\}$ b2005 , where
$\mathbb{I}$ is the identity operator; in other words,
$\mathcal{R}_{g}(\rho)=\max{(0,-\min_{\\{W\in\mathcal{W}\,|\,W\leq\mathbb{I}\\}}Tr(W\rho))}.$
(14)
A particular case of the Generalized Robustness is the Random Robustness,
where $\varphi$ in Eq.13 is taken to be the maximally mixed state
($\mathbb{I}/d$). In this case, the compact set of entanglement witnesses is
$\mathcal{M}=\\{W\in\mathcal{W}|Tr(W)=1\\}$. The Random Robustness
$\mathcal{R}_{r}(\rho)$ quantifies the resilience of the entanglement to white
noise, and is given by b2006 :
$d\times\mathcal{R}_{r}(\rho)=\max{(0,-\min_{\\{W\in\mathcal{W}\,|\,Tr(W)=1\\}}Tr(W\rho))}.$
(15)
The well known Negativity for bipartite states, which is the sum of the
negative eigenvalues of the partial transpose of the given state,
$\mathcal{N}(\rho)\equiv(\|\rho^{T_{A}}\|_{(1)}-1)/2$, can also be expressed
in terms of OEWs as b2005 :
$\mathcal{N}(\rho)=\max\\{0,-\min_{0\leq
W^{T_{A}}\leq\mathbb{I}}Tr(W\rho)\\}.$ (16)
The construction of entanglement witnesses is a hard problem. In an
interesting method proposed by Brandão and Vianna b2004 , the optimization of
entanglement witnesses is cast as a robust semidefinite program (RSDP).
Despite RSDP is computationally intractable, it is possible to perform a
probabilistic relaxation turning it into a semidefinite program(SDP), which
can be solved efficiently convexoptimization .
## IV Bounding geometric discord with witnessed entanglement
In this section we show that geometric discord, in any norm, is lower bounded
by entanglement. In particular, we show that norm-2 geometric discord is
bounded by negativity, but the relation is slightly different from that
conjectured by Girolami and Adesso [12].
For any two operators $A$,$B$$\in\mathcal{B}(\mathbb{C}^{d})$ and the Schatten
$p$-norm $\|A\|_{p}=Tr[(AA^{{\dagger}})^{p/2}]^{1/p}$, the following
inequality holds:
$\|A\|_{p}\|B\|_{q}\geq|Tr[AB^{{\dagger}}]|,$ (17)
where $1/q+1/p=1$.
The geometrical discord for a state
$\rho\in\mathcal{B}(\mathbb{C}^{d=d_{A}\times d_{B}})$ is:
$D_{p}(\rho)=\|\rho-\bar{\xi}\|_{p}^{p},$ (18)
where $\bar{\xi}$ is the closest non-discordant state. The witnessed
entanglement of $\rho$ can be written as:
$E_{w}(\rho)=\min\\{0,-Tr[W_{\rho}\rho]\\},$ (19)
where $W_{\rho}$ is the optimal entanglement witness of $\rho$. Plugging
$A=\|\rho-\bar{\xi}\|_{p}^{p}$ and $B=W_{\rho}$ in Eq.17, we get:
$\|\rho-\bar{\xi}\|_{p}\|W_{\rho}\|_{q}\geq|Tr[(\rho-\bar{\xi})W_{\rho}]|.$
(20)
If $\rho$ is entangled and $\bar{\xi}$ is separable we have
$|Tr[(\rho-\bar{\xi})W_{\rho}]|\geq|Tr[\rho W_{\rho}]|$, thus:
$\|\rho-\bar{\xi}\|_{p}\geq\frac{|Tr[\rho W_{\rho}]|}{\|W_{\rho}\|_{q}},$ (21)
which in terms of geometric discord (Eq.2) reads:
$D_{(p)}(\rho)\geq\Bigg{(}\frac{E_{w}(\rho)}{\|W_{\rho}\|_{q}}\Bigg{)}^{p}.$
(22)
Therefore, given any entanglement witness (it does not need to be optimal), we
have a bound for geometric discord in any norm. Note that Eq.6 is also valid
for multipartite states. For norm-1 and norm-2, Eq.6 reduces to:
$D_{(1)}(\rho)\geq\frac{E_{w}(\rho)}{\|W_{\rho}\|_{\infty}},$ (23)
$D_{(2)}(\rho)\geq\frac{E_{w}^{2}(\rho)}{Tr[W_{\rho}^{2}]}.$ (24)
If $W_{\rho}$ is the entanglement witness for the negativity,
$\mathcal{N}(\rho)=E_{w}(\rho)$ (see Eq.19), then $Tr[W_{\rho}^{2}]=n_{-}$,
where $n_{-}$ is the number of negative eigenvalues of the partial transpose
of $\rho$ ($\rho^{T_{A}}$). Thus, in norm-2, discord is lower bounded by
negativity as:
$D_{(2)}(\rho)\geq\frac{\mathcal{N}^{2}(\rho)}{n_{-}},$ (25)
where $0<n_{-}\leq d-1$ (remember $d=d_{A}\times d_{B}$). For norm-1 discord,
one has to calculate $\|W_{\rho}\|_{\infty}$, which is simply the largest
eigenvalue of $W_{\rho}$ in absolute value, and use Eq.23. Note that it is
easy. One has just to form a rank-$n_{-}$ projector with the eigenstates or
$\rho^{T_{A}}$ associated to the $n_{-}$ negative eigenvalues, then $W_{\rho}$
is the partial transpose of this projector.
## V Numerical applications
In this section we illustrate our results with numerical calculations on
maximally entangled pure staes, Werner states and bound entangled states. We
consider the negativity and random robustness.
Figure 1: (Color online) 1-norm geometric discord, negativity and random
robustness for $5\otimes 5$ Werner states.
### V.1 Werner states
Werner states ($d_{A}\otimes d_{A}$) werner , which are of the Bell diagonal
type, can be written as:
$\rho_{w}=\frac{d_{A}+k}{d_{A}^{3}-d}\mathbb{I}_{d}-\frac{d_{A}k-1}{d_{A}^{3}-d_{A}}|\psi^{+}\rangle\langle\psi^{+}|,$
(26)
where $|\psi^{+}\rangle=\sum_{i,j=0}^{d_{A}-1}(|ij\rangle+|ji\rangle)$. The
parameter $k$ is in the interval $[-1,1]$, and the state is entangled for
$k>0$.
In (Fig.1) we compare negativity, random robustness and 1-norm geometric
discord. Note that negativity and random robustness coincide in the entangled
region and are always less than the discord. Note also that the random
robustness in the non-entangled region ($Tr(\rho_{w}W_{\rho_{w}})\geq 0$) has
a functional behavior similar to the discord.
### V.2 Bound-entangled states
Bound entangled states have positive partial transpose ($ppt$) and are known
to be undistillable 3h98 . The negativity is useless in this case, but the
random robustness can give an interesting bound for the discord.
#### V.2.1 Horodecki’s $ppt$-entangled states
Consider a Hilbert space $\mathbb{C}^{3}\otimes\mathbb{C}^{3}$, and a
canonical orthonormal basis $\\{|i\rangle\\}_{0,1,2}$. Take the following
three states h97 ; 3h98 :
$Q=\mathbb{I}\otimes\mathbb{I}-\big{[}\sum_{i=0}^{2}|i\rangle\langle
i|\otimes|i\rangle\langle i|+|2\rangle\langle 2|\otimes|0\rangle\langle
0|\big{]},$ (27)
$|\psi\rangle=\frac{1}{3}\big{[}|0\rangle|0\rangle+|1\rangle|1\rangle+|2\rangle|2\rangle\big{]}$
(28)
and
$|\phi_{k}\rangle=|2\rangle\otimes\Big{[}\sqrt{\frac{1+k}{2}}|0\rangle+\sqrt{\frac{1-k}{2}}|2\rangle\Big{]},$
(29)
where $0\leq k\leq 1$. The following convex combination is a $ppt$-entangled
state for $0\leq k<1$, and is separable for $k=1$:
$\varrho_{k}=\frac{k}{8k+1}\big{[}3|\psi\rangle\langle\psi|+Q\big{]}+\frac{1}{8k+1}|\phi_{k}\rangle\langle\phi_{k}|.$
(30)
Note, in Fig.2a, that the most discordant state of this family is the less
entangled one, and vice-versa. However, it is not a general characteristic of
bound entangled states, as can be seen in the next example (Fig.2b).
#### V.2.2 UPB entangled states
In a bipartite Hilbert space
$\mathcal{H}=\mathbb{C}^{d_{A}}\otimes\mathbb{C}^{d_{B}}$, an orthogonal
product basis (PB) is an $l$-dimensional set of separable states spanning a
subspace $\mathcal{H}_{l}$ of $\mathcal{H}$. When the complement of this PB in
$\mathcal{H}$ has only entangled states, we say that the complete basis
containing PB is a unextendible product basis (UPB) upbprl .
Consider the following three classes of vectors in
$\mathcal{H}=\mathbb{C}^{4}\otimes\mathbb{C}^{4}$:
$\displaystyle|v_{j}\rangle$ $\displaystyle=$
$\displaystyle|j\rangle\otimes\frac{|(j+1)\mod 4\rangle-|(j+2)\mod
4\rangle}{\sqrt{2}},$ $\displaystyle|u_{j}\rangle$ $\displaystyle=$
$\displaystyle\frac{|(j+1)\mod 4\rangle-|(j+2)\mod
4\rangle}{\sqrt{2}}\otimes|j\rangle,$ $\displaystyle|w\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i,j=0}^{3}|i\rangle\otimes|j\rangle.$
Now define the following vectors: $|\psi_{k}\rangle=|v_{k}\rangle$ for
$k=0,1,2,3$, $|\psi_{k}\rangle=|u_{k\mod 4}\rangle$ for $k=4,5,6,7$, and
$|\psi_{k}\rangle=|w\rangle$ for $k=8$. Finally, a $ppt$-entangled state is
given by:
$\rho=\frac{1}{7}\Big{(}\mathbb{I}-\sum_{k=0}^{8}|\psi_{k}\rangle\langle\psi_{k}|\Big{)}.$
(31)
In (Fig.2b), we plotted random robustness and 1-norm geometric discord for the
convex mixture:
$\sigma=\frac{s}{16}\mathbb{I}+(1-s)\rho,$ (32)
which is separable for $s>0.169$. We see that the discord is always greater
than the entanglement, and the most discordant states are also the most
entangled.
Figure 2: (Color online) 1-norm geometric discord and random robustness for
$3\otimes 3$ (a) and $4\otimes 4$ (b) $ppt$-entangled states.
### V.3 Pure states versus Werner states
In Tabs. I and II, we compare the bounds for norm-1 and norm-2 in two
maximally entangled states, an $8\otimes 8$ Werner state, and a $2\otimes 8$
mixed state whose density matrix coincides with the $8\otimes 8$ Werner state.
The norm-1 bounds are much better than the norm-2 ones. Tab.III is an Erratum
for the published version of this paper.
| $Tr(\rho{W_{n}})$ | $Tr(\rho{W_{r}})$ | $Tr({W_{n}}^{2})$ | $Tr({W_{r}}^{2})$ | $\parallel W_{n}\parallel_{\infty}$ | $\parallel W_{r}\parallel_{\infty}$
---|---|---|---|---|---|---
$(d=2\otimes 2)\,\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ | -0.5000 | -0.5000 | 1 | 1.0000 | 0.5000 | 0.5000
$(d=4\otimes 4)\,\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ | -1.5000 | -0.2500 | 6 | 0.1677 | 1.5000 | 0.2503
$(d=8\otimes 8)\,\rho=\rho_{w}(8,-1)$ | -0.1250 | -0.1250 | 1 | 1.0000 | 0.1250 | 0.1250
$(d=2\otimes 32)\,\rho=\rho_{w}(8,-1)$ | -0.1786 | -0.0179 | 10 | 0.1013 | 0.5000 | 0.0600
Table 1: Entanglement and some properties of the corresponding entanglement witness. $W_{n}$ and $W_{r}$ are the entanglement witnesses for negativity and random robusteness, respectively. | $D_{2}$ | $\frac{Tr(\rho{W_{n}})^{2}}{Tr({W_{n}}^{2})}$ | $\frac{Tr(\rho{W_{r}})^{2}}{Tr({W_{r}}^{2})}$ | $D_{1}$ | $\frac{-Tr(\rho{W_{n}})}{\parallel W_{n}\parallel_{\infty}}$ | $\frac{-Tr(\rho{W_{r}})}{\parallel W_{r}\parallel_{\infty}}$
---|---|---|---|---|---|---
$(d=2\otimes 2)\,\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ | $0.5000$ | $0.2500$ | 0.2500 | 1.0000 | 1.0000 | 1.0000
$(d=4\otimes 4)\,\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ | $0.7500$ | $0.3750$ | 0.3727 | $1.5000$ | 1.0000 | 0.9988
$(d=8\otimes 8)\,\rho=\rho_{w}(8,-1)$ | $0.0179$ | $0.0156$ | 0.0156 | $1.0000$ | 1.0000 | 1.0000
$(d=2\otimes 32)\,\rho=\rho_{w}(8,-1)$ | $0.0102$ | $0.0032$ | $0.0032$ | $0.5714$ | $0.3580$ | $0.2983$
Table 2: Bounding geometric discord with witnessed entanglement. | $D_{2}$ | $Eq.21=\frac{\mathcal{N}^{2}}{d-1}$ | $D_{1}$ | $Eq.27=\frac{\mathcal{N}}{d}$ | $Eq.28=\frac{\mathcal{R}_{r}}{d}$
---|---|---|---|---|---
$(d=2\otimes 2)\,\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ | $0.5000$ | $0.0833$ | $1.0000$ | $0.1250$ | 0.5000
$(d=4\otimes 4)\,\rho=|\Phi^{+}\rangle\langle\Phi^{+}|$ | $0.7500$ | $0.1500$ | $1.5000$ | $0.0938$ | 0.2500
$(d=8\otimes 8)\,\rho=\rho_{w}(8,-1)$ | $0.0179$ | $0.0002$ | $1.0000$ | $0.0020$ | 0.0020
$(d=2\otimes 32)\,\rho=\rho_{w}(8,-1)$ | $0.0102$ | $0.0005$ | $0.5714$ | $0.0028$ | $0.0179$
Table 3: Errata for Eqs. 21, 27 and 28 in [Phys. Rev. A 86, 024302 (2012)]
## VI Conclusion
We obtained bounds for geometric discord, in any norm, in terms of
entanglement witnesses (EW). Many known measures of entanglement can be
expressed by Optimal EWs, which implies that our bounds are quite general. We
note that, in a previous work fernando2012 , we showed how to calculate
entanglement and geometric discord in systems of indistinguishable fermions,
and we checked that the geometric discord was also bounded by the witnessed
entanglement in that case.
Acknowledgments \- We thank Fernando G.S.L. Brandão for the discussions.
Financial support by the Brazilian agencies FAPEMIG, CNPq, and INCT-IQ
(National Institute of Science and Technology for Quantum Information).
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|
arxiv-papers
| 2012-07-05T17:01:36 |
2024-09-04T02:49:32.687997
|
{
"license": "Public Domain",
"authors": "Tiago Debarba, Thiago O. Maciel and Reinaldo O. Vianna",
"submitter": "Tiago Debarba",
"url": "https://arxiv.org/abs/1207.1298"
}
|
1207.1315
|
# An experimental study of exhaustive solutions for the Mastermind puzzle
Juan-J. Merelo-Guervós Antonio M. Mora Carlos Cotta Thomas P. Runarsson
Dept. of Computer Architecture and Technology, University of Granada, Spain,
email: {jmerelo,amorag}@geneura.ugr.es CITIC, http://citic.ugr.es Dept. of
Languages and Computer Sciences, University of Málaga,
email: ccottap@lcc.uma.es School of Engineering and Natural Sciences,
University of Iceland,
email: tpr@hi.is
###### Abstract
Mastermind is in essence a search problem in which a string of symbols that is
kept secret must be found by sequentially playing strings that use the same
alphabet, and using the responses that indicate how close are those other
strings to the secret one as hints. Although it is commercialized as a game,
it is a combinatorial problem of high complexity, with applications on fields
that range from computer security to genomics. As such a kind of problem,
there are no exact solutions; even exhaustive search methods rely on
heuristics to choose, at every step, strings to get the best possible hint.
These methods mostly try to play the move that offers the best reduction in
search space size in the next step; this move is chosen according to an
empirical score. However, in this paper we will examine several state of the
art exhaustive search methods and show that another factor, the presence of
the actual solution among the candidate moves, or, in other words, the fact
that the actual solution has the highest score, plays also a very important
role. Using that, we will propose new exhaustive search approaches that obtain
results which are comparable to the classic ones, and besides, are better
suited as a basis for non-exhaustive search strategies such as evolutionary
algorithms, since their behavior in a series of key indicators is better than
the classical algorithms.
###### keywords:
Mastermind, bulls and cows, logik, puzzles, games, combinatorial optimization,
search problems.
††journal: Computers and Operations Research
## 1 Introduction and description of the game of Mastermind
Mastermind [1] is a board game that has enjoyed world-wide popularity in the
last decades. Although its current version follows a design created in the 70s
by the Israeli engineer Mordecai Meirowitz [2], the antecedents of the game
can be traced back to traditional puzzles such as _bulls and cows_ [3] or the
so-called _AB_ [4] game played in the Far East. Briefly, MasterMind is a two-
player code-breaking game, or in some sense a single-player puzzle, where one
of the players –the _codemaker_ (CM)– has no other role in the game than
setting a hidden combination, and automatically providing hints on how close
the other player –the _codebreaker_ (CB)– has come to correctly guess this
combination. More precisely, the flow of the game is as follows:
* 1.
The CM sets and hides a length $\ell$ combination of $\kappa$ symbols.
Therefore, the CB is faced with $\kappa^{\ell}$ candidates for the hidden
combination, which is typically represented by an array of pegs of different
colors (but can also be represented using any digits or letter strings) and
hidden from the CB.
* 2.
The CB tries to guess this secret code by producing a combination (which we
will call move) with the same length, and using the same set of symbols. As a
response to that move, the CM acting as an oracle (which explains the
inclusion of this game in the category called oracle games) provides
information on the number of symbols guessed in the right position (black pegs
in the physical board game), and the number of symbols with the correct color,
but in an incorrect position (white pegs); this is illustrated in Table 1.
* 3.
The CB uses (or not, depending on the strategy he is following) this
information to produce a new combination, that is assessed in the same way. If
he correctly guesses the hidden combination in at most $N$ attempts, the CB
wins. Otherwise, the CM takes the game. $N$ usually corresponds to the
physical number of rows in the game board, which is equal to fifteen in the
first commercial version.
* 4.
CM and CB are then interchanged, and several rounds of the game are played,
this is a part of the game we do not consider. The player that is able to
obtain the minimal amount of attempts wins.
Table 1: Progress in a MasterMind game that tries to guess the secret combination ABBC. 2nd and 4th combinations are not consistent with the first one, not coinciding in two positions and one color with it. _Combination_ | _Response_
---|---
AABB | 2 black, 1 white
ABFE | 2 black
ABBD | 3 black
BBBE | 2 black
ABBC | 4 black
In this paper, we will consider the puzzle only from the point of view of the
CB , that is, a solver will be confronted with an oracle that will award, for
each move, a number of black and white pegs. We will make no assumptions on
the CM other that it will use random combinations (which in fact it does). A
human player will probably have some kind of bias, but we are interested in
finding the most general playing method without using that kind of
information.
From the point of view mentioned in the last paragraph this puzzle is, in
fact, a quite interesting combinatorial problem, as it relates to other
oracle-type problems such as the hacking of the PIN codes used in bank ATMs
[5, 6], uniquely identifying a person from queries to a genetic database [7],
or identifying which genotypes have interesting traits for selectively
phenotyping them [8]; this is just a small sample of possible applications of
the solution of the game of Mastermind. A more complete survey of applications
(up to 2005) can be found in our previous paper [9], but recent posts in the
questions-and-answers website StackOverflow are witness to the ongoing
interest in finding solutions to the game [10].
Mastermind is also, as has been proved recently, a complex problem,
paradigmatic of a whole class of search problems [11]; the problem of finding
the solution has been shown to be NP-complete under different formulations
[12, 13], and the problem of counting the number of solutions compatible with
a certain set of answers is #P-complete [14]. This makes the search for good
heuristic algorithms to solve it a challenge; besides, several issues remain
open, such as what is the lowest average number of guesses needed to solve the
problem for any given $\kappa$ and $\ell$. Associated to this, there arises
the issue of coming up with an efficient mechanism for finding the hidden
combination independently of the problem size, or at least, a method that
scales gracefully when the problem size increases.
In this paper we will mainly examine empirical exhaustive search algorithms
for sizes in which they are feasible and try to find out which are the main
factors that contribute to finding the hidden combination in a particular (and
small) number of moves; in this context, we mean by exhaustive algorithms
those that examine, in turn, all items in the search space, discarding a few
in each step until the solution (the secret code) is found. While most methods
of this kind consider as the only important factor the reduction of the search
space [15], we will prove that how different combinations, and in particular
the secret (and unknown) one is scored contributes also to finding the
solution faster. Taking into account how often the hidden combination appears
among the top scorers for a particular method, we will propose new exhaustive
search methods that are competitive with the best solutions for Mastermind
known so far. Besides, these methods do not add too much complexity to the
solution and present certain behaviors that are better than the empirical
solutions used so far.
It is obvious that exhaustive search methods can only be used for small sizes
in NP-hard problems such as this one. However, its analysis and the proposal
of new methods have traditionally been, and will also be used, to design
metaheuristic search methods (such as evolutionary algorithms) that are able
to search much further in the parameter space. Since that has been our line of
research for a long time [16, 17, 9], eventually we will use them in those
algorithms, but that is outside the scope of this paper.
The rest of the paper is organized as follows: we lay out the terminology and
explain the solutions to the game of Mastermind in Section 2; the state of the
art in solutions to it is presented next, in Section 3. Next we will analyze
the best performing Mastermind methods known so far in Section 4 and reach
some conclusions on its way of working; the conclusions extracted in this
analysis will be used in Section 5 to propose and analyze the new solutions
proposed in this paper, which we have called Plus and Plus2. This Section will
be followed by conclusions in the last Section 7.
## 2 The game of Mastermind
As mentioned in Section 1, a MasterMind problem instance is characterized by
two parameters: the number of colors $\kappa$ and the number of pegs $\ell$.
Let $\mathbb{N}_{\kappa}=\\{1,2,\cdots\,\kappa\\}$ be the set of symbols used
to denote the colors. Subsequently, any combination, either the hidden one or
one played by the CB, is a string $c\in\mathbb{N}_{\kappa}^{\ell}$. Whenever
the CB plays a combination $c_{p}$, a _response_
$h(c_{p},c_{h})\in\mathbb{N}^{2}$ is obtained from the CM, where $c_{h}$ is
the hidden combination. A response $\langle b,w\rangle$ indicates that the
$c_{p}$ matches $c_{h}$ in $b$ positions, and there exist other $w$ symbols in
$c_{p}$ present in $c_{h}$ but in different positions.
Let us define consistent combinations [11] in the following way: a combination
$c$ is _consistent_ with another played previously $c_{p}$ if, and only if,
$h(c,c_{p})=h(c_{p},c_{h})$, i.e., if $c$ has as many black and white pegs
with respect to the $c_{p}$ as $c_{p}$ has with respect to the hidden
combination. Intuitively, this captures the fact that $c$ might be a potential
candidate for secret code in light of the outcome of playing $c_{p}$. We can
easily extend this notion and denote a combination $c$ as consistent (or
feasible) if, and only if, it is consistent with all the combinations played
so far, i.e., $h(c,c^{i}_{p})=h(c^{i}_{p},c_{h})$ for $1\leqslant i\leqslant
n$, where $n$ is the number of combinations played so far, and $c^{i}_{p}$ is
the $i-$th combination played. Any consistent combination is a candidate
solution. A consistent set is the set of all consistent combinations at a
particular point in the game.
Consistent combinations are important because only a move using one of them
decreases the size of the consistent set, at least by one (the combination
itself). A non-consistent combination might or might not do that, depending on
its similarity to already played moves. So most Mastermind strategies play a
consistent combination. We will see next how these concepts are used to find
the secret combination in a minimum number of combinations.
## 3 State of the art
As should be obvious, once the set (or a subset) of consistent solutions has
been found, different methods have different heuristics to choose which
combination is played. A simple strategy, valid at all levels, is to play the
first consistent combination that is found (be it in an enumerative search, or
after random draws from the search space, or searching for it using
evolutionary algorithms[9]). In general, we will call this strategy Random; in
fact, the behavior of all these strategies is statistically indistinguishable,
and, even if it is valid, this strategy does not offer the best results.
Table 2: Table of partitions after two combinations have been played; this table is the result of comparing each combination against all the rest of the set, which is the set of consistent combinations in a game after two combinations have already been played. In boldface, the combinations which have the minimal worst set size (which happen to be in the 1b-1w column, but it could be any one); in this case, equal to ten. A strategy that tries to minimize worst case would play one of those combinations. The column 0b-1w with all values equal to 0 has been suppressed; column for combination 3b-1w, being impossible, is not shown either. Some rows have also been eliminated for the sake of compactness. Combination | Number of combinations in the partition with response
---|---
| 0b-2w | 0b-3w | 0b-4w | 1b-1w | 1b-2w | 1b-3w | 2b-0w | 2b-1w | 2b-2w | 3b-0w
AABA | 0 | 0 | 0 | 14 | 8 | 0 | 13 | 1 | 0 | 3
AACC | 8 | 0 | 0 | 10 | 5 | 0 | 8 | 4 | 1 | 3
AACD | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
AACE | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
AACF | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
ABAB | 8 | 0 | 0 | 10 | 5 | 0 | 8 | 4 | 1 | 3
ABAD | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
ABAE | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
ABAF | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
ABBC | 4 | 4 | 0 | 10 | 8 | 0 | 8 | 1 | 1 | 3
ABDC | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
ABEC | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
ABFC | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
ACAA | 0 | 0 | 0 | 14 | 8 | 0 | 13 | 1 | 0 | 3
ACCB | 4 | 4 | 0 | 10 | 8 | 0 | 8 | 1 | 1 | 3
ACDA | 0 | 0 | 0 | 16 | 10 | 2 | 5 | 3 | 0 | 3
BBAA | 8 | 0 | 0 | 10 | 5 | 0 | 8 | 4 | 1 | 3
BCCA | 4 | 4 | 0 | 10 | 8 | 0 | 8 | 1 | 1 | 3
BDCA | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
BECA | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
BFCA | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
CACA | 8 | 0 | 0 | 10 | 5 | 0 | 8 | 4 | 1 | 3
CBBA | 4 | 4 | 0 | 10 | 8 | 0 | 8 | 1 | 1 | 3
CBDA | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
CBEA | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
CBFA | 3 | 4 | 1 | 11 | 9 | 1 | 4 | 2 | 1 | 3
DACA | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
EBAA | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
FACA | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
FBAA | 6 | 2 | 0 | 11 | 6 | 1 | 4 | 5 | 1 | 3
And it does not do so because not all combinations in the consistent are able
to reduce its size in the next move in the same way. So other not so naïve
solutions concentrate on scoring all combinations in the consistent set
according to a heuristic method, and playing one of the combinations that
reaches the top score, a random one or the first one in lexicographical order.
This score is always based on the concept of Hash Collision Groups, HCG [4] or
partitions [15]. All combinations in the consistent set are compared with each
other, considering one the secret code and the other a candidate solution; all
combinations will be grouped in sets according to how they compare to a
particular one, as shown in Table 2. For instance, in such table there is a
big set of combinations whose response is exactly the same: ABDC, ABED, ABFC,
ABAD, ABAE, ABAF… All these combinations constitute a partition, and their
score will be exactly the same.
To formalize these ideas, let ${\vec{\Xi}}=\\{\Xi_{ibw}\\}$ be a three-
dimensional matrix that estimates the number $\Xi_{ibw}$ of combinations that
will remain feasible after combination $c_{i}$ is played and response $\langle
b,w\rangle$ is obtained from the CM. Then, the potential strategies for the CB
are:
1. 1.
Minimizing the worst-case partition [18]: pick
$c_{i}=\arg\min_{i}\\{\max_{b,w}(\Xi_{ibw})\\}$. For instance, in the set in
Table 2 this algorithm would play one of the combinations shown in boldface
(the first one in lexicographical order, since it is a deterministic
algorithm).
2. 2.
Minimizing the average-case partition [19, 20]: pick
$c_{i}=\arg\min_{i}\\{\sum_{b,w}p_{bw}\Xi_{ibw}\\}$, where $p_{bw}$ is the
prior probability of obtaining a particular outcome. If for instance we
compute $p_{bw}=\sum_{i}\Xi_{ibw}/\sum_{i,b,w}\Xi_{ibw}$, then AACD would be
the combination played among those in Table 2.
3. 3.
Maximizing the number of potential partitions [15]: pick
$c_{i}=\arg\max_{i}\\{|\\{\Xi_{ibw}>0\\}|\\}$, where $|C|$ is the cardinality
of set $C$. For example, a combination such as ABDC in Table 2 would result in
no empty partition. This strategy is also called _Most Parts_.
4. 4.
Maximizing the information gained [21, 22, 23]: pick
$c_{i}=\arg\max_{i}\\{H_{b,w}\left(\Xi_{ibw}\right)\\}$, where
$H_{b,w}(\Xi_{i[\cdot][\cdot]})$ is the entropy of the corresponding sub-
matrix. We will call this strategy Entropy.
1
2typedef Combination: vector$[1..\ell]$ of $\mathbb{N}_{\kappa}$;
3 procedure NextMove (in: $F$: List[Combination], out: $guess$: Combination);
4 var TopScorers: List[Combination];
5 Score( $F$ );
6 $guess\leftarrow$ RandomElement ( TopScorers );
Algorithm 1 Choosing the next move in the general case
All strategies based on partitions work as follows:
1. 1.
Score all combinations in the consistent set according to the method chosen
(Entropy, Most Parts, Best Expected, or Minimize Worst)
2. 2.
Play one of the combinations with the best score
3. 3.
Get the response from the codemaker, and unless it is all blacks, go to the
first step.
This is also represented more formally in Algorithm 1, where Score scores all
combinations according to the criterion chosen, and RandomElement extracts a
random element from the list passed as an argument using uniform distribution.
Exhaustive strategies have only (as far as we know) been examined and compared
for the base case of $\kappa=6,\ell=4$, for instance in [24]; restricted
versions of the game have been examined for other spaces, for instance in
[11]. The best results are obtained by Entropy and Most Parts, but its
difference is not statistically significant. All other results (including the
classical one proposed by Knuth [18]) are statistically worse. That is why in
this paper we will concentrate on these two strategies, which represent the
state of the art in exhaustive search. Bear in mind that all these strategies
are empirical, in the sense that they are based in an assumption of how the
reduction of the search space work; there is, for the time being, no other way
of proposing new strategies for the game of mastermind.
Besides, the size of the the search space is used as proxy for what is
actually the key to success of a strategy: the probability of drawing the
winning combination at each step. It is evident that reducing the number of
combinations will eventually result in drawing the secret one with probability
one. However, it should not be neglected that the probability of drawing it
even if it is not the only remaining combination is non-zero and care should
be taken so that this probability is either maximized, or at least considered
to minimize the number of moves needed to find it. So far, and to the best of
our knowledge, no study has been made of this probability; we will show in
this paper its influence on the success of a strategy.
Let us finally draw our attention to the first move. It is obviously an
important part of the game, and since, a priori, all combinations are
consistent a strategy would consist in using all combinations in the game and
playing one according to its score. However, this is not a sensible strategy
even for the smaller size; consistency does not make any sense in absence of
responses, since the partitions will not hold any information on the secret
code. So, in most cases, a fixed move is used, and the one proposed by Knuth
(according to its own Minimize Worst strategy) is most usually employed. Since
Knuth’s strategy [18] was created for $\kappa=6,\ell=4$ it is difficult to
extrapolate it to higher dimensions, so papers vary in which combination is
used. At any rate, the influence of the first move will be felt mainly in the
reduction of size achieved before the first empirical move is made (second
move of the game), but a good reduction will imply a significant change in the
average number of games. We will bear this in mind when testing the different
empirical strategies.
Essentially, then, Most Parts and Entropy are the state of the art in
exhaustive search strategies. These algorithms, along with all others used in
this paper, are written in Perl, released under a open source licence and can
be downloaded from http://goo.gl/G9mzZ. All parameters, experiment scripts,
data extraction scripts and R data files can be found within the app/IEEE-CIG
directory. Next we will examine how these algorithms work in two different
sizes, which are considered enough to assess its performance; in Section 5 we
will propose new methods and prove that they obtain a better average number of
moves making them the new state of the art in exhaustive search algorithms for
the game of Mastermind.
## 4 Analyzing exhaustive search methods
In this section, we will outline a methodology for analyzing exhaustive
solutions to the game of Mastermind and apply it to two strategies that are
usually considered the best, Entropy and Most Parts, for two different problem
sizes, keeping $\ell=4$ fixed and setting $\kappa=6$ (Subsection 4.1) and
$\kappa=8$ (Subsection 4.2).
### 4.1 $\ell=4,\kappa=6$
Let us look first at the smallest usual size, $\kappa=6$ and $\ell=4$. The
best two strategies in this case have been proved to be Entropy and Most Parts
[24, 19], in both cases starting using Knuth’s rule, ABCA [18]. An analysis of
the number of moves obtained by each method are shown in Table 3.
Table 3: Average (with error of the mean) and Maximum number of moves for two search strategies: Most Parts and Entropy. _Method_ | _Number of moves_
---|---
| _Average_ | _Maximum_ | _Median_
Entropy | 4.413 $\pm$ 0.006 | 6 | 4
Most Parts | 4.406 $\pm$ 0.007 | 7 | 4
To compute this average, 10 runs over the whole combination space were made.
In fact the difference in the number of moves is small enough to not be
statistically significant (using Wilcoxon test, p-value = 0.4577), but there
is a difference among them, the most striking being that, even if the maximum
number of moves is higher for Most Parts, Entropy has a higher average.
To check where that difference lies we will plot a histogram of the number of
moves needed to find the solution for both methods, see Fig. 1.
Figure 1: Histogram with the count of the number of times every method is able
to find the solution in every number of moves, for the Entropy method (black
and solid) and Most Parts (light or red, dashed). Figure 2: . Figure 3:
Difference in score among the number of games won by Entropy and Most Parts in
$x$ moves. Every $y$ value in the graph 2 has been multiplied by the number of
moves, resulting in the actual score the player would achieve. Remember that
less is better; positive values mean that Entropy beats Most Parts (for that
move), and vice versa.
As should be expected for the negligible difference in the average number of
moves, differences here are very small. It is noticeable, however, than Most
Parts finishes less times in 1 to 3 moves, and also 5 moves. Most parts only
finishes in less occasions than Entropy for 4 moves; quite obviously, too,
there are a few times in which Most Parts needs 7 moves, but only 8 out of the
total 12960 games (10 games times the total number of combinations, 1296).
Again, differences are not significant, but noticeable , so we will try to
seek what is its source; these differences must be the cause why eventually
Most Parts achieves a slightly higher average number of moves. Let us look
further into these differences by plotting the difference in the number of
games won by every one in a particular number of moves, let’s say $x$. That is
shown in Figure 2. For all moves, except 5 and 7 (by that move Entropy has
always finished), Most Parts has more games. Since both methods play the same
number of games, the difference is offset by the number of games more, around
150, that Entropy finishes by move number five; this is also seen clearly in
the histogram 1. However, the important issue is the difference in score, that
is, in number of moves brought by finishing each move. Differences in scores
are computed by multiplying the difference in the number of games by the move
number, and plotted in Figure 3. It obviously follows the same pattern than
Figure 2, however it gives us an idea of the contribution of differences to
total score. Since in this case the total score for Entropy is 57189 and for
Most Parts 57106, and Most Parts is always better (negative difference in
score) than Entropy, we see that for low number of moves Most Parts is
accumulating differences, although it is worsened by not being able to finish
as many times as Entropy in 5 moves. This is the key move, and the shape of
the plot indicates a change of regime in the game. It also shows that no
single strategy is better than the other, yielding the non-significant
difference in moves (and scant difference in score, less than 100 over 12960
games!). The main conclusion would be that , in fact, bot solutions are very
similar, and this is also supported by other experiments (not shown here) with
more games, that do not yield a significant difference either. However, we can
also see that the way they find the solution is different, that is why we will
study other aspects of the algorithm in the next paragraphs.
Table 4: Average and standard deviation of the number of combinations remaining after every move (or step). The numbers are the same after the first move (not shown here), since they are playing the same one. _Before move #_ | _Entropy_ | _Most Parts_
---|---|---
3 | 23 $\pm$ 14 | 24 $\pm$ 15
4 | 3.1 $\pm$ 1.8 | 3.4 $\pm$ 2.4
5 | 1.13 $\pm$ 0.35 | 1.17 $\pm$ 0.43
6 | 1 | 1.02 $\pm$ 0.17
7 | | 1
Since both methods try to reduce the size of the consistent set, we will look
at their size and how it changes with the number of moves. We will log the
size of the remaining consistent combinations at every step, and this is shown
in Fig. 4 and Table 4.
Figure 4: Average number of combinations (or moves) remaining after each move,
for the Entropy method (black and solid) and Most Parts (light or red). This
plot corresponds to the numbers represented in Table 4. The $y$ axis is
logarithmic for clarity, and the $x$ axis is shifted by one (1 means the
second move, size before the first move is always the whole space).
In effect, the Entropy method is more efficient in reducing the size of the
set of remaining solutions, which is the reason why usually it is presented as
the best method for solving Mastermind, since it achieves most effectively
what it is intended to achieve: reduction in search space size. At every step,
the difference is significant (using Wilcox test). This difference is of a
single combination at the beginning, and decreases with time while still being
significant; however, this reduction at the first stages of the game makes the
search simpler and would, in principle, imply an easy victory for the Entropy
technique, as would be expected. However, the result is a statistical draw,
with a slight advantage for the other method, Most Parts. Besides, this
reduction does not explain the advantage found in Figures 1 and 2: if the
number of solutions that remain is bigger (on average), why is Most Parts able
to find the solution at those same stages more often than Entropy, which seems
to be the key to success?
To discover why this happen, we will look at another result. As explained
above, all methods are based on scoring consistent combinations according to
the partitioning of the set they yield, and then playing randomly one of the
combinations with the top score. If, by chance, the winning combination is in
that set of top scorers, there is a non-zero possibility of playing it as next
move and thus winning the match. In effect, what we want is to maximize at
every step the probability of drawing the secret code. We will then look at
whether the winning combination effectively is or not among the top scorers at
each step. Results are shown in Fig. 5 and Table 5.
Table 5: Percentage of times the secret code is among the top scorers for each method. _In move #_ | _Entropy_ | _Most Parts_
---|---|---
2 | 0.1142857 | 0.3644788
3 | 0.2919495 | 0.5270987
4 | 0.7721438 | 0.8242563
5 | 0.9834515 | 0.9810066
6 | | 1
Figure 5: Chance of finding the secret code among the top scorers for the
Entropy method (black and solid) and Most Parts (light or red). This plot
corresponds to the numbers represented in Table 5.
Table 5 clearly shows that the probability of finding the hidden combination
among the top scorers increases with time, so that in the 5th move is
practically one. But it also shows that (in the first moves) there is almost
double the chance of having the winning combination among the top ones for
Most Parts than for Entropy, so, effectively, and obviously depending on the
size of the consistent set found at Table 4, the likelihood of playing the
winning combination is higher for Most Parts and its key to success. Making a
back of the envelope calculation, when the game arrives at the second move,
which roughly 11000 games do, a third of them will include the winning
combination among the top scorers, which again is roughly three thousand.
Figure 6: Chance of drawing the secret code for the Entropy method (black and
solid) and Most Parts (light or red).
This is not the whole picture, however. Every set of top scorers will have a
different size; even if the probability of finding the secret score among that
set is higher, the probability of drawing it will be different if the size is
smaller. The average of this probability has been drawn in Figure 6. While
initially the probability is very small (but different), Most Parts has a
better chance of obtaining the secret code at each move than Entropy, at least
until move number 6 (label 4). That change of regime is reflected in the
previous plots by a sudden increase in the number of games Entropy finishes
(see Figures 1 and 2). However, since by that time Most Parts has been able to
finish a good amount of times, the difference is not too big. Looking at
figure 4 and table 4, which plots the average size of the set of remaining
combinations, we see that by the 5th move (label 4) there is, almost always, a
single remaining combination, but it happens more often for Entropy than for
Most Parts; that is, in this move is when the reduction of the search space
effectively kicks in, accounting for the change in regime we mentioned before.
This, in turn, implies one of the main results of this paper: there are two
factors in the success of a method for playing Mastermind. The first is the
reduction it achieves on the search space size by playing combinations that
reduce it maximally, but there is a second and non-negligible factor: the
chance of playing the winning combination by having it among the top scorers.
It can be said that there is a particular number of moves, which in this case
is after the median, 4 moves, where a change of regime takes place. Before and
up to that number of games, if a method finishes it is mainly due to drawing,
by chance, the secret code. After the median the secret code is found because
it is the only remaining element in the search space.
Most studies so far, however, had only looked at the smallest size. Let us
study another problem size, with search space 4 times as big in the next
subsection.
### 4.2 $\ell=4,\kappa=8$
The way the solutions work will change when the problem size is increased, so
we will perform the same measurements for problem size $\kappa=8,\ell=4$.
Search space size is four times as big; time to solution grows faster than
lineally so it is not practical (although possible) to work with exhaustive
search for sizes bigger than that; in fact, $\ell=5,\kappa=6$ is the next step
that is feasible, but initial work has shown that the behavior is not
different from this case, and it takes around two times as much; while the
exhaustive solution to the smallest Mastermind size considered takes around
half a second, it takes around 5 seconds for $\kappa=8,\ell=4$, and 10 seconds
for $\kappa=6,\ell=5$. In practice, this means that instead of using the whole
search space 10 times over to compute this average, we will generate a
particular set of 5000 combinations, which includes all combinations at least
once and none more than two times. This instance set is available at the
method website (http://goo.gl/ONYLF). A solution is searched, then, for every
one of these combinations. We are playing ABCD as first move, as one
interpretation of Knuth’s [18] first move would say; that is, we play half the
alphabet and start again by the first letter (ABCA would play half the ABCDEF
alphabet and then start again).
The average number of moves is represented in Table 6. This confirms in parts
our above hypothesis: the better capability Entropy has to decrease the size
of the search space gives it an advantage in the average number of moves,
keeping at the same time the ability of solving it in less maximum number of
moves. However, for this number of experiments, the difference is significant
(Wilcoxon paired test = 0.052). We will check whether this difference has the
same origin as we hypothesized for the smaller search space size, by looking
at the same variables.
Table 6: Average (with error of the mean) and Maximum number of moves for two search strategies: Most Parts and Entropy for $\kappa=8,\ell=4$. _Method_ | _Number of moves_
---|---
| _Average_ | _Maximum_ | _Median_
Entropy | 5.132 $\pm$ 0.012 | 8 | 5
Most Parts | 5.167 $\pm$ 0.012 | 8 | 5
Figure 7: Frequency of the number of moves up to (and including) the secret
combination, for $\kappa=8,\ell=4$. As usual, red or light dashed line
represents Most Parts and solid black line Entropy.
The equivalent to Fig. 1 has been represented in Fig 7. The shape is similar,
but the solid line that represents Entropy is slightly below the dashed line
for Most Parts for the highest number of moves which accounts for the small
difference in average number of moves. Is this difference accounted for by the
size of the search space after each move? As previously, we will plot it in
Fig. 8 and Table 8. Differences for all moves are significant according to
Wilcoxon test, which means that actually Entropy is achieving what it was
designed for: reduce in a significant way the search space, until the secret
combination is found. However, in the same way as has been shown above, other
mechanisms are also at work to find the solution before that reduction.
Table 7: Average and standard deviation of the number of combinations remaining after every move. The number of combinations is the same after the first move (not shown here), since both are playing the same first move. _After move #_ | _Entropy (ABCD)_ | _Entropy (ABCA)_
---|---|---
2 | 98 $\pm$ 67 | 102 $\pm$ 69
3 | 13 $\pm$ 10 | 14 $\pm$ 12
4 | 2.4 $\pm$ 1.6 | 2.63 $\pm$ 1.96
5 | 1.21 $\pm$ 0.46 | 1.27 $\pm$ 0.52
6 | 1.07 $\pm$ 0.27 | 1.08 $\pm$ 0.31
7 | 1 | 1.25 $\pm$ 0.46
Figure 8: Average size of the consistent set, that is, the set of solutions
that have not been discarded at a point in the game for $\kappa=8,\ell=4$. As
usual, red or light dashed line represents Most Parts and solid black Entropy.
Figure 9: Difference among the number of games won by Entropy and Most Parts
in $x$ moves for $\kappa=8,\ell=4$. Figure 10: Difference in score among the
number of games won by Entropy and Most Parts in $x$ moves for
$\kappa=8,\ell=4$. Method is the same as for figure 3, that is, product of
difference and number of moves.
However, we will have to look in this case too at the differences in games and
score by both methods with is represented in Figures 9 and 10. The first one
represents the raw difference in number of times every one wins; again there
is a change in behavior or phase shift (which we have called before a change
of regime) when arriving at the fifth move, which is also around the median
(exactly the median, in this case; it was one less than the median in the
previous study). The scenario drawn in 10 is also similar, but shows that
Entropy is able to beat Most Parts mainly because it is able to finish more
times in less moves (around 5) than Most Parts (which accumulates lots of bad
games that need 6,7 and 8 moves). The final difference is around 200 points
for the 5000 games (25834 vs. 25662) which is very small but, in this case,
significant.
We mentioned in Subsection 4.1 that the probability of finding the secret code
among the top scorers was one of those features. It should be expected that
the probability of finding the secret code among the top scorers will change;
since the number of elements in the consistent set intuitively we should
expect it to decrease. But this intuition is wrong, as shown in Table 8 and
Fig. 11. In fact, if we compare these results with those shown in Table 5 we
see that, for the same move, the proportion of times in which the secret code
is among the top scorers is almost twice as big at the beginning for Most
Parts and almost three times as big for Entropy. Interestingly enough, this
also implies that, while for $\kappa=6,\ell=4$ this probability was three
times as high for Most Parts, it is only two times as high now.
Table 8: Percentage of times the secret code is among the top scorers for each method, $\kappa=8,\ell=4$. _In move #_ | _Entropy_ | _Most Parts_
---|---|---
2 | 0.3008602 | 0.6379276
3 | 0.3425758 | 0.6957395
4 | 0.4937768 | 0.7845089
5 | 0.8630084 | 0.9160276
6 | 0.9898990 | 1
Figure 11: Chance of finding the secret code among the top scorers for the
Entropy method (black and solid) and Most Parts (light or red). This plot
corresponds to the numbers represented in table 8 for $\kappa=8,\ell=4$.
Figure 12: Chance of drawing the secret code for the Entropy method (black and
solid) and Most Parts (light or red) for $\kappa=8,\ell=4$. Remember that the
$x$ axis is shifted by one, since the first move is fixed.
This decrease in the chance of finding the secret code might explain the
difference in the average number of moves needed to find the solution, which
for this size tilts the balance in the direction of Entropy. While for the
smaller size this probability was enough to compensate the superior capability
of the Entropy method in reducing the size of the search space, in this case
the difference is not so high, which makes the Entropy method find the
solution, on average, on less moves. This is actually only one of the factors
in the actual probability of drawing the secret code, which is shown in Figure
12. In the same fashion as the previous study (shown in Figure 6), Most Parts
is better in playing the secret combination up to the sixth move (label 5 in
the graphic). But, after that, probabilities are similar and the change in
regime takes place; in fact, looking at 9 and 8, it could be argued that the
fact that after the 6th move almost always there is only one combination left
accounts for the observed phase shift.
As we concluded the previous section, it is non-negligible the influence of
the probability of a score that is able to consistently give top score to the
secret combination; however, in this case the size of the search space implies
that the probability of playing the secret code is smaller (by the 3rd move in
$\kappa=6$ it is as high as the 4th move in $\kappa=8$), and thus its
influence is not enough to make Most Parts as good as Entropy.
At any rate, it is clear that, also for this size, the only determinant factor
is not the decrease in size brought by the method, but also the ability to
score correctly. This, in turn, can take us to design new heuristic methods
that are able to address both issues at the same time, and also study the
difference brought by a different initial move.
## 5 New exhaustive search methods
Our intention by proposing new methods is to use the two main factors that we
have proved have an influence in the performance of Mastermind-playing methods
to devise new strategies that could serve as another empirical proof of the
mechanism, and also, if possible, obtain methods that are able to obtain
better results (or, at least, not worse results).
A new method for Mastermind will somehow have to match the two capabilities
featured by Entropy and Most Parts: the first reduces competently the search
space, and the second is able to score better the actual secret combination,
which can then be selected for playing giving winning moves. So we will
propose several methods that will try to combine them with the objective of
achieving a better average number of moves.
1 typedef Combination: vector$[1..\ell]$ of $\mathbb{N}_{\kappa}$;
2 procedure NextPlus (in: $F$: List[Combination], out: $guess$: Combination);
3 var TopScorersEntropy, TopScorersMostParts, TopScorersAll:
List[Combination];
4 EntropyScore( $F$ );
5 MostPartsScore( $F$ );
6 TopScorersEntropy$\leftarrow$ TopScorersEntropy( $F$ );
7 TopScorersMostParts$\leftarrow$ TopScorersMostParts( $F$ );
8 TopScorersAll $\leftarrow$ TopScorersEntropy $\cap$ TopScorersMostParts;
9 $guess\leftarrow$ RandomElement ( TopScorersAll );
Algorithm 2 Choosing the next move in the Plus Mastermind solution method.
This algorithm is the new version we propose to the NextMove function
presented in Algorithm 1.
The first one, which we will call Plus, works as follows (please see Algorithm
2): the set of consistent combinations is scored according to both Most Parts
and Entropy. The sets of combinations with the top score according to both
methods are extracted. If its intersection is non-null, a random combination
from it is returned. If it is null, a random string of the union of both sets
is returned.
What we want to achieve with this method is a reduction of the consistent set
in the same way as Entropy, but, by intersecting it with the set of top
scorers for Most Part, the probability of finding the winning combination
among them is also increased.
1 typedef Combination: vector$[1..\ell]$ of $\mathbb{N}_{\kappa}$;
2 procedure NextPlus2 (in: $F$: List[Combination], out: $guess$: Combination);
3 var TopScorersEntropy, TopScorersMostParts, TopScorersAll:
List[Combination];
4 EntropyScore( $F$ );
5 MostPartsScore( $F$ );
6 TopScorersEntropy$\leftarrow$ TopScorersEntropy( $F$ );
7 TopScorersAll$\leftarrow$ TopScorersMostParts( TopScorersEntropy );
8 $guess\leftarrow$ RandomElement ( TopScorersAll );
Algorithm 3 Choosing the next move in the Plus Mastermind solution method.
Please note that up to line 6, this algorithm is identical to 2.
There is no single way of combining the two scoring methods. A second one
tested, which we will call Plus2 (and which is outlined in 3), works
similarly. It proceeds initially as the Entropy method, by scoring
combinations according to its partition entropy. But then, the top scorers of
this method are again scored according to Most Parts. Out of the subset of top
Entropy scorers with the highest Most Parts score, a random combination is
returned. Please note that, in order to compute the Most Parts score of the
top Entropy scorers, the whole Consistent Set $F$ must be scored. In 3, line 7
would extract the top scorers according to the Most Parts method from the set
of top scorers by the Entropy method. While in Plus what is played belongs to
a subset of Entropy and Most Parts only if their intersection is non-zero, in
Plus2 the new set of top scorers is always a subset of the top scorers for
Entropy, further scored using Most Parts.
As intended in Plus, Plus2 tries to reduce even more the consistent set size
by narrowing down the set of combinations to those that have only top scores
using both methods. In principle, the priority of Entropy and Most Parts can
be swapped, by first scoring according to Most Parts and then choosing those
with the best Entropy score. This method was also tested, but initial results
were not as good, so we will show only results for these two and compare them
with the traditional methods analyzed in the previous Section 4
Let us first look at the most important result: the number of moves. They are
shown in Table 9.
Table 9: Average and maximum number of moves for the new search strategies proposed, and comparison with the previous ones. _Method_ | _Number of moves: Average_ | _Number of moves: Maximum_
---|---|---
Entropy | 4.413 $\pm$ 0.006 | 6
Most Parts | 4.406 $\pm$ 0.007 | 7
Plus | 4.404$\pm$ 0.007 | 6
Plus2 | 4.41 $\pm$ 0.007 | 6
Difference is not statistically significant, but as indicated in the study
presented in the previous section, it is encouraging to see that results are,
at least, as good as previously, and maybe marginally better (for Plus, at
least); in this particular Mastermind competition, Plus would be the best;
however, it is clear that statistics dictate that it could happen otherwise in
a different one. At least the maximum number of moves is kept at the same
level as Entropy, which might indicate that it achieves the reduction in
search space size we were looking for. In fact, the size of the consistent set
is practically the same than for Entropy. However, there is some difference in
the size of the sets of top scorers, which is shown in Figure 13 and Table 10.
Table 10: Percentage of times the secret code is among the top scorers for each method, $\kappa=4,\ell=6$. Entropy column has been suppressed for clarity. _In move #_ | _Plus_ | _Plus2_ | _Most Parts_
---|---|---|---
2 | 0.364478 | 0.3644788 | 0.3644788
3 | 0.5395962 | 0.5414171 | 0.5270987
4 | 0.8397347 | 0.8484542 | 0.8242563
5 | 0.9877778 | 0.9862107 | 0.9810066
6 | | | 1
Figure 13: Chance of finding the secret code among the top scorers for the
Entropy method (black and solid) and Most Parts (light or red), to which we
have added Plus (dotted, blue) and Plus2 (dash-dotted, brown), for
$\kappa=6,\ell=4$. Figure 14: Chance of drawing the secret code for the
Entropy method (black and solid) and Most Parts (light or red) (same as in
Figure 6 to which we have added Plus and Plus2, practically one on top of the
other and represented with dark-blue and dotted (Plus) and blue dash-dotted
(Plus2).
Table and graph show that, as intended, Plus and Plus2 reduce the size of the
consistent set in the same proportion as Entropy does, but at the same time,
since the set of top scorers from which the move is randomly chosen is
smaller, the probability of finding the secret code among them is higher; in
fact, it is slightly higher than for Most Parts. These two facts, together,
explain the small edge in the number of moves, which corresponds also to the
small improvement in the secret-playing probability shown in Figure 14. In
fact, difference is significant from moves 1 to 3 (using Wilcoxon test) for
Plus2 against Most Parts, and from moves 2 and 3 for Plus. Difference in total
score is very small between the best (Plus, 57072) and the worst (Entropy,
57189), so, essentially, all methods could obtain the same results. However,
we have achieved to (significantly) increase the probability of obtaining the
secret combination at each step, which might account for the small difference.
Since this difference is offset by other random factors, however, no method is
significantly better than other, and the p-value comparing Plus and Entropy is
only 0.2780.
As we have seen before, the scenario is different at other sizes, so we will
have to experiment them over the $\kappa=8,\ell=6$ space. We use the same
instance of 5000 combinations as we did in the previous Subsection 4.2. This
size was big enough to find some differences among methods, and small enough
for being able to perform the whole experiment in a reasonable amount of time
(around 90 minutes for the whole set). The average number of moves is shown in
Table 11.
Table 11: Average and maximum number of moves for the two new search strategies, Plus and Plus2, along with the previously shown Most Parts and Entropy for $\kappa=8,\ell=4$. _Method_ | _Number of moves_
---|---
| _Average_ | _Maximum_ | _Median_
_Method_ | _Number of moves: Average_ | _Number of moves: Maximum_ |
Entropy | 5.132 $\pm$ 0.012 | 8 | 5
Most Parts | 5.167 $\pm$ 0.012 | 8 | 5
Plus | 5.154 $\pm$ 0.012 | 8 | 5
Plus2 | 5.139 $\pm$ 0.012 | 8 | 5
While previously the difference between Most Parts and Entropy was
statistically significant, the difference now between Most Parts and Plus2 is
significant with p= 0.08536. It is not significant the difference between
Plus/Plus2 and Entropy, Plus and Most Parts, and obviously between Plus and
Plus2. This is, indeed, an interesting result that shows that we have been
able to design a method that statistically is able to beat at least one of the
best classical methods, Most Parts, however, the edge obtained by them is not
enough to gain a clear victory over both of them.
Figure 15: Average size of the consistent set, that is, the set of solutions
that have not been discarded at a point in the game for $\kappa=8,\ell=4$. As
usual, red or light dashed line represents Most Parts and solid black Entropy,
dotted and blue for Plus and dash-dotted and brown for Plus2. Please note that
the $y$ axis is logarithmic; we have put it that way to highlight differences.
Figure 16: Chance of finding the secret code among the top scorers for the
Plus (dotted, blue), Plus2 (dash-dotted, brown), Entropy method (black and
solid) and Most Parts (light or red). Figure 17: Chance of finding the secret
code among the top scorers for the Entropy method (black and solid) and Plus2
(light or brown, dot-dashed). Plus has been suppressed for clarity; the $x$
axis is shifted by one, with $x=1$ representing the second move.
Let us check whether this difference stems from the design of the new
algorithms by looking, as before, at the size of the consistent sets (Fig. 15
and the probability of finding the secret code among the top scorers (Fig. 16.
Once again, the difference among the new methods and the old ones is very
small, and almost none between them. The average set size is virtually the
same as Entropy (98 vs. 99 and 100 after the second move, for instance), but
this was to be expected; they are, anyway, smaller than the size of the sets
for Most Parts.
The finding the secret code by chance probability is, as intended, more
similar to Most Parts (dotted and dot-dashed lines over the red dashed line);
that is, the probability of finding the secret code among the top scorers is
higher than for Entropy, also as intended. This mixed behavior (reduction of
search space as in Entropy, probability of finding the secret code among the
top scorers as in Most Parts) explain why the methods proposed in this paper
reach the results shown in Table 11, which improve, in a significant way, the
state of the art in solutions of the game of MasterMind. But looking more
precisely at the actual chance of drawing the secret code (not the binary
probability of finding it among the top scorers as in the previous graph),
represented in Figure 17 we find the situation very similar to the one shown
in 12, however, the probability of drawing the secret code is better for Plus2
than for Most Parts, and significantly so for all moves up to the 5th.
However, it is still worse for Plus2 than for Entropy for the third move
($x=2$), which accounts for its eventual tie with it. However, the difference
with respect to Most Parts seems to be enough to achieve a small difference at
the end.
As a conclusion to this section, we have proposed two new methods for solving
mastermind that try to improve the capability of drawing the secret
combination at the beginning of the game while, at the same time, decreasing
the size of the search space as Entropy does. The proposed methods obtain
results that are better than the worst of the previous methods (Entropy or
Most Parts), but not significantly better than the best method (Most Parts or
Entropy). However, they are robust in the sense than they are at least as good
(statistically) as the best for each size, so in a sense it could be said that
we have achieved a certain degree of success. Let us see if this conclusion
holds by slightly changing the circumstances of the game by using a different
initial combination.
## 6 Studying the effect of a different starting combination on the
differences among algorithms
To test whether, under different circumstances, the newly proposed methods
perform as well as the best one, and also check the influence of changing the
initial combination to a different interpretation of Knuth’s first move, we
will compare the two best algorithms seen before, Plus2 and Entropy for
$\kappa=8$ using ABCA as first move (instead of the previously used ABCD). In
principle, if we apply the partition score to the first move (which can be
done, even in the absence of a reduction of search space brought by a move) a
combination with different symbols (such as ABCD) obtains the maximum entropy
encore. However, in the absence of information about the secret combination it
is again a empirical exercise to test different initial moves, as has been
done in [19], for instance.
Table 12: Comparison of average and maximum number of moves using ABCA as first move and the best of the previous ones. _Method_ | _Number of moves_
---|---
| _Average_ | _Maximum_ | _Median_
Most Parts | 5.167 $\pm$ 0.012 | 8 | 5
Entropy (ABCD) | 5.132 $\pm$ 0.012 | 8 | 5
Plus2 (ABCD) | 5.139 $\pm$ 0.012 | 8 | 5
Entropy (ABCA) | 5.124 $\pm$ 0.012 | 8 | 5
Plus2 (ABCA) | 5.116 $\pm$ 0.012 | 8 | 5
The summary of the number of moves is again shown in Table 12. A priori, the
average number of moves is better than before. However, the only significant
difference (as usual, using paired Wilcoxon test) is between Entropy(ABCA) and
Plus2(ABCA) and Most Parts and Plus. The difference between the ABCA and ABCD
versions of both algorithms is not significant. The difference between Plus2
and Entropy using ABCA as first move is, once again, not significant, proving
that the new algorithm proposed, plus2, is as good as the best previous
algorithm available for the size (which, in this case, is Entropy).
We can also conclude from this experiment that, even if there is not a
significant difference for a particular algorithm in using ABCD or ABCA, it is
true that results using ABCA are significantly better for Entropy and Plus2
than for other algorithms such as Most Parts, with which there was no
significant difference using ABCD. We have not tested Most Parts and Plus in
this section since our objective was mainly comparing the best methods with a
new starting move with all methods using the other move; however, we can more
or less safely assume that results will be slightly, but not significantly,
better, and that they will be statistically similar to those obtained by
Entropy and Plus2.
Figure 18: Chance of finding the secret code among the top scorers for the Entropy method (black and solid) and Plus2 (light or brown, dot-dashed), both for ABCD as first move, and Entropy (black, dot-long-dash) and Plus2 (brown or light, long dash) for ABCA . Table 13: Average and standard deviation of the number of combinations remaining after every move for Entropy ABCD and ABCA (the quantities for Plus2 are practically the same). The number of combinations is the same after the first move (not shown here), since both are playing the same first move. _Before move #_ | _Entropy ABCD_ | _Entropy ABCA_
---|---|---
2 | 666 $\pm$ 310 | 706 $\pm$ 327
3 | 101 $\pm$ 67 | 99 $\pm$ 69
4 | 13 $\pm$ 10 | 13 $\pm$ 10
5 | 2.4 $\pm$ 1.5 | 2.28 $\pm$ 1.48
6 | 1.21 $\pm$ 0.46 | 1.23 $\pm$ 0.5
7 | 1.09 $\pm$ 0.29 | 1.19 $\pm$ 0.39
Figure 19: Difference in score among the number of games won by Entropy (ABCD)
and Entropy (ABCA) in $x$ moves for $\kappa=8,\ell=4$. Method is the same as
for figure 3, that is, product of difference and number of moves.
It is clear that, in this case, the advantage is mainly due to the changes in
search space from the first move, however, this will have an influence on the
probability of playing the secret combination, as shown in Figure 18. In both
cases (Entropy and Plus2) the probability is lower, however, the result is
(not significantly) better. The differences must be due, then, mainly to the
difference in consistent set size, which is shown in Table 13, but these
differences are not so clear-cut as should be expected, that is, smaller sizes
throughout all moves. The size is actually bigger for ABCA in the first move,
although smaller in the second. These two will actually have little influence
on the outcome (since, at that stage in games, drawing the secret code is
mainly the product of the scoring algorithm and the composition of the top
scorers). However, there is a significant difference before move 5th, which is
when in some cases the consistent set is reduced to one, which is also
reflected in the differences in score represented in Figure 19. Up to the
fifth move (label 2 in the probability graph 18, Entropy(ABCD) is better
(negative difference, Entropy(ABCD) $<$ Entropy(ABCA). However, it accumulates
score (remember that higher score is worse) in the 6th move, which eventually
implies the victory (which we should note is not significant) of Entropy
(ABCA).
At any rate, in this section we have proved that the two mechanisms we studied
in the previous sections is also at work to provoke the advantage of one
algorithm over other, and that studying mainly consistent set size and the
probability of drawing the secret code, and when one mechanism for finishing
the game takes over the other, is the best way of evaluating different
algorithms for playing the game of mastermind.
## 7 Conclusions and discussion
In this work we have analyzed the solutions to the game of Mastermind in a
novel way, taking into account not only the reduction of the size of the
search space brought by the combination played, but also how every method
scores the components of the search space and whether the secret combination
is among the top scorers, that is, the actual probability of drawing the
secret combination at each move. Using always combinations of length equal to
4, we have proved that the fact that Most Parts has an increased chance of
finding the secret combination among its top scorers counterbalances the
effective reduction of the set of consistent combinations brought by Entropy
for six colors; the difference among the probabilities for both method
decreases with the search space size (which we have proved for $\kappa=8$,
making Entropy beat Most Parts for that configuration.
Using that fact, we have proposed two new methods that effectively combine
Entropy and Most Parts by reducing search space size as the former, and having
a set of top scorers to choose at worst as bad as Most Parts. These two
methods, Plus and Plus2, behave statistically in the same way for both sizes
tested, are marginally (not significantly) better than Most Parts for
$\kappa=6,\ell=4$, and significantly better than Most Parts and marginally
better than Entropy for $\kappa=8,\ell=4$, this method being itself only
marginally better than Most Parts for that size. Besides, results obtained by
Plus/Plus2 are an improvement over the state of the art, published, for
instance, in [25] (which is comparable to that obtained by Entropy and, thus,
probably not statistically significant).
The two new methods presented do not present statistically significant
differences. This could be interpreted by stating that combining a scoring
strategy that reduces search space and gives top scores to the hidden
combination consistently, in general, is the most profitably course of action.
At the same time, doing to in different ways will not lead to significant
differences. However, although it is tempting to generalize this to spaces of
higher dimensions, heuristically it cannot be done. We can, however, affirm
that combining several scoring strategies, specially Entropy and Most Parts,
will yield better results even if using samples of the whole consistent set.
Besides, both methods, but specially Plus2, is always (for the two sizes
tested, which is considered enough) as good as the best previous method
described, which makes it more robust.
This paper also introduces and tests a methodology for testing different
algorithms for solving Mastermind. While the first approximation would be to
empirically measure the average number of moves, and the first would design
methods that maximize the reduction of search space every move, we have
introduced here the measurement of the probability of drawing the secret code
as a third empirical test to consider when designing a new method, since the
success of a solution depends, for the first moves, in that probability, to be
followed by the reduction of search space to a single element, as considered
so far.
In theory, these results and methodology could be extended to spaces with
bigger sizes; however, in practice, the increase in time complexity of the
algorithm bars it except from the simplest extensions. While solving
$\kappa=6,\ell=4$ takes around 0.5s, $\kappa=8,\ell=4$, whose search space is
only 4 times as big, needs around 10 times as much time, around 5 seconds. One
of the bottlenecks of the algorithm is the need to compare all elements in the
consistent set with each other, which is approximately quadratic in size; this
needs to be done for each step, besides, the number of steps also increases in
a complex way with the increase in search space size. However, in time it will
become possible to test in a reasonable amount of time whether these new
methods are still better than the classical ones, and in which proportion. It
is impossible to know in advance what will be the influence of finding the
secret code among the top scorers; in fact, it increases from $\kappa=6$ to
$\kappa=8$: the size of the consistent set increases, but the probability of
finding the secret code among them also increases (although the actual
probability of drawing that code decreases too); however, the actual size of
that set will also have an influence, with bigger sizes decreasing the actual
chance of playing the secret combination. How the three quantities will change
with the problem size (and, actually, with both dimensions of the problem:
number of colors $\kappa$ and combination length $\ell$) is beyond the scope
of this paper, and might not be easy to compute analytically.
It is more interesting, however, to use these results for methods that use a
sample of the set of consistent combinations, such as evolutionary algorithms
[26, 19]. Results obtained here can be used in two different ways: by taking
into account the size of the consistent set to set the sample size, which was
set heuristically [24] or fixed [19], and also by using a combination of Most
Parts and Entropy scores to compute the fitness of a particular combination.
This will allow to find solutions to Mastermind problems in bigger search
spaces. However, the limiting factor keeps on being the size of the consistent
set that will be used to score the combinations; since sample size increases
with problem size, eventually a non-feasible limit (in term of time or memory
usage) will be reached. However, using even a small size will extend the range
of problems with feasible solutions. Finding a way to score solutions that is
less-than-quadratic in time will also extend that range, although another
avenue, already explored years ago [16], would be to play non-consistent
combinations in some cases if enough time without finding consistent
combinations passes.
This leaves as future work improving the scoring used in evolutionary
algorithms using these results, and checking how far these solutions will go
into the search space. Managing to find a solution to the $\kappa=12,\ell=8$
in under one hour would be a good target, but it will need a good adjustment
of the evolutionary algorithm parameters, as well as tweaking the
implementation as far as it will allow (because implementation matters [27]).
Using either a distributed computing environment such as the Evolvable Agents
[28] or SofEA [29] will probably be needed to in order to shorten total time
to solution. However, an parallel version is not trivial, since things such as
the consistent set might have to be centralized.
## Acknowledgments
This work is supported by grants TIN2011-28627-C04-02, -01 and P08-TIC-03903.
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|
arxiv-papers
| 2012-07-05T18:09:59 |
2024-09-04T02:49:32.695408
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J. J. Merelo, Antonio M. Mora, Carlos Cotta, Thomas P. Runarsson",
"submitter": "Juan Juli\\'an Merelo-Guerv\\'os Pr.",
"url": "https://arxiv.org/abs/1207.1315"
}
|
1207.1538
|
# Dynamically stabilized decoherence-free states in non-Markovian open
fermionic systems
Heng-Na Xiong Department of Physics and Cenert for Quantum information
Science, National Cheng Kung University, Tainan 70101, Taiwan Wei-Min Zhang
wzhang@mail.ncku.edu.tw Department of Physics and Cenert for Quantum
information Science, National Cheng Kung University, Tainan 70101, Taiwan
Matisse Wei-Yuan Tu Department of Physics and Cenert for Quantum information
Science, National Cheng Kung University, Tainan 70101, Taiwan Daniel Braun
braun@irsamc.ups-tlse.fr Laboratoire de Physique Theorique, Universite Paul
Sabatier, Toulouse III, 118 route de Narbonne, F-31062 Toulouse, France
###### Abstract
Decoherence-free subspaces (DFSs) provide a strategy for protecting the
dynamics of an open system from decoherence induced by the system-environment
interaction. So far, DFSs have been primarily studied in the framework of
Markovian master equations. In this work, we study decoherence-free (DF)
states in the general setting of a non-Markovian fermionic environment. We
identify the DF states by diagonalizing the non-unitary evolution operator for
a two-level fermionic system attached to an electron reservoir. By solving the
exact master equation, we show that DF states can be stabilized dynamically.
††preprint: HEP/123-qed
## I Introduction
Quantum decoherence is a fundamental issue in open quantum systems and is also
the most important obstacle opposing the realization of a large scale quantum
computer. Unavoidable interactions with noisy environments typically render
initially prepared pure states mixed very rapidly. Nonetheless, a subspace of
Hilbert space can exist where a system undergoes a unitary evolution
irrespective of the interaction with its environment. Such a decoherence-free
subspace (DFS) can in principle provide a theoretically perfect strategy to
protect a system against quantum decoherence. The possibility of DFSs was
already pointed out by Zurek Zurek82 , who observed that if the interaction
Hamiltonian of the system with the environment has a degenerate eigenvalue,
then superpositions of the corresponding eigenstates remain coherent.
Generally, degenerate eigenvalues can notably arise if the coupling has a
certain symmetry, and the resulting protection against decoherence has been
observed early in the context of rotational tunneling Wilson35 ; Longuet-
Higgins63 ; Freed65 ; Stevens83 ; Haeusler85 ; Wuerger89 ; Braun93 ; Braun94 .
In the late 1990s, DFSs were rediscovered independently by several groups
Palma96 ; Zanardi97a ; Zanardi97b ; Duan97 ; Lidar98 ; Braun98 . The theory of
DFSs has already been extensively discussed in the context of the symmetries
of the Hamiltonian Zanardi97a ; Zanardi97b , semigroup dynamics in the
language of quantum master equation Lidar98 ; Braun98 ; Zanardi98 ; Lidar05 ;
Lidar99b ; Karasik08 , and the operator sum representation based on Kraus
operators Lidar99a . Meanwhile, the existence of DFSs has also been verified
experimentally with polarization-entangled photons Kwiat00 ; Altepeter04 ;
Yamamoto08 , trapped ions Kielpinski01 ; Langer05 , nuclear spins using
nuclear magnetic resonance techniques Viola01 ; Fortunato02 ; Ollerenshaw03 ;
Wei05 , and neutron interferometry Pushin11 . These experiments show that
encoding quantum information in a DFS can significantly prolong the storage
time. Therefore, DFS have attracted wide interest for applications in fault-
tolerant quantum computation Lidar99b ; Bacon00 ; Beige00 ; Beige00b ; Zhou04
; Wu05 ; Xue06 ; Monz09 , long distance quantum communication Xue08 , quantum
key distribution Yin08 , quantum teleportation Wei07 , quantum metrology
Roos06 , robust quantum repeaters Klein06 ; Klein08 , and coherent quantum
control Cappellaro07 ; Kiffner07 ; Wang10 . The requirement of a symmetry in
the coupling to the environment is not always necessary Braun00 . Recently, a
new Heisenberg-limited metrology protocol was proposed which exploits the
evolution of a DFS due to a collective change of the couplings to the
environment Braun11 .
According to Refs. Lidar98 ; Zanardi98 , a DFS in a system ruled by Markovian
dynamics is defined formally by the vanishing of the nonunitary Lindblad
(decohering) part in the Markovian master equation Lindblad76 , see Eq. (3).
This defines a Hilbert subspace in which dynamics is locally (in time)
unitary. Even so, decoherence can still arise if the Hamiltonian of the system
drives a state out of the DFS. This happens on the time scale of the system
Hamiltonian that is typically much longer than the microscopic decoherence
times Strunz01 . However, leaking out of the DFS can be suppressed by coupling
the system relatively strongly to the environment Beige00 ; Ognyan10 . An
alternative definition of a DFS in the Markovian context has been given in
Karasik08 . These authors not only derived necessary and sufficient conditions
for the vanishing of the nonunitary part of the master equation [see Eq. (4)],
but also found criteria for globally DF states, for which the dynamics
resulting from the full master equation (including the Hamiltonian part)
remains unitary.
The extension of such DFS criteria to non-Markovian dynamics is not
straightforward. The general non-Markovian master equation of the Nakajima-
Zwanzig form N58Z60 involves a complicated time-non-local memory integrand in
the nonunitary terms. However, the exact master equations that describe the
general non-Markovian dynamics have been recently developed for some class of
open quantum systems, including quantum Brownian motion Hu92 ; Karrlein97 ;
Kaake85 , entangled cavities with vacuum fluctuations AnZhang07 , coupled
harmonic oscillators Chou08 ; Paz0809 , quantum dot electronic systems in
nanosturctures TuZhang08 ; TuZhang09 , various nano devices with time-
dependent external control fields JinZhang10 , nanocavity systems including
initial system-reservoir correlations TanZhang11 , and photonic networks
imbedded in photonic crystals LeiZhang12 . These exact master equations can
all have the nonunitary Lindblad form, but the decoherence rates are time-
dependent and may become negative during the time evolution. This is different
from the Markovian case, where the decoherence rates are always positive.
Since the fact that all rates have the same sign plays an essential role in
showing the necessity of the DFS criterion, other possibilities of creating a
DF state may arise in non-Markovian cases.
In this work, we study the dynamics of an open fermionic system by solving the
exact non-Markovian fermionic master equation TuZhang08 ; TuZhang09 ;
JinZhang10 . We find a new type of pure DF states which arise from the fact
that certain time-dependent decoherence rates in the master equation switch
themselves off after the system reaches a stable state. We call these DF
states dynamically stabilized DF states. They are generated in particular for
non-Markovian environments. The mechanism how the DF states arise is therefore
very different from the known mechanism obtained from the Markovian master
equation. For a two-level fermionic system coupled to an electron reservoir,
we find two dynamically stabilized DF states that possess full quantum
coherence between the singly occupied states of the original two levels.
Practical applications of the dynamically stabilized DF states in electron
spin and charge qubits for quantum information processing are expected.
The paper is organized as follows. In Sec. II, we briefly review the general
criterion of DFSs based on the Markovian master equation formalism. In Sec.
III, we discuss the dynamics of electron systems in nanostructures via the
exact master equation. In particular, we consider a two-level electron system
whose two levels are coupled identically to an electron reservoir. We find
that the system Hilbert space can be split into two closed subspaces. Then in
Sec. IV, in terms of the full Lindblad generator, we discuss all the
possibilities how DF states can arise in the system. We find that the
vanishing of one of the two decoherence rates in the problem can give rise to
dynamically stabilized DF states. With only one decoherence term in the master
equation remaining, the well-known criterion of DFSs for Markovian decoherence
still provides a necessary and sufficient condition for DF states. In Sec. V,
we investigate under what conditions the dynamically stabilized DF states can
be reached by the same dissipative process that allows their existence. We
find that the initial state and the details of the dynamics given by time-
dependent non-equilibrium Green functions determines which DF state can be
reached. In Sec. VI, we discuss the generation of DF states in the Born-
Markovian (BM) dynamics within our exact framework. We show that the DF state
in the BM dynamics is a special case of the exact solution. Finally, a
conclusion is given in Sec. VI.
## II Criterion for DFS in Markovian case
In this section, we will briefly review the general criterion of DFS based on
the Markovian master equation following Ref. Karasik08 . Consider an open
system $S$ coupled to an external noisy environment $E$. One can write the
total Hamiltonian as
$H=H_{S}\otimes I_{E}+I_{S}\otimes H_{E}+H_{I},$ (1)
where $I$ is the identity operator and $H_{I}$ denotes the interaction
Hamiltonian between the system and its environment. If the dynamics of the
system-bath is Markovian and the system and the environment are initially
decoupled, the master equation for the density matrix of the system takes the
Lindbladian form Carmichael93 ; Breuer07 ; Weiss08
$\displaystyle\dot{\rho}\left(t\right)$
$\displaystyle=-i\left[\widetilde{H}_{S},\rho\left(t\right)\right]+L\left[\rho\left(t\right)\right],$
(2a) $\displaystyle L\left[\rho\right]$
$\displaystyle=\frac{1}{2}\sum_{\alpha=1}^{N}a_{\alpha}\left(2F_{\alpha}\rho
F_{\alpha}^{\dagger}-F_{\alpha}^{\dagger}F_{\alpha}\rho-\rho
F_{\alpha}^{\dagger}F_{\alpha}\right),$ (2b) where
$\widetilde{H}_{S}=H_{S}+\Delta$ is the renormalized system Hamiltonian with
$\Delta$ a possible hermitian contribution from the environment (”Lamb
shift”), $F_{\alpha}$ are orthogonal operators on the system Hilbert space
$\mathcal{H}_{S}$, and $a_{\alpha}$ denote real positive coefficients. Thus,
the commutator involving $\widetilde{H}_{S}$ in Eq. (2a) determines an
effectively unitary evolution of the system, while the decohering effect
induced by the environment is totally accounted for by the non-unitary term
$L\left[\rho\right]$.
For the Markovian master equation (2), the condition of an instantaneous DFS
at time $t$ amounts to the vanishing of $L\left[\rho\right]$, that is
Karasik08 ,
$L\left[\rho_{\text{DF}}\left(t\right)\right]=0.$ (3)
This ensures that $\rho_{\text{DF}}\left(t\right)$ obeys a unitary evolution
$\dot{\rho}_{\text{DF}}\left(t\right)=-i\left[\widetilde{H}_{S},\rho_{\text{DF}}\left(t\right)\right]$
at time $t$. This does not imply necessarily unitary evolution at all times,
as the evolution due to $\tilde{H}_{S}$ can drive the system out of the DFS.
In Lidar98 the DFS found from (3) was therefore called DFS to the first
order.
A sufficient and necessary condition for a state $|k_{\mathrm{DF}}\rangle$ to
be locally DF is given in terms of the operators $\left\\{F_{\alpha}\right\\}$
Karasik08 by
$\displaystyle F_{\alpha}|k_{\text{DF}}\rangle$
$\displaystyle=c_{\alpha}|k_{\text{DF}}\rangle,\forall\alpha,k_{\text{DF}}$
(4) $\displaystyle\mbox{ and
}\sum_{\alpha=1}^{N}a_{\alpha}F_{\alpha}^{\dagger}F_{\alpha}|k_{\mathrm{DF}}\rangle=\sum_{\alpha=1}^{N}a_{\alpha}|c_{\alpha}|^{2}|k_{\mathrm{DF}}\rangle$
Eq. (4) means that the DF states are degenerate eigenstates of all the
operators $\left\\{F_{\alpha}\right\\}$, and degenerate eigenstates of
$\sum_{\alpha=1}^{N}a_{\alpha}F_{\alpha}F_{\alpha}^{\dagger}$. A special case
is given by $c_{\alpha}=0$ $\forall\alpha$, in which case the second condition
in (4) is automatically fulfilled. In Karasik08 it was shown that the space
spanned by the $|k_{\mathrm{DF}}\rangle$ is a DFS at all times, if and only if
in addition to satisfying (4) it is also invariant under $\widetilde{H}_{S}$.
The generalization of (4) to non-Markovian master equations is not straight-
forward. However, it was recently shown Kossakowski10 that under certain
conditions the solution of a non-Markovian master equation N58Z60 with a
finite time memory kernel can be at the same time a solution of a local-in-
time non-Markovian master equation. In principle, local in time
generalizations of the Markovian master equation (2b) may be obtained by
making both the rates $a_{\alpha}$ and the operators $F_{\alpha}$ time-
dependent. Interestingly, the exact master equations describing the general
non-Markovian dynamics for a large class of bosonic and fermionic systems
AnZhang07 ; TuZhang08 ; TuZhang09 ; JinZhang10 ; TanZhang11 ; LeiZhang12 have
been developed recently. They all have the Lindbladian form of Eq. (2), but
the decoherence rates $a_{\alpha}$ in the nonunitary term (2b) are time-
dependent and local in time, whereas the operators $F_{\alpha}$ are time-
independent. The time-dependent decoherence rates are determined
microscopically and nonperturbatively by the retarded and correlation Green
functions in non-equilibrium Green function theory Sch61407 , where the back-
actions from reservoirs are fully taken into account. As a result, the non-
Markovian dynamics are fully characterized in terms of the time-non-local
retarded integrand in the Dyson equation, which governs the nonequilibrium
Green functions Kad62 ; Mahan00 . In addition to being time-dependent, the
decoherence rates can become negative for short times, representing in a
certain sense the back-flow of information from the environment to the system
TuZhang08 ; Piilo09 ; XiongZhang10 ; WuZhang10 ; LeiZhang11 . Since the fact
that all $a_{\alpha}$ have the same sign is an important requirement for
showing that (4) is necessary for a Markovian DF state (see the argument after
eq.(3.15) in Karasik08 ), additional DF states may arise in the non-Markovian
case. In the following, we will provide a new mechanism for generating
dynamically stabilized DF states in fermionic systems, based on the exact
fermionic master equation developed recently TuZhang08 ; TuZhang09 ;
JinZhang10 .
## III Exact Master Equation
We consider a general nanoelectronic system with $N$ energy levels coupled to
an electron reservoir. The Hamiltonian for the system, the electron reservoir,
and the interaction between them read
$\displaystyle H_{S}$
$\displaystyle=\sum_{i=1}^{N}\epsilon_{i}a_{i}^{\dagger}a_{i},~{}~{}~{}H_{B}=\sum_{k}\mathbf{\varepsilon}_{k}c_{k}^{\dagger}c_{k},$
$\displaystyle H_{I}$
$\displaystyle=\sum_{i=1}^{N}\sum_{k}\left(V_{ik}e^{i\phi_{i}}a_{i}^{\dagger}c_{k}+V_{ik}e^{-i\phi_{i}}c_{k}^{\dagger}a_{i}\right).$
(5)
Here $a_{i}^{\dagger}$ and $a_{i}$ are electron creation and annihilation
operators for $i$th level with energy $\epsilon_{i}$. The operators
$c_{k}^{\dagger}$ and $c_{k}$ denote electron creation and annihilation
operators for the energy level $\mathbf{\varepsilon}_{k}$ of the electron
reservoir. The coupling strength between the system and the reservoir is
described by the $V_{ik}\in\mathbb{R}$, with the phase $\phi_{i}$ of the
coupling made explicit. The total hamiltonian is $H=H_{S}+H_{B}+H_{I}$.
The exact master equation for such system was obtained in TuZhang08 ;
TuZhang09 ; JinZhang10 ,
$\dot{\rho}\left(t\right)=-i\left[\widetilde{H}_{S}\left(t\right),\rho\left(t\right)\right]+L\left[\rho\left(t\right)\right],$
(6a) where the renormalized system Hamiltonian
$\widetilde{H}_{S}\left(t\right)$ and the decoherence term
$L\left[\rho\left(t\right)\right]$ take the form
$\displaystyle\widetilde{H}_{S}(t)=\sum_{i,j=1}^{N}\widetilde{\epsilon}_{ij}(t)a_{i}^{\dagger}a_{j},$
(6b) $\displaystyle L[\rho(t)]=\sum_{i,j=1}^{N}\big{\\{}$
$\displaystyle\kappa_{ij}(t)\big{[}2a_{j}\rho(t)a_{i}^{\dagger}-a_{i}^{\dagger}a_{j}\rho(t)-\rho(t)a_{i}^{\dagger}a_{j}\big{]}$
$\displaystyle+\widetilde{\kappa}_{ij}(t)\big{[}2$ $\displaystyle
a_{i}^{\dagger}\rho(t)a_{j}-a_{j}a_{i}^{\dagger}\rho(t)-\rho(t)a_{j}a_{i}^{\dagger}\big{]}\big{\\}}.$
(6c) Here the shifted $\widetilde{\epsilon}_{ij}(t)$ and decoherence rates
$\kappa_{ij}(t)$ and $\widetilde{\kappa}_{ij}(t)$ are all time-dependent but
local in time. They are determined microscopically and nonperturbatively in
terms of the retarded and correlation Green functions by eliminating
completely all the reservoir degrees of freedom (i.e. tracing over all the
states of the environment). Their explicit forms are shown in Eq. (9) of Ref.
TuZhang08 .
To be more specific, let us consider the case of a nanoelectronic system with
$N=2$. Physically, such a system may be realized by a double quantum dot
system in which each dot has a single active energy (on-site) level, coupled
to electrodes with all spins polarized in both the dots and the electrodes.
Another example for $N=2$ is given by a single-level quantum dot coupled to
electrodes with allowed spin flips between two antiparallel directions.
Furthermore, we assume that the two energy levels of the system are
degenerate: $\epsilon_{1}=\epsilon_{2}=\epsilon_{0}$. Practically, the energy
degeneracy is easier to be realized in the second setting than in the first.
We also assume that both levels have the same coupling strength to the
electron reservoir, i.e., $V_{1k}=V_{2k}=V_{k}/\sqrt{2}$. Then we introduce
two effective fermion operators
$A_{+}=\frac{1}{\sqrt{2}}\left(a_{1}+e^{i\phi}a_{2}\right),~{}A_{-}=\frac{1}{\sqrt{2}}\left(-e^{-i\phi}a_{1}+a_{2}\right),$
with $\phi=\phi_{1}-\phi_{2}$. The Hamiltonian (5) can be rewritten in terms
of $A_{\pm}$ as follows
$\displaystyle H_{S}$
$\displaystyle=\epsilon_{+}A_{+}^{\dagger}A_{+}+\epsilon_{-}A_{-}^{\dagger}A_{-},H_{B}=\sum_{k}\varepsilon_{k}c_{k}^{\dagger}c_{k},$
$\displaystyle H_{I}$
$\displaystyle=\sum_{k}V_{k}\left[e^{i\phi_{1}}A_{+}^{\dagger}c_{k}+e^{-i\phi_{1}}c_{k}^{\dagger}A_{+}\right].$
(7)
where the effective energy levels $\epsilon_{\pm}=\epsilon_{0}$ are still
degenerate.
The system Hamiltonian is diagonalized in terms of $A_{\pm}$, such that the
original system is equivalent to an effective system which has two decoupled
energy levels $\epsilon_{\pm}$, out of which only one energy level
($\epsilon_{+}$) couples to the electron reservoir. As a result, the
corresponding exact master equation becomes
$\dot{\rho}\left(t\right)=-i\left[\widetilde{H}_{S}\left(t\right),\rho\left(t\right)\right]+L\left[\rho\left(t\right)\right],$
(8a) with the new expressions of $\widetilde{H}_{S}\left(t\right)$ and
$L\left[\rho\left(t\right)\right]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\widetilde{H}_{S}\left(t\right)=\widetilde{\epsilon}_{+}\left(t\right)A_{+}^{\dagger}A_{+}+\epsilon_{-}A_{-}^{\dagger}A_{-},$
(8b) $\displaystyle
L\left[\rho\left(t\right)\right]=\kappa\left(t\right)\left[2A_{+}\rho\left(t\right)A_{+}^{\dagger}-A_{+}^{\dagger}A_{+}\rho\left(t\right)-\rho\left(t\right)A_{+}^{\dagger}A_{+}\right]$
$\displaystyle~{}~{}~{}+\widetilde{\kappa}\left(t\right)\left[2A_{+}^{\dagger}\rho\left(t\right)A_{+}-A_{+}A_{+}^{\dagger}\rho\left(t\right)-\rho\left(t\right)A_{+}A_{+}^{\dagger}\right].$
(8c) The decoherence rates are
$\kappa\left(t\right)=\gamma\left(t\right)-\frac{\widetilde{\gamma}\left(t\right)}{2}$
and
$\widetilde{\kappa}\left(t\right)=\frac{\widetilde{\gamma}\left(t\right)}{2}$.
The renormalized energy level $\widetilde{\epsilon}_{+}\left(t\right)$ and the
rates $\gamma\left(t\right)$ and $\widetilde{\gamma}\left(t\right)$ are
determined exactly by
$\displaystyle\widetilde{\epsilon}_{+}(t)=-\mathrm{Im}[\dot{u}(t)u^{-1}(t)],$
(9a) $\displaystyle\gamma(t)=-\mathrm{Re}[\dot{u}(t)u^{-1}(t)],$ (9b)
$\displaystyle\widetilde{\gamma}(t)=\dot{v}(t)-2v(t)\mathrm{Re}[\dot{u}(t)u^{-1}(t)],$
(9c) and $u(t)$ and $v\left(t\right)$ are the retarded and correlation Green
functions in the Schwinger-Keldysh nonequilibrium Green function theory
JinZhang10 . They obey the integral-differential equations
$\displaystyle\frac{d}{dt}u\left(t\right)+i\epsilon_{+}u\left(t\right)+\int_{t_{0}}^{t}g\left(t-\tau\right)u\left(\tau\right)d\tau=0,$
(10a) $\displaystyle
v\left(t\right)=\int_{t_{0}}^{t}d\tau_{1}\int_{t_{0}}^{t}d\tau_{2}u\left(\tau_{1}\right)\widetilde{g}\left(\tau_{2}-\tau_{1}\right)u^{\ast}\left(\tau_{2}\right),$
(10b) subject to the boundary conditions $u\left(t_{0}\right)=1$ . The
integration kernels read
$g\left(\tau\right)=\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}J\left(\omega\right)e^{-i\omega\tau}$
and
$\widetilde{g}\left(\tau\right)=\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}J\left(\omega\right)f\left(\omega\right)e^{-i\omega\tau}$,
where the spectral density of the reservoir is given by
$J\left(\omega\right)=2\pi\sum_{k}\left|V_{k}\right|^{2}\delta\left(\omega-\omega_{k}\right)$,
and $f\left(\omega\right)=1/\left(e^{\beta\left(\omega-\mu\right)}+1\right)$
is the initial electron distribution of the reservoir.
Since the effective energy level $\epsilon_{-}$ is apparently decoupled from
the electron reservoir, one may naively think that the state generated by the
operator $A^{{\dagger}}_{-}$, namely $A^{{\dagger}}_{-}|0\rangle$ becomes
naturally a DF state. This is actually not true. The term with
$\widetilde{\kappa}(t)$ in (8c) will drive this state into another state since
$A^{{\dagger}}_{+}A^{{\dagger}}_{-}|0\rangle\neq 0$. However, there is an
occupation constant of motion in this system,
$\left[A_{-}^{\dagger}A_{-},H\right]=0$. This symmetry separates the system
Hilbert space into two closed subspaces
$\mathcal{H}_{+}=\left\\{|\mathsf{v}\rangle,|+\rangle\right\\}$ and
$\mathcal{H}_{-}=\left\\{|-\rangle,|\mathsf{d}\rangle\right\\}$, corresponding
to the occupation $N_{-}\equiv\langle A_{-}^{\dagger}A_{-}\rangle=0$ and $1$,
respectively. Here $|\mathsf{v}\rangle$ ($|\mathsf{d}\rangle$) is the vacuum
(doubly occupied) electron state, while
$|\pm\rangle=A_{\pm}^{\dagger}|\mathsf{v}\rangle$ are superpositions of singly
occupied states of the original two coupled levels,
$\displaystyle|+\rangle$
$\displaystyle=\left(|1\rangle+e^{-i\phi}|2\rangle\right)/\sqrt{2},$
$\displaystyle|-\rangle$
$\displaystyle=\left(-e^{i\phi}|1\rangle+|2\rangle\right)/\sqrt{2},$ (11)
and $|i\rangle=a_{i}^{\dagger}|\mathsf{v}\rangle$ ($i=1,2$). The two states
$|\pm\rangle$ are the superpositions of the two original energy levels with
the relative phases $\phi$ and $\phi+\pi$, where $\phi$ is arbitrary. As a
consequence, starting from any initial state in the closed subspace
$\mathcal{H}_{+}$ ($\mathcal{H}_{-}$), the system will be kept in this
subspace throughout the evolution process.
Eq.(8) has the standard form of Lindblad master equation, except that the
decoherence rates, $\kappa(t)$ and $\widetilde{\kappa}(t)$, can depend on time
and even become negative. They are local-in-time and determined
microscopically and nonperturbatively from Eq. (10). The DFS criterion of Eq.
(4) is then still sufficient, but may not be necessary. The operators
$A_{+}^{{\dagger}}$ and $A_{+}$ act on the four basis states
$\\{|$v$\rangle,|-\rangle,|+\rangle,|$d$\rangle\\}$ according to the following
relations:
$\begin{array}[c]{lll}A_{+}^{\dagger}|\mathsf{v}\rangle=|+\rangle,&{}{}&A_{+}|\mathsf{v}\rangle=0,\\\
A_{+}^{\dagger}|+\rangle=0,&&A_{+}|+\rangle=|\mathsf{v}\rangle,\\\
A_{+}^{\dagger}|-\rangle=|\mathsf{d}\rangle,&&A_{+}|-\rangle=0,\\\
A_{+}^{\dagger}|\mathsf{d}\rangle=0,&&A_{+}|\mathsf{d}\rangle=|-\rangle.\end{array}$
One checks that both $A_{+}^{\dagger}$ and $A_{+}$ have a four-fold degenerate
eigenvalue $c_{\alpha}=0$, but the corresponding eigenspaces are only two-
dimensional, given by $\\{|+\rangle,|$d$\rangle\\}$ for $A_{+}^{\dagger}$ and
$\\{|-\rangle,|$v$\rangle\\}$ for $A_{+}$. Since these two spaces do not
overlap, condition (4) is not satisfied, and as long as both $\kappa(t)$ and
$\widetilde{\kappa}(t)$ are positive, there is no DFS. In the next section, we
explore whether additional DF states can exist if $\kappa(t)$ and
$\widetilde{\kappa}(t)$ are not both positive, and show that a DFS can arise
dynamically, when at least one of the two rates $\kappa(t)$ and
$\widetilde{\kappa}(t)$ vanishes.
## IV DF states
### IV.1 Eigenstates of the Lindblad operator
Since, as discussed above, the Markovian DFS criterion (4) may not be
necessary for a non-Markovian system, we define a local (in time) DFS by the
vanishing of the non-unitary term $L[\rho(t)]$. Clearly, this leads to local
unitary time-evolution. In the following, we will discuss all possibilities
how $L[\rho(t)]=0$ can arise by calculating the eigenvalues of the full
Lindblad generator $L$ in Liouville space. If $L$ has a zero eigenvalue, the
corresponding eigenstate is DF. Conversely, a local DF state is by definition
an eigenstate of $L$ with eigenvalue zero. However, as the eigenstates of $L$
do not necessarily have all the properties of a density matrix, such states
may not be physical. One must therefore examine for each eigenstate whether it
is a physical state or can be combined with other eigenstates corresponding to
the same (degenerate) eigenvalue to form a physical state. In this way one can
find all physical DF states possible.
In the basis of
$\left\\{|\mathsf{v}\rangle,|+\rangle,|-\rangle,|\mathsf{d}\rangle\right\\}$,
the density matrix takes the general form
$\displaystyle\rho\left(t\right)=$
$\displaystyle\rho_{\mathsf{vv}}\left(t\right)|\mathsf{v}\rangle\langle\mathsf{v}|+\rho_{++}\left(t\right)|+\rangle\langle+|$
$\displaystyle+\rho_{+-}(t)|+\rangle\langle-|+\rho_{+-}^{\ast}\left(t\right)|-\rangle\langle+|$
$\displaystyle+\rho_{--}\left(t\right)|-\rangle\langle-|+\rho_{\mathsf{dd}}\left(t\right)|\mathsf{d}\rangle\langle\mathsf{d}|.$
(12)
Coherences between states with different particle numbers are not permitted
due to the particle number super-selection rule. Since there are only six
nonzero elements of $\rho\left(t\right)$, we can define the basis
$\\{|1\rangle\equiv|$v$\rangle\langle$v$|$,
$|2\rangle\equiv|+\rangle\langle+|$, $|3\rangle\equiv|+\rangle\langle-|$,
$|4\rangle\equiv|-\rangle\langle+|$, $|5\rangle\equiv|-\rangle\langle-|$,
$|6\rangle\equiv|$d$\rangle\langle$d$|\\}$ in Liouville space, which is
orthogonal with respect to the scalar product $\langle A|B\rangle={\rm
tr}{A^{\dagger}B}$. In this basis, the density matrix can be rewritten as a
column vector,
$|\rho\left(t\right)\rangle=\left[\rho_{\text{vv}}\left(t\right),\rho_{++}\left(t\right),\rho_{+-}\left(t\right),\rho_{-+}\left(t\right),\rho_{--}\left(t\right),\rho_{\text{dd}}\left(t\right)\right]^{T}$.
The Lindblad operator $L$ is represented by a matrix $L_{t}$, and its action
on a density matrix reduces to a simple matrix multiplication of $L_{t}$ with
$|\rho(t)\rangle$, where $L_{t}$ reads
$L_{t}=\left[\begin{array}[c]{cccccc}-2\widetilde{\kappa}\left(t\right)&2\kappa\left(t\right)&0&0&0&0\\\
2\widetilde{\kappa}\left(t\right)&-2\kappa\left(t\right)&0&0&0&0\\\
0&0&-\left(\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)\right)&0&0&0\\\
0&0&0&-\left(\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)\right)&0&0\\\
0&0&0&0&-2\widetilde{\kappa}\left(t\right)&2\kappa\left(t\right)\\\
0&0&0&0&2\widetilde{\kappa}\left(t\right)&-2\kappa\left(t\right)\end{array}\right].$
(13)
The eigenvalues and the corresponding eigenstates of $L_{t}$ are
$\displaystyle l_{1}$ $\displaystyle=0,\text{ \ \ \ \ \ \ \ \ }$
$\displaystyle l_{2}$
$\displaystyle=-2\left[\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)\right],$
$\displaystyle l_{3}$
$\displaystyle=-\left[\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)\right],\text{
}$ $\displaystyle l_{4}$
$\displaystyle=-\left[\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)\right],\text{
}$ $\displaystyle l_{5}$
$\displaystyle=-2\left[\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)\right],$
$\displaystyle l_{6}$ $\displaystyle=0,$ (14)
and
$\displaystyle|l_{1}\rangle$
$\displaystyle=\frac{1}{{\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)}}\left[\kappa\left(t\right)|1\rangle+\widetilde{\kappa}\left(t\right)|2\rangle\right],$
$\displaystyle|l_{2}\rangle$
$\displaystyle=\frac{1}{{2}}\left[-|1\rangle+|2\rangle\right],$
$\displaystyle|l_{3}\rangle$ $\displaystyle=|3\rangle,$
$\displaystyle|l_{4}\rangle$ $\displaystyle=|4\rangle,$
$\displaystyle|l_{5}\rangle$
$\displaystyle=\frac{1}{{2}}\left[-|5\rangle+|6\rangle\right],$
$\displaystyle|l_{6}\rangle$
$\displaystyle=\frac{1}{{\kappa\left(t\right)+\widetilde{\kappa}\left(t\right)}}\left[\kappa\left(t\right)|5\rangle+\widetilde{\kappa}\left(t\right)|6\rangle\right].$
(15)
We see that all eigenvalues come in pairs, and there are always at least two
eigenvalues equal to zero. In writing (15) we have assumed that
$\kappa(t)+\widetilde{\kappa}(t)\neq 0$. The case of
$\kappa(t)+\widetilde{\kappa}(t)=0$ will be discussed below. The normalization
used for $|l_{1}\rangle$ and $l_{6}\rangle$ is convenient, as in this way
these two states can be interpreted directly as density matrices if both
$\kappa(t)$ and $\widetilde{\kappa}(t)$ are positive. If one of the rates is
negative (and the other non-zero), both states become non-positive and
therefore cease to be physical states. Moreover, since they have orthogonal
support, no linear combination of them can bring about a positive state.
Similarly, states $|l_{2}\rangle$ and $|l_{5}\rangle$, as well as any linear
combination of them, are clearly non-positive. Finally, states $|l_{3}\rangle$
and $|l_{4}\rangle$ as well as any linear combination of them are traceless
and are therefore not physical states either. Since
$|l_{2}\rangle,\ldots,|l_{5}\rangle$ are independent of $\kappa(t)$ and
$\widetilde{\kappa}(t)$ (and are therefore never physical states, regardless
of the values of $\kappa(t)$ and $\widetilde{\kappa}(t)$), the only
possibility of having a physical DF state is through $|l_{1}\rangle$ and
$|l_{6}\rangle$ with both $\kappa(t)$ and $\widetilde{\kappa}(t)$ non-
negative, or one of them vanishing (in the latter case one may always choose
the eigenvector with positive global sign).
If $\kappa\left(t\right)=0$ and $\widetilde{\kappa}\left(t\right)\neq 0$,
$|l_{1}\rangle$ and $|l_{6}\rangle$ are two pure DF states $|2\rangle$ and
$|6\rangle$, that is, the subset of
$\left\\{|+\rangle,|\text{d}\rangle\right\\}$ contains all the possible DF
states in this case. Likewise, if $\widetilde{\kappa}\left(t\right)=0$ and
$\kappa\left(t\right)\neq 0$, $|l_{1}\rangle$ and $|l_{6}\rangle$ are the two
pure DF states $|1\rangle$ and $|5\rangle$, i.e.,
$\left\\{|\text{v}\rangle,|+\rangle\right\\}$. Below, by examining an explicit
example, we will show that the vanishing of one of the decoherence rates is
physically feasible after some time $t_{s}$, when the system reaches its
steady state, see Fig. 2.
If $\kappa\left(t\right)>0$ and $\widetilde{\kappa}\left(t\right)>0$, the two
eigenstates $|l_{1}\rangle$ and $|l_{6}\rangle$ are time-dependent mixed
states. They are decoherence free as much as they are locally stationary
states due to local detailed balance. From a perspective of application for
quantum information processing, these states are less interesting. They are
the analogues of thermal equilibrium states that are stationary under a
Markovian relaxation process.
It remains to consider the case $\kappa(t)+\widetilde{\kappa}(t)=0$. All
eigenvalues vanish, but as long as $\kappa(t)\neq 0$ the eigenvectors are
still given by Eq. (15), with the only difference that the normalization of
$|l_{1}\rangle$ and $|l_{6}\rangle$ through the prefactor
$1/(\kappa(t)+\widetilde{\kappa}(t))$ has to be removed. $|l_{1}\rangle$
becomes colinear with $|l_{2}\rangle$, and $|l_{6}\rangle$ colinear with
$|l_{5}\rangle$. The dimension of the space of eigenvectors of $L_{t}$ is
reduced to four, and $L_{t}$ can therefore not be fully diagonalized. As the
linearly independent eigenvectors $|l_{1}\rangle,\ldots,|l_{4}\rangle$ are
never physical states, this means that $L_{t}$ has no eigenstates that are
physically possible, and therefore no DF states exist.
If $\kappa(t)=\widetilde{\kappa}(t)=0$ at some time (for example, if the
retarded Green function $u\left(t\right)$ took a nonzero steady value, it
would be possible that both rates vanish for $t>t_{s}$), we have $L_{t}=0$,
and the whole Hilbert space becomes a dynamically stabilized DFS. From a
quantum information perspective this would be of course an ideal situation.
Unfortunately, in the fermionic system interacting with an electron reservoir
considered here, it appears that this situation does not arise, as shown in
Fig. 2.
In summary, as long as we restrict ourselves to pure states for our non-
Markovian master equation and discard the “trivial” case $L_{t}=0$, the DFS is
still given entirely by the Markovian criterion (4), as pure DF states only
exist if exactly one of the two terms (proportional to either $\kappa(t)$ or
$\widetilde{\kappa}(t)$) in the Lindblad superoperator remains, and the logic
of the proof of necessity of condition (4) remains intact in such a situation.
However, the time dependence of $\kappa(t)$ and $\widetilde{\kappa}(t)$ brings
about a new freedom, and allows for the dynamical stabilization of DF states
through the switching off of one of the decoherence rates.
### IV.2 Physical realization of DF states
In the following, we will discuss to what extent the DF states just discussed
can be reached through the same non-Markovian dynamics described by $L_{t}$.
We consider the double quantum dot system from above coupled to an electron
reservoir with a spectral density of the Lorentz form Mac06 ; Jin08 ;
TuZhang08 ,
$J\left(\omega\right)=\frac{\Gamma
d^{2}}{\left(\omega-\epsilon_{+}\right)^{2}+d^{2}},$ (16)
where $\Gamma$ is the system-reservoir coupling strength, and $d$ the
bandwidth of the effective reservoir spectrum. In the well-known wide-band
limit, i.e., $d\rightarrow\infty$, the spectral density approximately becomes
a constant one, $J(\omega)\rightarrow\Gamma$. This corresponds to the
Markovian limit.
With the above spectral density, the solution of $u(t)$ obeying Eq. (10a) can
be obtained analytically
$u\left(t\right)=\left\\{\begin{array}[c]{c}\frac{e^{-i\epsilon_{+}t}}{2d_{\Gamma}}\left[d_{\Gamma}^{+}e^{-\frac{d_{\Gamma}^{-}t}{2}}-d_{\Gamma}^{-}e^{-\frac{d_{\Gamma}^{+}t}{2}}\right],d\neq
2\Gamma,\\\
\left[1+\frac{dt}{2}\right]e^{-\left(i\epsilon_{+}+\frac{d}{2}\right)t},\text{
\ \ \ \ \ \ \ \ \ \ }d=2\Gamma,\end{array}\right.$ (17)
where $d_{\Gamma}=\sqrt{d^{2}-2\Gamma d}$ and $d_{\Gamma}^{\pm}=d\pm
d_{\Gamma}$. Obviously, after some time $t_{s}$, $u\left(t\right)$ always
decays to zero, as shown in Fig. 1 where the different behaviors of the
amplitude of $u\left(t\right)$ corresponds to different bandwidths $d$. For
the bandwidth $d\gtrsim 2\Gamma$ (weakly non-Markovian case), $u(t)$
exponentially decays to zero, which is a result similar to the Markovian
dynamics, see the discussion in Sec. VI. When the bandwidth $d<2\Gamma$, the
non-Markovian memory effect of the reservoir becomes significant, which
induces a short-time oscillation for $\left|u\left(t\right)\right|$.
Figure 1: The exact solution of $|u(t)|$ for an electron reservoir with the
Lorentz spectral density. Here we take $\epsilon_{0}=0.2\Gamma$. Note that
$|u(t)|$ decays exponentially for large bandwidth $d$, corresponding to the
weakly non-Markovian dynamics. For a small bandwidth $d<2\Gamma$, the strong
non-Markovian memory effect brings the short-time oscillations for $|u(t)|$.
Using the solution of Eq. (17), combined with Eqs. (10b) and (9), we can
easily calculate the decoherence rates $\kappa\left(t\right)$ and
$\widetilde{\kappa}\left(t\right)$ in eq. (8). The result is plotted in Fig. 2
for the cases of weakly ($d=10\Gamma$) and strongly ($d=0.5\Gamma$) non-
Markovian dynamics. In experiments, one can adjust the external bias ($\mu$)
to raise or lower the Fermi surface of the electron reservoir. Here we display
three cases where the bias is much higher ($\mu=10\Gamma$), relatively small
($\mu=0$), and much lower ($\mu=-10\Gamma$) than the quantum dot energy level
$\epsilon_{0}$. First, we see that when $d=0.5\Gamma$, as shown in Fig. 2 (c)
and (d), $\kappa\left(t\right)$ and $\widetilde{\kappa}\left(t\right)$ can
jump from a positive value to a negative value repeatedly during the evolution
process, which corresponds to the forth- and back-flows of the information
between the system and the environment, in evidence of the strong non-
Markovian dynamics.
We observe that, no matter what the values of the width $d$ and the bias $\mu$
are, the two decoherence rates $\kappa(t)$ and $\widetilde{\kappa}(t)$ never
satisfy $\kappa(t)+\widetilde{\kappa}(t)=0$. However, if we apply a large bias
to raise (or lower) the Fermi surface of the electron reservoir much higher
(or much lower) than the dot level, one of the two decoherence rates is
switched off after a time scale of a few $1/\Gamma$, which implies the
existence of a DFS.
As shown in Fig. 2 (a) and (b), in the weakly non-Markovian case, when
applying a positive bias $\mu=10\Gamma$, the decoherence rate
$\kappa\left(t\right)$ shows a positive peak in the beginning and then decays
rapidly to a zero steady value on a time scale of a few $1/\Gamma$, while
$\widetilde{\kappa}\left(t\right)$ turns negative first, and then climbs to a
nonzero steady value. For the strongly non-Markovian regime ($d=0.5\Gamma$),
the same situation happens, see the red curves in Fig. 2 (c) and (d). The
decoherence rate reaches $\kappa(t)=0$ on a time scale $t_{s}$ of a few
$1/\Gamma$, while $\widetilde{\kappa}(t)$ keeps jumping from positive values
to negative values repetitively. In this case, the dynamics for $t>t_{s}$ is
described by the following master equation
$\displaystyle~{}~{}~{}~{}~{}\dot{\rho}\left(t\right)=-i\left[\widetilde{H}_{S}\left(t\right),\rho\left(t\right)\right]+L\left[\rho\left(t\right)\right],$
(18a)
$\displaystyle~{}~{}\widetilde{H}_{S}\left(t\right)=\widetilde{\epsilon}_{+}\left(t\right)A_{+}^{\dagger}A_{+}+\epsilon_{-}A_{-}^{\dagger}A_{-},$
(18b) $\displaystyle
L\left[\rho\left(t\right)\right]=\widetilde{\kappa}\left(t\right)\left[2A_{+}^{\dagger}\rho\left(t\right)A_{+}-A_{+}A_{+}^{\dagger}\rho\left(t\right)-\rho\left(t\right)A_{+}A_{+}^{\dagger}\right].$
(18c)
Then the states $|+\rangle$ and $|\mathsf{d}\rangle$ become possible
dynamically stabilized DF states, since $A_{+}^{{\dagger}}|+\rangle=0$ and
also $A_{+}^{{\dagger}}|\mathsf{d}\rangle=0$ leads to the vanishing of the
nonunitary part $L[\rho(t)]$ after $t>t_{s}$. Therefore, the Markovian DFS
criterion of Eq. (4) still is both necessary and sufficient for a time-local
DFS when only one decoherence rate is turned on.
Figure 2: The exact solutions of decoherence rates $\kappa(t)$ and
$\widetilde{\kappa}(t)$ for different external bias voltage $\mu=eV$. (a)-(b)
for the weakly non-Markovian dynamics (with $d=10\Gamma$); (c)-(d) for the
strong non-Markovian dynamics (with $d=0.5\Gamma$). Here we set
$\epsilon_{0}=0.2\Gamma$, ${k_{B}}T=0.3\Gamma$. We note that generally,
$\kappa(t)$ and $\widetilde{\kappa}(t)$ are non-zero during the time
evolution. However, for a large positive (or negative) bias voltage, one of
them is switched off after some specific time.
On the other hand, if we apply a negative bias to the electron reservoir, e.g.
$\mu=-10\Gamma$, as shown by the black curves in Fig. 2, we find that for both
weakly and strongly non-Markovian dynamics, the decoherence rate
$\widetilde{\kappa}(t)$ goes to zero very quickly, while $\kappa(t)$ either
reaches a nonzero steady value (in the weakly non-Markovian regime), or keeps
jumping from a positive value to a negative value for all times (in the
strongly non-Markovian regime). Then the master equation for $t>t_{s}$ is
effectively given by
$\displaystyle~{}~{}~{}~{}~{}\dot{\rho}\left(t\right)=-i\left[\widetilde{H}_{S}\left(t\right),\rho\left(t\right)\right]+L\left[\rho\left(t\right)\right],$
(19a)
$\displaystyle~{}~{}\widetilde{H}_{S}\left(t\right)=\widetilde{\epsilon}_{+}\left(t\right)A_{+}^{\dagger}A_{+}+\epsilon_{-}A_{-}^{\dagger}A_{-},$
(19b) $\displaystyle
L\left[\rho\left(t\right)\right]=\kappa\left(t\right)\left[2A_{+}\rho\left(t\right)A_{+}^{\dagger}-A_{+}^{\dagger}A_{+}\rho\left(t\right)-\rho\left(t\right)A_{+}^{\dagger}A_{+}\right].$
(19c)
Again, the DFS criterion Eq. (4) is then both necessary and sufficient for DF
states. Since $A_{+}|-\rangle=0$ and $A_{+}|\mathsf{v}\rangle=0$, then we
obtain that $|-\rangle$ and $|\mathsf{v}\rangle$ are possible dynamically
stabilized DF states.
In summary, the above results show that the vanishing of one of the
decoherence rates implies that $\left\\{|+\rangle,|\mathsf{d}\rangle\right\\}$
of $\left\\{|-\rangle,|\mathsf{v}\rangle\right\\}$ become dynamically
stabilized DF states.
## V Physical realization for dynamically stabilized DF states
In the following, by examining the general exact solution of master Eq. (8),
we will prove that all pure dynamically stabilized DF states
$\\{|+\rangle,|\mathsf{d}\rangle\\}$ and $\\{|-\rangle,|\mathsf{v}\rangle\\}$
predicted above are physically realizable through the decoherence process
given by the same master equation. We also give the exact conditions for
generating these DF states in terms of the initial state, and a condition on
the Green’s functions $u(t)$ and $v(t)$.
First, by solving (8), we can exactly give the elements of
$\rho\left(t\right)$ in Eq. (12) in terms of the initial
$\rho\left(t_{0}\right)$ and the functions of $u\left(t\right)$ and
$v\left(t\right)$ as
$\displaystyle\rho_{\mathsf{vv}}\left(t\right)=[1-v\left(t\right)]\rho_{\mathsf{vv}}\left(t_{0}\right)+[1-v\left(t\right)-\left|u\left(t\right)\right|^{2}]\rho_{++}\left(t_{0}\right),$
$\displaystyle\rho_{++}\left(t\right)=v\left(t\right)\rho_{\mathsf{vv}}\left(t_{0}\right)+[v\left(t\right)+\left|u\left(t\right)\right|^{2}]\rho_{++}\left(t_{0}\right),$
$\displaystyle\rho_{+-}\left(t\right)=u\left(t\right)e^{i\epsilon_{-}t}\rho_{+-}\left(t_{0}\right),$
$\displaystyle\rho_{--}\left(t\right)=[1-v\left(t\right)]\rho_{--}\left(t_{0}\right)+[1-v\left(t\right)-\left|u\left(t\right)\right|^{2}]\rho_{\mathsf{dd}}\left(t_{0}\right),$
$\displaystyle\rho_{\mathsf{dd}}\left(t\right)=v\left(t\right)\rho_{--}\left(t_{0}\right)+[v\left(t\right)+\left|u\left(t\right)\right|^{2}]\rho_{\mathsf{dd}}\left(t_{0}\right).$
(20)
This expression for $\rho\left(t\right)$ is valid for an arbitrary spectral
density of the reservoir. This solution confirms the fact that
$\mathcal{H}_{+}=\left\\{|\mathsf{v}\rangle,|+\rangle\right\\}$ and
$\mathcal{H}_{-}=\left\\{|-\rangle,|\mathsf{d}\rangle\right\\}$ are two
independent closed subspaces. For any initial state in the subspace
$\mathcal{H}_{+}$ $\left(\mathcal{H}_{-}\right)$, the system will be
dynamically stabilized in this subspace.
The general solution (20) shows that if $v\left(t\right)=1$ is satisfied when
$t>t_{s}$, the initial vacuum state $|\mathsf{v}\rangle$ converges to the
stabilized state $|+\rangle$, while if
$v\left(t\right)+\left|u\left(t\right)\right|^{2}=1$ is reached when
$t>t_{s}$, the initial singly occupied state $|+\rangle$ converges to the
dynamically stabilized DF state $|+\rangle$. That is, the same dynamically
stabilized DF state $|+\rangle$ can be generated from different initial states
in the same subspace of $\mathcal{H}_{+}$ under different stabilization
conditions for $u(t)$ and $v(t)$. Similarly, one can obtain the remaining
dynamically stabilized DF states $|\mathsf{v}\rangle$, $|-\rangle$ and
$|\mathsf{d}\rangle$ from different initial states under different
stabilization condition, as shown in Table 1.
Table 1: The dynamically stabilized DF states for different choices of the initial state of the system, plus the different stabilization conditions. initial state | stabilization condition | DF state
---|---|---
$|\mathsf{v}\rangle$ | $v\left(t\right)=0$ (or $1$) | $|\mathsf{v}\rangle$ (or $|+\rangle$)
$|+\rangle$ | $v\left(t\right)+\left|u\left(t\right)\right|^{2}=0$ (or $1$) | $|\mathsf{v}\rangle$ (or $|+\rangle$)
$|-\rangle$ | $v\left(t\right)=0$ (or $1$) | $|-\rangle$ (or $|$d$\rangle$)
$|\mathsf{d}\rangle$ | $v\left(t\right)+\left|u\left(t\right)\right|^{2}=0$ (or $1$) | $|-\rangle$ (or $|$d$\rangle$)
Besides the initial states listed in Table 1, another more general initial
pure state is a superposition of $|+\rangle$ and $|-\rangle$, namely
$|\Phi\rangle=\alpha|+\rangle+\beta|-\rangle$ with
$|\alpha|^{2}+|\beta|^{2}=1$. In this case, as one can easily check from Eq.
(20), the resulting state becomes
$\displaystyle\rho_{\mathsf{vv}}\left(t\right)=[1-v\left(t\right)-\left|u\left(t\right)\right|^{2}]|\alpha|^{2},$
$\displaystyle\rho_{++}\left(t\right)=[v\left(t\right)+\left|u\left(t\right)\right|^{2}]|\alpha|^{2},$
$\displaystyle\rho_{+-}\left(t\right)=u\left(t\right)e^{i\epsilon_{-}t}\
\alpha\beta^{*},$
$\displaystyle\rho_{--}\left(t\right)=\left[1-v\left(t\right)\right]|\beta|^{2},$
$\displaystyle\rho_{\mathsf{dd}}\left(t\right)=v\left(t\right)|\beta|^{2}.$
(21)
In the case that both $\alpha$ and $\beta$ are nonzero, one has to have
$v(t)=0$ and $|u(t)|^{2}=1$ to generate a stabilized (pure) DF state
$|\Phi\rangle$. This is impossible unless the system is totally decoupled from
the environment from the beginning. In other words, except for a stabilized
mixed state, no stabilized DF state can be obtained. As a conclusion, Table 1
lists all the possible pure stabilized DF states in this system. The present
results confirm the statement in the last section that the only possible pure
dynamically stabilized DF states are $|v\rangle,|\pm\rangle$ and $|d\rangle$.
Figure 3: The exact solution of $v(t)$ and $v(t)+|u(t)|^{2}$ for an electron
reservoir with varying the external bias voltage $\mu=eV$. (a)-(b) for the
weakly non-Markovian dynamics (with $d=10\Gamma$); (c)-(d) for the strongly
non-Markovian dynamics (with $d=0.5\Gamma$ ). Here we set
$\epsilon_{0}=0.2\Gamma$, ${K_{B}}T=0.3\Gamma$. By applying a large positive
(or negative) bias voltage, the steady value of $v(t)$ and $v(t)+|u(t)|^{2}$
will approach 1 (or 0).
For the Lorentz spectral density, we display the functions of $v(t)$ and
$v(t)+|u(t)|^{2}$ in Fig. 3. Interestingly, by comparing Fig. 3 (a)-(b) and
(c)-(d), we find that in the strongly non-Markovian case ($d=0.5\Gamma$),
applying a relatively small bias voltage (for example
$\left|\mu\right|=2\Gamma$) makes the steady values of $v\left(t\right)$ and
$v\left(t\right)+\left|u\left(t\right)\right|^{2}$ quickly approach $1$ or
$0$, whereas this is not the case in the weakly non-Markovian regime
($d=10\Gamma$), where $v(t)\to 0.9$ for $\mu=2\Gamma$ (and $v(t)\to 0.1$ for
$\mu=-2\Gamma$). This indicates that the back-flow of information from the
reservoir, due to a strongly non-Markovian memory of the reservoir, helps the
stabilization of the system in the states of
$\\{|+\rangle,|\mathsf{d}\rangle\\}$ or $\\{|\mathsf{v}\rangle,|-\rangle\\}$.
This facilitates the physical realization of the dynamically stabilized DF
states.
Furthermore, we can now prove that under the stabilization conditions listed
in Table 1, one of the decoherence rates vanishes when $t>t_{s}$. In fact,
$\kappa(t)$ and $\widetilde{\kappa}(t)$ in Eq. (9) can be further simplified
as
$\displaystyle\kappa\left(t\right)$
$\displaystyle=\frac{\left|u\left(t\right)\right|^{2}}{2}\frac{d}{dt}\frac{1-v\left(t\right)}{\left|u\left(t\right)\right|^{2}},$
(22a) $\displaystyle\widetilde{\kappa}\left(t\right)$
$\displaystyle=\frac{\left|u\left(t\right)\right|^{2}}{2}\frac{d}{dt}\frac{v\left(t\right)}{\left|u\left(t\right)\right|^{2}}.$
(22b)
It clearly shows that the condition $v(t)=1$ (or ($v(t)+|u(t)|^{2}=1$) implies
$\kappa(t)=0$, and the condition $v(t)=0$ (or $v(t)+|u(t)|^{2}=0$) indicates
$\widetilde{\kappa}(t)=0$, namely, one decoherence rate is turned off for
$t>t_{s}$. In this case, the Markovian DFS criterion still provides a
necessary and sufficient condition for generating dynamically stabilized DF
states. However, for the non-Markovian dynamics, we have shown in Table 1 that
which final state is realized depends both on the initial state and the
details of the dynamics given by the Green functions $u(t)$ and $v(t)$.
## VI Comparison to the Born-Markov dynamics
In the previous Sections, using the exact master equation, we have shown the
generation of DF states for a system coupled to a non-Markovian reservoir. For
comparison, in the following, we discuss the corresponding results in terms of
the BM master equation.
The BM dynamics usually corresponds to the case where the coupling strength
between the system and the electron reservoir is very weak, and the
characteristic correlation time of the electron reservoir is sufficiently
shorter than that of the system. In such a case, the electron reservoir has no
memory effect on the evolution of the system. Then the solution of
$u\left(t\right)$, $v\left(t\right)$ are reduced to TuZhang08 ; TuZhang09 ;
JinZhang10
$\displaystyle u_{\text{BM}}\left(t\right)$
$\displaystyle=e^{-i\widetilde{\epsilon}_{+}t-\frac{1}{2}J\left(\epsilon_{+}\right)t},$
$\displaystyle v_{\text{BM}}\left(t\right)$
$\displaystyle=\big{[}1-e^{-J\left(\epsilon_{+}\right)t}\big{]}f\left(\epsilon_{+}\right),$
(23)
where
$\widetilde{\epsilon}_{+}=\epsilon_{+}+\left(\delta\epsilon_{+}\right)_{\text{BM}}$
with the energy shift
$\left(\delta\epsilon_{+}\right)_{\text{BM}}=\mathcal{P}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\frac{\Gamma\left(\omega\right)}{\omega-\epsilon_{+}}$.
Note that for the wide-band limit, we simply have
$\widetilde{\epsilon}_{+}=\epsilon_{+}$ and $J(\epsilon_{+})=\Gamma$.
Substituting Eq. (23) into Eq. (9), we obtain the constant rates
$\displaystyle\kappa\left(t\right)$
$\displaystyle=\frac{1}{2}J\left(\epsilon_{+}\right)\left[1-f\left(\epsilon_{+}\right)\right],$
$\displaystyle\widetilde{\kappa}\left(t\right)$
$\displaystyle=\frac{1}{2}J\left(\epsilon_{+}\right)f\left(\epsilon_{+}\right).$
(24)
Here $f\left(\epsilon_{+}\right)$ is the fermion distribution function of the
electron reservoir at the frequency $\epsilon_{+}$, i.e.,
$f\left(\epsilon_{+}\right)=\frac{1}{e^{\left(\epsilon_{+}-\mu\right)/K_{B}T}+1}.$
(25)
Thus the exact master equation is reduced to the BM master equation, where the
decoherence rates are time-independent. This gives the standard Lindblad form
for the Markovian dynamics. Based on the general DFS criterion of Eq. (4), we
see that there is in general no DFS for this system in the BM limit.
However, if we apply a large positive bias $\mu=eV$ such that,
$\left(\mu-\epsilon_{+}\right)/k_{B}T\gg 1$, then
$f\left(\epsilon_{+}\right)\rightarrow 1$, which leads to $\kappa\rightarrow
0$. $\widetilde{H}_{S}$ leaves the states $|+\rangle$ and $|$d$\rangle$
invariant during the time evolution. The relation $A_{+}^{\dagger}|+\rangle=0$
(or $A_{+}^{\dagger}|$d$\rangle=0$) guarantees
$L\left[\rho\left(t\right)\right]=0$ for these states. The states
$\left\\{|+\rangle,|\text{d}\rangle\right\\}$ are therefore DF states in the
BM limit under large positive bias. Likewise, applying a negative large bias
to the electron reservoir such that $\left(\epsilon_{+}-\mu\right)/K_{B}T\gg
1$, then $f\left(\epsilon_{+}\right)\rightarrow 0$ and
$\widetilde{\kappa}(t)\rightarrow 0$. In this case, the states
$\left\\{|\text{v}\rangle,|-\rangle\right\\}$ are the DF states in the BM
limit.
In conclusion, the states $\left\\{|+\rangle,|\text{d}\rangle\right\\}$ and
$\left\\{|\text{v}\rangle,|-\rangle\right\\}$ are also DF in the BM limit if
one of the decoherence rates is switched off by properly tuning the bias
voltage on the electron reservoir. This result is consistent with the result
in the non-Markovian case discussed above. The apparent difference is that the
DF states in the BM limit seem to exist without the dynamical stabilization
processes. However, this difference is not crucial in reality. As is well-
known Carmichael93 ; Breuer07 , the BM master equation with the constant
decoherence rates, Eq. (24), is derived under the condition $t\gg\tau_{r}$
where $\tau_{r}$ is the characteristic time of the reservoir Carmichael93 . In
other words, a stabilization time scale $t_{s}$ has implicitly been used in
deriving the BM master equation, such that the decoherence rates become time-
independent for $t>t_{s}\gg\tau_{r}$ . Therefore, the concept of the
dynamically stabilized DF states, based on the exact non-Markovian master
equation, gives the generalized picture of DFS. It contains the Markovian DFS
as a special case.
## VII Conclusion
In summary, we have investigated the DFS of a non-Markovian fermionic open
systems, based on an exact non-Markovian master equation developed recently.
The master equation has a nonunitary term of the standard Lindblad form, but
the corresponding decoherence rates are time-dependent and local in time. They
are determined microscopically and nonperturbatively from the Schwinger-
Keldysh nonequilibrium Green functions and fully account for the non-Markovian
memory effect.
As concrete example we have studied a fermionic system with two degenerate
energy levels coupled identically to a fermionic reservoir. We find that the
whole Hilbert space is split into two closed subspaces. For any initial state
in one of the subspaces, the system will remain in this subspace forever. By
diagonalizing the full Lindblad operator we found that physical DF states
exist if and only if one of the two relevant decoherence rates switches itself
off dynamically. Such a situation can be achieved physically. Two of the DF
states are coherent superpositions with an arbitrary relative phase between
the two original energy levels, which may be of physical interest for quantum
computation.
Which DF state is reached as result of the dissipative dynamics depends both
on the initial state and the details of the dynamics, as expressed by the
time-dependent non-equilibrium Green’s function. We show this explicitly by
solving exactly the non-Markovian master equation. Interestingly, we find that
the strongly non-Markovian memory can help to stabilize the DF states compared
to the Markovian case.
###### Acknowledgements.
This work is supported by the National Science Council of ROC under Contract
No. NSC-99-2112-M-006-008-MY3, the National Center for Theoretical Science of
Taiwan.
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|
arxiv-papers
| 2012-07-06T07:25:43 |
2024-09-04T02:49:32.712574
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Heng-Na Xiong, Wei-Min Zhang, Matisse Wei-Yuan Tu, Daniel Braun",
"submitter": "Heng-Na Xiong",
"url": "https://arxiv.org/abs/1207.1538"
}
|
1207.1543
|
# A Note on Inextensible Flows of Curves in $E^{n}$
Önder Gökmen Yıldız, Murat Tosun, Sıddıka Özkaldı Karakuş 1 Bilecik
University, Faculty of Sciences and Arts, Department of Mathematics, 11210,
Bilecik, Turkey. ogokmen.yildiz@bilecik.edu.tr siddika.karakus@bilecik.edu.tr
2 Sakarya, Faculty of Sciences and Arts, Department of Mathematics, Sakarya,
Turkey tosun@sakarya.edu.tr
###### Abstract.
In this paper, we investigate the general formulation for inextensible flows
of curves in $E^{n}$. The necessary and sufficient conditions for inextensible
curve flow are expressed as a partial differential equation involving the
curvatures.
###### Key words and phrases:
Curvature flows, inextensible, Euclidean n-space.
2000 Mathematics Subject Classification: 53C44, 53a04, 53A05, 53A35.
## 1\. Introduction
It is well known that many nonlinear phenomena in physics, chemistry and
biology are described by dynamics of shapes, such as curves and surfaces. The
evolution of curve and surface has significant applications in computer vision
and image processing. The time evolution of a curve or surface generated by
its corresponding flow in -for this reason we shall also refer to curve and
surface evolutions as flows throughout this article- is said to be
inextensible if, in the former case, its arclength is preserved, and in the
latter case, if its intrinsic curvature is preserved [8]. Physically,
inextensible curve flows give rise to motions in which no strain energy is
induced. The swinging motion of a cord of fixed length, for example, or of a
piece of paper carried by the wind, can be described by inextensible curve and
surface flows. Such motions arise quite naturally in a wide range of a
physical applications. For example, both Chirikjian and Burdick [1] and
Mochiyama et al. [10] study the shape control of hyper-redundant, or snake-
like robots.
Inextinsible curve and surface flows also arise in the context of many
problems in computer vision [7][9] and computer animation [2], and even
structural mechanics [11].
There have been a lot of studies in the literature on plane curve flows,
particularly on evolving curves in the direction of their curvature vector
field (referred to by various names such as “curve shortening”, flow by
curvature” and “heat flow”). Particularly relevant to this paper are the
methods developed by Gage and Hamilton [3] and Grayson [5] for studying the
shrinking of closed plane curves to circle via heat equation.
In this paper, we develop the general formulation for inextensible flows of
curves in $E^{n\text{ }}$. Necessary and sufficient conditions for an
inextensible curve flow are expressed as a partial differential equation
involving the curvatures.
## 2\. Preliminary
To meet the requirements in the next sections, the basic elements of the
theory of curves in the Euclidean n-space $E^{n}$ are briefly presented in
this section (A more complete elementary treatment can be found in [4][6]).
Let $\alpha:I\subset R\mathbb{\longrightarrow}E^{n\text{ }}$be an arbitrary
curve in $E^{n\text{ }}$ Recall that the curve $\alpha$ is said to be a unit
speed curve (or parameterized by arclength functions) if
$\left\langle\alpha^{\prime}(s),\alpha^{\prime}(s)\right\rangle=1$, where
$\left\langle.,.\right\rangle$ denotes the standard inner product of given by
$\left\langle X,Y\right\rangle={\displaystyle\sum\limits_{i=1}^{n}}x_{i}y_{i}$
for each
$X=\left(x_{1},x_{2},...,x_{n}\right),Y=\left(y_{1},y_{2},...,y_{n}\right)\in
R^{n}$. In particular, norm of a vector $X\in R^{n}$ is given by
$\left\|X\right\|=\sqrt{\left\langle X,Y\right\rangle}$. Let
$\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the moving Frenet frame along the
unit speed curve $\alpha$, where $V_{i}\left(i=1,2,...,n\right)$ denotes the
$i^{th}$ Frenet vector field. Then Frenet formulas are given by
$\left[\begin{array}[c]{c}V_{1}^{\prime}\\\ V_{2}^{\prime}\\\
V_{3}^{\prime}\\\ \vdots\\\ V_{n-2}^{\prime}\\\ V_{n-1}^{\prime}\\\
V_{n}^{\prime}\end{array}\right]=\left[\begin{array}[c]{cccccccc}0&k_{1}&0&0&\cdots&0&0&0\\\
-k_{1}&0&k_{2}&0&\cdots&0&0&0\\\ 0&-k_{2}&0&k_{3}&\cdots&0&0&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&0&0&0&\cdots&0&k_{n-2}&0\\\ 0&0&0&0&\cdots&-k_{n-2}&0&k_{n-1}\\\
0&0&0&0&\cdots&0&-k_{n-1}&0\end{array}\right]\left[\begin{array}[c]{c}V_{1}\\\
V_{2}\\\ V_{3}\\\ \vdots\\\ V_{n-2}\\\ V_{n-1}\\\ V_{n}\end{array}\right]$
where $k_{i}\left(i=1,2,...,n\right)$ denotes the $i^{th}$ curvature function
of the curve [4][6]. If all of the curvatures $k_{i}\left(i=1,2,...,n\right)$
of the curve vanish nowhere in $I\subset R$, it is called a non-degenerate
curve.
## 3\. Inextinsible Flows of Curve in $E^{n}$
Throughout this paper, we suppose that
$\alpha:\left[0,l\right]\times\left[0,w\right)\mathbb{\longrightarrow}E^{n\text{
}}$
is a one parameter family of smooth curves in $E^{n\text{ }}$, where $l$ is
the arclength of the initial curve. Let $u$ be the curve parameterization
variable, $0\leq u\leq l$. If the speed curve $\alpha$ is denoted by
$v=\left\|\frac{d\alpha}{du}\right\|$ then the arclength of $\alpha$ is
$S(u)={\displaystyle\int\limits_{0}^{u}}\left\|\frac{\partial\alpha}{\partial
u}\right\|du={\displaystyle\int\limits_{0}^{u}}vdu.$
The operator $\frac{\partial}{\partial s}$ is given with respect to $u$ by
$\frac{\partial}{\partial s}=\frac{1}{v}\frac{\partial}{\partial u}.$ (3.1)
Thus, the arclength is $ds=vdu$.
###### Definition 3.1.
Any flow of the curve can be expressed following form:
$\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$
where $f_{i}$ denotes the $i^{th}$ scalar speed of the curve Let the arclength
variation be
$S(u,t)={\displaystyle\int\limits_{0}^{u}}vdu.$
In the Euclidean space the requirement that the curve not be subject to any
elongation or compression can be expressed by the condition
$\frac{\partial}{\partial
t}S(u,t)={\displaystyle\int\limits_{0}^{u}}\frac{\partial v}{\partial
t}du=0,\text{ \ }u\in\left[0,l\right].$ (3.2)
###### Definition 3.2.
A curve evolution $\alpha(u,t)$ and its flow $\frac{\partial\alpha}{\partial
t}$ are said to be inextensible if
$\frac{\partial}{\partial t}\left\|\frac{\partial\alpha}{\partial
u}\right\|=0.$
Now, we research the necessary and sufficient condition for inelastic curve
flow. For this reason, we need to the following Lemma.
###### Lemma 3.3.
Let $\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$ be a smooth flow of the
curve $\alpha$. The flow is inextensible if and only if
$\frac{\partial v}{\partial t}=\frac{\partial f_{1}}{\partial u}-f_{2}vk_{1}.$
(3.3)
###### Proof.
Since $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial t}$ commute
and $v^{2}=\left\langle\frac{\partial\alpha}{\partial
u},\frac{\partial\alpha}{\partial u}\right\rangle,$ we have
$\displaystyle 2v\frac{\partial v}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
t}\left\langle\frac{\partial\alpha}{\partial u},\frac{\partial\alpha}{\partial
u}\right\rangle$ $\displaystyle=2\left\langle\frac{\partial\alpha}{\partial
u},\frac{\partial}{\partial
u}\left({\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}\right)\right\rangle$
$\displaystyle=2\left\langle
vV_{1},{\displaystyle\sum\limits_{i=1}^{n}}\frac{\partial f_{i}}{\partial
u}V_{i}+{\displaystyle\sum\limits_{i=1}^{n}}f_{i}\frac{\partial
V_{i}}{\partial u}\right\rangle$ $\displaystyle=2\left\langle
vV_{1},\frac{\partial f_{1}}{\partial u}V_{1}+f_{1}\frac{\partial
V_{1}}{\partial u}+...+\frac{\partial f_{n}}{\partial
u}V_{n}+f_{n}\frac{\partial V_{n}}{\partial u}\right\rangle$
$\displaystyle=2\left\langle vV_{1},\frac{\partial f_{1}}{\partial
u}V_{1}+f_{1}vk_{1}V_{2}+...+\frac{\partial f_{n}}{\partial
u}V_{n}-f_{n}vk_{n-1}V_{n-1}\right\rangle$
$\displaystyle=2\left(\frac{\partial f_{1}}{\partial u}-f_{2}vk_{1}\right).$
Thus, we reach
$\frac{\partial v}{\partial t}=\frac{\partial f_{1}}{\partial u}-f_{2}vk_{1}.$
∎
###### Theorem 3.4.
Let $\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the moving Frenet frame of the
curve $\alpha$ and $\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$ be a differentiable flow of
$\alpha$ in $E^{n\text{ }}$.Then the flow is inextensible if and only if
$\frac{\partial f_{1}}{\partial s}=f_{2}k_{1}.$ (3.4)
###### Proof.
Suppose that the curve flow is inextensible. From equations (3.2) and (3.3)
for $u\in\left[0,l\right]$, we see that
$\frac{\partial}{\partial
t}S(u,t)={\displaystyle\int\limits_{0}^{u}}\frac{\partial v}{\partial
t}du={\displaystyle\int\limits_{0}^{u}}\left(\frac{\partial f_{1}}{\partial
u}-f_{2}vk_{1}\right)du=0.$
Thus, it can be see that
$\frac{\partial f_{1}}{\partial u}-f_{2}vk_{1}=0.$
Considering the last equation and (3.1), we reach
$\frac{\partial f_{1}}{\partial s}=f_{2}k_{1}.$
Conversely, following similar way as above, the proof is completed.
Now, we restrict ourselves to arclength parameterized curves. That is, $v=1$
and the local coordinate $u$ corresponds to the curve arclength $s$. We
require the following Lemma
###### Lemma 3.5.
Let $\left\\{V_{1},V_{2},...,V_{n}\right\\}$ be the moving Frenet frame of the
curve $\alpha$. Then, the differentions of
$\left\\{V_{1},V_{2},...,V_{n}\right\\}$ with respect to $t$ is
$\displaystyle\frac{\partial V_{1}}{\partial t}$
$\displaystyle=\left[{\displaystyle\sum\limits_{i=2}^{n-1}}\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-f_{i+1}k_{i}\right)V_{i}\right]+\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial s}\right)V_{n},$ $\displaystyle\frac{\partial V_{j}}{\partial
t}$ $\displaystyle=-\left(f_{j-1}k_{j-1}+\frac{\partial f_{j}}{\partial
s}-f_{j+1}k_{j}\right)V_{1}+\left[{\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\
k\neq i\end{subarray}}^{n}}\Psi_{kj}V_{k}\right],\text{ \ }1<j<n,$
$\displaystyle\frac{\partial V_{n}}{\partial t}$
$\displaystyle=-\left(f_{n-1}k_{n-1}+\frac{\partial f_{n}}{\partial
s}\right)V_{1}+\left[{\displaystyle\sum\limits_{k=2}^{n-1}}\Psi_{kn}V_{k}\right],$
where $\Psi_{kj}=\left\langle\frac{\partial V_{j}}{\partial
t},V_{k}\right\rangle$ and $\Psi_{kn}=\left\langle\frac{\partial
V_{n}}{\partial t},V_{k}\right\rangle$.
###### Proof.
For $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial s}$ commute,
it seen that
$\displaystyle\frac{\partial V_{1}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial t}\left(\frac{\partial\alpha}{\partial
s}\right)=\frac{\partial}{\partial s}\left(\frac{\partial\alpha}{\partial
t}\right)=\frac{\partial}{\partial
s}\left({\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}\right)={\displaystyle\sum\limits_{i=1}^{n}}\frac{\partial
f_{i}}{\partial
s}V_{i}+{\displaystyle\sum\limits_{i=1}^{n}}f_{i}\frac{\partial
V_{i}}{\partial s}$ $\displaystyle=\frac{\partial f_{1}}{\partial
s}V_{1}+f_{1}\frac{\partial V_{1}}{\partial s}+\frac{\partial f_{2}}{\partial
s}V_{2}+f_{2}\frac{\partial V_{2}}{\partial s}+...+\frac{\partial
f_{n}}{\partial s}V_{n}+f_{n}\frac{\partial V_{n}}{\partial s}$
$\displaystyle=\frac{\partial f_{1}}{\partial
s}V_{1}+f_{1}k_{1}V_{2}+\frac{\partial f_{2}}{\partial
s}V_{2}+f_{2}\left(-k_{1}V_{1}+k_{2}V_{3}\right)+...+\frac{\partial
f_{n}}{\partial s}V_{n}-f_{n}k_{n-1}V_{n-1}.$
Substituting the equation (3.4) into the last equation and using Theorem 3.4.,
we have
$\frac{\partial V_{1}}{\partial
t}=\left[{\displaystyle\sum\limits_{i=2}^{n-1}}\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-f_{i+1}k_{i}\right)V_{i}\right]+\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial s}\right)V_{n}.$
Now, let us differentiate the Frenet frame with respect to $t$ for $1<j<n$ as
follows;
$\displaystyle 0$ $\displaystyle=\frac{\partial}{\partial t}\left\langle
V_{1},V_{j}\right\rangle=\left\langle\frac{\partial V_{1}}{\partial
t},V_{j}\right\rangle+\left\langle V_{1},\frac{\partial V_{j}}{\partial
t}\right\rangle$ $\displaystyle=\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial s}-f_{i+1}k_{i}\right)+\left\langle V_{1},\frac{\partial
V_{j}}{\partial t}\right\rangle.$ (3.5)
From (3.5), we have obtain
$\frac{\partial V_{j}}{\partial t}=-\left(f_{j-1}k_{j-1}+\frac{\partial
f_{j}}{\partial
s}-f_{j+1}k_{j}\right)V_{1}+\left[{\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\
k\neq j\end{subarray}}^{n}}\Psi_{kj}V_{k}\right].$
Lastly, considering $\left\langle V_{1},V_{n}\right\rangle=0$ and following
similar way as above, we reach
$\frac{\partial V_{n}}{\partial t}=-\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial
s}\right)V_{1}+\left[{\displaystyle\sum\limits_{k=2}^{n-1}}\Psi_{kn}V_{k}\right].$
###### Theorem 3.6.
Suppose that the curve flow $\frac{\partial\alpha}{\partial
t}={\displaystyle\sum\limits_{i=1}^{n}}f_{i}V_{i}$ is inextensible. Then the
following system of partial differential equations holds:
$\displaystyle\frac{\partial k_{1}}{\partial t}$
$\displaystyle=f_{2}k_{1}^{2}+f_{1}\frac{\partial k_{1}}{\partial
s}+\frac{\partial^{2}f_{2}}{\partial s^{2}}-2\frac{\partial f_{2}}{\partial
s}k_{2}-f_{3}\frac{\partial k_{2}}{\partial s}-f_{2}k_{2}^{2}-f_{4}k_{3}k_{2}$
$\displaystyle\frac{\partial k_{i-1}}{\partial t}$
$\displaystyle=-\frac{\partial\Psi_{(i-1)i}}{\partial s}-\Psi_{(i-2)i}k_{i-2}$
$\displaystyle\frac{\partial k_{i}}{\partial t}$
$\displaystyle=\frac{\partial\Psi_{(i-1)i}}{\partial s}-\Psi_{(i+2)i}k_{i+2}$
$\displaystyle\frac{\partial k_{n-1}}{\partial t}$
$\displaystyle=-\frac{\partial\Psi_{(n-1)n}}{\partial
s}-\Psi_{(n-2)n}k_{n-2}.$
∎
∎
###### Proof.
Since $\frac{\partial}{\partial s}\frac{\partial V_{1}}{\partial
t}=\frac{\partial}{\partial t}\frac{\partial V_{1}}{\partial s}$, we get
$\displaystyle\frac{\partial}{\partial s}\frac{\partial V_{1}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
s}\left[{\displaystyle\sum\limits_{i=2}^{n-1}}\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial
s}-f_{i+1}k_{i}\right)V_{i}+\left(f_{n-1}k_{n-1}+\frac{\partial
f_{n}}{\partial s}\right)V_{n}\right]$
$\displaystyle={\displaystyle\sum\limits_{i=2}^{n-1}}\left[\left(\frac{\partial
f_{i-1}}{\partial s}k_{i-1}+f_{i-1}\frac{\partial k_{i-1}}{\partial
s}+\frac{\partial^{2}f_{i}}{\partial s^{2}}-\frac{\partial f_{i+1}}{\partial
s}k_{i}-f_{i+1}\frac{\partial k_{i}}{\partial s}\right)V_{i}\right]$
$\displaystyle+{\displaystyle\sum\limits_{i=2}^{n-1}}\left[\left(f_{i-1}k_{i-1}+\frac{\partial
f_{i}}{\partial s}-f_{i+1}k_{i}\right)\frac{\partial V_{i}}{\partial
s}\right]$ $\displaystyle+\left(\frac{\partial f_{n-1}}{\partial
s}k_{n-1}+f_{n-1}\frac{\partial k_{n-1}}{\partial
s}+\frac{\partial^{2}f_{n}}{\partial
s^{2}}\right)V_{n}+\left(f_{n-1}k_{n-1}+\frac{\partial f_{n}}{\partial
s}\right)\frac{\partial V_{n}}{\partial s}$
while
$\frac{\partial}{\partial t}\frac{\partial V_{1}}{\partial
s}=\frac{\partial}{\partial t}\left(k_{1}V_{2}\right)=\frac{\partial
k_{1}}{\partial t}V_{2}+k_{1}\frac{\partial V_{2}}{\partial t}.$
Thus, from the both of above two equations, we reach
$\frac{\partial k_{1}}{\partial t}=f_{2}k_{1}^{2}+f_{1}\frac{\partial
k_{1}}{\partial s}+\frac{\partial^{2}f_{2}}{\partial s^{2}}-2\frac{\partial
f_{3}}{\partial s}k_{2}-f_{3}\frac{\partial k_{2}}{\partial
s}-f_{2}k_{2}^{2}-f_{4}k_{3}k_{2}.$
For $1<i<n$, noting that $\frac{\partial}{\partial s}\frac{\partial
V_{i}}{\partial t}=\frac{\partial}{\partial t}\frac{\partial V_{i}}{\partial
s}$, it is seen that
$\displaystyle\frac{\partial}{\partial s}\frac{\partial V_{i}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
s}\left[-\left(f_{i-1}k_{i-1}+\frac{\partial f_{i}}{\partial
s}-f_{i+1}k_{i}\right)V_{1}+{\displaystyle\sum\limits_{k=2}^{n}}\Psi_{kj}V_{k}\right]$
$\displaystyle=-\left(\frac{\partial f_{i-1}}{\partial
s}k_{i-1}+f_{i-1}\frac{\partial k_{i-1}}{\partial
s}+\frac{\partial^{2}f_{i}}{\partial s^{2}}-\frac{\partial f_{i+1}}{\partial
s}k_{i}-f_{i+1}\frac{\partial k_{i}}{\partial s}\right)V_{1}$
$\displaystyle+\left(f_{i-1}k_{i-1}+\frac{\partial f_{i}}{\partial
s}-f_{i+1}k_{i}\right)\frac{\partial V_{1}}{\partial
s}+{\displaystyle\sum\limits_{\begin{subarray}{c}k=2\\\ k\neq
i\end{subarray}}^{n}}\left(\frac{\partial\Psi_{ki}}{\partial
s}V_{k}+\Psi_{ki}\frac{\partial V_{k}}{\partial s}\right)$
while
$\frac{\partial}{\partial t}\frac{\partial V_{i}}{\partial
s}=\frac{\partial}{\partial
t}\left(-k_{i-1}V_{i-1}+k_{i}V_{i+1}\right)=-\frac{\partial k_{i-1}}{\partial
t}V_{i-1}-k_{i-1}\frac{\partial V_{i-1}}{\partial t}+\frac{\partial
k_{i}}{\partial t}V_{i+1}+k_{i}\frac{\partial V_{i+1}}{\partial t}.$
Thus, we obtain
$\frac{\partial k_{i-1}}{\partial t}=-\frac{\partial\Psi_{(i-1)i}}{\partial
s}-\Psi_{(i-2)i}k_{i-2}$
and
$\frac{\partial k_{i}}{\partial t}=\frac{\partial\Psi_{(i+1)i}}{\partial
s}-\Psi_{(i+2)i}k_{i+1}.$
Lastly, considering $\frac{\partial}{\partial s}\frac{\partial V_{n}}{\partial
t}=\frac{\partial}{\partial t}\frac{\partial V_{n}}{\partial s}$ and following
similar way as above, we reach
$\frac{\partial k_{n-1}}{\partial t}=-\frac{\partial\Psi_{(n-1)n}}{\partial
s}-\Psi_{(n-2)n}k_{n-2}.$
∎
## References
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* [2] M. Desbrun, M.-P. Cani-Gascuel, Active implicit surface for animation, in: Proc. Graphics Interface Canadian Inf. Process. Soc., 143–150 (1998).
* [3] M. Gage, R.S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23, 69–96 (1986).
* [4] H. Gluk, Higher curvatures of curves in Euclidean space, Amer. Math. Month. 73(1966), 699-704.
* [5] M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26, 285–314 (1987).
* [6] H. H. Hacisalihoğlu, Differential Geometry, University of İnönü Press, Malatya, 1983.
* [7] M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models, in: Proc. 1st Int. Conference on Computer Vision, 259–268 (1987).
* [8] D. Y. Kwon, F.C. Park, D.P. Chi, Inextensible flows of curves and developable surfaces, Appl. Math. Lett. 18 (2005) 1156-1162.
* [9] H.Q. Lu, J.S. Todhunter, T.W. Sze, Congruence conditions for nonplanar developable surfaces and their application to surface recognition, CVGIP, Image Underst. 56, 265–285 (1993).
* [10] H. Mochiyama, E. Shimemura, H. Kobayashi, Shape control of manipulators with hyper degrees of freedom, Int. J. Robot.Res., 18, 584–600 (1999).
* [11] D.J. Unger, Developable surfaces in elastoplastic fracture mechanics, Int. J. Fract. 50, 33–38 (1991).
* [12] O. G. Yildiz, S. Ersoy, M. Masal, A Note on Inextensible Flows of Curves on Oriented Surface, arXiv:1106.2012v1.
* [13] O. G. Yildiz, Minkowski Uzayında Yüzey Üzerinde Eğrilerin elastik Olmayan Hareketleri, Sakarya Üniversitesi Fen Bilimleri Enstitüsü Yüksek Lisans Tezi, 2011.
|
arxiv-papers
| 2012-07-06T07:40:28 |
2024-09-04T02:49:32.722347
|
{
"license": "Public Domain",
"authors": "\\\"Onder G\\\"okmen Y{\\i}ld{\\i}z, Murat Tosun and S{\\i}dd{\\i}ka \\\"O.\n Karaku\\c{s}",
"submitter": "\\\"Onder G\\\"okmen Y{\\i}ld{\\i}z",
"url": "https://arxiv.org/abs/1207.1543"
}
|
1207.1602
|
# Finite-time scaling of dynamic quantum criticality
Shuai Yin zsuyinshuai@163.com Xizhou Qin Chaohong Lee
lichaoh2@mail.sysu.edu.cn Fan Zhong stszf@mail.sysu.edu.cn State Key
Laboratory of Optoelectronic Materials and Technologies, School of Physics and
Engineering, Sun Yat-sen University, Guangzhou 510275, People’s Republic of
China
###### Abstract
We develop a theory of finite-time scaling for dynamic quantum criticality by
considering the competition among an external time scale, an intrinsic
reaction time scale and an imaginary time scale arising respectively from an
external driving field, the fluctuations of the competing orders and thermal
fluctuations. Through a successful application in determining the critical
properties at zero temperature and the solution of real-time Lindblad master
equation near a quantum critical point at nonzero temperatures, we show that
finite-time scaling offers not only an amenable and systematic approach to
detect the dynamic critical properties, but also a unified framework to
understand and explore nonequilibrium dynamics of quantum criticality, which
shows specificities for open systems.
###### pacs:
64.70.Tg, 64.60.Ht, 75.10.Pq
Detecting quantum phase transitions (QPTs) and understanding their real-time
dynamics are of great importance sachdev ; Coleman ; sachdevpt ; Dziarmaga ;
polrmp . Recent experimental breakthrough in ultracold atoms Greiner promises
new tools to study the quantum critical dynamics Zhang . In the nonequilibrium
critical dynamics of QPTs at zero temperature in which a controlling parameter
is changed with time through a critical point Dziarmaga ; polrmp , the Kibble-
Zurek mechanism (KZM), which was first introduced in cosmology by Kibble
kibble1 and then in condensed matter physics by Zurek zurek1 , has been found
to describe the dynamics of QPTs well Dziarmaga ; polrmp ; Zurek . In this
adiabatic–impulse–adiabatic approximation of the KZM, the system considered is
assumed to cease evolving in the impulse regime within which adiabaticity
breaks down due to critical slowing down binder . Yet, dynamical scaling has
been reported just within this regime deng and confirmed in both integrable
and nonintegrable systems grandi ; kolodrubetz . In classical critical
dynamics, an explanation based on coarsening has been developed biroli ,
however in quantum phase transitions, a systematic understanding of the full
scaling behavior is still lacking.
On the other hand, natural systems and their measurements exist inevitably in
nonzero temperatures, though probably only initial thermal states need
considering in ultracold atoms Polkovnikov . Thermal effects on a quantum
critical state can give rise to a variety of exotic behavior in the famous
quantum critical regime (QCR) Chakravarty as exhibits in a wide range of
strongly correlated systems sachdev ; Coleman ; sachdevpt ; Broun . Yet, as
both phases exhibit complex long-range quantum entanglement near the quantum
critical point and are violently excited thermally, it is a great challenge to
describe quantum critical dynamics at finite temperatures, let alone
nonequilibrium real-time effects patane ; chandran . Indeed, none of the
analytic, semiclassical, or numerical methods of condensed-matter physics
yields accurate results for dynamics in the QCR except for some special
systems in 1D sachdevpt . Even in isolated situations it is difficult to study
the time evolution of nonequilibrium systems with many degrees of freedom
Kinoshita ; Hofferberth ; rigol ; Dziarmaga ; polrmp . Therefore, systematic
approaches have to be invoked.
Figure 1: (color online) Schematic phase diagram under a sweep of $g$ near its
critical value 0. Two equilibrium phases (light grey/green) dominated by the
reaction time $\tau_{s}$ are separated by two crossover domains fanning out
from the quantum critical point $O$. One domain is the QCR (dark/blue)
controlled by the imaginary time scale $\tau_{T}$. The other is the new FTS
(grey/red) regime governed by the driving time scale $\tau_{d}$.
Time plays a fundamental role in quantum criticality owing to the interplay of
static and dynamic behaviors. Specifically, by varying the distance to the
critical point $g$ at a time rate $R$, a continuous QPT at a finite
temperature $T$ is characterized mainly by three time scales. The first one is
a reaction time $\tau_{s}$ that arises from the fluctuations of the competing
orders and blows up as $\tau_{s}\sim|g|^{-\nu z}$ with the standard critical
exponents $\nu$ and $z$ as $g$ vanishes sachdev . The second one is an
“imaginary” time scale $\tau_{T}=1/T$ (the Plank and the Boltzmann constants
have been set to 1) due to the finite $T$, since the real time is its
analytical continuation to imaginary numbers through a Feynman path integral
representation sachdev . The third one is an externally imposed driving time
scale $\tau_{d}$ that results from the driving and grows as $\tau_{d}\sim
R^{-z/r}$ with a rate exponent $r$ that is related to $z$ and other static
critical exponents Gong . It is the competition among $\tau_{s}$, $\tau_{T}$,
and $\tau_{d}$ that lead to a diversity of equilibrium and dynamic universal
phenomena near a quantum critical point.
Here we study systematically the competition among the three characteristic
time scales according to the theory of finite-time scaling (FTS) Gong . As
seen in Fig. 1, besides the usual equilibrium regimes and the QCR which are
respectively dominated by $\tau_{s}$ and $\tau_{T}$, our most important result
is that a new nonequilibrium FTS domain is created. In this domain, $\tau_{d}$
is the shortest time among the trio and thus dominates, just as the well-known
regime of finite-size scaling in which the characteristic size $L$ of the
system is shorter than its correlation length. At $T=0$, this indicates the
FTS domain overlaps just the impulse regime of the KZM for sweeping $g$. As a
consequence, although the system falls out off equilibrium, the state does not
cease evolving; rather, it evolves according to the imposed time scale
$\tau_{d}$ instead of $\tau_{s}$ with nonadiabatic excitations obeying FTS.
Therefore, FTS improves the understanding of KZM on its dark impulse regime
and produces naturally scaling forms suggested in deng ; grandi ; kolodrubetz
. In addition, FTS enables us to the study within the same framework other
driving dynamics than the KZ protocols chandran , which focus on changing non-
symmetry breaking terms like $g$. We shall show that these provides a
convenient method to determine the critical points and exponents, which were
invoked as input for scaling collapses reported in deng ; grandi ; kolodrubetz
. Similarly, at $T\neq 0$, in the FTS regime, there are now nonadiabatic
thermal excitations controlled again by $\tau_{d}$ and thus obeying again FTS.
Further, from Fig. 1, the FTS regime pushes the QCR to higher temperatures
since only then $\tau_{T}$ dominates. Consequently, FTS enables one to probe
directly the quantum critical point and its scaling behavior at nonzero
temperatures as $T$ becomes subordinate and just a perturbation. Thus, it
shows a ‘dynamic cooling’ effect that enables one to probe the zero-
temperature scaling at nonzero-temperatures while keeping $T$ subsidiary. This
offers us an extra approach to detect and study quantum criticality at finite
temperatures. Owing to its conceptual simplicity and accessability, FTS
therefore provides a unified framework not only to detect the dynamic critical
properties, but also to understand and explore the nonequilibrium dynamics of
quantum criticality both at $T=0$ and $T\neq 0$.
As another important result, we shall show that in nonequilibrium quantum
critical dynamics of open systems one must include an additional variable such
as the coupling to a heat bath to the intrinsic quantum dynamics note . This
is an important difference from the classical case and must be considered when
extending nonequilibrium quantum critical dynamics to finite temperatures
patane . We shall see that the master equation in the Lindblad form just
offers such a variable and is thus an appropriate platform to study real-time
nonequilibrium quantum criticality.
We start with an open many-body quantum system interacting with a heat bath
patane to study the interplay of quantum and thermal fluctuations. The state
of such a system can be described by a density matrix operator $\rho$
according to quantum statistical physics. For weak system-environment
couplings, after assuming Markovian and tracing over the bath variables, one
obtains the master equation for $\rho$ in the Lindblad form Lindblad ; attal ;
Mai ,
$\partial\rho/\partial t=-i[\mathcal{H},\rho]+c\mathcal{L}\rho,$ (1)
where $\mathcal{L}\rho=-\sum_{i=1,j\neq i}^{N_{E}}\beta_{j}(V_{i\rightarrow
j}^{{\dagger}}V_{i\rightarrow j}\rho+\rho V_{i\rightarrow
j}^{{\dagger}}V_{i\rightarrow j}-2V_{i\rightarrow j}\rho V_{i\rightarrow
j}^{{\dagger}})/2$, $c$ is the dissipation rate and measures the coupling
strength between the system and the bath, $N_{E}$ is the total number of
energy levels,
$\beta_{i}=\textrm{exp}(-E_{i}/T)/\textrm{Tr}\textrm{exp}(-\mathcal{H}/T)$
with $E_{i}$ being the $i$th eigenvalue of $\mathcal{H}$, and $V_{i\rightarrow
j}$ is the thermal jump matrix whose element at the $j$th row and $i$th column
is one or zero in the energy representation. $V_{i\rightarrow j}$ fulfills
$\beta_{i}\rho_{E}V_{i\rightarrow j}=\beta_{j}V_{i\rightarrow j}\rho_{E}$ with
the equilibrium density matrix operator
$\rho_{E}\equiv\textrm{exp}(-\mathcal{H}/T)/\textrm{Tr}\textrm{exp}(-\mathcal{H}/T)$
whose eigenvalues are $\beta_{i}$. This can be regarded as a detailed balance
condition in equilibrium. The Lindblad equation (1) is a real-time dynamical
equations which integrates both the quantum and the thermal contributions. It
has been widely used in quantum optics orszag and relaxation processes in
open quantum systems attal ; znidaric . Although for large couplings, Eq.(1)
may be inapplicable wgwang , this equation gives a reasonable description in
the weak coupling limit, for instance, for the time-independent $\mathcal{H}$,
the steady solution of Eq. (1) is $\rho_{E}$ independent of $c$ Lindblad ;
attal ; Mai , this is consistent with the foundation of statistical mechanics
Schrodinger . In the following, we focus on the weak coupling limit and
consider the scaling properties of the Lindblad equation.
The theory of FTS Gong takes explicitly into account the rate $R$, which
plays a role similar to $L^{-1}$ since it imposes on a system an additional
time scale that manipulates its evolution. In classical critical dynamics, the
nonequilibrium dynamic scaling can be generalized directly from the
equilibrium ones as confirmed by the renormalization-group theory Gong .
However, in the nonequilibrium quantum criticality, as pointed out, a coupling
strength must be considered as an independent scaling variable. In the weak
coupling limit, this strength can be reduced to the dissipation rate $c$.
Accordingly, for a length rescaling of factor $b$, an order parameter $M$
transforms as
$\displaystyle
M(t,g,h_{z},T,L,c,R)=b^{-\beta/\nu}M(tb^{-z},gb^{1/\nu},h_{z}b^{\beta\delta/\nu},\quad$
$\displaystyle Tb^{z},L^{-1}b,cb^{z},Rb^{r}),\quad$ (2)
where the two critical exponents $\beta$ and $\delta$ are defined as usual in
classical critical phenomena by $M\propto g^{\beta}$ in the absence of an
external probe field $h_{z}$ conjugate to $M$ and $M\propto h_{z}^{\delta}$ at
$g=0$, respectively. In the weak coupling limit, $c$ is small thus one can
expect its scaling behavior is controlled by the the fixed point corresponding
to the critical point at $c=0$, thus the dimension of $c$ is identical with
$t^{-1}$ as can be inspected from Eq. (1) Mai . This is checked latter by the
numerical solution of Eq. (1).
With Eq. (2), one can describe in a unified framework different kinds of
driven dynamics via changing $g$, $h_{z}$ or $T$ and readily define different
regimes and their crossovers. Taking $g=Rt$ for instance, neglecting $h_{z}$,
suppressing one independent variable, and choosing $b$ such that $Rb^{r}$
becomes a constant, one finds an FTS scaling form
$\displaystyle M(g,T,L,c,R)=R^{\beta/\nu r}f_{1}(gR^{-1/\nu
r},TR^{-z/r},\quad$ $\displaystyle L^{-1}R^{-1/r},cR^{-z/r}),\ $ (3)
where $r=z+1/\nu$ obtained from $g=Rt$ and its rescaling Gong and the
function $f_{i}$ with an integer $i$ denotes a scaling function. FTS dominates
when $|g|R^{-1/\nu r}\ll 1$, $TR^{-z/r}\ll 1$, $L^{-1}R^{-1/r}\ll 1$, and
$cR^{-z/r}\ll 1$. The first gives $\tau_{d}\sim R^{-z/r}\ll|g|^{-\nu
z}\sim\tau_{s}$, the second $\tau_{d}\ll 1/T=\tau_{T}$ as they ought to be.
Crossovers to other regimes occur near $|\hat{g}|\sim R^{1/\nu r}$ and
$\hat{T}\sim R^{z/r}$ as depicted in Fig. 1 and similar ones for $L$ and $c$.
The first gives $|\hat{t}|\sim R^{-\nu z/(1+\nu z)}$ because
$\hat{g}=R\hat{t}$. This is just the scaling of the KZM upon identifying
$\hat{t}$ with the freeze-out time instant Dziarmaga ; polrmp ; Zurek for a
closed system $c=0$ in the thermodynamic limit ($L\rightarrow\infty$) and at
$T=0$.
Several remarks are in order here. (a) Equation (3) is different from the
similar scaling form for finite temperatures in chandran because $c$ must be
included to introduce the thermal fluctuation in the nonequilibrium situation.
(b) To return to the equilibrium scaling form at finite-temperatures sachdev ,
the scaling function $f_{i}$ must satisfy a constraint of $\partial
f_{i}/\partial c=0$ for $R=0$. (c) Beside recovering the full scaling forms of
finite-size for closed system in grandi ; kolodrubetz by fixing $c=0$ and
$T=0$, the nonequilibrium dissipation scaling for spontaneous emissions in
zero-temperature open quantum systems can also be studied by fixing $T=0$ in
Eq. (3). (d) Note that $c$ should be small in the weak coupling limit and thus
the regime dominated by $c$ may be inaccessible.
Instead of sweeping $g$, when $h_{z}=R_{z}t$, one obtains similarly the order
parameter
$\displaystyle M_{h}=R_{z}^{\beta/\nu r_{z}}f_{2}(gR_{z}^{-1/\nu
r_{z}},h_{z}R_{z}^{-\beta\delta/\nu r_{z}},\qquad\quad$ $\displaystyle
TR_{z}^{-z/r_{z}},L^{-1}R_{z}^{-1/r_{z}},cR^{-z/r_{z}})\\!\\!\\!$ (4)
with $r_{z}=z+\beta\delta/\nu$. Different regimes and their crossovers can
also be readily defined. Different from sweeping $g$ through the critical
point as the ordinary KZM protocols deng ; grandi ; kolodrubetz ; chandran ,
here we fix $g$ and change the symmetry breaking field $h_{z}$. This provides
a method to determine the critical point from distinct critical behaviors for
$g=0$ and $g\neq 0$, a method which we shall utilize below and may also be
realizable experimentally. Note that in this protocol, the form of $\tau_{d}$
remains remarkably if $R$ and $r$ are replaced with their counterparts.
However, in addition to the fixed $\tau_{s}$ for the fixed $g$, there exists
another reaction time diverging with $|h_{z}|^{-\nu z/\beta\delta}$. These
result in new competitions but act only as corrections in the FTS regime,
showing an advantage of FTS.
Now we show that FTS can provide methods to detect quantum critical properties
such as the critical point and critical exponents. For simplicity, we consider
$T=0$ and $c=0$ in the thermodynamic limit $L\rightarrow\infty$. According to
Eq. (4), at $h_{z}=0$, $M_{h}$ reduces to
$M_{0}(g,R_{z})=R_{z}^{\beta/\nu r_{z}}f_{3}(gR_{z}^{-1/\nu r_{z}}),$ (5)
while the field at $M_{h}=0$, denoted by $h_{z0}$, scales as
$h_{z0}(g,R_{z})=R_{z}^{\beta\delta/\nu r_{z}}f_{4}(gR_{z}^{-1/\nu r_{z}}).$
(6)
Differentiating $M_{h}$ with respect to $h_{z}$ in Eq. (4), one obtains the
susceptibility at zero field,
$\chi(g,R_{z})=R_{z}^{\beta(1-\delta)/\nu r_{z}}f_{5}(gR_{z}^{-1/\nu r_{z}}).$
(7)
To fix the critical point, we can define a cumulant
$C(g,R_{z})\equiv M_{0}/(h_{z0}\chi)=f_{6}(gR_{z}^{-1/\nu r_{z}})$ (8)
similar to the Binder cumulant in finite-size scaling binder . As $C$ is a
function of only one independent variable, its curves for different $R_{z}$
intersect at the critical point $g=0$ at which $C$ becomes a constant
$f_{6}(0)$ independent on $R_{z}$. This gives the critical point with which
all the critical exponents can then be estimated. For example, $\beta/\nu
r_{z}$ and $\beta\delta/\nu r_{z}$ can be estimated respectively from Eqs. (5)
and (6) by fitting $M_{0}$ and $h_{z0}$ for a series of $R_{z}$ at $g=0$.
Similarly, from Eq. (3) at $c=0$, $T=0$, and $L\rightarrow\infty$, $\beta/\nu
r$ can be estimated by fitting $M$ for a series of $R$ at $g=0$. From these
three exponent ratios and the scaling law sachdev $\beta(\delta+1)=(d+z)\nu$
with the space dimensionality $d$, one can determine all the critical
exponents.
As an example of the FTS method to determine critical properties, we consider
the one-dimensional (1D) transverse-field Ising model whose Hamiltonian is
sachdev
$\mathcal{H}=-h_{x}\sum\limits_{n=1}^{N}\sigma_{n}^{x}-\sum\limits_{n=1}^{N-1}\sigma_{n}^{z}\sigma_{n+1}^{z},$
(9)
and has been realized in CoNb2O6 experimentally Coldea , where
$\sigma_{n}^{x}$ and $\sigma_{n}^{z}$ are the Pauli matrices, $h_{x}$ is the
transverse field, and the Ising coupling has been set to unity as our energy
unit. The model exhibits a continuous QPT from a ferromagnetic phase to a
quantum paramagnetic phase at a critical point $h_{xc}$ (and so
$g=h_{x}-h_{xc}$) at $T=0$ sachdev . The order parameter is the magnetization
$M=\sum_{n=1}^{N}\langle\sigma_{n}^{z}\rangle/N$ for the $N$ spins with the
angle brackets denoting the quantum and/or thermal average. As a method to
probe the transition, we add to $\mathcal{H}$ a symmetry-breaking term
$-h_{z}\sum_{n=1}^{N}\sigma_{n}^{z}$.
We illustrate our approach at $T=0$ and $c=0$ at which Eq. (1) is same to
Schödinger’s equation and some exact results are available for comparison. We
solve the model using the time-evolving block-decimation algorithm Vidal ,
which is capable of treating large system sizes. We determine the critical
point in Fig. 2 and apply it purposely to determine the critical exponents in
Fig. 3. The good agreement of the results collected in Table 1 shows the power
of FTS.
Figure 2: (color online) Estimation of quantum critical point. Curves of the
cumulant $C$ for different $R_{z}$ intersect at the critical point $h_{xc}$ or
$g=0$. Owing to possible errors from the truncation of the singular values in
the Schmidt decomposition Vidal , however, the intersections are slightly
scattered as shown in the inset. Nevertheless, the average of all the
intersections is $h_{xc}^{N}=0.999(2)$, a good estimate of the exact value
$h_{xc}=1$. We choose a lattice size of $L=2000$, which has been checked to
produce a negligible size effect.
Figure 3: (color online) Estimation of critical exponents. Our solutions with $h_{xc}^{N}=0.999$, $T=0$, and $L=2000$ yield $\beta/\nu r_{z}=0.0436$, $\beta\delta/\nu r_{z}=0.651$, and $\beta/\nu r=0.0622$ from power-law fits according to the scaling forms (5), (6) and (3), respectively. We then obtain all the critical exponents listed in Table 1 with their exact results for comparison. As the statistical errors of the fits are tiny, we fit data at $h_{xc}=0.997$ and $1.001$ and the largest difference in each exponent is used as an estimate of the error given also in Table 1. Table 1: Critical point and exponents for the 1D transverse-field Ising model | $h_{xc}$ | $\beta$ | $\delta$ | $\nu$ | $z$
---|---|---|---|---|---
Numerical | 0.999(2) | 0.125(11) | 14.9(6) | 0.98(4) | 1.01(3)
Exact sachdev | 1 | 0.125 | 15 | 1 | 1
Figure 4: (color online) Nonequilibrium scaling at nonzero temperatures. (a)
Data of $M_{h}$ versus $T$ plotted in the inset for the three different sets
of $R_{z}$, $c$, and $L$ so choosing as to fix the value of
$L^{-1}R_{z}^{-1/r_{z}}$ and $cR_{z}^{-z/r_{z}}$ collapse as expected onto a
single curve for the fixed $LR_{z}^{1/r_{z}}=1.166$ and
$cR_{z}^{-z/r_{z}}=3.603$ according to the FTS (4) at $g=0$ ($h_{x}=0.999$),
$h_{z}=0$. (b) If, instead of $cR_{z}^{-z/r_{z}}$, we fix all $c=0.7$, the
value for $L=6$, and keep others, the rescaled curves then do _not_ collapse.
Having successfully demonstrated FTS at $T=0$, we now turn to $T\neq 0$ at
which most experiments operate. To examine the general nonequilibrium FTS (4)
for $T\neq 0$, we solve numerically Eq. (1) for the Hamiltonian (9) along with
the field $h_{z}$ by a finite difference method to second order with periodic
boundary conditions. We find that $M_{h}$ can now saturate correctly with the
thermal fluctuations. Moreover, Fig. 4 shows clearly the validity of the FTS
form (4). Further, upon comparing (a) with (b) in Fig. 4, it is obvious that
$c$ must enter into the scaling forms with a scaling dimension $z$. Note that
although here we only solve directly Eq. (1) for small lattices, the results
show that it is suitable for describing the nonequilibrium behavior at finite
temperatures near the quantum critical point. Moreover, the rapidly developing
numerical renormalization-group methods cirac , for example, seem quite
promising to solve the equation for larger lattice sizes cirac2 .
In conclusion, FTS not only provides a unified understanding of the driving
dynamics in general and lights up the dark impulse regime of KZM at zero
temperature in particular, but also sheds light on the QCR at nonzero
temperatures by establishing its own regime. It offers a powerful unified
approach amenable to both numerics and experiments to study equilibrium and
nonequilibrium dynamics of quantum criticality. We have shown that in the
latter in open systems one must include the dissipation rate as an independent
scaling variable and the Lindblad equation can be a valuable framework for
such studies. Although we have studied a simple model for illustration, our
approach should be applicable to more complex systems as well. In addition, as
our results indicate that the classical theory of FTS with proper
modifications can well describe quantum criticality, new physics may be in
action Coleman if it is violated.
We thank Junhong An, Peter Drummond, and Xiwen Guan for their valuable
comments and discussions. Y.S. and F.Z. were supported by NNSFC (10625420) and
FRFCUC. C.L. was supported by NNSFC (11075223), NBRPC (2012CB821300
(2012CB821305)), NCETPC (NCET-10-0850). We acknowledge use of some source
codes for TEBD from http://physics.mines.edu/downloads/software/tebd/.
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* (33) R. Coldea, et al. Science 327, 177 (2010).
* (34) G. Vidal, Phys. Rev. Lett. 93, 040502 (2004).
* (35) F. Verstraete, V. Murg, and J. I. Cirac, Adv. Phys 57, 143 (2010).
* (36) F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Phys. Rev. Lett. 93, 207204 (2004); M. Zwolak and G. Vidal, Phys. Rev. Lett. 93, 207205 (2004).
|
arxiv-papers
| 2012-07-06T12:28:19 |
2024-09-04T02:49:32.730344
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuai Yin, Xizhou Qin, Chaohong Lee and Fan Zhong",
"submitter": "Shuai Yin",
"url": "https://arxiv.org/abs/1207.1602"
}
|
1207.1633
|
# Turbulent mixing driven by mean-flow shear and internal gravity waves in
oceans and atmospheres
Helmut Z. Baumert Institute for Applied Marine and Limnic Studies, Hamburg,
Germany
###### Abstract
This study starts with balances deduced by Baumert and Peters (2004, 2005)
from results of stratified-shear experiments made in channels and wind tunnels
by Itsweire (1984) and Rohr and Van Atta (1987), and of free-decay experiments
in a resting stratified tank by Dickey and Mellor (1980). Using a modification
of Canuto’s (2002) ideas on turbulence and waves, these balances are merged
with an (internal) gravity-wave energy balance presented for the open ocean by
Gregg (1989), without mean-flow shear. The latter was augmented by a linear
(viscous) friction term. Gregg’s wave-energy source is interpreted on its
long-wave spectral end as internal tides, topography, large-scale wind, and
atmospheric low-pressure actions. In addition, internal eigen waves, generated
by mean-flow shear, and the aging of the wave field from a virginal (linear)
into a saturated state are taken into account. Wave packets and turbulence are
treated as particles (vortices, packets) by ensemble kinetics so that the loss
terms in all three balances have quadratic form. Following a proposal by
Peters (2008), the mixing efficiency of purely wave-generated turbulence is
treated as a universal constant, as well as the turbulent Prandtl number under
neutral conditions. It is shown that: (i) in the wind tunnel, eigen waves are
switched off, (ii) due to remotely generated long waves or other non-local
energy sources, coexistence equilibria of turbulence and waves are stable even
at Richardson numbers as high as $10^{3}$; (iii) the three-equation system is
compatible with geophysically shielded settings like certain stratified
laboratory flows. The agreement with a huge body of observations surprises.
Gregg’s (1989) wave-model component and the a.m. universal constants taken
apart, the equations contain only one additional dimensionless parameter for
the eigen-wave closure, estimated as $Y\approx 1.35.$
Baumert Turbulence by shear & waves .. 35pc H. Z. Baumert, IAMARIS, Bei den
Mühren 69A,
D-20457 Hamburg, Germany (baumert@iamaris.org) Report on the SHEWAMIX project,
Grant N62909-10-1-7050, ONR-Global.
## 1 Introduction
### 1.1 General
In geophysical flows, turbulence is ubiquitous. Today turbulent engineering
flows may already be simulated using DNS, i.e. fully resolving scales down to
the smallest ones, provided they are large compared with molecular cluster
scales. However, for ocean and atmospheric flows this will remain impossible
even in the foreseeable future. Here particularly the dominating stably
stratified flows, i.e. all forms of coexistence of turbulence and internal
waves, represent a specific observational and theoretical challenge. Without a
better understanding of the fundamental physics of these processes all our
regional to global models of weather and climate remain incomplete.
Oceanic, atmospheric and stellar turbulence has challenged a number of
prominent scientists and research groups since long, in the recent decade
namely Woods (2002), Galperin et al. (2007), Canuto et al. (2008), and
Zilitinkevich et al. (2008), also Kantha and Carniel (2009). They all
emphasized a major contradiction between observational experience and existing
theories:
* (i)
For controlled stratified shear flows in the laboratory exists undoubtly a
critical gradient Richardson number of $R_{g}^{b}\leq{\cal
O}(\hbox{$\,{}^{1}\\!/_{4}$})$ above which turbulence dies out (Itsweire,
1984; Itsweire et al., 1986; Rohr and Van Atta, 1987; Rohr et al., 1987; Rohr
et al., 1988a, b; Van Atta, 1999). Also in the field the qualitative and
strongly (but not totally) limiting role of $R_{g}^{b}$ is without doubt
(Peters et al., 1988) and became specifically visible with the advent of
Lagrangian floaters in the Lagrangian time spectra of turbulence and wave-like
fluctuations where $\Omega=N$ marks a sharp divide: to the left a flat wave
spectrum, to the right a Kolmogorov time spectrum (D’Asaro and Lien, 2000a,
b). Also the existing theories (Richardson, 1923; Miles, 1961; Howard, 1961;
Hazel, 1972; Thorpe, 1973, Abarbanel et al., 1984; Baumert and Peters, 2004)
point all into the same direction.
* (ii)
Geophysical flows are reported to exhibit not always, but more than often
significant stable turbulence levels and mixing capabilities at
$R_{g}\gg\hbox{$\,{}^{1}\\!/_{4}$}$ (Peters et al., 1988; Canuto, 2002; Poulos
et al., 2002; Nakamura and Mahrt, 2005; Grachev et al., 2005, 2006). Also
Peters and Baumert (2007) report problems in validating a $K$-$\varepsilon$
turbulence closure against comprehensive estuarine microstructure measurements
in the Hudson river. For tidal phases with weak shear the observed turbulence
levels exceeded the model values by 2 to 3 orders of magnitude. The dynamic
time lag of the turbulent state variables in the M2 tide (about one hour, see
Baumert and Radach, 1992) can be excluded as a source of the deviations
because the model naturally contains this effect.
The text below makes an attempt to resolve the sketched contradiction using
ideas of Woods (2002 and literature cited therein) and Canuto (2002) regarding
the role of waves and billow turbulence. We identify internal gravity waves
(so far neglected in most other studies) as the potentially responsible
phenomenon.
For a better understanding of the language used in this text we first
introduce notational conventions and then continue with an overview of major
physical processes within the world of stratified shear turbulence and
internal gravity waves.
Below, we always means the author and the dear reader in a dialogue.
### 1.2 Setup and notation
For simplicity we focus on a simple non-trivial situation, a spatially one-
dimensional “channel” flow with velocity component $U$ in horizontal ($x$)
direction, with variation of $U$ along the vertical, $z$ (pointing upwards),
and with Eulerian density and icopycnal coordinate fluctuations.
The decomposition of our flow field into mean and fluctuations reads as
follows:
$\displaystyle U(z,t)$ $\displaystyle=$ $\displaystyle\langle
U\rangle+\tilde{u}(z,t)+u^{\prime}(z,t)\,,$ (1) $\displaystyle W(z,t)$
$\displaystyle=$ $\displaystyle\langle
W\rangle+\tilde{w}(z,t)+w^{\prime}(z,t)\,.$ (2)
Variables with tilde are small-scale short-wave components111actually wave
packets, present only under stratified conditions. Primed variables denote
turbulent fluctuations. Both fluctuating components vanish in the mean.
The turbulent kinetic energy, TKE or $\cal K$, is defined as follows,
${\cal K}=\frac{1}{2}\langle u^{\prime 2}+v^{\prime 2}+w^{\prime 2}\rangle.$
(3)
A wave’s total energy, $\cal E$, is the sum of potential, $\langle
N^{2}\,\tilde{\zeta}^{2}\rangle/2$, and kinetic energy,
$\langle\tilde{u}^{2}+\tilde{w}^{2}\rangle/2$ (Gill, 1982):
${\cal
E}=\frac{1}{2}\langle\tilde{u}^{2}+\tilde{w}^{2}+N^{2}\,\tilde{\zeta}^{2}\rangle.$
(4)
Here $\tilde{u}$ points into the direction of wave propagation so that in a
plane wave $\tilde{v}=0$. Average $\langle.\rangle$ is taken at least over one
wave period which “by definition” is longer than the characteristic turbulent
time scale. In the following the dimensionless gradient Richardson number,
$R_{g}$, plays a central role. In a stratified shear flow as above, shear is
given by
$\langle S\rangle=\frac{\partial\langle U\rangle}{\partial z}\,,$ (5)
while stratification is characterized by the Brunt-Väisälä frequency squared,
$\langle
N^{2}\rangle=-\frac{g}{\langle\rho\rangle}\,\frac{d\langle\rho\rangle}{dz}.$
(6)
$\langle\rho\rangle$ is the background density field.
The gradient Richardson number, $R_{g}$, characterizes the dimensionless ratio
of the two aspects, shear and stratification:
$R_{g}=\langle N^{2}\rangle/\langle S\rangle^{2}\,.$ (7)
Below the averaging operators are mostly omitted for brevity of notation. But
fluctuations are consequently labeled either by tilde (wave-like) or prime
(turbulent).
Based on $R_{g}$, under controlled laboratory conditions where the linear
eigen waves leave the experimental site before quadratic saturation and a feed
back into the TKE pool can happen (wind tunnel of Van Atta), the following
hydrodynamic regimes for horizontally homogeneous flows are found (Baumert and
Peters, 2004, 2005):
* (a)
$R_{g}\leq R_{g}^{a}\equiv 0$: unstable and neutral stratification,
convective turbulence, no internal waves at all.
* (b)
$0\equiv R_{g}^{a}<R_{g}<R_{g}^{b}\equiv\hbox{$\,{}^{1}\\!/_{4}$}$: stable
stratification, shear-dominated growing turbulence, coexistence of turbulence
and internal waves.
* (c)
$\hbox{$\,{}^{1}\\!/_{4}$}\equiv
R_{g}^{b}<R_{g}<R_{g}^{c}\equiv\hbox{$\,{}^{1}\\!/_{2}$}$: stable
stratification,
wave-dominated decaying turbulence, coexistence of turbulence and internal
waves.
* (d)
$\hbox{$\,{}^{1}\\!/_{2}$}\equiv R_{g}^{c}<R_{g}$: stable stratification, no
coexistence of turbulence and waves, waves-only regime.
Under those conditions the turbulent Prandtl number $\sigma$ is (Baumert and
Peters, 2004, 2005)
$\sigma=\frac{\mu}{\nu}=\frac{\sigma_{0}}{1-(N/\Omega)^{2}}$ (8)
The above values for the critical numbers $R_{g}^{a},R_{g}^{b}$ and
$R_{g}^{c}$ hold for the asymptotic case $Re\rightarrow\infty$.
The situation is different if we leave the laboratory wind tunnel and consider
the open ocean, stratified rivers or the stably stratified atmosphere where
turbulence is not only locally generated through local mean-flow shear but
also through the action of space-filling (non-local) spectra of internal
gravity waves (IGWs). These are generated e.g. by tidal forces, possibly at
remote places, arriving at our point of interest along various pathways.
## 2 Major physical interactions in stably stratified oceans and atmospheres
Fig. 1 schematically presents the major interactions between the energies of
mean222Mean-flow kinetic energy, $\hbox{$\,{}^{1}\\!/_{2}$}\,\langle
U\rangle^{2}$, MKE; Internal tides and fluctuating motion components333TKE,
$\cal K$; r.m.s. vorticity, $\Omega$; WKE.. Fig. 2 does the same for r.m.s.
vorticity and turbulent viscosity, $\nu$. The latter connects $\cal K$ and
$\Omega$ with the mean flow through the so-called Kolmogorov-Prandtl relation
in the following form (Baumert and Peters, 2004, Baumert, 2012):
$\nu={{\cal K}}/{\pi\,\Omega}\,.$ (9)
The TKE dissipation rate, $\varepsilon$, transforming TKE into heat, is
defined as (Baumert and Peters, 2004, Baumert, 2012)
$\varepsilon={{\cal K}\,\Omega}/{\pi}=\nu\,\Omega^{2}\,.$ (10)
The buoyancy flux, $B$, transforming TKE into background potential energy,
$PE_{b}$, may be expressed using the eddy diffusivity, $\mu$, as follows444The
buoyancy flux (11) refers to purely shear-generated turbulence (for details
see Subsection 5.1.2 and Baumert and Peters, 2004, 2005).,
$B=-\frac{g}{\langle\,\rho\rangle}\langle
w^{\prime}\,\rho^{\prime}\rangle=\mu\,N^{2}\,,$ (11)
where $\mu$ is related with the eddy viscosity $\nu$ through $\sigma$, the
turbulent Prandtl number function:
$\mu=\nu/\sigma\,\quad\quad\mbox{or}\quad\quad\sigma=\nu/\mu\,.$ (12)
Finally, the molecular heat flux, $\Phi_{i}$, transforms internal energy into
background potential energy, $PE_{b}$.
There is further the rate $\Pi$, which transforms the energy of internal tides
and wind-generated large-scale internal motions over a long chain of various
friction-poor wave-wave interactions into short-wave internal-wave energy, and
there is the rate $\tilde{P}$, which transforms the energy of internal-waves
spectra into TKE, mainly by breaking (a shot noise process), and to a low
degree by wave shear.
The mean-flow shear, $S$, controls TKE production through $P-\Psi$ and
internal-wave generation through $\Psi$. In the (conventional) neutrally
stratified case, $N^{2}=0$, the total loss term in the mean-flow kinetic-
energy balance (MKE) is $-P$, which is at the same time the only source of
small-scale energy:
$P=\nu\,S^{2}\,.$ (13)
But for $N^{2}>0$ small-scale energy means the sum of turbulence and waves,
$P=(P-\Psi)+\Psi.$
The shear governs not only TKE and IGW production but also the generation of
r.m.s. vorticity. But vorticity is also influenced by short internal-gravity
waves through the term $\tilde{S}$: in the shearless IGW-dissipation case
(e.g. Gregg, 1989) we have also vorticity generation.
Figure 1: Interaction of major forms of fluid-mechanical energy in stratified
oceanic, atmospheric or stellar shear flows. $\Psi$ is the flux of eigen-wave
energy mentioned in the text. $\Pi$ is the energy flow which corresponds to
remotely generated waves. The wind tunnel allows to cut off the flux
$\tilde{P}$ from the the wave-energy to the TKE pool because the wave spectrum
cannot reach a saturated state. In the long-term equilibrium holds
$\tilde{P}=(1-f)(\Psi+\Pi)$ where $f$ is the fraction of direct wave
dissipation and tends to zero if the wave spectrum approaches saturation. $f$
may become relevant for short wave ages. Figure 2: Interaction of major
vorticity-controlling motion components in stratified oceanic and atmospheric
shear flows. The r.m.s. vorticity $\Omega$ is governed by the mean-flow shear,
$S$, and a wave-induced pseudo shear, $\tilde{S}$. The latter is controlled by
eigen waves ($Y$) and remotely generated long waves ($X$). For details see
text.
When we use the word energy in the present context, we mostly mean for brevity
TKE ($\cal K$), i.e. turbulent kinetic energy. If we talk here about vorticity
we similarly mean for brevity the r.m.s. turbulent vorticity. For its detailed
mechanical interpretation we refer to Baumert (2012).
The precise meaning of $\Omega$ is actually the module of the vorticity
frequency. The module of the vorticity itself, $\omega$, is
$\omega=2\,\pi\,\Omega$. The enstrophy is ${\omega^{2}}/{2}$.
## 3 Balances for small-scale motions
Before we start to write down transport equations for primary variables we
first discuss their nature and their balances. For this aim we look again at
Fig. 1 and there namely on the two boxes in the middle row with the names TKE
and Internal waves.
The left box, TKE, is fed by two components: by mean-flow shear in the form of
$P-\Psi$, and by internal-wave breaking, $\tilde{P}$, see (15) below. But it
generates heat by $\varepsilon$ via (10), exports buoyancy $B$ via
(11)555generating thus background potential energy, $PE_{b}$., and generates
by $\tilde{\Psi}$ in the course of aging and subsequent collapse short
internal waves, when its time scale approaches the internal-wave period
(Baumert and Peters, 2005).
Turbulence collapsing into waves is an event mostly bound to certain special
conditions (e.g. Dickey and Mellor, 1980; D’Asaro and Lien, 2000a, b). Under
smooth and almost-equilibrium conditions the TKE may collapse and generate
waves which then after aging saturate and feed their energy back into the TKE
pool, as part of $\tilde{P}$. In the following $\tilde{\Psi}$ is therefore
mostly neglected.
The right box, Internal waves, is fed by three components: by large-scale wave
sources, $\Pi$, by the rate of eigen-wave generation, $\Psi$, and by the
collapse rate discussed already above and neglected further below. This box
further exhibits two relevant losses: the linear wave friction, $\tilde{L}$,
$\tilde{L}=c_{1}\,{\cal E}\,,$ (14)
and the quadratic wave friction, $\tilde{P}$,
$\tilde{P}=c_{2}\,{\cal E}^{2}\,.$ (15)
The latter is mainly caused by wave breaking and dominates the wave-energy
balance, but not completely. The loss term666see also the wave-energy balance
(18) below (14) and (15) follows arguments developed by Gregg (1989) which we
augment as follows.
From a purely theoretical point of view the phenomenon of wave breaking tells
us nothing about its kinetics. It is the processes before breaking occurs
which form the bottleneck. If we accept the idea that not only vortices
(actually: vortex-dipole filaments, see Baumert, 2012) are particles which
move in space until collision, then the interpretation of wave-energy dynamics
in terms of particle dynamics does not come as a surprise, in particular in
view of the billow turbulence discussed by Woods (2002).
Due to the general particle-wave dualism of field theories, which is a well
established concept in classical continuum mechanics777It became most famous
in quantum mechanics and has been somewhat monopolized there., also wave
packets can be treated as particles moving with their group velocity until
collision. After collision they either move ahead, or change their paths, or,
with low probability, their energy is dissipated by breaking in dissipative
patches (billows). In contrast to vortex kinetics, here the probabilities are
not symmetric. In both cases (vortex dipoles and wave packets) the collision
events are highly intermittent.
Besides its phenomenological basis, the quadratic term in (15) might therefore
have deeper roots or at least analogies in the kinetics of reactive particle
“gases” or molecular reactions in fluids where particle-collision
probabilities follow the product of their spatial densities. For collisions
between particles of same kind, the quadratic collision term is thus a logical
consequence.
Fig. 2 shows the major feedback loop between the small-scale motions and the
mean flow which eventually smoothes the flow through the turbulent viscosity.
The central left box, r.m.s. vorticity, is fed by two components: by the mean-
flow shear, $S$, and by the internal-wave field through the pseudo shear,
$\tilde{S}$. The vorticity $\Omega$ itself controls together with the TKE
$\cal K$ the turbulent viscosity via (9) which eventually smoothes the flow.
The spectral signatures of shear-generated fluctuations and wave-wave
interactions differ qualitatively. While shear influences vorticity directly
by prescribing a time scale $\propto S^{-1}$, the long-wave sources ($\Pi$) of
IGW spectra do it more indirectly via a longer chain of wave-wave interactions
cascading down to critical frequencies around $N$. This implies that each of
the two mechanisms needs “his” closure.
For conditions of homogeneous888in horizontal and vertical direction shear,
stratification and wave fields the above three balances can be so far
formulated as follows (compare with Figs. 1 and 2 ):
$\displaystyle\frac{d\Omega}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left(\frac{S^{2}}{2}-\Omega^{2}+\tilde{S}^{2}\right),$
(16) $\displaystyle\frac{d{\cal K}}{dt}$ $\displaystyle=$
$\displaystyle(P-\Psi)-B-\varepsilon+\tilde{P},$ (17)
$\displaystyle\frac{d\cal E}{dt}$ $\displaystyle=$
$\displaystyle(\Pi+\Psi)-\tilde{L}-\tilde{P}.$ (18)
With the exception of $\Pi$, all variables in (16, 17, 18) have purely local
character.
In (17) we used the relation
$B+\Psi=\nu\,N^{2}/\sigma_{0}\,,$ (19)
derived and discussed in greater detail by Baumert and Peters (2004, 2005).
Here $\sigma_{0}=\hbox{$\,{}^{1}\\!/_{2}$}$ is the value of the turbulent
Prandtl number function $\sigma$ for the case of neutral stratification
($R_{g}=0$ or $\Omega\rightarrow\infty$). With (8) we have
$\Psi=\frac{\varepsilon}{\sigma_{0}}\,\left(\frac{N}{\Omega}\right)^{4}.$ (20)
Note that the system (16 – 18) is not one of the common three-equation models
used in traditional turbulence modeling. The focus of this modeling branch is
directed on higher and higher orders of the closure equations, e.g. equations
for second and third moments and so forth, all derived from the Navier-Stokes
equation.
In later sections we will see that the system (16 – 18) is dynamically stiff.
Under spatially homogeneous conditions well-defined steady-state solutions
exist, but perturbations of this state are connected with very different
characteristic relaxation times of the systems components. In particular,
$\Omega$ is the component relaxing fastest into what we call structural
equilibrium. The TKE ($\cal K$) relaxes significantly slower into a new state
while the wave energy, $\cal E$, relaxes extremely slowly. The wave-energy
pool needs a longer spin-up time because the flux of shear-generated wave
energy, $\Psi$, consists of random linear wave packets which simply need time
“to meet and break”, of course by chance.
Figure 3: Linear wave packet, hypothetically generated by shear. Breaking
occurs by chance superpositions and results in dissipative patches or billows.
For later use we introduce here the “viscous fraction” $f$ of the total energy
loss of the wave pool towards the heat pool,
$f=\frac{\tilde{L}}{\tilde{L}+\tilde{P}}$ (21)
with the obvious property
$f+\frac{\tilde{P}}{\tilde{L}+\tilde{P}}=1.$ (22)
The fraction $1-f$ describes a “wave age”, i.e. the relative wave-energy loss
into the TKE pool. Clearly,
$0\leq f<1.$ (23)
Figure 4: Solid: laminar (linear) loss fraction of wave energy in the course
of saturation, $f(t)=\tilde{L}(t)/(\tilde{L}(t)+\tilde{P}(t))$; dashed: “wave
age”, $1-f(t)$.
We note in passing that the steady state is not the only dynamically invariant
state of TKE and waves. Also the state of exponential evolution (Van Atta,
1999) in wind tunnels belongs to this class. This state, taken as a reference,
has also the property that perturbations relax back towards reference.
In our present situation where we deal with relaxation times orders of
magnitudes apart, the use of the so-called Tikhonov principle seems to be
helpful. It means to concentrate on processes with moderate relaxation times.
(16) is so fast that it can be taken as being always in structural
equilibrium. (18) is so slow that its time derivative is small compared with
the source and sink terms at the right-hand side and can be neglected. We are
thus left with only one differential and two algebraic equations. However, in
the case of very stiff algebraic equations it is sometimes more useful to
apply the method of non-stationary embedding. Here it would mean to re-
establish the character of (16) and (18) as differential equations and to seek
the stationary solution via relaxation to the stationary state.
## 4 Special cases
Below we discuss some special cases of our general system(16 –18): the
neutrally stratified case ($N=0$), the stratified ($N^{2}>0$) but
geophysically shielded ($\Pi=0$) case, and the stably stratified wind tunnel.
### 4.1 Neutral stratification, $N=0$
Homogeneous shear means constant shear along the horizontal and vertical axes.
The assumption $N=0$ means neutral stratification. Internal gravity waves of
any kind are not supported by the fluid so that all terms in (18) vanish,
together with this equation. Further, the terms $\Psi,\;\;\Pi$, $\tilde{P}$,
$\tilde{S}$ are zero such that eventually (16, 17) look as follows:
$\displaystyle\frac{d\Omega}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left(\frac{S^{2}}{2}-\Omega^{2}\right),$ (24)
$\displaystyle\frac{d{\cal K}}{dt}$ $\displaystyle=$ $\displaystyle
P-\varepsilon.$ (25)
These equations correspond to a clear mechanistic interpretation of turbulence
as dipole chaos in the sense of a two-fluid approach (excitons in form of
quasi-rigid vortex tubes made of inviscid fluid, and the materially identical
inviscid but not excited fluid between the tubes) derived by Baumert (2005 –
2012). As a byproduct, this theory gives von Karman’s number as
$\kappa=1/\sqrt{2\,\pi}=0.399$.
### 4.2 Stable stratification, $N^{2}>0$, $\Pi=0$
Here we mean stratified shear flows under idealized laboratory conditions
where the role of tides and external geophysical influences are excluded, i.e.
$\Pi=0$, which implies also $\tilde{S}=0$, to be discussed later. These
conditions are called below “geophysically shielded”.
Homogeneity means here constant $N^{2}$ and $S^{2}$ in the horizontal plane
and on the vertical axis, which is not easy to realize in a laboratory. But on
a simplistic theoretical level we can get some insight when we consider only
the stationary system (16, 17, 18). The vorticity balance gives trivially
$\Omega^{2}=S^{2}/2$ and is therefore omitted for brevity. It remains the
following:
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
P-\Psi-B-\varepsilon+\tilde{P},$ (26) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\Psi-\tilde{L}-\tilde{P}\,.$ (27)
We neglect the linear molecular friction $\tilde{L}$, rewrite (27) to get
$\tilde{P}\approx\Psi$; wie insert this into (26) and get the following:
$P\approx B+\varepsilon\,.$ (28)
Remarkably, $\Psi$ cancels out of (26, 27) so that finally the oceanographer’s
standard balance formulation (28) is obtained. We come back to this point
later.
In the past experiments with stratified wind tunnels gave deep insights into
the nature of the turbulence-wave interactions (Rohr et al., 1988; Van Atta,
1999; Baumert and Peters 2004, 2005). However, they do not support relation
(28). This is the contradiction we mentioned in the Introduction and will be
discussed next.
### 4.3 The stratified wind tunnel, $\Pi=0$, $R_{g}>0$
#### 4.3.1 The system.
This important case is an example of strong horizontal inhomogeneity and non-
equilibrium conditions. In the entrance facility of the tunnel the forced flow
passes a fine grid which leaves an initial high-frequency short-wave
turbulence and internal-wave signature in a locally homogeneous fluid body of
limited size. We consider this fluid body as moving with the mean flow in a
plug-flow sense. The small-scale properties then evolve during the trip within
the fluid body along the homogeneously sheared and stratified tunnel until its
end, where the body leaves the tunnel, including its small-scale properties.
Typically an exponential evolution of TKE is observed along the longitudinal
axis, either exponential growth or exponential decay (Van Atta, 1999). The
waves were not recorded but it must be hypothesized that they did reach
saturation level.
This situation is artificial. In a natural hydrodynamic system the growth of
turbulence is limited because it would somewhere begin to reduce its own
source (shear) by mixing the mean flow (see Fig. 2) so that an equilibrium
will sooner or later be reached, corresponding to the large-scale energy input
to the flow. But this feedback from the turbulence-wave system to the mean-
flow system takes time and the travel time through the tunnel is too short.
Exactly this sort of decoupling of processes is what the tunnel
experimentalists aim at. They wish in particular to cut off the feedback
$\tilde{P}$ from shear-generated wave-energy into TKE, as it contaminates the
clear and simple picture. In other words, they want to see the naked
interrelations in stratified shear turbulence as studied already by Richardson
(1920), Howard (1961) and Miles (1961). Those three authors neglected shear-
generated waves and their subsequent feedback as a result of spectral
saturation.
#### 4.3.2 Advection-dispersion-reaction (ADR) and the plug-flow concept.
At a first glance the sheared flow of a stratified wind tunnel seems to
represent a major problem for detailed analyses. However, the Taylor-Aris
theory of shear dispersion (Taylor, 1953; Aris, 1956; Baumert, 1973; Fischer
et al., 1979) allows to compute an effective longitudinal dispersion
coefficient, $D_{L}$, and to cast the transport equations corresponding to
(16, 17, 18) in the following general form of an advection-dispersion-reaction
equation:
$\frac{\partial Y}{\partial t}+\frac{\partial}{\partial x}\left(\langle
U\rangle\,Y-D_{L}\,\frac{\partial Y}{\partial x}\right)=-Y/\hat{\tau}.$ (29)
Here $Y$ is a placeholder for the variables $\Omega$, $\cal K$, and $\cal E$,
and $\hat{\tau}$ is an effective time constant of a hypothetically decaying
(or growing, when $\hat{\tau}<0$) variable $Y$.
As long as the Peclet number of the problem, $Pe=\left|\langle
U\rangle^{2}\hat{\tau}/(4\,D_{L})\right|\gg 1$, it can be shown (e.g. Baumert,
1973) that the stationary form of (29, with ${\partial Y}/{\partial t}=0$) can
be simplified into a so-called plug-flow description:
$\langle U\rangle\,\frac{dY}{dx}\approx-Y/\hat{\tau}.$ (30)
In a wind tunnel this concept is a useful approximation because the velocity
$\langle U\rangle$ and thus the above similarity number are typically high
enough. In a stationary plug-flow sense we thus have :
$\displaystyle\langle U\rangle\frac{\partial\Omega}{\partial x}$
$\displaystyle=$
$\displaystyle\frac{1}{\pi}\left(\frac{S^{2}}{2}-\Omega^{2}\right),$ (31)
$\displaystyle\langle U\rangle\frac{\partial\cal K}{\partial x}$
$\displaystyle=$ $\displaystyle P-\Psi-B-\varepsilon+\tilde{P},$ (32)
$\displaystyle\langle U\rangle\frac{\partial\cal E}{\partial x}$
$\displaystyle=$ $\displaystyle\Psi-\tilde{L}-\tilde{P}.$ (33)
Now we introduce the dimensionless travel-time coordinate along the wind-
tunnel axis,
$\hat{t}=x\cdot S/\langle U\rangle.$ (34)
We further introduce dimensionless variables via
$\displaystyle\hat{\Omega}$ $\displaystyle=$
$\displaystyle\Omega\,\sqrt{2}/S\,,$ (35) $\displaystyle\hat{\cal K}$
$\displaystyle=$ $\displaystyle{\cal K}/{\cal K}_{0}\,,$ (36)
$\displaystyle\hat{\cal E}$ $\displaystyle=$ $\displaystyle{\cal E}/{\cal
E}_{0}\,.$ (37)
Here ${\cal K}_{0}$ and ${\cal E}_{0}$ are the initial conditions so that
$\hat{\cal K}_{t=0}=\hat{\cal E}_{t=0}=1$. Due to the initiation of the flow
by the grid the vorticity typically begins with high initial values,
$\hat{\Omega}_{t=0}\gg 1$.
#### 4.3.3 General case.
The above conventions allow to rewrite the transport equations (31, 32, 33)
with some algebra in dimensionless form:
$\displaystyle\frac{d\hat{\Omega}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\left(1-\hat{\Omega}^{2}\right),$ (38)
$\displaystyle\frac{d\hat{\cal K}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\left(2-\frac{2R_{g}}{\sigma_{0}}-\hat{\Omega}^{2}\right)\frac{\hat{\cal
K}}{\pi\,\hat{\Omega}\,\sqrt{2}}+\frac{\tilde{P}}{S\cdot{\cal K}_{0}},$ (39)
$\displaystyle\frac{d\hat{\cal E}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\frac{{\cal K}_{0}}{{\cal
E}_{0}}\left(\frac{\sqrt{8}}{\pi\,\sigma_{0}}\,R_{g}^{2}\;\frac{\hat{\cal
K}}{\hat{\Omega}^{3}}-\frac{\tilde{P}}{S\cdot{\cal K}_{0}}\right).$ (40)
Here we neglected the linear molecular friction term $\tilde{L}=c_{1}\,\cal E$
and replaced according to (19) $\Psi$ with $\nu\,N^{2}/\sigma_{0}-B$. For $B$
we used (11) and (12).
Figure 5: Wind-tunnel turbulence-waves model in dimensionless variables.
$\hat{\Omega}$ (solid blue) converges soon ($\hat{t}\approx 10$) to its
structural-equilibrium value, $\hat{\Omega}_{\infty}=1$. $\hat{K}$ (solid red)
goes through a minimum and starts at $\hat{t}\approx 10$ a phase of
exponential growth. $\hat{\cal E}$ (solid green) rests long time close to its
initial condition $\hat{\cal E}_{0}=1$ and enters at $\hat{t}\approx 10$ into
a phase of exponential growth. The ratio $\tilde{P}/\Psi$ (dashed blue,
$0<\tilde{P}/\Psi<1$) remains initially very small but jumps then around
$\hat{t}\approx 1500$ to its asymptotic value $(\tilde{P}/\Psi)_{\infty}=1.$
In this example $R_{g}=0.16$, $\hat{\alpha}=10^{-6}$, and $\beta=1$. Notice
the double-logarithmic character of the presentation.
We now abbreviate $\alpha={\cal E}_{0}/S$, $\beta={\cal E}_{0}/{\cal K}_{0}$
and $\hat{\alpha}=\alpha\times c_{2}$ so that
$\displaystyle\frac{d\hat{\Omega}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\,\pi}\left(1-\hat{\Omega}^{2}\right),$ (41)
$\displaystyle\frac{d\hat{\cal K}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\left(2-\frac{2R_{g}}{\sigma_{0}}-\hat{\Omega}^{2}\right)\frac{\hat{\cal
K}}{\pi\,\hat{\Omega}\,\sqrt{2}}+\hat{\alpha}\,\beta\,\hat{\cal E}^{2},$ (42)
$\displaystyle\frac{d\hat{\cal E}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\left(\frac{\sqrt{8}}{\pi\,\sigma_{0}}\,R_{g}^{2}\;\frac{\hat{\cal
K}}{\hat{\Omega}^{3}}-\,\hat{\alpha}\,\beta\,\hat{\cal E}^{2}\right).$ (43)
According to Gregg (1989), in the ocean we have $c_{2}=7.4\times 10^{-5}$ s
m-2. $\alpha$ and $c_{2}$ are the only parameters or variables in (41 – 43)
which are not dimensionless. But the two appear only in form of the product
$\hat{\alpha}=\alpha\times c_{2}$, which is again dimensionless and was
already used in (42) and (43). To make an order of magnitude estimate of
$\hat{\alpha}$ we take for $S$ the Garret-Munk value, $S_{GM}=0.0036$ s-1, and
$\hat{\cal E}\approx 10^{-4}$ m2s-2. This gives a characteristic guess of
$\hat{\alpha}\approx{\cal O}(10^{-6}$). This value was used to compute the
data in Fig. 5.
The system (41 – 43) can be solved numerically by standard methods if its high
stiffness is adequately taken into account999Numerical overflows may occur. If
automatic stiffness techniques are applied the solution may begin to switch
periodically between different methods. It is always helpful to reduce the
maximum time step as far as possible.. The first phase of the evolution based
on (41 – 43) is shown in Fig. 5 where we may identify three regimes with two
separating breakpoints. The first regime is the initialization or spin-up
regime. The first breakpoint labels its end at $\hat{t}\approx 10$ and is
associated with the transition into structural equilibrium. The second regime
may be called the typical wind-tunnel regime (an artificially decoupled or
naked turbulence-wave system in exponential growth, $\tilde{P}\approx 0$). Its
end is labeled by the second breakpoint at $\hat{t}\approx 1500$. The third
regime can be called a hyper-equilibrium regime characterized by
$\tilde{P}=\Psi$ and exponential growth, too. Both exponential growth regimes
exhibit the property
$\displaystyle\frac{1}{\hat{K}}\frac{d\hat{K}}{d\hat{t}}$ $\displaystyle=$
$\displaystyle\mbox{constant}\,,$ (44) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{d}{d\hat{t}}\left(\frac{1}{\hat{K}}\frac{d\hat{K}}{d\hat{t}}\right)\,.$
(45)
We use for regime 3 the name hyper equilibrium because it has the condition
$\tilde{P}=\Psi$ in common with the natural equilibrium (oceanographer’s
regime) but differs with respect to the dynamic state: while in the natural
case all first time derivatives vanish, in the hyper case they are constants
$>0$, but the second derivatives vanish.
The simulations show that in an ideal stratified wind tunnel of infinite
length the hyper-equilibrium regime can in principle be reached, e.g. for
$\hat{t}\gg 1500$. But such a length can hardly be achieved in practice. Thus
the most interesting and scientifically unique part in a stratified wind
tunnel (or a salt-stratified channel flow) is the section between the lower
and the upper breakpoint.
In a somewhat sloppy form we may summarize: the total travel time through the
wind tunnel is too short for the shear-generated waves to develop a saturated
spectrum which would almost equate the breaking term $\tilde{P}$ with the
generation term $\Psi$.
## 5 Stably stratified natural shear flows: $R_{g}>0$,
$\Pi>0,\tilde{S}^{2}>0$
We now come back to the system (16, 17, 18) and study its the TKE balance
where the waves and the vorticity are assumed to stay in a steady state:
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
S^{2}/2-\Omega^{2}+\tilde{S}^{2},$ (46) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\Pi+\Psi-\tilde{L}-\tilde{P}\,,$ (47) $\displaystyle\frac{d\cal
K}{dt}$ $\displaystyle=$ $\displaystyle P+\tilde{P}-\Psi-B-\varepsilon\,.$
(48)
### 5.1 TKE balance and mixing efficiencies
The definition (21) allows now to rewrite the TKE balance (48) as follows,
$\frac{d\cal K}{dt}=(1-f)\Pi+P-B-\varepsilon-f\,\Psi\,.$ (49)
Shear and occasionally overturning waves are qualitatively different
generation mechanisms with differing spectral signatures, differing mixing
efficiencies and differing buoyancy fluxes. Both mechanisms are associated
with buoyancy fluxes which add up (parallel circuitry??) to the total buoyancy
flux $B$:
$\displaystyle B$ $\displaystyle=$ $\displaystyle B^{\prime}+\tilde{B}\,.$
(50)
The so-called mixing efficiency, $\Gamma$, is generally defined as
$\Gamma=B/\varepsilon$. Due to the additive nature of $B=B^{\prime}+\tilde{B}$
also $\Gamma$ is additive:
$\Gamma=\frac{B}{\varepsilon}\;=\frac{B^{\prime}+\tilde{B}}{\varepsilon}\;=\;\frac{B^{\prime}}{\varepsilon}+\frac{\tilde{B}}{\varepsilon}=\Gamma^{\prime}+\tilde{\Gamma}\,.$
(51)
With these preliminaries we may rewrite (49) as follows:
$\frac{d\cal K}{dt}=(1-f)\,\Pi+P-B^{\prime}-\tilde{B}-\varepsilon-f\,\Psi\,.$
(52)
#### 5.1.1 Mixing efficiency of purely shear-generated turbulence:
$\Gamma^{\prime}=\Gamma^{\prime}(R_{g});\;\tilde{B}=0,\;\tilde{P}=0,\;\Pi=0,\;\tilde{S}^{2}=0.$
Using exclusively definitions like (10) and (8) given in the previous sections
and subsections, we begin here to specify $B^{\prime}$ as follows:
$B^{\prime}=\varepsilon\;\Gamma^{\prime}=\mu\,N^{2}=\frac{\nu}{\sigma}\,N^{2}=\frac{\nu}{\sigma_{0}}\left[1-\left(\frac{N}{\Omega}\right)^{2}\right].$
(53)
We solve (53) for $\Gamma^{\prime}$ and get the following:
$\Gamma^{\prime}=\frac{\nu}{\sigma}\,\frac{N^{2}}{\varepsilon}=\frac{1}{\sigma}\left(\frac{N}{\Omega}\right)^{2}=\frac{1}{\sigma_{0}}\left(\frac{N}{\Omega}\right)^{2}\left[1-\left(\frac{N}{\Omega}\right)^{2}\right].$
(54)
In the pure shear-generated case, $\Pi=\tilde{S}=0$, we thus have
$\Omega^{2}=S^{2}/2$, and with $R_{g}=N^{2}/S^{2}$ (54) rewrites as follows:
$\Gamma^{\prime}=\frac{2}{\sigma_{0}}R_{g}\left(1-2\,R_{g}\right).$ (55)
This function is presented in Fig. 6 and is well supported by observations as
shown in an earlier study by Baumert and Peters (2004, 2005).
Figure 6: The mixing efficiency $\Gamma^{\prime}$ according to equation (55)
for shear-generated turbulence.
#### 5.1.2 Mixing efficiency of purely wave-generated turbulence:
$\tilde{\Gamma}=\tilde{\Gamma}_{0}=0.2;\;P=\Psi=0,\;S=0.$
The mixing situation is qualitatively different in the case without mean-flow
shear where turbulence is exclusively generated by wave spectra fed eventually
by large-scale long-wave external sources. According to the state of the art
(Osborne, 1980; Oakey, 1982; Gregg et al., 1986; Peters et al. 1988), the
mixing efficiency may still be taken to be a universal constant,
$\tilde{\Gamma}=\tilde{\Gamma}_{0}=0.2$. Clearly, in purely shear-generated
turbulence a gradient-Richardson number is not available so that
$\tilde{\Gamma}$ cannot be a function of it. Further, in this case this number
is not even defined because the shear is zero.
### 5.2 The remote-wave closure and oceanographer’s balance:
$P=\Psi=0,\;S=0,\;\Pi>0,\;\tilde{S}^{2}>0$
Here we will derive an approximate relation between the large-scale long-wave
source of wave-generated turbulence (“remote waves”), $\Pi$, and the
correspondingly induced component $\tilde{S}^{2}$ in the vorticity balance
(46). We choose a flow without mean-flow shear, $S=0$ and $B^{\prime}=0$, such
that, according to (46), $\Omega^{2}=\tilde{S}^{2}$. The TKE balance (52)
reads in this case
$\frac{d\cal K}{dt}=(1-f)\,\Pi-\tilde{B}-\varepsilon\,,$ (56)
and in the steady state:
$(1-f)\,\Pi=\tilde{B}+\varepsilon.$ (57)
The remaining rate $f\,\Pi$ describes the linear or molecular energy loss of
the wave field.
We replace $\tilde{B}$ with $\tilde{\Gamma}\,\varepsilon$ and get
$(1-f)\,\Pi=\tilde{\gamma}\,\varepsilon=\tilde{\gamma}\,\nu\,\Omega^{2}=\tilde{\gamma}\,\nu\,\tilde{S}^{2}$
(58)
so that the following is finally the closure of our problem:
$\tilde{S}^{2}=\frac{(1-f)\,\Pi}{\tilde{\gamma}\,\nu}\,.$ (59)
## 6 Natural coexistence equilibria
We consider now the general natural coexistence of waves and turbulence where
$0\leq f<1$, $\Pi>0$, $P>0$ etc.
### 6.1 Vorticity
We insert the approximate closure relation (59) in the vorticity balance (46)
and get with the abbreviation $\tilde{\gamma}=1+\tilde{\Gamma}$
$\Omega^{2}=\frac{S^{2}}{2}+\frac{(1-f)\,\Pi}{\tilde{\gamma}\;\nu}\,.$ (60)
For convenience reasons we introduce $\eta$ and choose for (60) the following
presentation,
$\eta=\frac{\Omega^{2}}{S^{2}}=\frac{1}{2}+\frac{X}{\tilde{\gamma}},$ (61)
where for our convenience
$X=(1-f)\,\frac{\Pi}{\nu\,S^{2}}=(1-f)\,\frac{\Pi}{P}.$ (62)
### 6.2 TKE
Now we discuss the steady-state version of the general TKE balance (52):
$(1-f)\,\Pi+P=B^{\prime}+\tilde{B}+\varepsilon+f\,\Psi\,.$ (63)
According to our previous discussions, it may be written as follows:
$(1-f)\,\Pi+\nu\,S^{2}=\frac{\nu}{\sigma}\,N^{2}+\tilde{\gamma}\,\varepsilon+\frac{f}{\sigma_{0}}\left(\frac{N}{\Omega}\right)^{4}\varepsilon.$
(64)
We rewrite this equation identically as
$(1-f)\,\Pi+\nu\,S^{2}=\frac{\nu}{\sigma}\,N^{2}+\left[\tilde{\gamma}+\,\frac{f}{\sigma_{0}}\left(\frac{N}{\Omega}\right)^{4}\right]\nu\,\Omega^{2},$
(65)
and divide both sides of (65) by $\nu\,S^{2}$ to get with (62)
$1+X=\frac{R_{g}}{\sigma}+\left[\tilde{\gamma}+\,\frac{f}{\sigma_{0}}\left(\frac{N}{\Omega}\right)^{4}\right]\eta\;.$
(66)
We now expand $(N/\Omega)^{4}$ into
$(N/S)^{4}/(\Omega/S)^{4}=R_{g}^{2}/\eta^{2}$, replace $\eta$ with (61) and
get finally
$\left(\frac{N}{\Omega}\right)^{4}=\left(\frac{R_{g}}{\eta}\right)^{2}=\left(\frac{2\,\tilde{\gamma}\,R_{g}}{\tilde{\gamma}+2\,X}\right)^{2}.$
(67)
Finally, with the helpf of (61), (67) and with the abbreviation (66) can be
brought into the following final form:
$\displaystyle
1+X=\frac{R_{g}}{\sigma}+\left(\tilde{\gamma}+\,\frac{f}{\sigma_{0}}\;\frac{R_{g}^{2}}{\eta^{2}}\right)\,\eta\,,$
(68)
so that
$\displaystyle
1+X=\frac{R_{g}}{\sigma}+\tilde{\gamma}\,\eta+\,\frac{f}{\sigma_{0}}\;\frac{R_{g}^{2}}{\eta}\,,$
(69)
where we remember that $\eta=1/2+X/\tilde{\gamma}.$ We solve (69) for
$\sigma$:
$\displaystyle\sigma_{(1)}=\frac{R_{g}}{1-\tilde{\gamma}/2-4\,\tilde{\gamma}\,f\,R_{g}^{2}/(\tilde{\gamma}+2\,X)}\,.$
(70)
This function should also satisfy the definition (8) of the turbulent Prandtl
number. This means that
$\sigma_{(2)}=\frac{\sigma_{0}}{1-(N/\Omega)^{2}}=\frac{\sigma_{0}\,\eta}{\eta-
Rg}$ (71)
and with (61) we get
$\sigma=\frac{(\tilde{\gamma}+2\,X)\,\sigma_{0}}{\tilde{\gamma}+2\,X-2\,\tilde{\gamma}\,Rg}=\sigma_{(2)}(X,R_{g})\,.$
(72)
Here $\sigma_{0}$ and $\tilde{\gamma}=1+\tilde{\Gamma}=1.2$ are a universal
constants.
Now our unknown $X=(1-f)\,\Pi/P$ is easily determined by equating (69) and
(72):
$\sigma_{(1)}(f,X,R_{g})=\sigma_{(2)}(X,R_{g})\,.$ (73)
This gives the solution
$X=X(f,R_{g})\,.$ (74)
Now the knowledge of $X$ as function of $f$ and $R_{g}$ allows to present (72)
in the following form:
$\sigma=\sigma_{(1)}(X_{(f,R_{g})},R_{g})=\sigma_{(2)}(f,R_{g})\,.$ (75)
The last expression is a family of curves giving us, in the way we are used
to, for each value of $f\in(0,1)$ one curve $\sigma=\sigma(R_{g})$ as function
of the gradient Richardson number.
### 6.3 Validity limits
For simplicity we analyze the system’s behavior for $f=0$ and solve the
following equation,
$\sigma_{(1)}(f=0,X,R_{g})=\sigma_{(2)}(X,R_{g})\,,$ (76)
and find
$\frac{2\,R_{g}}{2-\tilde{\gamma}}=\frac{(\gamma+2X)\sigma_{0}}{\tilde{\gamma}+2(X-\tilde{\gamma}Rg)}\,,$
(77)
which is easily solved for $X$:
$X=\frac{1}{2}\;\frac{R_{g}-2\,\tilde{\gamma}\,R_{g}^{2}-\sigma_{0}\,(2-\tilde{\gamma})\,\tilde{\gamma}}{\sigma_{0}\,(2-\tilde{\gamma})/2-R_{g}}\,.$
(78)
Fig. 7 illustrates the function $X=X(R_{g})$ around the singular point,
$R_{g}=R_{g}^{(1)}$:
$R_{g}^{(1)}=(2-\tilde{\gamma})\frac{\sigma_{0}}{2}=(1-\tilde{\Gamma})\frac{\sigma_{0}}{2}=0.2\,.$
(79)
We see that
$X=\left\\{\begin{array}[]{ll}<0&\mbox{if}\quad R_{g}<R_{g}^{(1)},\\\
>0&\mbox{if}\quad R_{g}>R_{g}^{(1)}.\\\ \end{array}\right.$ (80)
We remember that according to (62) negative $X\propto\Pi/P$ such that $X<0$
mean sucking internal-wave energy out of the wave-energy pool. We accept
therefore that for $R_{g}<R_{g}^{(1)}$ a physically reasonable solution for an
equilibrium coexistence of waves and turbulence does not exist.
The function $X$ in Fig. 7 exhibits an obvious minimum. We take (78),
differentiate and find the zero of $dX/dR_{g}$ here:
$R_{g}^{(2)}=(2-\tilde{\gamma})\,\sigma_{0}=(1-\tilde{\Gamma})\,\sigma_{0}=0.4.$
(81)
Considering this point in Fig. 7, the solution on the left is physically
unrealistic because a decreasing $R_{g}$ would lead to an increase in $X$.
With decreasing $R_{g}$ into a region left of the minimum in $X$ we approach a
zone where the model is simply no longer correct. The reason is surely the
vorticity balance which has been deduced from wind-tunnel experiments and then
combined with the new remote-wave closure (59). Consequently its work is
guaranteed only in two cases:
* •
purely shear-generated turbulence with exclusion of eigen-wave feedback, i.e.
$\tilde{P}=0$ (wind tunnel);
* •
shear-generated turbulence with inclusion of eigen-wave feedback,
$\tilde{P}=(1-f)(\Pi+\Psi)>0$, but dominant presence of remote waves, i.e.
$X\propto\Pi/P\gg 1$ (ocean, atmosphere).
Unfortunately, the important case with of eigen-wave feedback,
$\tilde{P}=(1-f)\Psi>0$, but without remote wave-energy source,
$X\propto\Pi/P=0$, is not understood so far. It plays a role when a flow setup
is shielded from outer influences. This may play a role in technical systems
like circulating cooling ponds where the waves stem not from geophysical
sources and where they evolve into a sufficiently saturated spectrum. This
will be discussed in the next Section 7.
### 6.4 Asymptotic of $\sigma$ for $R_{g}\rightarrow\infty.$
We take the expression (78) for $X$ and look at very high $R_{g}$ where in
$R_{g}$ quadratic term dominate constants and linear terms such that
$\lim_{R_{g}\rightarrow\infty}X(R_{g})=\frac{-2\,\tilde{\gamma}\,R_{g}^{2}}{-2\,R_{g}}={\tilde{\gamma}\,R_{g}}\,.$
(82)
The study of the Prandtl number function at high $R_{g}$ is still more easy.
We take the left-hand side of (77),
$\sigma_{(1)}=2\,R_{g}/(2-\tilde{\gamma})$, and use the definition
$\tilde{\gamma}=1+\tilde{\Gamma}=1.2$ given already previously. Obviously we
have
$\lim_{R_{g}\rightarrow\infty}\sigma(R_{g})=\frac{2}{1-\tilde{\Gamma}}\,R_{g}=2.5\,R_{g}\;.$
(83)
For comparison with the numerical solution and with observational data we
refer to Fig. 9.
Figure 7: The relative wave-energy input to the TKE pool, $X=(1-f)\,\Pi/P$, as
a function of the gradient Richardson number, $R_{g}$. Negative $X$ indicate
withdrawal of energy from the wave-energy pool and are physically irrelevant.
## 7 Geophysically shielded systems
These systems have no external sources of internal-wave energy but they are in
coexistence equilibrium of turbulence and saturated waves with significant
eigen-wave feedback, $f\ll 1$. Here $\Pi=0$ and the steady-state TKE balance
reads in our notation as follows:
$P=B^{\prime}+\tilde{B}+\varepsilon\,+f\,\Psi\,=B^{\prime}+\tilde{\gamma}\,\varepsilon\,+f\,\Psi\,.$
(84)
The vorticity balance is
$\eta=\frac{\varepsilon}{P}=\frac{\Omega^{2}}{S^{2}}=\frac{1}{2}+(1-f)\,\frac{Y}{\tilde{\gamma}}\,,$
(85)
where for young waves, $f\approx 1$, the effect of $Y$ is negligible and the
situation is close to the wind-tunnel case. Not so for older wave spectra.
We combine (84) and (85) and get after some algebra the following steady-state
condition,
$2=4R_{g}\left(1-\frac{2\,\tilde{\gamma}\,R_{g}}{\tilde{\gamma}+2\,(1-f)\,Y}\right)+\tilde{\gamma}+2\,(1-f)\,Y\,,$
(86)
where $Y$ appears as a function of the steady-state Richardson number
$R_{g}^{s}$, and of the ‘wave age’ $1-f$. The solution $Y=Y(R_{g}^{s})$ of
(88) is presented in Fig. 8 for $f=0$.
The above means that with $Y$ we have a tunable parameter which allows us to
adjust our model value for $R_{g}^{s}$ according to measurements or
observations. Unfortunately these are rare for shielded conditions described
above so that we are inclined to choose according to tradition the value
$R_{g}^{s}=1/4$, which corresponds to $Y=0.136$. Another choice would be the
minimum value of $Y$ which is 0.135 and corresponds to $R_{g}^{s}=0.265$. This
somewhat arbotrary situation probably explains the large scatter in the
measurements of $\sigma$ and underlines umso mehr the necessity of dedicated
experiments and observational programmes.
We note in closing this Section that the more general form of the steady-state
vorticity balance is
$\eta=\frac{\varepsilon}{P}=\frac{\Omega^{2}}{S^{2}}=\frac{1}{2}+\frac{X+(1-f)\,Y}{\tilde{\gamma}}\,,$
(87)
where the external forces $X\propto\Pi/P$ appear together with the feedback
via eigen waves, $Y$. Our values of $Y$ are situated well below the validity
limit of $X$. In the geophysically shielded case ($X=0$) and young waves
($f\approx 1$) the action of $Y$ is screened and we have $\eta=1/2$, which is
the wind tunnel situation. In the same case with adult spectra ($f\ll 1$) we
have the classical shielded case.
Figure 8: The parameter $Y$ as a function of the steady-state gradient
Richardson number $R_{g}^{st}$ and for $f=0$. The horizontal and the right
vertical dashed gray lines cross at the minimum of the function $Y$. The left
vertical dashed line labels $R_{g}^{st}=\hbox{$\,{}^{1}\\!/_{4}$}$. Values
$Y\approx 0.135\dots 0.15$ would correspond to mathematically admissible
values $R_{g}^{st}\approx 0.15\dots 0.3$.
## 8 Application
In the previous Sections of this report we have looked at special physical
situations which we knew and understood sufficiently well. Now we put these
pieces together and write down the full equation system for applications also
to unknown situations. We choose here the philosophy of non-stationary
embedding. At a first glance the associated evolution equations (88 – 99) look
voluminous compared with the lean system of their purely algebraic steady-
state counterparts, but non-stationary embedding avoids stiffness problems
right on the most fundamental level and is thus substantially more robust in
the computational practice.
### 8.1 Generalized equations
For an effective notation we define the following differential operator,
$\displaystyle{\cal D}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial}{\partial t}-\frac{\partial}{\partial
z}\,\nu\,\frac{\partial}{\partial z}\right),$ (88) $\displaystyle\tilde{\cal
D}$ $\displaystyle=$ $\displaystyle\left(\frac{\partial}{\partial
t}-\frac{\partial}{\partial z}\,\tilde{\nu}\,\frac{\partial}{\partial
z}\right),$ (89)
so that the general set of balances can be written for a stratified water
column as follows:
$\displaystyle{\cal D}\,\Omega$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\left[\frac{S^{2}}{2}+\frac{1-f}{1+\tilde{\Gamma}}\left(\frac{\Pi}{P}+Y\right)S^{2}-\Omega^{2}\right],$
(90) $\displaystyle\tilde{\cal D}\,{\cal K}$ $\displaystyle=$
$\displaystyle\Pi+P-\nu\left[f\,\Psi+\frac{N^{2}}{\sigma}+(1+\tilde{\Gamma})\Omega^{2}\right]$
(91) $\displaystyle\tilde{\cal D}\,{\cal E}$ $\displaystyle=$
$\displaystyle\Pi+\Psi-c_{1}\,{\cal E}-c_{2}\,{\cal E}^{2}\;,$ (92)
$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle\frac{c_{1}\,{\cal
E}}{c_{1}\,{\cal E}+c_{2}\,{\cal E}^{2}}\,,$ (93) $\displaystyle\sigma$
$\displaystyle=$ $\displaystyle\frac{\sigma_{0}}{1-N^{2}/\Omega^{2}}\;,$ (94)
$\displaystyle P$ $\displaystyle=$ $\displaystyle\nu\,S^{2}\;,$ (95)
$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\frac{P}{\sigma_{0}}\,\frac{S^{2}}{\Omega^{2}}\;R_{g}^{2}\,,$
(96) $\displaystyle\nu$ $\displaystyle=$ $\displaystyle{\cal
K}/{\pi\,\Omega}\,,$ (97) $\displaystyle\mu$ $\displaystyle=$
$\displaystyle{\nu}/\sigma\,,$ (98) $\displaystyle R_{g}$ $\displaystyle=$
$\displaystyle N^{2}/S^{2}\,.$ (99)
To apply this theory in form of a numerical model it needs to be combined with
a scheme which provides us with the mean-flow variables from which we may
derive the shear $S$ and the Brunt-Väisälä frequency, $N$. Furthermore we need
initial conditions for all variables. But the hydrodynamic system is highly
dissipative and ‘forgets’ the initial conditions soon such that here
“reasonable guesses” would suffice.
### 8.2 The parameters
#### 8.2.1 Overview.
$\tilde{\Gamma}=0.2$ and $\sigma_{0}=1/2$ are universal constants. $f$ is a
function of time. Its final equilibrium value, $f_{\infty}$, depends on the
molecular-viscosity parameter $c_{1}$. In many cases it is sufficient to set
$f_{\infty}=0$.
With the exception of $\tilde{\nu}$, the whole system contains only one
tunable paramter, $Y\sim{\cal O}(10^{-1})$. The inner physical structure of
$Y$ (and $\tilde{\Gamma}$) we have not yet fully understood. I.e. we are not
able to derive its value from other than pragmatic arguments like ‘it works’,
because it gives the right gradient Richardson number for the geophysically
shielded steady-state.
About the spatial ‘diffusivity’ of wave packets, $\tilde{\nu}$, we only know
that it scales with the characteristic group velocity $\langle c_{g}\rangle$
of the wave packets, multiplied by a characteristic length scale which is
possibly the characteristic wave lenght of a packet:
$\displaystyle\tilde{\nu}$ $\displaystyle\propto$ $\displaystyle\langle
c_{g}\rangle\times\cal L\,.$ (100)
The last open problem to be discussed is the role of $\Pi$. According to Gregg
(1989) it can be estimated from the r.m.s. 10-meter wave shear, $\langle
S_{10}^{2}\rangle$, as function of time and of the region of the world ocean
under study. This will be done in the next Subsection.
### 8.3 Wave-induced dissipation: the ocean case
The above results apply to stratified oceanic and atmospheric flows as well.
The following estimator of the long-wave, non-local (“external”) energy source
$\Pi$ is based on extensive studies in the world oceans and is thus not
automtically applicable to atmospheric conditions. For the latter a comparable
result is unknown.
Gregg (1989) presented a summary of comprehensive, extensive and direct
dissipation and stadardized shear observations ($S_{10}$) made in ocean waters
around the globe, where the following conditions applied at least
approximately:
* •
There was almost no mean-flow shear, $S\approx 0$.
* •
The IGW field was almost perfectly saturated, $f\approx 0$.
* •
The observations were done for conditions of quasi-steady state,
$d\Omega/dt=d{\cal K}/dt=d{\cal E}/dt=0$.
Gregg established the following empirical relation between the wave-induced
dissipation rate, $\tilde{\varepsilon}$, the 10-meter high-pass filtered
vertical shear, $S_{10}=\Delta U/10\,\mbox{m}$, and the effective Brunt-
Väisälä frequency, $\langle N^{2}\rangle^{1/2}$:
$\displaystyle\tilde{\varepsilon}$ $\displaystyle=$ $\displaystyle
a_{1}\,\frac{\langle N^{2}\rangle}{N_{0}^{2}}\;\frac{\langle
S_{10}^{4}\rangle}{S_{GM}^{4}}\;.$ (101)
Very low frequencies have been removed from $S_{10}$ by filtering. The average
in $\langle S_{10}^{4}\rangle$ is taken over longer observation periods.
$S_{GM}$ used in (101) is the so-called Garrett-Munk shear:
$\displaystyle S_{GM}^{4}$ $\displaystyle=$ $\displaystyle
a_{2}\,\frac{\langle N^{2}\rangle^{2}}{N_{0}^{4}},$ (102)
with the following empirical parameters:
$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle 7\times
10^{-10}\;\mbox{m${}^{2}$\,s${}^{-3}$}\,,$ (103) $\displaystyle a_{2}$
$\displaystyle=$ $\displaystyle 1.66\times 10^{-10}\;\mbox{s${}^{-4}$}\,,$
(104) $\displaystyle N_{0}$ $\displaystyle=$ $\displaystyle 5.2\times
10^{-3}\;\mbox{s${}^{-1}$}\,.$ (105)
We insert (102) in (101) and get
$\displaystyle\tilde{\varepsilon}$ $\displaystyle=$ $\displaystyle
a_{3}\,\frac{\langle S_{10}^{4}\rangle}{\langle N^{2}\rangle}\,,$ (106)
where
$\displaystyle a_{3}$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{a_{2}}\,N_{0}^{2}\;\;=\;\;1.14\times
10^{-4}\;\mbox{m}^{2}\,\mbox{s}^{-1}\,.$ (107)
We further take into account that (see Gregg, 1989)
$\displaystyle\langle S_{10}^{4}\rangle$ $\displaystyle=$ $\displaystyle
2\,\langle S_{10}^{2}\rangle^{2}$ (108)
such that (106) may be written as follows,
$\displaystyle\tilde{\varepsilon}$ $\displaystyle=$ $\displaystyle
a_{4}\,\frac{\langle S_{10}^{2}\rangle^{2}}{\langle
N^{2}\rangle}\;=\;a_{4}\,\frac{\langle S_{10}^{2}\rangle}{\tilde{R}_{g}}\,,$
(109)
with the wave-based gradient Richardson number,
$\displaystyle\tilde{R}_{g}$ $\displaystyle=$ $\displaystyle\frac{\langle
S_{10}^{2}\rangle}{\langle N^{2}\rangle}.$ (110)
The latter needs to be unterschieden von the mean-field based gradient
Richardson number $R_{g}=\langle N^{2}\rangle/\langle S\rangle^{2}$. The
parameter $a_{4}$ reads as follows:
$\displaystyle a_{4}$ $\displaystyle=$ $\displaystyle
2\,a_{3}\;\;=\;\;2.3\times 10^{-4}\;\mbox{m${}^{2}$\,s${}^{-1}$}\,.$ (111)
With (109) we have a solid estimate of $\Pi$ as
$\displaystyle\Pi=$ $\displaystyle=$
$\displaystyle(1+\tilde{\Gamma})\,\varepsilon\;=\;\;a_{4}\,(1+\tilde{\Gamma})\,\frac{\langle
S_{10}^{2}\rangle}{\tilde{R}_{g}}\,,$ (112)
so that herewith the model system is completely closed and we can begin with
simulations, provided we have a certain knowledge about the wave-field
parameters $\langle S_{10}^{2}\rangle$ and $\tilde{R}_{g}$ for our ocean or
atmosphere region of interest.
## 9 Comparison with observations
During the about 15-years walk towards the system (88 – 99) partial solutions
were step by step confronted with corresponding observational and experimental
data. The totality of these theory-observation comparisons cannot be repeated
here. We refer for the wind-tunnel experiments to Baumert and Peters (2004,
2005), for the Monin-Obukhov boundary layer to Baumert (2005a), for the
neutrally stratified case to Baumert (2005b, 2012). These specific cases are
well described by subsets of (88 – 99), corresponding individually to the
special physical situation studied.
With respect to a theory-data comparison the present article thus concentrates
on the case $R_{g}\gg 1$ where without mentioning it $R_{g}$ means here always
an equilibrium or steady-state value. We look namely on the relation
$\sigma=\sigma(R_{g})$, analyzed and summarized by Zilitinkevich et al.
(2008). These authors used more or less the same comprehensive data base for
their discussions like Galperin et al. (2007), Canuto et al. (2008) and Kantha
and Carniel (2009). In the core these are the CASES-99 (stable nocturnal BL,
see Poulos et al., 2002) and the SHEBA experiments (arctic BL over ice; see
Grachev et al., 2005, 2006).
Zilitinkevich et al. (2008) conclude that the class-wise average of the data
is best described by $\sigma=0.8+5\,R_{g}$ shown in full red in Fig. 9. In our
view this line is somewhat above the CASES-99 data and our alternative
theoretically derived relation, $\sigma=(1+{5}\,R_{g})/2$, fits better. It is
given in full green in Fig. 9 and located somewhat below the full-red curve.
Of course, in view of the enormous scatter of the original data, the red and
the green lines are well within the huge overall scatter range.
The two blue lines (full and dashed blue) in Fig. 9 represent our simulation
results in good agreement with the observations of CASES-99. At the same time
they might illustrate why natural data exhibit such a strong scatter. While
the dashed blue line is the case $f=0.5\,\%$, the full blue line stands for
$f=0$. I.e. very small absolute variations in $f$ cause large variations in
the solutions. Generally the huge scatter might be caused by the varying age
(or degree of saturation) of the waves involved. These waves are mostly of
non-local origin so that in principle all points of the distant space are
candidates for their birth places and any age of arriving waves can be
expected at our study site. On the one hand structural equilibrium needs to be
achieved to avoid such a scatter. This appears to be difficult under the
action of waves which also modify the flow. But still more important seems to
be the different age of the incoming wave spectra.
According to our theory Fig. 9 shows also the two new critical gradient
Richardson numbers for the wave-turbulence coexistence in form of two vertical
thin black dashed lines at $R_{g}^{(1)}=(1-\tilde{\Gamma})/4=0.2$ and
$R_{g}^{(2)}=(1-\tilde{\Gamma})/2=0.4$. The higher value labels the lower
applicability limit of our theory.
Figure 9: Turbulent Prandtl number as function of the gradient Richardson
number for various coexistence equilibria of turbulence and internal waves.
The theory in dashed blue contains the relative viscous energy loss of the
wave field, $f$, taken here as $0.5\,\%$. Zilitinkevich et al. (2008)
recommend as a phenomenological rule the relation $\sigma\approx
0.8+5\times\,R_{g}$ (full, red), which does not contain the pole visible in
the CASES-99 data and our $f=0.5\;\%$ case. Only in the case $f=0$ our theory
gives a straight line (full blue) which gives asymptotically
$\sigma=(1+5\times R_{g})/2$ (full green). The thin black dashed vertical
lines indicate the positions 0.2 and 0.4. The thin black horizontal lines
label 0.5 and 1.
## 10 Discussion
This report may be seen as an experiment to draw a (dotted) theoretico-
physical bottom line under more than 50 years of ambitious observational
programs and experimental laboratory research into stratified small-scale
geophysical and engineering turbulence and mixing processes. Looking
backwards, today these programs appear as a single planned initiative wherein
mutually supplementing pieces fit perfectly together. On the water side the
efforts took place in the United States and the United Kingdom. On the
atmospheric side also multi-national efforts need to be acknowledged. Our
results may encourage those who believe that the “turbulence problem” (Hunt,
2011) is eventually solvable.
###### Acknowledgements.
This study is a continuation of a long-term research engagement in turbulence
and mixing in oceans and inland waters, initially supported by the European
Union s projects PROVESS (MAS3-CT97-0159) and CARTUM (MAS3-CT98-0172), later
by the U.S. National Science Foundation (OCE-9618287, OCE-9796016,
OCE-98195056). The actual last phase was partially supported by the Department
of the Navy Grant N62909-10-1-7050 issued by the Office of Naval Research
Global. The United States Government has thus a royalty-free license
throughout the world in all copyrightable material contained herein. Further
partial support by the German BMBF, Ministry for Research and Education in the
context oft he WISDOM-2 project, is gratefully acknowledged. The author
further highly acknowledges the cooperation with Dr. Hartmut Peters from Earth
and Space Research in Seattle, USA. His constant advice, corrective comments,
interest and help made this study possible.
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|
arxiv-papers
| 2012-07-05T15:01:27 |
2024-09-04T02:49:32.738269
|
{
"license": "Public Domain",
"authors": "Helmut Z. Baumert",
"submitter": "Helmut Z. Baumert",
"url": "https://arxiv.org/abs/1207.1633"
}
|
1207.1705
|
# Updown Categories: Generating Functions and Universal Covers
Michael E. Hoffman
Dept. of Mathematics, U. S. Naval Academy
Annapolis, MD 21402 USA
meh@usna.edu
(July 6, 2012
Keywords: category, poset, differential poset, universal cover, partition,
rooted tree
MR Classifications: Primary 18B35, 06A07; Secondary 57M10, 05A17, 05C05)
###### Abstract
A poset can be regarded as a category in which there is at most one morphism
between objects, and such that at most one of
$\operatorname{Hom}(c,c^{\prime})$ and $\operatorname{Hom}(c^{\prime},c)$ is
nonempty for $c\neq c^{\prime}$. If we keep in place the latter axiom but
allow for more than one morphism between objects, we have a sort of
generalized poset in which there are multiplicities attached to the covering
relations, and possibly nontrivial automorphism groups. We call such a
category an “updown category.” In this paper we give a precise definition of
such categories and develop a theory for them. We also give a detailed account
of ten examples, including updown categories of integer partitions, integer
compositions, planar rooted trees, and rooted trees.
## 1 Introduction
Suppose we have a collection of combinatorial objects, naturally graded, so
that any object of rank $n$ can be built up in $n$ steps from a single object
in rank 0. Further, for any object $p$ of rank $n$ and $q$ of rank $n+1$,
there are some number $u(p;q)$ of ways to build up $p$ to make $q$, and
$d(p;q)$ ways to break down $q$ to get $p$. Here $u(p;q)$ and $d(p;q)$ are
nonnegative integers, possibly unequal, though we require that $u(p;q)\neq 0$
if and only if $d(p;q)\neq 0$. For example (as in [6]) our collection might be
the set of rooted trees, with $u(p;q)$ the number of vertices of the rooted
tree $p$ to which a new edge and terminal vertex can be added to get $q$, and
$d(p;q)$ the number of terminal vertices (and incoming edges) that can be
removed from $q$ to get $p$.
We can obtain a natural definition of such a situation by modifying the
categorical definition of a poset. A poset is usually thought of as a category
with at most one morphism between objects, and at most one of the sets
$\operatorname{Hom}(p,q)$ and $\operatorname{Hom}(p,q)$ nonempty when $p\neq
cq$. If we keep in place the second condition but permit
$\operatorname{Hom}(p,q)$ to have more than one element, we allow for
multiplicities (if $p\neq q$) and automorphisms (if $p=q$). If in addition the
object set is graded, we call such a category (precisely defined in §2 below)
an “updown category.” For an updown category $\mathcal{C}$, there are
nonnegative integers $u(p;q)$ and $d(p;q)$ for
$p,q\in\operatorname{Ob}\mathcal{C}$ with $q$ having rank one greater than
$p$, such that
$u(p;q)|\operatorname{Aut}q|=d(p;q)|\operatorname{Aut}p|.$
Then the set $\operatorname{Ob}\mathcal{C}$ has a natural graded poset
structure, and the operators $U$ and $D$ on the free vector space
$\Bbbk(\operatorname{Ob}\mathcal{C})$ defined by equations
$Up=\sum_{\text{q covers p}}u(p;q)q\quad\text{and}\quad Dp=\sum_{\text{p
covers q}}d(q;p)q.$
are adjoint for the inner product on $\Bbbk(\operatorname{Ob}\mathcal{C})$
given by $\langle p,q\rangle=|\operatorname{Aut}p|\delta_{p,q}$.
For any updown category $\mathcal{C}$, there are associated two generating
functions, defined in $\S\ref{S:ecgf}$: the object generating function and the
morphism generating function. If $\mathcal{C}$ is a univalent updown category
(i.e., $u(p;q)=d(p;q)$ for all $p,q\in\operatorname{Ob}\mathcal{C}$), then the
former is the rank-generating function of the graded poset
$\operatorname{Ob}\mathcal{C}$. Computation of these generating functions is
facilitated if $\mathcal{C}$ is evenly up-covered (i.e., $\sum_{\text{$q$
covers $p$}}u(p,q)$ depends only on the grade $|p|$ of $p$) or evenly down-
covered ($\sum_{\text{$p$ covers $q$}}d(q;p)$ only depends on $|p|$).
Univalent updown categories admit a natural definition of universal covers. In
[5] the author developed a theory of universal covers for weighted-relation
posets, i.e., ranked posets in which each covering relation has a single
number $n(x,y)$ assigned to it. The universal cover of a weighted-relation
poset $P$ is the “unfolding” of $P$ into a usually much larger weighted-
relation poset $\widetilde{P}$, so that the Hasse diagram of $\widetilde{P}$
is a tree and all covering relations of $\widetilde{P}$ have multiplicity 1.
Although $\widetilde{P}$ had a natural description in each of the seven
examples considered in [5], the general construction of $\widetilde{P}$ given
in [5, Theorem 3.3] was somewhat unsatisfactory since it involved many
arbitrary choices. In §4 we show that univalent updown categories are
essentially “categorified” weighted-relation posets and give a functorial
definition of universal covers for them (Theorem 4.3 below). We also give a
functorial description of two univalent updown categories
$\mathcal{C}^{\uparrow}$ and $\mathcal{C}^{\downarrow}$ associated with an
updown category $\mathcal{C}$.
In §5 we offer ten examples, which encompass all those given in [5]. These
include updown categories whose objects are the subsets of a finite set,
monomials, necklaces, integer partitions, integer compositions, planar rooted
trees, and rooted trees. For each example we compute the object and morphism
generating functions and describe the associated covering spaces.
The author takes pleasure in acknowledging the support he has received from
various institutions during the extended gestation of this paper. The basic
idea was conceived during the academic year 2002-2003, when the author was
partially supported by the Naval Academy Research Council (NARC). During the
following academic year, while on sabbatical leave, he wrote a first draft [7]
during a stay at the Max-Planck-Institut für Mathematik (MPIM) in Bonn, and
subsequently presented its contents to the research group of Prof. Lothar
Gerritzen at Ruhr-Universität Bochum during a visit supported by the German
Academic Exchange Service (DAAD). The author thanks Prof. Gerritzen and Dr.
Ralf Holtkamp for helpful discussions. During the academic years 2004-2005 and
2005-2006, the author received partial support from the NARC. Finally, he
again enjoyed the hospitality of the MPIM during 2012.
## 2 Updown categories
We begin by defining an updown category.
###### Definition 2.1.
An updown category is a small category $\mathcal{C}$ with a rank functor
$|\cdot|:\mathcal{C}\to\mathbb{N}$ (where $\mathbb{N}$ is the ordered set of
natural numbers regarded as a category) such that
* A1.
Each rank $\mathcal{C}_{n}=\\{p\in\operatorname{Ob}\mathcal{C}:|p|=n\\}$ is
finite.
* A2.
The zeroth rank $\mathcal{C}_{0}$ consists of a single object $\hat{0}$, and
$\operatorname{Hom}(\hat{0},p)$ is nonempty for all objects $p$ of
$\mathcal{C}$.
* A3.
For objects $p,p^{\prime}$ of $\mathcal{C}$,
$\operatorname{Hom}(p,p^{\prime})$ is always finite, and
$\operatorname{Hom}(p,p^{\prime})=\emptyset$ unless $|p|<|p^{\prime}|$ or
$p=p^{\prime}$. In the latter case, $\operatorname{Hom}(p,p)$ is a group,
denoted $\operatorname{Aut}(p)$.
* A4.
Any morphism $p\to p^{\prime}$, where $|p^{\prime}|=|p|+k$, factors as a
composition $p=p_{0}\to p_{1}\to\dots\to p_{k}=p^{\prime}$, where
$|p_{i+1}|=|p_{i}|+1$;
* A5.
If $|p^{\prime}|=|p|+1$, the actions of $\operatorname{Aut}(p)$ and
$\operatorname{Aut}(p^{\prime})$ on $\operatorname{Hom}(p,p^{\prime})$ (by
precomposition and postcomposition respectively) are free.
Given an updown category, we can define the multiplicities mentioned in the
introduction as follows.
###### Definition 2.2.
For any two objects $p,p^{\prime}$ of an updown category $\mathcal{C}$ with
$|p^{\prime}|=|p|+1$, define
$u(p;p^{\prime})=\left|\operatorname{Hom}(p,p^{\prime})/\operatorname{Aut}(p^{\prime})\right|=\frac{|\operatorname{Hom}(p,p^{\prime})|}{|\operatorname{Aut}(p^{\prime})|}$
and
$d(p;p^{\prime})=\left|\operatorname{Hom}(p,p^{\prime})/\operatorname{Aut}(p)\right|=\frac{|\operatorname{Hom}(p,p^{\prime})|}{|\operatorname{Aut}(p)|}.$
It follows immediately from these definitions that
$u(p;p^{\prime})|\operatorname{Aut}(p^{\prime})|=d(p;p^{\prime})|\operatorname{Aut}(p)|.$
(1)
We note two extreme cases. First, suppose $\mathcal{C}_{n}$ is empty for all
$n>0$. Then $\mathcal{C}$ is essentially the finite group
$\operatorname{Aut}\hat{0}$. Second, suppose that every set
$\operatorname{Hom}(p,p^{\prime})$ has at most one element. Then $\mathcal{C}$
is a graded poset with least element $\hat{0}$.
Two important special types of updown categories are defined as follows.
###### Definition 2.3.
An updown category $\mathcal{C}$ is univalent if $\operatorname{Aut}(p)$ is
trivial for all $p\in\operatorname{Ob}\mathcal{C}$. An updown category
$\mathcal{C}$ is simple if $\operatorname{Hom}(c,c^{\prime})$ has at most one
element for all $c,c^{\prime}\in\operatorname{Ob}\mathcal{C}$, and the
factorization in A4 is unique, i.e., for $|c^{\prime}|>|c|$ any
$f\in\operatorname{Hom}(c,c^{\prime})$ has a unique factorization into
morphisms between adjacent ranks.
Of course simple implies univalent, but not conversely. A univalent updown
category is the “categorification” of a weighted-relation poset in the sense
of [5]; see §4 below for details.
If $\mathcal{C}$ and $\mathcal{D}$ are updown categories, their product
$\mathcal{C}\times\mathcal{D}$ is the usual one, i.e.
$\operatorname{Ob}(\mathcal{C}\times\mathcal{D})=\operatorname{Ob}\mathcal{C}\times\operatorname{Ob}\mathcal{D}$
and
$\operatorname{Hom}_{\mathcal{C}\times\mathcal{D}}((c,d),(c^{\prime},d^{\prime}))=\operatorname{Hom}_{\mathcal{C}}(c,c^{\prime})\times\operatorname{Hom}_{\mathcal{D}}(d,d^{\prime}).$
The rank is defined on $\mathcal{C}\times\mathcal{D}$ by $|(c,d)|=|c|+|d|$. We
have the following result.
###### Proposition 2.1.
If $\mathcal{C}$ and $\mathcal{D}$ are updown categories, then so is their
product $\mathcal{C}\times\mathcal{D}$.
###### Proof.
Since
$(\mathcal{C}\times\mathcal{D})_{n}=\coprod_{i+j=n}\mathcal{C}_{i}\times\mathcal{D}_{j},$
axiom A1 is clear; and evidently
$\hat{0}=(\hat{0}_{\mathcal{C}},\hat{0}_{\mathcal{D}})$ satisfies A2. Checking
A3 is routine, and for A4 we can combine factorizations
$c=c_{0}\to c_{1}\to\dots\to c_{k}=c^{\prime}\quad\text{and}\quad d=d_{0}\to
d_{1}\to\dots\to d_{l}=d^{\prime}$
into
$(c,d)\to(c_{1},d)\to\dots\to(c^{\prime},d)\to(c^{\prime},d_{1})\to\dots\to(c^{\prime},d^{\prime}).$
Finally, for A5 note that, e.g.,
$\operatorname{Hom}((c,d),(c^{\prime},d))\cong\operatorname{Hom}(c,c^{\prime})\times\operatorname{Aut}(d),$
and the action of
$\operatorname{Aut}(c,d)\cong\operatorname{Aut}(c)\times\operatorname{Aut}(d)$
on this set is free if and only if the action of $\operatorname{Aut}(c)$ on
$\operatorname{Hom}(c,c^{\prime})$ is free. ∎
We note that the product of two univalent updown categories is univalent, but
the product of simple updown categories need not be simple: see Example 1 in
§6 below.
We now define a morphism of updown categories.
###### Definition 2.4.
Let $\mathcal{C},\mathcal{D}$ be updown categories. A morphism from
$\mathcal{C}$ to $\mathcal{D}$ is a functor $F:\mathcal{C}\to\mathcal{D}$ with
$|F(p)|=|p|$ for all $p\in\operatorname{Ob}\mathcal{C}$, and such that, for
any $p,q\in\operatorname{Ob}\mathcal{C}$ with $|q|=|p|+1$, the induced maps
$\operatorname{Aut}(p)\to\operatorname{Aut}(F(p)),$
$\coprod_{\\{q^{\prime}:F(q^{\prime})=F(q)\\}}\operatorname{Hom}(p,q^{\prime})/\operatorname{Aut}(p)\to\operatorname{Hom}(F(p),F(q))/\operatorname{Aut}(F(p)),$
and
$\coprod_{\\{q^{\prime}:F(q^{\prime})=F(q)\\}}\operatorname{Hom}(p,q^{\prime})/\operatorname{Aut}(q^{\prime})\to\operatorname{Hom}(F(p),F(q))/\operatorname{Aut}(F(q))$
are injective.
We have the following result.
###### Proposition 2.2.
Suppose $F:\mathcal{C}\to\mathcal{D}$ is a morphism of updown categories. If
$\mathcal{D}$ is univalent, then so is $\mathcal{C}$; if $\mathcal{D}$ is
simple, then $\mathcal{C}$ is also simple and $F$ is injective as a function
on object sets.
###### Proof.
It follows immediately from Definition 2.4 that $\mathcal{C}$ must be
univalent when $\mathcal{D}$ is. Now suppose $\mathcal{D}$ is simple. Then
$\mathcal{C}$ is univalent, and it follows from Definition 2.4 that the
induced function
$\coprod_{\\{q^{\prime}:F(q^{\prime})=F(q)\\}}\operatorname{Hom}(p,q^{\prime})\to\operatorname{Hom}(F(p),F(q))$
is injective when $|q|=|p|+1$: but $\operatorname{Hom}(F(p),F(q))$ is (at
most) a one-element set, so $F$ must be injective on object sets and
$\operatorname{Hom}(p,q)$ can have at most one object. But then unique
factorization of morphisms in $\mathcal{C}$ follows from that in
$\mathcal{D}$, so $\mathcal{C}$ is simple. ∎
There is a morphism of updown categories
$\mathcal{C}\to\mathcal{C}\times\mathcal{D}$ given by sending
$c\in\operatorname{Ob}\mathcal{C}$ to $(c,\hat{0}_{\mathcal{D}})$ whenever
$\mathcal{C}$ and $\mathcal{D}$ are updown categories; similarly there is a
morphism $\mathcal{D}\to\mathcal{C}\times\mathcal{D}$. We denote the $n$-fold
cartesian power of $\mathcal{C}$ by $\mathcal{C}^{n}$.
Let $\Bbbk$ be a field of characteristic 0,
$\Bbbk(\operatorname{Ob}\mathcal{C})$ the free vector space on
$\operatorname{Ob}\mathcal{C}$ over $\Bbbk$. We now define “up” and “down”
operators on $\Bbbk(\operatorname{Ob}\mathcal{C})$.
###### Definition 2.5.
For an updown category $\mathcal{C}$, let
$U,D:\Bbbk(\operatorname{Ob}\mathcal{C})\to\Bbbk(\operatorname{Ob}\mathcal{C})$
be the the linear operators given by
$Up=\sum_{|p^{\prime}|=|p|+1}u(p;p^{\prime})p^{\prime}$
and
$Dp=\begin{cases}\sum_{|p^{\prime}|=|p|-1}d(p^{\prime};p)p^{\prime},&|p|>0,\\\
0,&p=\hat{0},\end{cases}$
for all $p\in\operatorname{Ob}\mathcal{C}$.
###### Theorem 2.1.
The operators $U$ and $D$ are adjoint with respect to the inner product on
$\Bbbk(\operatorname{Ob}\mathcal{C})$ defined by
$\langle p,p^{\prime}\rangle=\begin{cases}|\operatorname{Aut}(p)|,&\text{if
$p^{\prime}=p$},\\\ 0,&\text{otherwise.}\end{cases}$
###### Proof.
Since $\langle Up,p^{\prime}\rangle=\langle p,Dp^{\prime}\rangle=0$ unless
$|p^{\prime}|=|p|+1$, it suffices to consider that case. Then
$\langle Up,p^{\prime}\rangle=u(p;p^{\prime})\langle
p^{\prime},p^{\prime}\rangle=u(p;p^{\prime})|\operatorname{Aut}(p^{\prime})|$
while
$\langle p,Dp^{\prime}\rangle=d(p;p^{\prime})\langle
p,p\rangle=d(p;p^{\prime})|\operatorname{Aut}(p)|,$
and the two agree by equation (1). ∎
Now we extend the definitions of $u(p;p^{\prime})$ and $d(p;p^{\prime})$ to
any pair $p,p^{\prime}\in\operatorname{Ob}\mathcal{C}$ by setting
$u(p;p^{\prime})=d(p;p^{\prime})=0$ if
$\operatorname{Hom}(p,p^{\prime})=\emptyset$ and
$u(p;p^{\prime})=\frac{\langle
U^{|p^{\prime}|-|p|}(p),p^{\prime}\rangle}{|\operatorname{Aut}(p^{\prime})|},\quad
d(p;p^{\prime})=\frac{\langle
U^{|p^{\prime}|-|p|}(p),p^{\prime}\rangle}{|\operatorname{Aut}(p)|}$ (2)
otherwise. It is immediate that equation (1) still holds, and that
$U^{k}(p)=\sum_{|p^{\prime}|=|p|+k}u(p;p^{\prime})p^{\prime}$
and
$D^{k}(p)=\sum_{|p^{\prime}|=|p|-k}d(p;p^{\prime})p^{\prime}$
for any $p\in\operatorname{Ob}\mathcal{C}$. (However, it is no longer true
that $u(p;q)$ and $d(p;q)$ have any simple relation to
$|\operatorname{Hom}(p,q)|$ when $|q|-|p|>1$.) An important special case of
the extended equation (1) is
$\frac{d(\hat{0};p)}{u(\hat{0};p)}=\frac{|\operatorname{Aut}(p)|}{|\operatorname{Aut}\hat{0}|}$
(3)
for any object $p$ of $\mathcal{C}$. If $\operatorname{Aut}\hat{0}$ is
trivial, equation (3) gives the order of the automorphism group of
$p\in\operatorname{Ob}\mathcal{C}$ as a ratio of multiplicities (cf.
Proposition 2.6 of [6]). We also have the following result.
###### Theorem 2.2.
If $|p|\leq k\leq|q|$, then
$u(p;q)=\sum_{|p^{\prime}|=k}u(p;p^{\prime})u(p^{\prime};q),$
and similarly for $u$ replaced by $d$.
###### Proof.
Using equation (2), we can write $u(p,q)$ as
$\frac{\langle
U^{|q|-|p|}p,q\rangle}{|\operatorname{Aut}(q)|}=\frac{1}{|\operatorname{Aut}(q)|}\langle
U^{k-|p|}U^{|q|-k}(p),q\rangle=\frac{1}{|\operatorname{Aut}(q)|}\sum_{|p^{\prime}|=k}u(p;p^{\prime})\langle
U^{k-|p|}p^{\prime},q\rangle\\\
=\frac{1}{|\operatorname{Aut}(q)|}\sum_{|p^{\prime}|=k}u(p;p^{\prime})u(p^{\prime};q)|\operatorname{Aut}(q)|=\sum_{|p^{\prime}|=k}u(p;p^{\prime})u(p^{\prime};q),$
and the proof for $d$ is similar. ∎
###### Definition 2.6.
For an updown category $\mathcal{C}$, define the induced partial order on
$\operatorname{Ob}\mathcal{C}$ by setting $p\preceq q$ if and only if
$\operatorname{Hom}(p,q)\neq\emptyset$.
It follows from Theorem 2.2 that $p\preceq q\iff u(p;q)\neq 0\iff d(p;q)\neq
0$. Henceforth we write $p\lhd q$ if $q$ covers $p$ in the induced partial
order. Of course different updown categories can have the same induced poset:
see Examples 5 and 6 of §5 below. If the updown category $\mathcal{C}$ was a
poset to start with (thought of as a category in the usual way), then all the
weights $u(p;q)=d(p;q)$ assigned to the covering relations in
$(\operatorname{Ob}\mathcal{C},\preceq)$ are 1. We call such an updown
category unital. Evidently simple $\implies$ unital $\implies$ univalent.
In the univalent case, equation (3) is trivial since $u(p;q)=d(p;q)$ for all
$p$ and $q$. Nevertheless, we have the following interpretation of the
multiplicity in this case.
###### Theorem 2.3.
Let $\mathcal{C}$ be a univalent updown category. For
$p,q\in\operatorname{Ob}\mathcal{C}$ with $|q|-|p|=n>0$, $u(p;q)=d(p;q)$ is
the number of distinct strings $(h_{1},\dots,h_{n})$ such that each $h_{i}$ is
a morphism between adjacent ranks and $h_{n}h_{n-1}\cdots h_{1}$ is a morphism
from $p$ to $q$.
###### Proof.
We use induction on $n$. The result is immediate if $n=1$, since in a
univalent updown category
$u(p;q)=d(p;q)=|\operatorname{Hom}(p,q)|$
when $|q|=|p|+1$. Now if $N(p,q)$ denotes the number of strings
$(h_{1},\dots,h_{n})$ as in the statement of the proposition, it is evident
that, for $|q|>|p|+1$,
$N(p,q)=\sum_{r\lhd q}N(p,r)N(r,q).$
But then the inductive step follows from Theorem 2.2. ∎
## 3 Even Covering and Generating Functions
In this section we introduce the even covering properties, which are satisfied
in many of the examples of updown categories given in §6 below. We also define
several generating functions associated with any updown category.
###### Definition 3.1.
Let $\mathcal{C}$ be an updown category. Then $\mathcal{C}$ is evenly up-
covered if there is a sequence of numbers $u_{0},u_{1},\dots$ so that, for any
$p\in\mathcal{C}_{n}$,
$\sum_{q\rhd p}u(p;q)=u_{n}.$
Dually, $\mathcal{C}$ is evenly down-covered if there is a sequence of numbers
$d_{1},d_{2},\dots$ so that, for any $p\in\mathcal{C}_{n}$,
$\sum_{q\lhd p}d(q;p)=d_{n}.$
We note that any simple updown category is evenly down-covered with $d_{n}=1$
for all $n$. Another special case occurs often enough that we make the
following definition.
###### Definition 3.2.
We call $\mathcal{C}$ factorial if it is evenly down-covered with
$\sum_{q\lhd p}d(q;p)=|p|$
for all $p\in\operatorname{Ob}\mathcal{C}$.
If $\mathcal{C}$ is evenly up-covered, then by induction using Theorem 2.2 it
follows that
$\sum_{|c|=n}u(\hat{0};c)=u_{0}u_{1}\cdots u_{n-1}$ (4)
for $c\in\mathcal{C}_{n}$. On the other hand, if $\mathcal{C}$ is evenly down-
covered then one has $D^{n}c=d_{n}d_{n-1}\cdots d_{1}\hat{0}$ for
$c\in\mathcal{C}_{n}$, and so
$d(\hat{0};c)=\frac{\langle\hat{0},D^{n}c\rangle}{|\operatorname{Aut}\hat{0}|}=d_{n}d_{n-1}\cdots
d_{1}$ (5)
for such $c$. In particular, $d(\hat{0};c)=|c|!$ for all
$c\in\operatorname{Ob}\mathcal{C}$ if $\mathcal{C}$ is factorial.
Although the even covering properties are not generally preserved under
products, we do have the following result.
###### Theorem 3.1.
If $\mathcal{C}$ and $\mathcal{D}$ are factorial updown categories, then so is
$\mathcal{C}\times\mathcal{D}$.
###### Proof.
Any object covered by
$(c,d)\in\operatorname{Ob}(\mathcal{C}\times\mathcal{D})$ must have the form
$(c^{\prime},d)$ with $c^{\prime}\lhd c$, or $(c,d^{\prime})$, with
$d^{\prime}\lhd d$. Thus
$\sum_{p\lhd(c,d)}d(p;(c,d))=\sum_{c^{\prime}\lhd
c}d((c^{\prime},d);(c,d))+\sum_{d^{\prime}\lhd d}d((c,d^{\prime});(c,d))\\\
=\sum_{c^{\prime}\lhd c}d(c^{\prime};c)+\sum_{d^{\prime}\lhd
d}d(d^{\prime};d)=|c|+|d|=|(c,d)|.$
∎
Neither of the two even covering properties implies the other. Examples 8 and
10 of §5 below are evenly up-covered but not evenly down-covered, and it is
easy to construct simple updown categories that are not evenly up-covered. For
simple updown categories that are evenly up-covered, we have the following
result.
###### Theorem 3.2.
Suppose $\mathcal{C}$ and $\mathcal{D}$ are simple updown categories that are
both evenly up-covered with the same sequence $\\{u_{n}\\}$. Then
$\mathcal{C}$ and $\mathcal{D}$ are isomorphic as updown categories.
###### Proof.
It suffices to give a functor $F:\mathcal{C}\to\mathcal{D}$ that is bijective
on the object sets such that $F(c)\lhd F(c^{\prime})$ for all
$c,c^{\prime}\in\operatorname{Ob}\mathcal{C}$. We proceed by induction on
rank. For rank 0 we set $F(\hat{0}_{\mathcal{C}})=\hat{0}_{\mathcal{D}}$.
Suppose $F$ has been defined through rank $n$. For each $p\in\mathcal{C}_{n}$,
choose a bijection $\phi_{p}$ from
$C^{+}(p)=\\{p^{\prime}\in\mathcal{C}_{n+1}|p\lhd p^{\prime}\\}$ to
$C^{+}(F(p))$ (which is possible since both sets have $u_{n}$ elements). Then
for $q\in\mathcal{C}_{n+1}$, set $F(q)=\phi_{p}(q)$, where $p$ is the unique
element of $\mathcal{C}_{n}$ that $q$ covers. ∎
Now we turn to generating functions.
###### Definition 3.3.
Let $\mathcal{C}$ be an updown category. The object generating function of
$\mathcal{C}$ is
$O_{\mathcal{C}}(t)=\sum_{p\in\operatorname{Ob}\mathcal{C}}\frac{t^{|p|}}{|\operatorname{Aut}(p)|}=\sum_{n\geq
0}\sum_{p\in\mathcal{C}_{n}}\frac{t^{n}}{|\operatorname{Aut}(p)|},$
and the morphism generating function of $\mathcal{C}$ is
$M_{\mathcal{C}}(t)=\sum_{p,q\in\operatorname{Ob}\mathcal{C},\ p\lhd
q}\frac{u(p;q)t^{|p|+|q|}}{|\operatorname{Aut}(p)|}=\sum_{n\geq
0}\sum_{p\in\mathcal{C}_{n}}\sum_{q\in\mathcal{C}_{n+1}}\frac{u(p;q)t^{2n+1}}{|\operatorname{Aut}(p)|}.$
(6)
Both $O_{\mathcal{C}}(t)$ and $M_{\mathcal{C}}(t)$ are elements of the formal
power series ring $\mathbb{Q}[[t]]$. If $\mathcal{C}$ is univalent, then
$O_{\mathcal{C}}(t)=\sum_{n\geq 0}|\mathcal{C}_{n}|t^{n}$
and
$M_{\mathcal{C}}(t)=\sum_{n\geq
0}\sum_{p\in\mathcal{C}_{n}}\sum_{q\in\mathcal{C}_{n+1}}|\operatorname{Hom}(p,q)|t^{2n+1}$
are elements of $\mathbb{Z}[[t]]$. In view of equation (1), the morphism
generating function can be written
$M_{\mathcal{C}}(t)=\sum_{p,q\in\operatorname{Ob}\mathcal{C},\ p\lhd
q}\frac{d(p;q)t^{|p|+|q|}}{|\operatorname{Aut}(q)|}.$ (7)
###### Definition 3.4.
For an updown category $\mathcal{C}$, the formal series of $\mathcal{C}$ is
$S_{\mathcal{C}}(t)=\sum_{p\in\operatorname{Ob}\mathcal{C}}\frac{pt^{|p|}}{|\operatorname{Aut}(p)|}\in\Bbbk(\operatorname{Ob}\mathcal{C})[[t]].$
These definitions are related by the following result.
###### Theorem 3.3.
If the inner product $\langle,\rangle$ of Theorem 2.1 is extended to
$\Bbbk(\operatorname{Ob}\mathcal{C})[[t]]$, then
$\langle S_{\mathcal{C}}(t),S_{\mathcal{C}}(t)\rangle=O_{\mathcal{C}}(t^{2})$
(8)
and
$\langle US_{\mathcal{C}}(t),S_{\mathcal{C}}(t)\rangle=\langle
S_{\mathcal{C}}(t),DS_{\mathcal{C}}(t)\rangle=M_{\mathcal{C}}(t).$ (9)
###### Proof.
Immediate from Theorem 2.1 and the definitions. ∎
The generating functions of a product can be obtained from those of its
factors as follows.
###### Corollary 3.1.
For updown categories $\mathcal{C}$ and $\mathcal{D}$,
$O_{\mathcal{C}\times\mathcal{D}}(t)=O_{\mathcal{C}}(t)O_{\mathcal{D}}(t)$
(10)
and
$M_{\mathcal{C}\times\mathcal{D}}(t)=M_{\mathcal{C}}(t)O_{\mathcal{D}}(t^{2})+O_{\mathcal{C}}(t^{2})M_{\mathcal{D}}(t).$
(11)
###### Proof.
We have $S_{\mathcal{C}\times\mathcal{D}}(t)=S_{\mathcal{C}}(t)\otimes
S_{\mathcal{D}}(t)$ under the evident identification of
$\Bbbk(\operatorname{Ob}(\mathcal{C}\times\mathcal{D}))$ with
$\Bbbk(\operatorname{Ob}\mathcal{C})\otimes\Bbbk(\operatorname{Ob}\mathcal{D})$.
Hence
$\langle
S_{\mathcal{C}\times\mathcal{D}}(t),S_{\mathcal{C}\times\mathcal{D}}(t)\rangle=\langle
S_{\mathcal{C}}(t)\otimes S_{\mathcal{D}}(t),S_{\mathcal{C}}(t)\otimes
S_{\mathcal{D}}(t)\rangle=\langle
S_{\mathcal{C}}(t),S_{\mathcal{C}}(t)\rangle\langle
S_{\mathcal{D}}(t),S_{\mathcal{D}}(t)\rangle,$
and equation (10) follows using equation (8). Similarly, we have
$\langle
US_{\mathcal{C}\times\mathcal{D}}(t),S_{\mathcal{C}\times\mathcal{D}}(t)\rangle=\langle
U(S_{\mathcal{C}}(t)\otimes S_{\mathcal{D}}(t)),S_{\mathcal{C}}(t)\otimes
S_{\mathcal{D}}(t)\rangle=\\\ \langle US_{\mathcal{C}}(t)\otimes
S_{\mathcal{D}}(t)+S_{\mathcal{C}}(t)\otimes
US_{\mathcal{D}}(t),S_{\mathcal{C}}(t)\otimes S_{\mathcal{D}}(t)\rangle=\\\
\langle US_{\mathcal{C}}(t),S_{\mathcal{C}}(t)\rangle\langle
S_{\mathcal{D}}(t),S_{\mathcal{D}}(t)\rangle+\langle
S_{\mathcal{C}}(t),S_{\mathcal{C}}(t)\rangle\langle
US_{\mathcal{D}}(t),S_{\mathcal{D}}(t)\rangle$
from which equation (11) follows via equation (9). ∎
_Remark._ It follows from the preceding result that
$O_{\mathcal{C}^{n}}(t)=(O_{\mathcal{C}}(t))^{n}\quad\text{and}\quad
M_{\mathcal{C}^{n}}(t)=nM_{\mathcal{C}}(t)(O_{\mathcal{C}}(t^{2}))^{n-1}$
where $\mathcal{C}^{n}$ is the $n$-fold product of $\mathcal{C}$.
If $\mathcal{C}$ is evenly up-covered or evenly down-covered, there is a
direct relation between the object and morphism generating functions.
###### Theorem 3.4.
Let $\mathcal{C}$ be an updown category with
$O_{\mathcal{C}}(t)=\sum_{n\geq 0}a_{n}t^{n}.$
* 1.
If $\mathcal{C}$ is evenly up-covered, then
$M_{\mathcal{C}}(t)=\sum_{n\geq 0}a_{n}u_{n}t^{2n+1}.$
* 2.
If $\mathcal{C}$ is evenly down-covered, then
$M_{\mathcal{C}}(t)=\sum_{n\geq 1}a_{n}d_{n}t^{2n-1}.$
###### Proof.
Immediate from equations (6) and (7) respectively. ∎
_Remark._ Two consequences of the second part are: (i) if $\mathcal{C}$ is
simple, then $O_{\mathcal{C}}(t^{2})=1+tM_{\mathcal{C}}(t)$; and (ii) if
$\mathcal{C}$ is factorial, then
$M_{\mathcal{C}}(t)=tO_{\mathcal{C}}^{\prime}(t^{2})$.
If the updown category $\mathcal{C}$ is both evenly up-covered and evenly
down-covered, the preceding result gives two expressions for
$M_{\mathcal{C}}(t)$ which must agree. This gives us the following result.
###### Corollary 3.2.
Suppose the updown category $\mathcal{C}$ is both evenly up-covered (with
sequence $\\{u_{n}\\}$) and evenly down-covered (with sequence $\\{d_{n}\\}$).
Then $a_{n}u_{n}=a_{n+1}d_{n+1}$ for all $n\geq 0$, where
$O_{\mathcal{C}}(t)=\sum_{n\geq 0}a_{n}t^{n}$. In particular, if $\mathcal{C}$
is evenly up-covered and factorial, then
$a_{0}=|\operatorname{Aut}\hat{0}|^{-1}$ and
$a_{n}=\frac{u_{0}u_{1}\cdots u_{n-1}}{n!|\operatorname{Aut}\hat{0}|},\quad
n\geq 1.$
## 4 Univalent Updown Categories, Weighted-relation Posets, and Universal
Covers
Let $\mathfrak{U}$ be the category of updown categories,
$\mathfrak{U}\mathfrak{U}$ the full subcategory of univalent updown
categories. For a functor $F$ between univalent updown categories
$\mathcal{C}$, $\mathcal{D}$, Definition 2.4 reduces to the requirement that
$F$ preserve rank and that the induced function
$\coprod_{\\{q^{\prime}:F(q^{\prime})=F(q)\\}}\operatorname{Hom}(p,q^{\prime})\to\operatorname{Hom}(F(p),F(q))$
(12)
be injective whenever $p,q\in\operatorname{Ob}\mathcal{C}$ with $|q|=|p|+1$.
The notion of a weighted-relation poset was defined in [5]. This consists of a
ranked poset
$P=\bigcup_{n\geq 0}P_{n}$
with a least element $\hat{0}\in P_{0}$, together with nonnegative integers
$n(x,y)$ for each $x,y\in P$ so that $n(x,y)=0$ unless $x\preceq y$, and
$n(x,y)=\sum_{|z|=k}n(x,z)n(z,y)$ (13)
whenever $|x|\leq k\leq|y|$. A morphism of weighted-relation posets $P,Q$ is a
rank-preserving map $f:P\to Q$ such that
$n(f(t),f(s))\geq\sum_{s^{\prime}\in f^{-1}(f(s))}n(t,s^{\prime})$ (14)
for any $s,t\in P$ with $|s|=|t|+1$. Let $\mathfrak{W}$ be the category of
weighted-relation posets.
Given an updown category $\mathcal{C}$, it follows from Theorem 2.2 that the
weight functions $n(x,y)=u(x;y)$ and $n(x,y)=d(x;y)$ on the poset
$\operatorname{Ob}\mathcal{C}$ (with the partial order defined by Definition
2.6) both satisfy equation (13). So we have two weighted-relation posets based
on $\operatorname{Ob}\mathcal{C}$ corresponding to these two sets of weights.
In fact, we can describe them functorially.
If $\mathcal{C}$ is an updown category, we can form a univalent updown
category $\mathcal{C}^{\uparrow}$ with
$\operatorname{Ob}\mathcal{C}^{\uparrow}=\operatorname{Ob}\mathcal{C}$, and
with $\operatorname{Hom}_{\mathcal{C}^{\uparrow}}(p,p^{\prime})$ defined as
follows. We declare
$\operatorname{Hom}_{\mathcal{C}^{\uparrow}}(p,p)=\operatorname{Aut}_{\mathcal{C}^{\uparrow}}(p)$
trivial for all $p$, and for $|p^{\prime}|>|p|$ define
$\operatorname{Hom}_{\mathcal{C}^{\uparrow}}(p,p^{\prime})$ as the set of
equivalence classes in $\operatorname{Hom}_{\mathcal{C}}(p,p^{\prime})$ under
the relation $f_{n}f_{n-1}\cdots
f_{1}\sim\alpha_{n}f_{n}\cdots\alpha_{1}f_{1}$, where each $f_{i}$ is a
morphism between adjacent ranks and
$\alpha_{i}\in\operatorname{Aut}(\operatorname{trg}f_{i})$. It is routine to
check that $\mathcal{C}^{\uparrow}$ satisfies the axioms of an updown
category, and for $p,p^{\prime}\in\operatorname{Ob}\mathcal{C}$ with
$|p^{\prime}|=|p|+1$ the multiplicity is
$|\operatorname{Hom}_{\mathcal{C}^{\uparrow}}(p,p^{\prime})|=\left|\operatorname{Hom}_{\mathcal{C}}(p,p^{\prime})/\operatorname{Aut}_{\mathcal{C}}(p^{\prime})\right|=u(p;p^{\prime}).$
Of course $\mathcal{C}^{\uparrow}$ coincides with $\mathcal{C}$ if
$\mathcal{C}$ is univalent.
Similarly, for any updown category $\mathcal{C}$ there is a univalent updown
category $\mathcal{C}^{\downarrow}$ with
$\operatorname{Ob}\mathcal{C}^{\downarrow}=\operatorname{Ob}\mathcal{C}$,
trivial automorphisms, and
$\operatorname{Hom}_{\mathcal{C}^{\downarrow}}(p,p^{\prime})$ defined as the
set of equivalence classes in $\operatorname{Hom}_{\mathcal{C}}(p,p^{\prime})$
under the relation $f\sim f_{n}\beta_{n}f_{n-1}\cdots f_{1}\beta_{1}$ for
$f=f_{n}f_{n-1}\cdots f_{1}$ a factorization of
$f\in\operatorname{Hom}_{\mathcal{C}}(p,p^{\prime})$ into morphisms between
adjacent ranks and $\beta_{i}\in\operatorname{Aut}(\operatorname{src}f_{i})$.
Then
$|\operatorname{Hom}_{\mathcal{C}^{\downarrow}}(p,p^{\prime})|=\left|\operatorname{Hom}_{\mathcal{C}}(p,p^{\prime})/\operatorname{Aut}_{\mathcal{C}}(p)\right|=d(p;p^{\prime})$
for $p,p^{\prime}\in\operatorname{Ob}\mathcal{C}$ with $|p^{\prime}|=|p|+1$.
We have the following result.
###### Theorem 4.1.
There are two functors $\mathfrak{U}\to\mathfrak{U}\mathfrak{U}$, taking an
updown category $\mathcal{C}$ to $\mathcal{C}^{\uparrow}$ and
$\mathcal{C}^{\downarrow}$ respectively.
###### Proof.
We first consider the “up” functor. For a morphism
$F:\mathcal{C}\to\mathcal{D}$ of updown categories, there is an induced
functor $F^{\uparrow}:\mathcal{C}^{\uparrow}\to\mathcal{D}^{\uparrow}$ of
univalent updown categories: $F^{\uparrow}(p)=F(p)$ for
$p\in\operatorname{Ob}\mathcal{C}$, and $F^{\uparrow}$ sends the equivalence
class $[f]$, where $f\in\operatorname{Hom}_{\mathcal{C}}(p,q)$, to the
equivalence class
$[F(f)]\in\operatorname{Hom}_{\mathcal{D}^{\uparrow}}(F(p),F(q))$. Now
Definition 2.4 requires that $F$ preserve rank and that the induced function
$\coprod_{\\{q^{\prime}:F(q^{\prime})=F(q)\\}}\operatorname{Hom}_{\mathcal{C}}(p,q^{\prime})/\operatorname{Aut}_{\mathcal{C}}(p^{\prime})\to\operatorname{Hom}_{\mathcal{D}}(F(p),F(q))/\operatorname{Aut}_{\mathcal{D}}(F(q))$
be injective for all $p,q\in\operatorname{Ob}\mathcal{C}$ with $|q|=|p|+1$.
This is exactly the statement that the induced functor $F^{\uparrow}$ is a
morphism of univalent updown categories. The proof for the “down” functor is
similar. ∎
Note that the functors of the preceding result respect products, e.g.,
$(\mathcal{C}\times\mathcal{D})^{\uparrow}$ can be naturally identified with
$\mathcal{C}^{\uparrow}\times\mathcal{D}^{\uparrow}$. Note also that
$\mathcal{C}^{\uparrow}$ is evenly up-covered if $\mathcal{C}$ is, and
$\mathcal{C}^{\downarrow}$ is evenly-down covered if $\mathcal{C}$ is. Now we
pass from univalent updown categories to weighted-relation posets.
###### Theorem 4.2.
There is a functor $Wrp:\mathfrak{U}\mathfrak{U}\to\mathfrak{W}$, sending a
univalent updown category $\mathcal{C}$ to the set
$\operatorname{Ob}\mathcal{C}$ with the partial order of Definition 2.6 and
the weight function $n(x,y)=u(x;y)=d(x;y)$.
###### Proof.
The only thing to check is the morphisms. Suppose
$F:\mathcal{C}\to\mathcal{D}$ is a morphism of $\mathfrak{U}\mathfrak{U}$.
Then $F$ defines a function on the object sets, and the function (12) is
injective. Hence
$\sum_{\\{q^{\prime}:F(q^{\prime})=F(q)\\}}|\operatorname{Hom}(p,q^{\prime})|\leq|\operatorname{Hom}(F(p),F(q))|$
and so (since, e.g., $n(p,q^{\prime})=|\operatorname{Hom}(p,q^{\prime})|$),
inequality (14) holds and $F$ induces a morphism of weighted-relation posets.
∎
As defined in [5], a morphism $f:P\to Q$ of weighted-relation posets is a
covering map if $f$ is surjective and the inequality (14) is an equality. A
universal cover $\widetilde{P}$ of $P$ is a cover $\widetilde{P}\to P$ such
that, if $P^{\prime}\to P$ is any other cover, then there is a covering map
$\widetilde{P}\to P^{\prime}$ so that the composition $\widetilde{P}\to
P^{\prime}\to P$ is the cover $\widetilde{P}\to P$. In [5] such a universal
cover was constructed for any weighted-relation poset $P$.
In fact, the construction of [5] can be made considerably simpler and more
natural if we work instead with univalent updown categories. We first
categorify the definition of covering map.
###### Definition 4.1.
A morphism $\pi:\mathcal{C}^{\prime}\to\mathcal{C}$ of univalent updown
categories is a covering map if $\pi$ is surjective on the object sets and the
induced function
$\coprod_{\\{q^{\prime}:\pi(q^{\prime})=\pi(q)\\}}\operatorname{Hom}(p,q^{\prime})\to\operatorname{Hom}(\pi(p),\pi(q))$
(15)
is a bijection for all $p,q\in\operatorname{Ob}\mathcal{C}^{\prime}$ with
$|q|=|p|+1$. A covering map $\pi:\widetilde{\mathcal{C}}\to\mathcal{C}$ is
universal if for any other covering map
$\phi:\mathcal{C}^{\prime}\to\mathcal{C}$ there is a covering map
$\psi:\widetilde{\mathcal{C}}\to\mathcal{C}^{\prime}$ with $\pi=\phi\psi$.
Then we have the following result.
###### Theorem 4.3.
Every univalent updown category $\mathcal{C}$ has a universal cover
$\widetilde{\mathcal{C}}$.
###### Proof.
We define $\widetilde{\mathcal{C}}$ to be the category whose rank-$n$ objects
are strings $(f_{1},f_{2},\dots,f_{n})$ of morphisms
$f_{i}\in\operatorname{Hom}(c_{i-1},c_{i})$, where $c_{i}\in\mathcal{C}_{i}$,
and whose morphisms are just inclusions of strings. It is straightforward to
verify that $\widetilde{\mathcal{C}}$ is a univalent updown category (with
$\hat{0}_{\widetilde{\mathcal{C}}}$ the empty string). Define the functor
$\pi:\widetilde{\mathcal{C}}\to\mathcal{C}$ by sending the empty string to
$\hat{0}\in\operatorname{Ob}\mathcal{C}$, the nonempty string
$(f_{1},\dots,f_{n})$ of $\widetilde{\mathcal{C}}$ to the target of $f_{n}$ in
$\operatorname{Ob}\mathcal{C}$, and the inclusion
$(f_{1},\dots,f_{j})\subset(f_{1},\dots,f_{n})$ to the morphism
$f_{n}f_{n-1}\cdots f_{j+1}\in\operatorname{Hom}(c_{j},c_{n})$. That the
induced function (15) is a bijection is a tautology.
Now let $P:\mathcal{C}^{\prime}\to\mathcal{C}$ be another cover of
$\mathcal{C}$. To define a covering map
$F:\widetilde{\mathcal{C}}\to\mathcal{C}^{\prime}$ with $\pi=PF$, we proceed
by induction on rank. Start by sending the empty string in
$\widetilde{\mathcal{C}}_{0}$ to the element $\hat{0}$ of
$\mathcal{C}^{\prime}$. Now suppose $F$ is defined through rank $n-1$, and
consider a rank-$n$ object $(f_{1},\dots,f_{n})$ of $\widetilde{\mathcal{C}}$.
Let $c_{n}=\pi(f_{1},\dots,f_{n})$. By the induction hypothesis we have
$c_{n-1}^{\prime}=F(f_{1},\dots,f_{n-1})\in\operatorname{Ob}\mathcal{C}^{\prime}$,
and $c_{n-1}=P(c_{n-1}^{\prime})$ is the target of $f_{n-1}$, hence the source
of $f_{n}$. Since
$P:\coprod_{\\{c^{\prime}:p(c^{\prime})=c_{n}\\}}\operatorname{Hom}(c_{n-1}^{\prime},c^{\prime})\to\operatorname{Hom}(c_{n-1},c_{n})$
is a bijection, there is a unique morphism $g$ of $\mathcal{C}^{\prime}$ with
$\operatorname{src}(g)=c_{n-1}^{\prime}$ sent to $f_{n}:c_{n-1}\to c_{n}$. We
define $F(f_{1},\dots,f_{n})$ to be $\operatorname{trg}(g)$, and the image of
the inclusion of $(f_{1},\dots,f_{n-1})$ in $(f_{1},\dots,f_{n})$ to be $g$.
This actually defines the functor $F$ through rank $n$, since by the induction
hypothesis $F$ assigns to the inclusion of any proper substring
$(f_{1},\dots,f_{k})$ in $(f_{1},\dots,f_{n-1})$ a morphism $h$ from
$F(f_{1},\dots,f_{k})$ to $c_{n-1}^{\prime}$ in $\mathcal{C}^{\prime}$; then
$F$ sends the inclusion of $(f_{1},\dots,f_{k})$ in $(f_{1},\dots,f_{n})$ to
$gh$. ∎
_Remark._ Thinking of the functor $\pi:\widetilde{\mathcal{C}}\to\mathcal{C}$
as a function on the object sets, the number of elements of
$\widetilde{\mathcal{C}}_{n}$ which $\pi$ sends to $p\in\mathcal{C}_{n}$ is
$|\pi^{-1}(p)|=u(\hat{0};p)=d(\hat{0};p)=\frac{\langle\hat{0},D^{n}p\rangle}{|\operatorname{Aut}\hat{0}|},$
(16)
as follows from Theorem 2.3. In particular, if $\mathcal{C}$ is factorial then
$|\pi^{-1}(p)|=n!$ and $|\widetilde{\mathcal{C}}_{n}|=n!|\mathcal{C}_{n}|$.
Also, it follows from equation (16) that
$|\widetilde{\mathcal{C}}_{n}|=\sum_{p\in\mathcal{C}_{n}}u(\hat{0};p).$ (17)
If $\mathcal{C}$ is evenly up-covered, this equation and equation (4) imply
that
$|\widetilde{\mathcal{C}}_{n}|=u_{0}u_{1}\cdots u_{n-1}.$ (18)
The construction of $\widetilde{\mathcal{C}}$ in the the proof of Theorem 4.3
is functorial: given a morphism $F:\mathcal{C}\to\mathcal{D}$ of univalent
updown categories, we have a morphism
$\widetilde{F}:\widetilde{\mathcal{C}}\to\widetilde{\mathcal{D}}$ given by
$\widetilde{F}(f_{1},f_{2},\dots,f_{n})=(F(f_{1}),F(f_{2}),\dots,F(f_{n})).$
Also, the updown category $\widetilde{\mathcal{C}}$ is evidently simple. Thus,
if $\mathfrak{S}\mathfrak{U}$ is the full subcategory of simple updown
categories in $\mathfrak{U}$, then there is a functor
$\mathfrak{U}\mathfrak{U}\to\mathfrak{S}\mathfrak{U}$ taking $\mathcal{C}$ to
$\widetilde{\mathcal{C}}$. In fact, we have the following result.
###### Proposition 4.1.
The functor $\mathfrak{U}\mathfrak{U}\to\mathfrak{S}\mathfrak{U}$ taking
$\mathcal{C}$ to $\widetilde{\mathcal{C}}$ is right adjoint to the inclusion
functor $\mathfrak{S}\mathfrak{U}\to\mathfrak{U}\mathfrak{U}$.
###### Proof.
It suffices to show that
$\operatorname{Hom}_{\mathfrak{U}\mathfrak{U}}(\mathcal{C},\mathcal{D})\cong\operatorname{Hom}_{\mathfrak{S}\mathfrak{U}}(\mathcal{C},\widetilde{\mathcal{D}})$
for any simple updown category $\mathcal{C}$ and univalent updown category
$\mathcal{D}$. A morphism $F:\mathcal{C}\to\mathcal{D}$ of univalent updown
categories gives rise to
$\widetilde{F}:\widetilde{\mathcal{C}}\to\widetilde{\mathcal{D}}$, and since
$\mathcal{C}$ is simple there is a natural identification
$\mathcal{C}\cong\widetilde{\mathcal{C}}$, giving us a morphism
$\mathcal{C}\to\widetilde{\mathcal{D}}$. To go back the other way, just
compose with the covering map $\pi:\widetilde{\mathcal{D}}\to\mathcal{D}$. ∎
The universal cover functor
$\mathfrak{U}\mathfrak{U}\to\mathfrak{S}\mathfrak{U}$ does not respect
products: in fact, $\widetilde{\mathcal{C}}\times\widetilde{\mathcal{D}}$ is
generally not simple. (This does not contradict the preceding result, because
our product is not a categorical product in $\mathfrak{U}\mathfrak{U}$.) We do
have the following result.
###### Proposition 4.2.
If $\mathcal{C}$ and $\mathcal{D}$ are univalent updown categories, then the
number of rank-$n$ objects in $\widetilde{\mathcal{C}\times\mathcal{D}}$ is
$\sum_{k=0}^{n}\binom{n}{k}|\widetilde{\mathcal{C}}_{k}||\widetilde{\mathcal{D}}_{n-k}|.$
###### Proof.
Using the generating functions of the preceding section, equation (17) can be
written
$O_{\widetilde{\mathcal{C}}}(t^{2})=\langle(1-tU)^{-1}\hat{0},S_{\mathcal{C}}(t)\rangle.$
Then
$\displaystyle O_{\widetilde{\mathcal{C}\times\mathcal{D}}}(t^{2})=$
$\displaystyle\langle(1-tU)^{-1}(\hat{0}_{\mathcal{C}}\otimes\hat{0}_{\mathcal{D}}),S_{\mathcal{C}}(t)\otimes
S_{\mathcal{D}}(t)\rangle$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}t^{n}\sum_{k=0}^{n}\binom{n}{k}\langle U^{k}\hat{0}_{\mathcal{C}}\otimes
U^{n-k}\hat{0}_{\mathcal{D}},S_{\mathcal{C}}(t)\otimes
S_{\mathcal{D}}(t)\rangle$ $\displaystyle=$ $\displaystyle\sum_{n\geq
0}t^{n}\sum_{k=0}^{n}\binom{n}{k}\langle
U^{k}\hat{0}_{\mathcal{C}},S_{\mathcal{C}}(t)\rangle\langle
U^{n-k}\hat{0}_{\mathcal{D}},S_{\mathcal{D}}(t)\rangle$ $\displaystyle=$
$\displaystyle\sum_{n\geq
0}t^{n}\sum_{k=0}^{n}\binom{n}{k}t^{k}|\widetilde{\mathcal{C}}_{k}|t^{n-k}|\widetilde{\mathcal{D}}_{n-k}|,$
from which the conclusion follows. ∎
## 5 Examples
In this section we present ten examples of updown categories. Many of the
associated weighted-relation posets appear in the last section of [5]. For the
convenience of the reader we have included a cross-reference to [5] at the
beginning of each example where it applies.
###### Example 1.
(Subsets of a finite set; [5, Ex. 1], [13, Ex. 2.5(b)], [3, Ex. 6.2.6].)
First, let $\mathcal{A}$ be an updown category such that
$\mathcal{A}_{0}=\\{\hat{0}\\}$, $\mathcal{A}_{1}=\\{\hat{1}\\}$,
$\mathcal{A}_{n}=\emptyset$ for $n\neq 0,1$, and
$\operatorname{Hom}(\hat{0},\hat{1})$ has a single element. The groups
$\operatorname{Aut}(\hat{0})$ and $\operatorname{Aut}(\hat{1})$ are trivial
since they act freely on the one-element set
$\operatorname{Hom}(\hat{0},\hat{1})$. The object and morphism generating
functions are evidently
$O_{\mathcal{A}}(t)=1+t\quad\text{and}\quad M_{\mathcal{A}}(t)=t.$
Evidently $\mathcal{A}$ is simple and factorial.
Now let $\mathcal{B}=\mathcal{A}^{n}$. Since $\mathcal{A}$ is factorial,
$\mathcal{B}$ is factorial by Theorem 3.1. Objects of $\mathcal{B}$ can be
identified with subsets of $\\{1,2,\dots,n\\}$: an $n$-tuple
$(c_{1},\dots,c_{n})$ corresponds to the set $\\{i:c_{i}=\hat{1}\\}$. The
induced partial order is inclusion of sets, and in fact $\mathcal{B}$ is
unital (but not simple for $n\geq 2$). In [5] it is shown that the universal
cover $\widetilde{\mathcal{B}}$ is the simple updown category whose rank-$m$
elements are linearly ordered $m$-element subsets of $\\{1,\dots,n\\}$, and
whose morphisms are inclusions of initial segments. This makes it obvious that
$|\pi^{-1}(b)|=m!$ for all $b\in\mathcal{B}_{m}$, which also follows from
equation (16).
From the remark following Corollary 3.1, the generating functions are
$O_{\mathcal{B}}(t)=(1+t)^{n}\quad\text{and}\quad
M_{\mathcal{B}}(t)=nt(1+t^{2})^{n-1}.$
###### Example 2.
(Monomials; [5, Ex. 2], [4, Ex. 2.2.1].) Let $\SS$ be the category with
$\SS_{n}=\\{[n]\\}$, where $[n]=\\{1,2,\dots,n\\}$ (and $[0]=\emptyset$), and
let $\operatorname{Hom}([m],[n])$ be the set of injective functions from $[m]$
to $[n]$. Then the axioms are easily seen to hold, with
$\operatorname{Aut}[n]=\Sigma_{n}$, the symmetric group on $n$ letters. Since
$\operatorname{Hom}([n],[n+1])$ has $(n+1)!$ elements, we have
$u([n];[n+1])=1$ and $d([n];[n+1])=n+1$ (so $\SS$ is factorial). The
generating functions are
$O_{\SS}(t)=\sum_{n\geq 0}\frac{t^{n}}{n!}=e^{t}\quad\text{and}\quad
M_{\SS}(t)=\sum_{n\geq 0}\frac{t^{2n+1}}{n!}=te^{t^{2}}.$ (19)
Now let $\mathcal{M}=\SS^{n}$. Objects of $\mathcal{M}$ can be identified with
monomials in $n$ commuting indeterminates $t_{1},\dots,t_{n}$. The
automorphism group of $t_{1}^{i_{1}}t_{2}^{i_{2}}\cdots t_{n}^{i_{n}}$ is
$\Sigma_{i_{1}}\times\Sigma_{i_{2}}\times\dots\times\Sigma_{i_{n}}$, and a
monomial $u$ precedes a monomial $v$ in the induced partial order if $u$ is a
factor of $v$. By Theorem 3.1 $\mathcal{M}$ is factorial, so
$d(1;t_{1}^{i_{1}}\cdots t_{n}^{i_{n}})=(i_{1}+\dots+i_{n})!$
by equation (5). Hence by equation (3)
$u(1;t_{1}^{i_{1}}\cdots
t_{n}^{i_{n}})=\frac{(i_{1}+\dots+i_{n})!}{i_{1}!\cdots i_{n}!}.$
Then it follows (using equation (17)) that
$|\widetilde{\mathcal{M}^{\uparrow}}_{m}|=\sum_{i_{1}+\dots+i_{n}=m}\binom{m}{i_{1}\
i_{2}\ \cdots i_{n}}=n^{m}$
and
$|\widetilde{\mathcal{M}^{\downarrow}}_{m}|=\sum_{i_{1}+\dots+i_{n}=m}m!=n(n+1)\cdots(n+m-1).$
The weighted-relation poset $Wrp(\mathcal{M}^{\uparrow})$ appears in [5],
where it is shown that the universal cover
$\widetilde{\mathcal{M}^{\uparrow}}$ can be identified with the simple updown
category whose objects are monomials in $n$ noncommuting indeterminates
$T_{1},\dots,T_{n}$, and whose morphisms are inclusions as left factors; the
covering map $\pi:\widetilde{\mathcal{M}^{\uparrow}}\to\mathcal{M}^{\uparrow}$
sends $T_{i}$ to $t_{i}$ (e.g.,
$\pi^{-1}(t_{1}^{2}t_{2})=\\{T_{1}^{2}T_{2},T_{1}T_{2}T_{1},T_{2}T_{1}^{2}\\}$).
A similar description of $\widetilde{\mathcal{M}^{\downarrow}}$ can be
obtained by reworking the construction of Theorem 4.3 as follows. Objects in
$\widetilde{\mathcal{M}^{\downarrow}}$ are those monomials in the noncommuting
indeterminates $\\{T_{ij}:1\leq i\leq n,j\geq 1\\}$ such that (a) no
indeterminate is repeated; and (b) if $T_{ij}$ occurs, then so does $T_{ik}$
for $k<j$. The covering map
$\pi:\widetilde{\mathcal{M}^{\downarrow}}\to\mathcal{M}^{\downarrow}$ sends
$T_{ij}$ to $t_{i}$ (e.g.,
$\pi^{-1}(t_{1}^{2}t_{2})=\\{T_{11}T_{12}T_{21},T_{12}T_{11}T_{21},T_{11}T_{21}T_{12},T_{12}T_{21}T_{11},T_{21}T_{11}T_{12},T_{21}T_{12}T_{11}\\}$).
For any object $w$ of $\widetilde{\mathcal{M}^{\downarrow}}$, there are $n$
permutations $\sigma_{1},\sigma_{2}\dots,\sigma_{n}$ that can be extracted
from the second subscripts: e.g., for $T_{13}T_{21}T_{11}T_{12}$ the
permutations are $\sigma_{1}=312$ and $\sigma_{2}=1$. The partial order on
objects of $\widetilde{\mathcal{M}^{\downarrow}}$ is given by having the
monomial $wT_{ij}$ cover $w^{\prime}$ if $w^{\prime}$ has the same sequence of
first subscripts as $w$, and $wT_{ij}$ has the same associated permutations as
$w^{\prime}$ except that $\sigma_{i}$ for $wT_{ij}$ covers $\sigma_{i}$ for
$w^{\prime}$ in the sense of the preceding example. For example,
$T_{13}T_{21}T_{11}T_{12}$ generates the order ideal
$T_{13}T_{21}T_{11}T_{12}\rhd T_{12}T_{21}T_{11}\rhd T_{11}T_{21}\rhd
T_{11}\rhd 1.$
By equations (19) and the remark following Corollary 3.1, the generating
functions are
$O_{\mathcal{M}}(t)=e^{nt}\quad\text{and}\quad
M_{\mathcal{M}}(t)=nte^{nt^{2}}.$
###### Example 3.
Let $\mathcal{G}$ be the category whose objects are isomorphism classes of
finite graphs. Then $\mathcal{G}$ is graded by the number of vertices, with
$\hat{0}$ the empty graph. A morphism from $H$ to $G$ is an injective function
$f:v(H)\to v(G)$ on the vertex sets such that $f(v_{1})$ and $f(v_{2})$ are
connected in $G$ if and only if $v_{1}$ and $v_{2}$ are connected in $H$. If
$G\rhd H$, then there is a vertex $v$ of $G$ so that $G-\\{v\\}$ is isomorphic
to $H$. Evidently $\mathcal{G}$ is factorial, since for any
$G\in\mathcal{G}_{n}$
$\sum_{|H|=n-1}d(H;G)=n.$
But $\mathcal{G}$ is also uniformly up-covered, any $G$ covering
$H\in\mathcal{G}_{n}$ can be obtained from $H$ by adjoining a new vertex and
edges between that vertex and some subset of the $n$ vertices of
$\mathfrak{H}$: thus
$\sum_{|G|=n+1}u(H;G)=2^{n}.$
It follows from Corollary 3.2 that
$O_{\mathcal{G}}(t)=\sum_{n\geq 0}\frac{2^{\binom{n}{2}}}{n!}t^{n},$
and thus from Theorem 3.4 that
$M_{\mathcal{G}}(t)=\sum_{n\geq 1}\frac{2^{\binom{n}{2}}}{(n-1)!}t^{2n-1}.$
Objects of the universal cover $\widetilde{\mathcal{G}}^{\uparrow}$ can be
identified with graphs whose vertices are labelled by the positive integers;
morphisms of $\mathcal{G}^{\uparrow}$ preserve labels. From equation (18)
follows $|\widetilde{\mathcal{G}}_{n}^{\uparrow}|=2^{\binom{n}{2}}$. On the
other hand, an element $(\emptyset,G_{1},G_{2},\dots,G_{n})$ of
$\widetilde{\mathcal{G}}_{n}^{\downarrow}$ can be specified by giving a
bijection
$f:\\{1,2,\dots,n\\}\to v(G_{n})$
such that each $G_{i}$ is the full subgraph of $G_{n}$ on the vertices
$\\{f(1),\dots,f(i)\\}$. This makes it evident that $|\pi^{-1}(G_{n})|=n!$, in
accordance with the remark following Theorem 4.3.
###### Example 4.
(Necklaces; [5, Ex. 3].) For a fixed positive integer $c$, let
$\mathcal{N}_{m}$ be the set of $m$-bead necklaces with beads of $c$ possible
colors. More precisely, a rank-$m$ object of $\mathcal{N}$ is an equivalence
class of functions $f:\mathbf{Z}/m\mathbf{Z}\to[c]$, where $f$ is equivalent
to $g$ if there is some $n$ so that $f(a+n)=g(a)$ for all
$a\in\mathbf{Z}/m\mathbf{Z}$. Thus, for $c=2$ the equivalence class
$\\{(1,1,2,2),(2,1,1,2),(2,2,1,1),(1,2,2,1)\\}\quad\text{represents the
necklace}\hskip
21.68121pt{\psset{dotstyle=*}\psdots(7.11317pt,4.2679pt)(0.0pt,11.38109pt)}{\psset{dotstyle=o}\psdots(-7.11317pt,4.2679pt)(0.0pt,-2.84526pt)}{\psarc(0.0pt,4.2679pt){7.11317pt}{10.0}{80.0}}{\psarc(0.0pt,4.2679pt){7.11317pt}{100.0}{170.0}}{\psarc(0.0pt,4.2679pt){7.11317pt}{190.0}{260.0}}{\psarc(0.0pt,4.2679pt){7.11317pt}{280.0}{350.0}}\hskip
14.45377pt.$
A morphism from the equivalence class of $f$ in $\mathcal{N}_{m}$ to the
equivalence class of $g$ in $\mathcal{N}_{n}$ is an injective function
$h:\mathbf{Z}/m\mathbf{Z}\to\mathbf{Z}/n\mathbf{Z}$ with $f(a)=gh(a)$ for all
$a\in\mathbf{Z}/m\mathbf{Z}$, and such that $h$ preserves the cyclic order,
i.e., if we pick representatives of the $h(i)$ in $\mathbf{Z}$ with $0\leq
h(i)\leq n-1$, then some cyclic permutation of $(h(0),h(1),\dots,h(m-1))$ is
an increasing sequence. Informally, $u(p;q)$ is the number of ways to insert a
bead into necklace $p$ to get necklace $q$, and $d(p;q)$ is the number of
different beads of $q$ that can be deleted to give $p$.
Note that $\mathcal{N}$ is factorial (there are $m$ different beads that can
be removed from $p\in\mathcal{N}_{m}$) and also evenly up-covered with
$u_{m}=mc$ for $m\geq 1$ (in a necklace with $m\geq 1$ beads there are $m$
places that a bead of $c$ possible colors can be inserted); of course
$u_{0}=c$. Thus, by Corollary 3.2
$a_{n}=\begin{cases}1,&\text{if $n=0$;}\\\ \frac{c^{n}}{n},&\text{if $n\geq
1$;}\end{cases}$
and so $O_{\mathcal{N}}(t)=1-\log(1-ct)$. Again using the fact that
$\mathcal{N}$ is factorial (and Theorem 3.4), we have
$M_{\mathcal{N}}(t)=\frac{ct}{1-ct^{2}}.$
We have $|\widetilde{\mathcal{N}}_{m}^{\uparrow}|=(m-1)!c^{m}$ by equation
(18): cf. the discussion in [5], where the same result is obtained by
identifying elements of rank $\widetilde{\mathcal{N}}_{m}^{\uparrow}$ with
necklaces of $m$ beads in $c$ colors in which the beads are labelled by
$1,2\dots,m$. On the other hand, an element of
$\widetilde{\mathcal{N}}_{m}^{\downarrow}$ can be regarded as an equivalence
class of pairs $(f,\sigma)$, where $f:\mathbf{Z}/m\mathbf{Z}\to[c]$ and
$\sigma$ is a permutation of $\\{0,1,\dots,m-1\\}$. The equivalence relation
is that $(f,\sigma)\sim(g,\tau)$ if $f\neq g$ and there is some $0\leq n\leq
m-1$ with $g(x)=f(x+n)$ and $\tau(x)=\sigma(x+n)$ for all
$x\in\mathbf{Z}/m\mathbf{Z}$. Evidently there are $m!$ such equivalence
classes for a given $[f]\in\mathcal{N}_{m}$, in accord with the factoriality
of $\mathcal{N}$.
###### Example 5.
(Integer partitions with unit weights; [5, Ex. 5], [12], [4, Ex. 1.6.8].) Let
$\mathcal{Y}$ be the category with $\operatorname{Ob}\mathcal{Y}$ the set of
integer partitions, i.e., finite sequences
$(\lambda_{1},\lambda_{2},\dots,\lambda_{k})$ of positive integers with
$\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{k}$. The rank of a partition
is $|\lambda|=\lambda_{1}+\lambda_{2}+\dots+\lambda_{k}$; we write
$\ell(\lambda)$ for the length (number of parts) of $\lambda$. The set of
morphisms $\operatorname{Hom}(\lambda,\mu)$ contains a single element if and
only if $\lambda_{i}\leq\mu_{i}$ for all $i$. Then $\mathcal{Y}$ is evidently
unital but not simple.
Since $\mathcal{Y}_{n}$ is the set of partitions of $n$, the object generating
function
$O_{\mathcal{Y}}(t)=\sum_{n\geq
0}|\mathcal{Y}_{n}|t^{n}=\frac{1}{(1-t)(1-t^{2})(1-t^{3})\cdots}$
is familiar. The morphism generating function is
$M_{\mathcal{Y}}(t)=\sum_{n\geq
0}|\\{(\lambda,\mu):\>\lambda\in\mathcal{Y}_{n},\>\lambda\lhd\mu\\}|t^{2n+1}$
since $\mathcal{Y}$ is unital. Using the case $k=1$ of [12, Theorem 3.2], it
follows that
$M_{\mathcal{Y}}(t)=\frac{t}{1-t^{2}}O_{\mathcal{Y}}(t^{2})=\frac{t}{(1-t^{2})^{2}(1-t^{4})(1-t^{6})\cdots}.$
In [5] it is shown that the universal cover $\widetilde{\mathcal{Y}}$ is the
poset of standard Young tableaux, so $u(\hat{0};\lambda)=d(\hat{0};\lambda)$
is the number of standard Young tableaux of shape $\lambda$.
###### Example 6.
Let $\mathcal{K}$ be the category with $\operatorname{Ob}\mathcal{K}$ the set
of integer partitions, and $\operatorname{Hom}(\lambda,\mu)$ defined as
follows. Let $\lambda=(\lambda_{1},\dots,\lambda_{n})$ and
$\mu=(\mu_{1},\dots,\mu_{m})$, always written in decreasing order. Then a
morphism from $\lambda$ to $\mu$ is an injective function $f:[n]\to[m]$ such
that $\lambda_{i}\leq\mu_{j}$ whenever $f(i)=j$.
The partial order induced on
$\operatorname{Ob}\mathcal{K}=\operatorname{Ob}\mathcal{Y}$ is the same as
that of the preceding example: the difference is that we now have nontrivial
automorphism groups and weights on covering relations. The automorphism group
of $\lambda=(\lambda_{1},\dots,\lambda_{k})$ is the subgroup of $\Sigma_{k}$
consisting of those permutations $\sigma$ such that $\lambda_{i}=\lambda_{j}$
whenever $\sigma(i)=j$. If we let $m_{i}(\lambda)$ be the number of parts of
$\lambda$ of size $i$, this means that
$|\operatorname{Aut}(\lambda)|=m_{1}(\lambda)!m_{2}(\lambda)!\cdots.$
For partitions $\lambda,\mu$ with $|\mu|=|\lambda|+1$,
$\operatorname{Hom}(\lambda,\mu)$ is nonempty exactly when (i) $\mu$ comes
from $\lambda$ by adding a part of size 1; or (ii) $\mu$ comes by replacing a
size-$k$ part of $\lambda$ by a part of size $k+1$. In case (i) we put
$u(\lambda;\mu)=1$ and $d(\lambda;\mu)=m_{1}(\mu)$, while in case (ii) we put
$u(\lambda;\mu)=m_{k}(\lambda)$ and $d(\lambda;\mu)=m_{k+1}(\mu)$. The weights
$d(\lambda;\mu)$ appear implicitly in [9] and explicitly in [8], where they
are referred to as “Kingman’s branching”: see especially Figure 4 of [8].
The object generating function can be computed as follows:
$O_{\mathcal{K}}(t)=\sum_{\lambda\in\operatorname{Ob}\mathcal{K}}\frac{t^{|\lambda|}}{|\operatorname{Aut}(\lambda)|}=\sum_{m_{1},m_{2},\dots\geq
0}\frac{t^{m_{1}+2m_{2}+\dots}}{m_{1}!m_{2}!\cdots}=\\\ \left(\sum_{m_{1}\geq
0}\frac{t^{m_{1}}}{m_{1}!}\right)\left(\sum_{m_{2}\geq
0}\frac{t^{2m_{2}}}{m_{2}!}\right)\cdots=\exp(t+t^{2}+\cdots)=\exp\left(\frac{t}{1-t}\right).$
To find the morphism generating function
$M_{\mathcal{K}}(t)=\sum_{\lambda\in\operatorname{Ob}\mathcal{K}}\frac{t^{2|\lambda|+1}}{|\operatorname{Aut}(\lambda)|}\sum_{\lambda\lhd\mu}u(\lambda;\mu)$
(20)
we first observe that
$\sum_{\lambda\lhd\mu}u(\lambda;\mu)=1+\ell(\lambda)=1+m_{1}(\lambda)+m_{2}(\lambda)+\cdots,$
since (using the description of $u(\lambda;\mu)$ above) this is the number of
ways to obtain a partition covering $\lambda$: we can increase by one any of
the $\ell(\lambda)$ parts of $\lambda$, or add a new part of size 1. Thus
equation (20) is
$M_{\mathcal{K}}(t)=\sum_{m_{1},m_{2},\dots\geq
0}\frac{t^{1+2m_{1}+4m_{2}+\cdots}}{m_{1}!m_{2}!\cdots}(1+m_{1}+m_{2}+\cdots)\\\
=(t+t^{3}+t^{5}+\cdots)O_{\mathcal{K}}(t^{2})=\frac{t}{1-t^{2}}\exp\left(\frac{t^{2}}{1-t^{2}}\right).$
The universal cover $\widetilde{\mathcal{K}}^{\uparrow}$ can be described in
terms of set partitions: elements of $\widetilde{\mathcal{K}}_{n}^{\uparrow}$
are partitions of the set $[n]$, with
$\pi:\widetilde{\mathcal{K}}^{\uparrow}\to\mathcal{K}^{\uparrow}$ sending each
partition to the integer partition of $n$ given by its block sizes. Thus
$|\widetilde{\mathcal{K}}_{n}^{\uparrow}|$ is the $n$th Bell number [11,
A000110]. We can identify set partitions with the construction of Theorem 4.3
as follows. For convenience we write a set partition as $(P_{1},\dots,P_{k})$
with $|P_{1}|\geq|P_{2}|\geq\dots\geq|P_{k}|$ and, if $|P_{i}|=|P_{j}|$ for
$i<j$, then $\max P_{i}<\max P_{j}$. Assign the unique partition of $[1]$ to
the morphism from $\hat{0}$ to $(1)$, and suppose inductively that we have
assigned an ordered partition $P=(P_{1},\dots,P_{k})$ of $[n]$ to the chain
$(h_{1},\dots,h_{n})$ of morphisms between adjacent ranks of
$\mathcal{K}^{\uparrow}$ from $\hat{0}$ to
$\operatorname{trg}(h_{n})=(\lambda_{1},\dots,\lambda_{k})\in\operatorname{Ob}\mathcal{K}_{n}^{\uparrow}$
so that $\lambda_{i}=|P_{i}|$. Let
$f\in\operatorname{Hom}_{\mathcal{K}}(\lambda,\mu)$ be a representative of the
equivalence class
$h_{n+1}\in\operatorname{Hom}_{\mathcal{K}^{\uparrow}}(\lambda,\mu)$, where
$|\mu|=n+1$. If $\mu$ has length $k+1$, assign $(P_{1},\dots,P_{k},\\{n+1\\})$
to the chain $(h_{1},\dots,h_{n},h_{n+1})$. Otherwise, $\mu$ has length $k$
and there is a unique $i\in[k]$ such that $\lambda_{i}<\mu_{f(i)}$: in this
case, assign to $(h_{1},\dots,h_{n+1})$ the rearrangement of
$(P_{1}^{\prime},\dots,P_{k}^{\prime})$, where
$P_{j}^{\prime}=\begin{cases}P_{j}\cup\\{n+1\\},&\text{if $j=i$},\\\
P_{j},&\text{otherwise,}\end{cases}$
so that $P_{i}^{\prime}$ immediately follows $P_{m}^{\prime}$, where
$m=\max\\{j<i:|P_{j}^{\prime}|\geq|P_{i}^{\prime}|\\}$. Evidently the set
partition assigned to $(h_{1},\dots,h_{n+1})$ projects to $\mu$ in either
case.
Rank-$n$ objects of the universal cover $\widetilde{\mathcal{K}}^{\downarrow}$
can be described as sequences $s=(a_{1},\dots,a_{n})$ such that $m_{1}(s)\geq
m_{2}(s)\geq\cdots$, where $m_{i}(s)$ is the number of occurrences of $i$ in
$s$; the covering map sends $s$ to $(m_{1}(s),m_{2}(s),\dots)$. See [11,
A005651]. As in the preceding paragraph, we can proceed inductively to
identify these objects with the construction of Theorem 4.3. Start by
assigning $s=(1)$ to the morphism from $\hat{0}$ to $(1)$. Suppose now we have
assigned $s=(a_{1},\dots,a_{n})$ to a chain of morphisms $(h_{1},\dots,h_{n})$
between adjacent ranks of $\mathcal{K}^{\downarrow}$ from $\hat{0}$ to
$\lambda=(\lambda_{1},\dots,\lambda_{k})\in\mathcal{K}_{n}^{\downarrow}$ so
that $m_{i}(s)=\lambda_{i}$ for $1\leq i\leq k$, and let
$h_{n+1}\in\operatorname{Hom}_{\mathcal{K}^{\downarrow}}(\lambda,\mu)$ where
$|\mu|=n+1$. Now a representative
$f\in\operatorname{Hom}_{\mathcal{K}}(\lambda,\mu)$ of $h_{n+1}$ must be
“almost an automorphism” exchanging parts of equal size with just one
exception: there is a unique $i\in[\ell(\mu)]$ such that either $i$ is not in
the image of $f$ (in which case $\mu_{i}=1$), or else
$\lambda_{f^{-1}(i)}<\mu_{i}$ (in which case $\mu_{i}=\lambda_{f^{-1}(i)}+1$).
Let $S=\\{j>i:\lambda_{f^{-1}(j)}=\mu_{i}\\}$: note that $S$ is independent of
the choice of $f$. Now define a permutation $\sigma$ of $[\ell(\mu)]$ as
follows. If $S=\emptyset$, let $\sigma$ be the identity; otherwise, if
$S=\\{i+1,\dots,l\\}$, let $\sigma(a)=a+1$ for $i\leq a\leq l-1$,
$\sigma(l)=i$, and $\sigma(a)=a$ for $a\notin\\{i,\dots,l\\}$. We then assign
the sequence $s^{\prime}=(\sigma(a_{1}),\dots,\sigma(a_{k}),i)$ to the chain
$(h_{1},\dots,h_{n},h_{n+1})$. If $i\notin\operatorname{im}f$, then
$\mu_{j}=1$ for $j\geq i$ and either $i=\ell(\mu)=k+1$ (if $S$ is empty) or
$l=\ell(\mu)=k+1$ (if it isn’t): either way $\mu$ differs by $\lambda$ by
having 1 inserted in the $i$th position, and $s^{\prime}$ projects to $\mu$.
If $\mu_{i}=\lambda_{f^{-1}(i)}+1$, then we must have $\lambda_{j}=\mu_{j}$
for $j<i$, and $\mu$ differs from $\lambda$ in having a part of size
$\mu_{i}-1$ increased by 1. If $S$ is empty, $\lambda_{f^{-1}(i)}=\lambda_{i}$
and $\mu_{i}=m_{i}(s^{\prime})=m_{i}(s)+1=\lambda_{i}+1$. Otherwise,
$\mu_{i}=m_{i}(s^{\prime})=m_{l}(s)+1=\lambda_{f^{-1}(l)}$ and
$m_{j+1}(s^{\prime})=m_{j}(s)$ for $i\leq j\leq l-1$. Either way, $s^{\prime}$
again projects to $\mu$.
###### Example 7.
(Integer compositions; [5, Ex. 6].) Let $\mathcal{C}_{n}$ be the set of
integer compositions of $n$, i.e. sequences $I=(i_{1},\dots,i_{p})$ of
positive integers with $a_{1}+\dots+a_{m}=n$; as with partitions we write
$\ell(I)$ for the length of $I$. A morphism from
$(i_{1},\dots,i_{p})\in\mathcal{C}_{n}$ to
$(j_{1},\dots,j_{q})\in\mathcal{C}_{m}$ is an order-preserving injective
function $f:[p]\to[q]$ such that $i_{a}\leq j_{f(a)}$ for all $a\in[p]$. Then
$\mathcal{C}$ is a univalent updown category (but not simple).
The object generating function is
$O_{\mathcal{C}}(t)=\sum_{n\geq 0}|\mathcal{C}_{n}|t^{n}=1+\sum_{n\geq
1}2^{n-1}t^{n}=\frac{1-t}{1-2t}.$
Now for any composition $I$,
$\sum_{I\lhd J}u(I;J)=\ell(I)+\ell(I)+1=2\ell(I)+1$
since we can get a composition covering $I$ either by increasing each of its
$\ell(I)$ parts, or by inserting a part of size 1 into one of $\ell(I)+1$
possible positions. Thus, the morphism generating function is
$M_{\mathcal{C}}(t)=\sum_{n\geq
0}t^{2n+1}\sum_{k=1}^{n}|\mathcal{C}_{n,k}|(2k+1),$
where $\mathcal{C}_{n,k}$ is the set of compositions of $n$ with $k$ parts.
Evidently $|\mathcal{C}_{n,k}|=\binom{n-1}{k-1}$, so
$M_{\mathcal{C}}(t)=\sum_{n\geq
0}t^{2n+1}\sum_{k=1}^{n}\binom{n-1}{k-1}(2k+1)=\sum_{n\geq
0}(n+2)2^{n-1}t^{2n+1}=\frac{t-t^{3}}{(1-2t^{2})^{2}}.$
The universal cover $\widetilde{\mathcal{C}}$ is constructed in [5] using
Cayley permutations as defined in [10]: a Cayley permutation of rank $n$ is a
length-$n$ sequence $s=(a_{1},\dots,a_{n})$ of positive integers such that any
positive integer $i<j$ appears in $s$ whenever $j$ does. See [11, A00679]. The
covering map $\pi:\widetilde{\mathcal{C}}\to\mathcal{C}$ sends a sequence $s$
to the composition $(m_{1}(s),m_{2}(s),\dots)$. To relate this to the
construction of Theorem 4.3, we again proceed inductively. Send the morphism
from $\hat{0}$ to $(1)$ to the Cayley permutation $(1)$, and suppose we have
assigned to a chain $(h_{1},h_{2},\dots,h_{n})$ of morphisms between
consecutive ranks of $\mathcal{C}$ from $\hat{0}$ to
$I=(i_{1},\dots,i_{k})\in\mathcal{C}_{n}$ a Cayley permutation
$s=(a_{1},\dots,a_{n})$ that projects to $I$: note that
$\max\\{a_{1},\dots,a_{n}\\}=k$. Now let $h_{n+1}\in\operatorname{Hom}(I,J)$
with $J\in\mathcal{C}_{n+1}$. Then either $\ell(J)=k$ and $h_{n+1}$ is the
identity function on $[k]$, or $\ell(J)=k+1$. In the first case, there is
exactly one position $q$ where $J$ differs from $I$: assign to
$(h_{1},\dots,h_{n+1})$ the Cayley permutation
$s^{\prime}=(a_{1},\dots,a_{n},q)$. Then
$m_{q}(s^{\prime})=m_{q}(s)+1=i_{q}+1$ and $m_{i}(s^{\prime})=m_{i}(s)$ for
$i\neq q$, so $s^{\prime}$ projects to $J$. In the second case, there is
exactly one element $q\in[k+1]$ that $h_{n+1}$ misses: assign
$s^{\prime}=(h_{n+1}(a_{1}),\dots,h_{n+1}(a_{n}),q)$ to
$(h_{1},\dots,h_{n+1})$. Then
$\pi(s^{\prime})=(m_{1}(s^{\prime}),m_{2}(s^{\prime}),\dots)$ differs from $I$
only in having an additional 1 inserted in the $q$th place, and so must be
$J$.
###### Example 8.
(Planar rooted trees; [5, Ex. 4].) Let $\mathcal{P}_{n}$ consist of functions
$f:[2n]\to\\{-1,1\\}$ so that the partial sums $S_{i}=f(1)+\cdots+f(i)$ have
the properties that $S_{i}\geq 0$ for all $1\leq i\leq 2n$, and $S_{2n}=0$. We
declare $\operatorname{Aut}(f)$ to be trivial for all objects $f$ of
$\mathcal{P}$, and define a morphism from $f\in\mathcal{P}_{n}$ to
$g\in\mathcal{P}_{n+1}$ to be an injective, order-preserving function
$h:[2n]\to[2n+2]$ such that the two values of $[2n+2]$ not in the image of $h$
are consecutive, and $f(i)=gh(i)$ for $1\leq i\leq 2n$. Then $\mathcal{P}$ is
a univalent updown category. Using the well-known identification of balanced
bracket arrangements with planar rooted trees, e.g.
$(1,1,-1,1,1,-1,-1,-1)\quad\text{is identified
with}\quad{\psline{*-*}(7.11317pt,0.0pt)(14.22636pt,14.22636pt)}{\psline{*-*}(14.22636pt,14.22636pt)(21.33955pt,0.0pt)}{\psline{*-*}(21.33955pt,0.0pt)(21.33955pt,-14.22636pt)}\hskip
28.90755pt,$
we can think of $\mathcal{P}$ as the updown category of planar rooted trees;
the rank is the count of non-root vertices. (The empty bracket arrangement
$\emptyset$ is identified with the tree $\bullet$ consisting of the root
vertex.) In view of the well-known enumeration of planar rooted trees by
Catalan numbers, the object generating function is simply
$O_{\mathcal{P}}(t)=\sum_{n\geq 0}|\mathcal{P}_{n}|t^{n}=\sum_{n\geq
0}\frac{1}{n+1}\binom{2n}{n}t^{n}=\frac{1-\sqrt{1-4t}}{2t}.$
Since there are $2n+1$ possibilities for order-preserving injections
$[2n]\to[2n+2]$ that miss two consecutive values, $\mathcal{P}$ is evenly up-
covered with $u_{n}=2n+1$ and by Theorem 3.4 the morphism generating function
is
$M_{\mathcal{P}}(t)=\sum_{n\geq
0}\frac{2n+1}{n+1}\binom{2n}{n}t^{2n+1}=\sum_{n\geq
0}\binom{2n+1}{n+1}t^{2n+1}=\frac{1-\sqrt{1-4t^{2}}}{2t\sqrt{1-4t^{2}}}.$
From equation (18) we have $|\widetilde{\mathcal{P}}_{n}|=(2n-1)!!$ for $n\geq
1$. In [5] the universal cover of $Wrp(\mathcal{P})$ is described as the
weighted-relation poset whose rank-$n$ elements are permutations
$(a_{1},a_{2},\dots,a_{2n})$ of the multiset $\\{1,1,2,2,\dots,n,n\\}$ such
that, if $a_{i}>a_{j}$ with $i<j$, then there is some $k<j$, $k\neq i$, such
that $a_{k}=a_{i}$. (The covering map sends a sequence
$s=(a_{1},\dots,a_{2n})$ to a sequence of 1’s and $-1$’s by sending the first
occurrence of $i$ in $s$ to 1 and the second to $-1$.) This construction can
be identified with $\widetilde{\mathcal{P}}$ as described in Theorem 4.3 in an
obvious way. For example, the morphism from $\emptyset$ to $(1,1,-1,1,-1,-1)$
given by the composition $h_{3}h_{2}h_{1}$, where $h_{1}=\emptyset$,
$h_{2}=\\{(1,1),(2,4)\\}$ and $h_{3}=\\{(1,1),(2,2),(3,3),(4,6)\\}$, can be
coded by the sequence $(1,2,2,3,3,1)$.
###### Example 9.
(Rooted trees; [5, Ex. 7].) Let $\mathcal{T}_{n}$ consist of partially ordered
sets $P$ such that (i) $P$ has $n+1$ elements; (ii) $P$ has a greatest
element; and (iii) for any $v\in P$, the set of elements of $P$ exceeding $v$
forms a chain. The Hasse diagram of such a poset $P$ is a tree with the
greatest element (the root vertex) at the top. A morphism of $\mathcal{T}$
from $P\in\mathcal{T}_{m}$ to $Q\in\mathcal{T}_{n}$ is an injective order-
preserving function $f:P\to Q$ that sends the root of $P$ to the root of $Q$,
and which preserves covering relations (i.e., if $v\lhd w$ in the partial
order on $P$, then $f(v)\lhd f(w)$ in the partial order on $Q$). Then
$\mathcal{T}$ is an updown category.
The updown category $\mathcal{T}$ was studied extensively in [6], though
without using the categorical language. To see that the construction of the
preceding paragraph gives the same multiplicities as in [6], consider a
morphism from $P\in\mathcal{T}_{n}$ to $Q\in\mathcal{T}_{n+1}$. Any such
morphism misses only some terminal vertex $v\in Q$, so we can think of it as
identifying $P$ with $Q-\\{v\\}$. Elements of
$\operatorname{Hom}(P,Q)/\operatorname{Aut}(Q)$
amount to different choices for the parent of $v$ in $Q$, i.e., different
choices for terminal vertices of $P$ to which a new edge and vertex can be
attached to form $Q$: this is $n(P;Q)$ as defined in [6]. On the other hand,
elements of
$\operatorname{Hom}(P,Q)/\operatorname{Aut}(P)$
amount to different choices of $v$, and thus to different choices for an edge
of $Q$ that when cut leaves $P$: this is $m(P;Q)$ as defined in [6].
The object generating function
$O_{\mathcal{T}}(t)=\sum_{n\geq
0}t^{n}\sum_{P\in\mathcal{T}_{n}}\frac{1}{|\operatorname{Aut}(P)|}$
can be evaluated using a result of [2]. First, we note from [1] (cf. the
discussion in [5]) that
$u(\bullet;P)=n(\bullet;P)=\frac{(|P|+1)!}{P!|\operatorname{Aut}(P)|},$
where $P!$ is the “tree factorial,” i.e., the product
$\prod_{v\ \text{is a vertex of $P$}}(|P_{v}|+1)!$
where $P_{v}$ is the subtree of $P$ having $v$ as its root. Thus
$O_{\mathcal{T}}(t)=\sum_{n\geq
0}t^{n}\sum_{P\in\mathcal{T}_{n}}\frac{u(\bullet;P)P!}{(n+1)!}.$
From §5.3 of [2] we have
$\sum_{P\in\mathcal{T}_{n}}u(\bullet;P)P!=(n+1)^{n},$
so
$O_{\mathcal{T}}(t)=\sum_{n\geq 0}\frac{(n+1)^{n}}{(n+1)!}t^{n}.$
(We note that $tO_{\mathcal{T}}(t)$ is the functional inverse of $te^{-t}$:
see [14, §5.3].) Now $P\in\mathcal{T}_{n}$ has a total of $n+1$ vertices to
which new edges can be added, so $\mathcal{T}$ is evenly up-covered with
$u_{n}=n+1$ and by Theorem 3.4 the morphism generating function is
$M_{\mathcal{T}}(t)=\sum_{n\geq 0}\frac{(n+1)^{n}}{n!}t^{2n+1}.$
In [5] the weighted-relation poset $Wrp(\mathcal{T}^{\uparrow})$ is discussed,
and it is shown that rank-$n$ objects of the universal cover
$\widetilde{\mathcal{T}}^{\uparrow}$ can be described as permutations of
$[n]$. (The partial order on permutations in
$\widetilde{\mathcal{T}}^{\uparrow}$ is as follows: a permutation $\tau$ of
$[n+1]$ covers the permutation $\tau\iota_{\tau^{-1}(n+1)}^{n}$ of $[n]$,
where $\iota_{m}^{n}$ is the order-preserving injection from $[n]$ to $[n+1]$
that misses $m$.) On the other hand, objects of
$\widetilde{\mathcal{T}}_{n}^{\downarrow}$ can be thought of as pairs $(P,f)$,
where $P\in\mathcal{T}_{n}$ and
$f:\\{0,1,2,\dots,n\\}\to P$
is a bijection such that $f(i)$ exceeds $f(j)$ (in the partial order on $P$)
whenever $i>j$. (Cf. the remark following [6, Prop. 2.5].)
###### Example 10.
(Binary rooted trees) Let $\mathcal{B}_{n}$ be the set of rooted binary trees
with $n+1$ terminal vertices (leaves). That is, an element of
$\mathcal{B}_{n}$ is a rooted tree in which each vertex has two daughters or
none (in which case it is a leaf). Any $P\in\mathcal{B}_{n}$ defines a metric
on its set $L(P)$ of leaves: the distance $\delta(p,q)$ from leaf $p$ to leaf
$q$ is the number of non-terminal vertices contained in the unique shortest
path from $p$ to $q$. An automorphism of $P\in\mathcal{B}_{n}$ is a bijection
$f$ on $L(P)$ such that $\delta(f(p),f(q))=\delta(p,q)$ for all $p,q$. A
morphism from $P\in\mathcal{B}_{n}$ to $Q\in\mathcal{B}_{n+1}$ is an injection
$f:L(P)\to L(Q)$ such that $\delta(f(p_{1}),f(p_{2}))\geq\delta(p_{1},p_{2})$
for all $p_{1},p_{2}\in L(P)$, and the only $r\in L(Q)$ with
$r\notin\operatorname{im}f$ is distance 1 from some $s\in\operatorname{im}f$.
For example, if
$P={\psline{*-*}(7.11317pt,0.0pt)(14.22636pt,14.22636pt)}{\psline{*-*}(14.22636pt,14.22636pt)(21.33955pt,0.0pt)}{\psline{*-*}(21.33955pt,0.0pt)(14.22636pt,-14.22636pt)}{\psline{*-*}(21.33955pt,0.0pt)(28.45274pt,-14.22636pt)}\hskip
36.135pt\text{and}\hskip
21.68121ptQ={\psline{*-*}(5.69054pt,0.0pt)(14.22636pt,14.22636pt)}{\psline{*-*}(14.22636pt,14.22636pt)(22.76219pt,0.0pt)}{\psline{*-*}(22.76219pt,0.0pt)(17.07164pt,-14.22636pt)}{\psline{*-*}(22.76219pt,0.0pt)(28.45274pt,-14.22636pt)}{\psline{*-*}(5.69054pt,0.0pt)(0.0pt,-14.22636pt)}{\psline{*-*}(5.69054pt,0.0pt)(11.38109pt,-14.22636pt)}$
then $|\operatorname{Aut}(P)|=2$, $|\operatorname{Aut}(Q)|=8$, and
$|\operatorname{Hom}(P,Q)|=8$: hence $u(P;Q)=1$ and $d(P;Q)=4$. (If we call a
pair of leaves distance 1 apart together with their common parent a “bud”,
then $u(P;Q)$ is the number of leaves of $P$ that can be replaced by a bud to
get $Q$, and $d(P;Q)$ is twice the number of buds of $Q$ that can be replaced
by a leaf to get $P$.)
Let $R\in\mathcal{B}_{n}$, and let $T$ be a particular realization of $R$ in
the plane, i.e., a planar binary rooted tree. Since $T$ has $n$ non-terminal
vertices, the group $G=\mathbf{Z}_{2}^{n}$ (where $\mathbf{Z}_{2}$ is the
group of order 2) acts on $T$ by rotations around each such vertex: the
isotropy group of $T$ is $\operatorname{Aut}R$. Then the number of distinct
planar binary rooted trees $T$ that can represent $R$ is
$\left|G/\operatorname{Aut}R\right|=\frac{2^{n}}{|\operatorname{Aut}R|}.$
Now there are $C_{n}$ distinct planar binary rooted trees with $n$ non-
terminal vertices, where $C_{n}$ is the $n$th Catalan number, so
$\sum_{P\in\mathcal{B}_{n}}\frac{2^{n}}{|\operatorname{Aut}R|}=C_{n}$
and the object generating function is
$O_{\mathcal{B}}(t)=\sum_{R\in\operatorname{Ob}\mathcal{B}}\frac{t^{|R|}}{|\operatorname{Aut}R|}=\sum_{n\geq
0}\frac{C_{n}}{2^{n}}t^{n}=\frac{1-\sqrt{1-2t}}{t}.$
Since $\mathcal{B}$ is evenly up-covered with $u_{n}=n+1$, by Theorem 3.4 the
morphism generating function is
$M_{\mathcal{B}}(t)=\sum_{n\geq
0}\frac{C_{n}(n+1)}{2^{n}}t^{2n+1}=\frac{t}{\sqrt{1-2t^{2}}}.$
Since $\mathcal{B}$ is evenly up-covered, equation (18) implies
$|\widetilde{\mathcal{B}}_{n}^{\uparrow}|=n!$, and by Theorem 3.2
$\widetilde{\mathcal{B}}^{\uparrow}$ must be isomorphic to
$\widetilde{\mathcal{T}}^{\uparrow}$. In fact, there is a natural way to
associate a permutation of $[n]$ to any
$c\in\widetilde{\mathcal{B}}_{n}^{\uparrow}$. Given
$(c_{0},c_{1},\dots,c_{n})\in\widetilde{\mathcal{B}}_{n}^{\uparrow}$, there is
a corresponding planar binary root tree with labelled non-terminal vertices: a
node gets label $i$ if $c_{i-1}\to c_{i}$ involves adding a bud at that node.
Put another set of labels $0,1,\dots,n$ on the leaves, running left to right.
For example, the two elements of $\widetilde{\mathcal{B}}_{2}^{\uparrow}$ are
$\rput(0.0pt,21.33955pt){{\circlenode{A}{1}}}\rput(-14.22636pt,0.0pt){{\circlenode{B}{2}}}\rput(28.45274pt,-21.33955pt){\rnode{C}{{\psframebox{2}}}}\rput(-28.45274pt,-21.33955pt){\rnode{D}{{\psframebox{0}}}}\rput(0.0pt,-21.33955pt){\rnode{E}{{\psframebox{1}}}}\psset{nodesep=1.0pt}{\ncline{-}{A}{B}}{\ncline{-}{A}{C}}{\ncline{-}{B}{D}}{\ncline{-}{B}{E}}\hskip
72.26999pt\text{and}\hskip
72.26999pt\rput(0.0pt,21.33955pt){{\circlenode{A}{1}}}\rput(14.22636pt,0.0pt){{\circlenode{B}{2}}}\rput(-28.45274pt,-21.33955pt){\rnode{C}{{\psframebox{0}}}}\rput(0.0pt,-21.33955pt){\rnode{D}{{\psframebox{1}}}}\rput(28.45274pt,-21.33955pt){\rnode{E}{{\psframebox{2}}}}\psset{nodesep=1.0pt}{\ncline{-}{A}{B}}{\ncline{-}{A}{C}}{\ncline{-}{B}{D}}{\ncline{-}{B}{E}}\hskip
72.26999pt.$
Now define a permutation of $[n]$ by sending $i\in[n]$ to the label on the
last common ancestor of the leaves labelled $i-1$ and $i$. For example, our
two labelled trees above correspond respectively to the permutation exchanging
1 and 2, and to the identity permutation.
An element $U\in\widetilde{\mathcal{B}}_{n}^{\downarrow}$ with
$\pi(U)=V\in\mathcal{B}_{n}^{\downarrow}$ can be thought of as $V$ equipped
with an appropriate set of labels on its edges so that exactly one of each
pair of edges coming out of a non-terminal vertex carries a label. More
precisely, let $\hat{V}$ be the set of non-terminal vertices of $V$: then
$U\in\pi^{-1}(V)$ can be identified with a pair of functions $(g,h)$, where
$g:[n]\to\hat{V}$ is a bijection such that $g(v)>g(w)$ in $U$ when $v>w$, and
$h:\hat{V}\to\\{L,R\\}$ (so that there are $2^{n}$ possiblities for $h$). In
this way one sees, e.g., that for the binary trees $P,Q$ above one has
$|\pi^{-1}(P)|=1\cdot 2^{2}=4$ and $|\pi^{-1}(Q)|=2\cdot 2^{3}=16$.
A summary of our examples (U=univalent, UC=evenly up-covered, F=factorial):
# | Description | Object g.f. | Morphism g.f. | U | UC | F
---|---|---|---|---|---|---
1 | Subsets of $[n]$ | $(1+t)^{n}$ | $nt(1+t^{2})^{n-1}$ | yes | yes | yes
2 | Monomials | $e^{nt}$ | $nte^{nt^{2}}$ | no | yes | yes
3 | Finite graphs | $\sum_{n\geq 0}2^{\binom{n}{2}}\frac{t^{n}}{n!}$ | $\sum_{n\geq 1}2^{\binom{n}{2}}\frac{t^{2n-1}}{(n-1)!}$ | no | yes | yes
4 | Necklaces | $1-\log(1-ct)$ | $\frac{ct}{1-ct^{2}}$ | no | yes | yes
5 | Partitions | $\prod_{n=1}^{\infty}\frac{1}{1-t^{n}}$ | $\frac{t}{1-t^{2}}\prod_{n=1}^{\infty}\frac{1}{1-t^{2n}}$ | yes | no | no
6 | Partitions | $\exp(\frac{t}{1-t})$ | $\frac{t}{1-t^{2}}\exp(\frac{t^{2}}{1-t^{2}})$ | no | no | no
7 | Compositions | $\frac{1-t}{1-2t}$ | $\frac{t(1-t^{2})}{(1-2t^{2})^{2}}$ | yes | no | no
8 | Planar rtd. trees | $\frac{1-\sqrt{1-4t}}{2t}$ | $\frac{1-\sqrt{1-4t^{2}}}{2t\sqrt{1-4t^{2}}}$ | yes | yes | no
9 | Rooted trees | $\sum_{n\geq 0}\frac{(n+1)^{n}}{(n+1)!}t^{n}$ | $\sum_{n\geq 0}\frac{(n+1)^{n}}{n!}t^{2n+1}$ | no | yes | no
10 | Binary rtd. trees | $\frac{1-\sqrt{1-2t}}{t}$ | $\frac{t}{\sqrt{1-2t^{2}}}$ | no | yes | no
## References
* [1] D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra, _J. Symbolic Comput._ 27 (1999), 581-600.
* [2] C. Brouder, Runge-Kutta methods and renormalization, _Eur. Phys. J. C_ 12, (2000), 521-534.
* [3] K. Engel, _Sperner Theory_ , Encyclopedia of Mathematics and its Applications 65, Cambridge University Press, New York, 1997.
* [4] S. Fomin, Duality of graded graphs, _J. Algebraic Combin._ 3 (1994), 357-404.
* [5] M. E. Hoffman, An analogue of covering space theory for ranked posets, _Electron. J. Combin._ 8 (2001), res. art. 32.
* [6] M. E. Hoffman, Combinatorics of rooted trees and Hopf algebras, _Trans. Amer. Math. Soc._ 355 (2003), 3795-3811.
* [7] M. E. Hoffman, Updown categories, MPIM preprint 2004-11; also available as arXiv math.CO/0402450.
* [8] S. Kerov, The boundary of Young lattice and random Young tableaux, _Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ 1994)_ , DIMACS Series in Discrete Mathematics and Theoretical Computer Science 24, American Mathematical Society, Providence, RI, 1996, pp. 133-158.
* [9] J. F. C. Kingman, Random partitions in population genetics, _Proc. Roy. Soc. London Ser. A_ 361 (1978), 1-20.
* [10] M. Mor and A. S. Fraenkel, Cayley permutations, _Disc. Math._ 48 (1984), 101-112.
* [11] N. J. A. Sloane, _The On-Line Encyclopedia of Integer Sequences_ , http://oeis.org.
* [12] R. P. Stanley, Differential posets, _J. Amer. Math. Soc._ 1 (1988), 919-961.
* [13] R. P. Stanley, Variations on differential posets, _Invariant Theory and Tableaux (Minneapolis, MN 1988)_ , IMA Volumes in Mathematics and its Applications 19, Springer-Verlag, New York, 1990, pp. 145-165.
* [14] R. P. Stanley, _Enumerative Combinatorics_ , vol. 2, Cambridge University Press, New York, 1999.
|
arxiv-papers
| 2012-07-06T18:58:55 |
2024-09-04T02:49:32.752415
|
{
"license": "Public Domain",
"authors": "Michael E. Hoffman",
"submitter": "Michael E. Hoffman",
"url": "https://arxiv.org/abs/1207.1705"
}
|
1207.1756
|
# Derivations of Siegel modular forms from connections
Enlin Yang Department of Mathematical Science, Tsinghua University, Beijing,
P. R. China 100084 yangenlin0727@126.com and Linsheng Yin Department of
Mathematical Science, Tsinghua University, Beijing, P. R. China 100084
lsyin@math.tsinghua.edu.cn
###### Abstract.
We introduce a method in differential geometry to study the derivative
operators of Siegel modular forms. By determining the coefficients of the
invariant Levi-Civita connection on a Siegel upper half plane, and further by
calculating the expressions of the differential forms under this connection,
we get a non-holomorphic derivative operator of the Siegel modular forms. In
order to get a holomorphic derivative operator, we introduce a weaker notion,
called modular connection, on the Siegel upper half plane than a connection in
differential geometry. Then we show that on a Siegel upper half plane there
exists at most one holomorphic modular connection in some sense, and get a
possible holomorphic derivative operator of Siegel modular forms.
###### Key words and phrases:
Levi-Civita connection, Siegel modular form, differential operator
## Introduction
In this paper, we introduce a differential geometric method to study the
derivative operators of Siegel modular forms, which, theoretically, may be
applied to the study of the derivative operators of any automorphic form. Our
idea comes from the observation on the two derivative operators of the
classical modular forms constructed by combinations. It is well-known [13]
that if $f$ is a modular forms of weight $2k$, then
$D_{k}f:=\frac{df}{dz}-\frac{\sqrt{-1}k}{y}f$ is a non-holomorphic modular
forms of weight $2k+2$, and $D_{k}f:=\frac{df}{dz}-\sqrt{-1}kG_{2}(z)f$, due
to J.-P. Serre [9], is a holomorphic modular forms of weight $2k+2$, where
$G_{2}(z)$ is the Eisenstein series of weight 2. We notice that the first
operator can be constructed by the Levi-Civita connection corresponding to the
invariant metric in the classical upper half plane, but the second can not be
constructed from any connection. However, if we loosen some condition in the
definition of the connection and define a concept called modular connection,
we can get Serre’s holomorphic derivative from the unique holomorphic modular
connection on the upper half plane. In this paper we extend these results to
Siegel upper half planes and Siegel modular forms. We determine the
coefficients of the Levi-Civita connection corresponding to the invariant
metric in a Siegel upper half plane, and compute the expressions of the
differential forms under the connection, which give us a non-holomorphic
derivative operator of Siegel modular forms. Our main results are as follows.
Let $\mathbb{H}_{g}$ be the Siegel upper plane of degree $g$,
$\\{dZ_{ij}:1\leq i,j\leq g\\}$ a series of coordinates on $\mathbb{H}_{g}$,
$\Gamma_{g}=\mbox{\rm Sp}(2g,\mathbb{Z})$ the full Siegel modular group which
acts on $\mathbb{H}_{g}$ naturally, $M_{k}=M_{k}(\Gamma_{g})$ the vector space
of the classical (or scalar-valued) Siegel modular forms of weight $k$,
$\widetilde{M}_{k}=\widetilde{M}_{k}(\Gamma_{g})$ the
$\mathbb{C}^{\infty}$-Siegel modular forms of weight $k$. Put
$\frac{\partial}{\partial Z}=(\partial_{ij})_{g\times
g}\qquad\text{and}\qquad\partial_{ij}=\frac{1}{2^{1-\delta(i,j)}}\cdot\frac{\partial}{\partial
Z_{ij}},$
where $\delta(i,j)=1$ if $i=j$ and $\delta(i,j)=0$ if $i\neq j$.
###### Theorem 0.1 (See theorem 2.7).
Let $f\in M_{2k}(\Gamma_{g})$. Then
$\det\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]f\right)\in\widetilde{M}_{2gk+2}(\Gamma_{g}).$
Here $\Gamma_{g}$ can be replaced by any congruence subgroup and the weight
$2k$ can also be replaced by any positive integer, with a little modification
of our proofs.
To get a holomorphic derivative operator, we introduce the notion of modular
connections on the Siegel upper half plane, whose condition is weaker than the
classical definition of the connections. Then we show the following result.
###### Theorem 0.2 (See theorem 2.9).
Any symmetric $g\times g$ matrix $G(Z)=(G_{ij}(Z))$ consisting of
$\mathbb{C}^{\infty}$ functions on $\mathbb{H}_{g}$, which satisfies the
transformation formula
$(CZ+D)^{-1}G(\gamma(Z))=G(Z)\cdot(CZ+D)^{t}+2C^{t}$
for any $\gamma=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\mbox{\rm Sp}(2g,\mathbb{Z})$, gives a unique modular
connection $\mathbb{D}$ such that for any $\mathbb{C}^{\infty}$-function $f$
on $\mathbb{H}_{g}$
$\mathbb{D}(dZ_{rs})=-\sum_{i,j=1}^{g}G_{ij}dZ_{si}dZ_{rj}\quad\text{and}\quad\mathbb{D}(f(\det(dZ)^{k})=\mbox{\rm
Tr}\left(\left[\frac{\partial}{\partial Z}-kG\right]fdZ\right)(\det(dZ))^{k},$
and thus gives a derivative operator $M_{2k}\rightarrow\widetilde{M}_{2kg+2}$
by $f\mapsto\det\left(\left[\frac{\partial}{\partial Z}-kG\right]f\right)$.
Furthermore, there exists at most one holomorphic symmetric matrix $G$ to
satisfy the transformation formula. If such a $G$ exists, the operator
corresponding to $G$ is holomorphic.
In the classical case of $g=1$, the function $\sqrt{-1}G_{2}(z)$ is the unique
holomorphic function on the upper half plane satisfying the condition, which
gives Serre’s derivative. But when $g\geq 2$ we are not able to construct such
a matrix function $G$.
H. Maass has constructed a non-holomorphic derivative operator of Siegel
modular forms by invariant differential operators. For Siegel modular forms
$f$ of weight $k$, Maass ([7], P317) defines the operator
$D_{k}f(Z)=\det(Y)^{\kappa-k-1}\det\left(\frac{\partial}{\partial
Z}\right)[\det(Y)^{k+1-\kappa}f(Z)],$
where $\kappa=(g+1)/2$ and the determinant of $\frac{\partial}{\partial Z}$ is
taken first, and shows that the differential operator $D_{k}$ acts on the
$\mathbb{C}^{\infty}$-Siegel modular forms and maps $\widetilde{M}_{k}$ to
$\widetilde{M}_{k+2}$. We do not know the relation between our operator in
Theorem 0.1 and Maass’. Compared to our operator, $D_{k}$ is linear with
respect to $f$. Moreover, our operator is a combination of degree 1 partial
derivatives of $f$, but $D_{k}$ is a combination of degree $g$ partial
derivatives. G. Shimura [8] considers the compositions
$D_{r}^{k}=D_{r+2k-2}\cdots D_{r+2}D_{r}$ of Maass’ operator, which maps
$\widetilde{M}_{r}$ to $\widetilde{M}_{r+2k}$. For our operator one can also
consider the compositions and then construct the Rankin-Cohen brackets. We
wish that Maass’ operator could be got in this way.
The paper is organized as follows. In section one, we introduce the concept of
modular connection on a Siegel upper plane, and show several lemmas on it. In
section two, we compute the expressions of the differential forms under the
modular connection, and prove the two theorems above. Finally in section
three, we show Lemma 2.1 which explicitly gives the connection coefficients of
the Levi-Civita connection on a Siegel upper half plane.
Our calculations in sections 2 and 3 are tested by matlab in the cases $g=2$
and $g=3$.
## 1\. Modular Connections
In this section we first recall the definition of connections in differential
geometry. Then we introduce the notion of modular connection on a Siegel upper
half plane, and show several lemmas about it.
### 1.1. Connections in differential geometry
For the backgrounds and notations on differential geometry, especially on
connections, we refer to the books [2] and [4]. Here we just recall some basic
definitions and results on connections. Suppose $E$ is a $q$-dimensional real
vector bundle on a smooth manifold $M$, and $\Gamma(E)$ is the set of smooth
sections of $E$ on $M$. Let $T^{*}(M)$ be the cotangent space of $M$. A
connection on the vector bundle $E$ is a map
$D:\quad\Gamma(E)\longrightarrow\Gamma(T^{*}(M)\otimes E),$
which satisfies the following conditions
1. (1)
For any $s_{1},s_{2}\in\Gamma(E)$,
$D(s_{1}+s_{2})=D(s_{1})+D(s_{2}).$
2. (2)
For any $s\in\Gamma(E)$ and any $\alpha\in\mathbb{C}^{\infty}(M)$,
$D(\alpha s)=d\alpha\otimes\alpha D(s).$
If $M$ has a generalized Riemannian metric $G=\sum_{i,j}g_{ij}du^{i}du^{j}$,
by the fundamental theorem of Riemannian geometry, $M$ has a unique torsion-
free and metric-compatible connection, called Levi-Civita connection of $M$.
The coefficients $\Gamma_{ij}^{k}$ of the Levi-Civita connection are given by
$\Gamma_{ij}^{k}=\frac{1}{2}\sum_{l}g^{kl}\left(\frac{\partial
g_{il}}{\partial u^{j}}+\frac{\partial g_{jl}}{\partial u^{i}}-\frac{\partial
g_{ij}}{\partial u^{l}}\right),$ (1.1)
where $g^{ij}$ are elements of the matrix $(g^{ij}):=(g_{ij})^{-1}$.
The following lemma is useful in the application of connections to automorphic
forms.
###### Lemma 1.1.
Let $\Gamma$ be a group, $(M,G)$ a Riemannian manifold and $D$ the Levi-Civita
connection on $M$. If $\Gamma$ has a smooth left action on $M$ such that
$G(\sigma_{\star}X,\sigma_{\star}Y)=G(X,Y)$ for all $\sigma\in\Gamma,X,Y\in
T(M)$, then
$\sigma D=D\sigma\qquad(\sigma\in\Gamma).$
Moreover, if $M$ is a complex manifold such that $\Gamma$ maps $(r,s)$ forms
to $(r,s)$ forms, and put $D=D^{1,0}+D^{0,1}$, where $D^{1,0}$ is the
holomorphic part, then for $\sigma\in\Gamma$
$\sigma D^{1,0}=D^{1,0}\sigma\quad\text{ and }\quad\sigma
D^{0,1}=D^{0,1}\sigma.$
###### Proof.
$D$ is the unique torsion free connection which preserves the Riemannian
metric $G$. Since $G$ is $\Gamma$-invariant, the connection
$\sigma^{-1}D\sigma$ also preserves the Riemannian metric and is torsion free
for any $\sigma\in\Gamma$, hence $\sigma D=D\sigma$. For more detail, see
([10], P35). ∎
### 1.2. Seigel upper half plane
We first fix some notations. The Siegel upper half plane of degree $g\geq 1$
is defined to be the $g(g+1)/2$ dimensional open complex variety
$\mathbb{H}_{g}:=\\{Z=X+\sqrt{-1}Y\in M(g,\mathbb{C})\mid Z^{t}=Z,Y>0\\}.$
Write $Z=(Z_{ij})$. Set $\Omega=\\{(i,j)\mid 1\leq i\leq j\leq g\\}$ with the
dictionary order. If $I=(i,j)\in\Omega$, we define $Z_{I}:=Z_{ij}$. Fix a
series of coordinates $\\{dZ_{I},d\bar{Z}_{I}\mid I\in\Omega\\}$ on
$\mathbb{H}_{g}$. The symplectic group of degree $g>0$ over $\mathbb{R}$ is
the group
$\mbox{\rm Sp}(2g,\mathbb{R})=\left\\{M\in
GL(2g,\mathbb{R})\,\middle|\,MJM^{t}=J\right\\},$
where $J=\left(\begin{array}[]{cc}0&I_{g}\\\ -I_{g}&0\end{array}\right)$. We
usually write an element of $\mbox{\rm Sp}(2g,\mathbb{R})$ in the form
$\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)$, where $A,B,C$ and $D$
are $g\times g$ blocks. The symplectic group $\mbox{\rm Sp}(2g,\mathbb{R})$
acts on $\mathbb{H}_{g}$ by the rule:
$\gamma(Z):=(AZ+B)(CZ+D)^{-1},\quad
Z\in\mathbb{H}_{g},\quad\gamma=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\mbox{\rm Sp}(2g,\mathbb{R}).$
By Maass ([6], P98), $d(\gamma
Z)=(ZC^{t}+D^{t})^{-1}dZ(CZ+D)^{-1}:=(d\tilde{Z}_{ij})$. Let
$(d\tilde{Z}_{11},d\tilde{Z}_{12},\cdots,d\tilde{Z}_{1g},\cdots,\cdots,d\tilde{Z}_{gg})=(dZ_{11},dZ_{12},\cdots,dZ_{1g},\cdots,\cdots,dZ_{gg})\cdot
S(\gamma,Z)$
where $S:=S(\gamma,Z)$ is a $\frac{g(g+1)}{2}\times\frac{g(g+1)}{2}$ matrix of
holomorphic functions on $\mbox{\rm Sp}(2g,\mathbb{Z})\times\mathbb{H}_{g}$.
From Lemma 1.1, one can see that the connection matrix $\omega$ consisting of
the connection coefficients of the Levi-Civita connection associated to the
invariant metric $ds^{2}=\mbox{\rm Tr}(Y^{-1}dZ\cdot Y^{-1}d\bar{Z})$ given by
Siegel ([6], P8) on the Siegel upper plane $\mathbb{H}_{g}$ satisfies
$\gamma(\omega)=-S^{-1}\cdot dS+S^{-1}\cdot\omega\cdot S$
for all $\gamma\in\mbox{\rm Sp}(2g,\mathbb{R})$. Refer also to the proof of
Lemma 1.5 below. But in the studying of modular forms, we only need that the
equality holds for all $\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$, the Siegel
modular group. So we need to introduce a weaker notion to study modular forms.
Now we recall the definition of Siegel modular forms, for more details, see
[1] and [3].
###### Definition 1.2.
A (classical) Siegel modular form of weight $k$ (and degree $g$) is a
holomorphic function $f:\mathbb{H}_{g}\rightarrow\mathbb{C}$ such that
$f(\gamma(Z))=\det(CZ+D)^{k}f(Z)$
for all $\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\\\
\end{array}\right)\in\mbox{\rm Sp}(2g,\mathbb{Z})$ (with the usual
holomorphicity requirement at $\infty$ when $g=1$).
### 1.3. Modular connections
The notations are the same as those above.
###### Definition 1.3 (Modular Connection Coefficients (MCC)).
The modular connection coefficient on $\mathbb{H}_{g}$ is a series of
$\mathbb{C}^{\infty}$-functions $\\{\Gamma_{IJ}^{K}\mid I,J,K\in\Omega\\}$
such that for all $\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$,
$\gamma(\omega)=-S^{-1}\cdot dS+S^{-1}\cdot\omega\cdot S,\quad\text{or}\quad
S\cdot\gamma(\omega)=\omega\cdot S-dS,$
where $\omega=(\omega_{I}^{J})$ and
$\omega_{I}^{J}=\sum_{K\in\Omega}\Gamma_{IK}^{J}dZ_{K}$. Here $I$ and $J$ are
the row and column indices respectively. When $\\{\Gamma_{IJ}^{K}\\}$ are
holomorphic, we call it holomorphic MCC (HMCC). The matrix $\omega$ is called
the modular connection matrix.
In the following, $\mathbb{C}^{\infty}(\mathbb{H}_{g})$ is the set of
$\mathbb{C}^{\infty}$ functions on $\mathbb{H}_{g}$, and
$\text{Hol}(\mathbb{H}_{g})$ is the set of holomorphic functions on
$\mathbb{H}_{g}$.
###### Definition 1.4 (Modular Connection).
Let $\\{\Gamma_{IJ}^{K}\\}$ be a MCC (resp. HMCC) on $\mathbb{H}_{g}$ and
$\Omega^{\infty}$ be the commutative
$\mathbb{C}^{\infty}(\mathbb{H}_{g})$-algebra (resp.
$\text{Hol}(\mathbb{H}_{g})$-algebra) generated by $\\{dZ_{I}\\}_{I\in\Omega}$
with the relations $dZ_{I}dZ_{J}=dZ_{J}dZ_{I}$ for any $I,J\in\Omega$. The
linear operator
$D:\Omega^{\infty}\longrightarrow\Omega^{\infty}$
is uniquely defined by the following two relations
$D(dZ_{K})=-\sum_{I,J\in\Omega}\Gamma_{IJ}^{K}dZ_{I}dZ_{J}$
and
$D(fdZ_{K_{1}}dZ_{K_{2}}\cdots dZ_{K_{r}})=df\cdot dZ_{K_{1}}\cdots
dZ_{K_{r}}+\sum_{i=1}^{r}fdZ_{K_{1}}dZ_{K_{2}}\cdots D(dZ_{K_{i}})\cdots
dZ_{K_{r}},$
and we call it the modular connection associated to $\\{\Gamma_{IJ}^{K}\\}$.
One can easily show that $D(dZ_{K})=-\sum_{I\in\Omega}\omega_{I}^{K}\cdot
dZ_{I}$ and
$(D(dZ_{11}),D(dZ_{12}),\cdots,\cdots,\cdots,D(dZ_{gg}))=-(dZ_{11},dZ_{12},\cdots,\cdots,\cdots,dZ_{gg})\cdot\omega$
When $\\{\Gamma_{IJ}^{K}\\}$ is holomorphic, we also call $D$ a holomorphic
modular connection. Compared with the definition of connections in
differential geometry, except for the weaker conditions, the modular
connection also ignore the part on $\\{d\bar{Z}_{I}\\}_{I\in\Omega}$.
### 1.4. Basic lemmas on modular connections
The following two lemmas are basic to our application of modular connections
to the Siegel modular forms. For the modular connections, we have the similar
result to Lemma 1.1.
###### Lemma 1.5.
Let $D$ be a modular connection on $\mathbb{H}_{g}$. Then $\gamma D=D\gamma$
for any $\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$. Moreover, if $f$ is a Siegel
modular forms of weight $2k$, then $D\left(f(\det(dZ))^{k}\right)$ is
invariant under the action of $\Gamma_{g}=\mbox{\rm Sp}(2g,\mathbb{Z})$.
###### Proof.
Let $\alpha=(dZ_{11},\cdots,dZ_{1g},dZ_{22},\cdots,dZ_{2g},\cdots,dZ_{gg})$.
Then $D\alpha=-\alpha\omega$. On one side,
$\gamma(D\alpha)=-\gamma(\alpha)\gamma(\omega)=-\alpha
S(-S^{-1}dS\cdot+S^{-1}\cdot\omega\cdot S)=\alpha(dS-\omega\cdot S).$
On the other side,
$D(\gamma\alpha)=D(\alpha\cdot S)=D(\alpha)\cdot S+\alpha\cdot
dS=-\alpha\omega\cdot S+\alpha dS=\alpha(-\omega\cdot S+dS).$
For $f\in M_{2k}(\Gamma_{g})$, we see $f(\det(dZ))^{k}$ is invariant under the
action of $\Gamma_{g}$, and so is $D(f(\det(dZ))^{k})$. ∎
The following lemma directly from Lemma 1.1 gives a modular connection.
###### Lemma 1.6.
The holomorphic part $D^{1,0}$ of the Levi-Civita connection associated to the
invariant metric $ds^{2}=\mbox{\rm Tr}(Y^{-1}dZ\cdot Y^{-1}d\bar{Z})$ on the
Siegel upper plane $\mathbb{H}_{g}$ is a modular connection.
We will give the explicit expression of $D^{1,0}(f\det(dZ)^{k})$ in
Proposition 2.4.
### 1.5. Modular connections in the classical case
Let us consider the classical upper half plane $\mathbb{H}=\mathbb{H}_{1}$ to
look for what condition of the coefficient
$\Gamma:=\Gamma_{(1,1),(1,1)}^{(1,1)}$ of a modular connection should satisfy.
Let $\omega=\Gamma dz$. One can easily check that for $\gamma\in\mbox{\rm
SL}(2,\mathbb{Z})$
$\gamma(\omega)=-S^{-1}dS+S^{-1}\omega
S\Longleftrightarrow\frac{\gamma(\Gamma)}{(cz+d)^{2}}=\Gamma+\frac{2c}{cz+d}.$
Recall that ([5], P113)
$\frac{1}{(cz+d)^{2}}\cdot\frac{\sqrt{-1}}{\text{Im}(\gamma
z)}=\frac{\sqrt{-1}}{\text{Im}(z)}+\frac{2c}{cz+d}\quad\text{and}\quad\frac{\sqrt{-1}G_{2}(\gamma
z)}{(cz+d)^{2}}=\sqrt{-1}G_{2}(z)+\frac{2c}{cz+d},$
where
$G_{2}(z)=\frac{1}{2\pi}\left(\sum_{n\not=0}\frac{1}{n^{2}}+\sum_{m\not=0}\sum_{n\in\mathbb{Z}}\frac{1}{(mz+n)^{2}}\right).$
So $\frac{\sqrt{-1}}{y}$ and $\sqrt{-1}G_{2}(z)$ give us two modular
connections on $\mathbb{H}$, which we denote by $D_{1}$ and $D_{2}$
respectively. The later is holomorphic. We have
$D_{1}(fdz)=\left(\frac{df}{dz}-\frac{\sqrt{-1}}{y}f\right)dzdz\quad\text{and}\quad
D_{2}(fdz)=\left(\frac{df}{dz}-\sqrt{-1}G_{2}(z)f\right)dzdz.$
By the expressions of $D_{1}(f(dz)^{k})$ and $D_{2}(f(dz)^{k})$ and by Lemma
1.5, we have
###### Corollary 1.7.
Let $f$ be a modular form of weight $2k$. Then
$\frac{df}{dz}-\frac{\sqrt{-1}k}{y}f$ and $\frac{df}{dz}-\sqrt{-1}kG_{2}(z)f$
are modular forms of weight $2k+2$. They are non-holomorphic and holomorphic,
respectively.
In fact, the modular connection $D_{1}$ comes from the Levi-Civita connection
$D$ associated to the invariant metric $ds^{2}=\frac{dz\,d\bar{z}}{y^{2}}$.
###### Lemma 1.8.
$D(dz)=-\frac{\sqrt{-1}}{y}dz\,dz\quad\text{and}\quad
D(d\bar{z})=\frac{\sqrt{-1}}{y}d\bar{z}\,d\bar{z}$. So the coefficients of $D$
give the modular connection $D_{1}$.
###### Proof.
Since $ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}=\frac{dz\,d\bar{z}}{y^{2}}$, using
the coordinates ${dz,d\overline{z}}$ and the equality (1.1), we have
$\Gamma_{1,1}^{1}=\frac{\sqrt{-1}}{y},\ \Gamma_{2,1}^{1}=\Gamma_{1,2}^{1}=0,\
\Gamma_{2,1}^{2}=\Gamma_{1,2}^{2}=0$ and
$\Gamma_{2,2}^{2}=-\frac{\sqrt{-1}}{y}$. ∎
The connection $D_{2}$ is not from differential geometry. It is unique.
###### Lemma 1.9 (Uniqueness Lemma).
$\sqrt{-1}G_{2}(z)$ is the unique holomorphic function $\Gamma$ satisfying
$\frac{\gamma(\Gamma)}{(cz+d)^{2}}=\Gamma+\frac{2c}{cz+d}~{}~{}\text{for
all}~{}~{}\gamma=\left(\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}\right)\in\mbox{\rm SL}(2,\mathbb{Z}),$
and so $D_{2}$ is the unique holomorphic modular connection on $\mathbb{H}$.
###### Proof.
We have
$\gamma(\sqrt{-1}G_{2}(z)-\Gamma)=(cz+d)^{2}(\sqrt{-1}G_{2}(z)-\Gamma)$ for
all $\gamma\in\mbox{\rm SL}(2,\mathbb{Z})$. So $\sqrt{-1}G_{2}(z)-\Gamma$ is a
modular form of weight 2 and hence must be zero ([5], P117). ∎
We will generalize these results to $\mathbb{H}_{g}$.
## 2\. Derivative Operators of Siegel Modular Forms
In this section, we first state the result determining the coefficients of the
invariant Levi-Civita connection on a Siegel upper half plane, whose proof we
put in the last section, then we compute the expressions of the differential
forms under this connection. Finally we get a non-holomorphic derivative
operator and a possible holomorphic derivative operator.
### 2.1. Coefficients of Levi-Civita connection
The notations are the same as those in section 1. For $I=(i,j)\in\Omega$, put
$N(I)=\frac{(i-1)(2g-i)}{2}+j,$
which gives a one to one and order keeping correspondence between $\Omega$ and
$\\{1,2,\cdots,\frac{g(g+1)}{2}\\}$. Write $u^{N(I)}=Z_{ij}$ and
$u^{N(g,g)+N(i,j)}=\overline{Z}_{ij}$. For $Z=X+\sqrt{-1}Y\in\mathbb{H}_{g}$,
let $R=(R_{ij})=Y^{-1}$. For $1\leq s\leq g$, we set
$\Omega_{s}=\\{(1,s),(2,s),\cdots,(s,s),(s,s+1),\cdots,(s,g)\\}\subset\Omega.$
Let $K=(r,s)\in\Omega$. Assume that the elements of $Z$ in the column
including $Z_{K}=Z_{rs}$ are
$u^{a_{1}}=Z_{1s},u^{a_{2}}=Z_{2s},\cdots,u^{a_{s}}=Z_{ss},u^{a_{s+1}}=Z_{s+1,s}=Z_{s,s+1},\cdots,u^{a_{g}}=Z_{gs}=Z_{sg},$
and the elements in the row including $Z_{rs}$ are
$u^{b_{1}}=Z_{r1}=Z_{1r},u^{b_{2}}=Z_{r2}=Z_{2r},\cdots,u^{b_{r}}=Z_{rr},u^{b_{r+1}}=Z_{r,r+1},\cdots,u^{b_{g}}=Z_{rg}.$
For $I\times J\in\Omega_{s}\times\Omega_{r}$, assume $Z_{I}=u^{a_{i}}$ and
$Z_{J}=u^{b_{j}}$. Similarly do it for $J\times
I\in\Omega_{s}\times\Omega_{r}$. We define
$\Gamma_{IJ}^{K}\,(=\Gamma_{JI}^{K}):=\left\\{\begin{array}[]{ll}\frac{\sqrt{-1}R_{ij}}{2^{(1-\delta(r,s))(1-\delta(a_{i},b_{j}))}}&\mbox{
if }I\times J\text{ or }J\times I\in\Omega_{s}\times\Omega_{r}\\\ 0&\text{ if
}I\times J\text{ and }J\times
I\not\in\Omega_{s}\times\Omega_{r},\end{array}\right.$ (2.1)
where $\delta(r,s)$ denotes the Kronecker delta symbol. For example,
$\Gamma_{(1,i)(1,j)}^{(1,1)}=\sqrt{-1}R_{ij}$ and $\Gamma_{IJ}^{(1,1)}=0$ if
$I$ or $J\not\in\Omega_{1}$.
Notice that if we use the coordinates $\\{u^{a_{i}},u^{b_{j}}\\}$, then the
coefficients of the Levi-Civita connection satisfy
${}_{Z}\\!\Gamma_{IJ}^{K}={}_{u}\\!\Gamma_{N(I)N(J)}^{N(K)}.$
In this paper we will use these two kinds of coordinates alternately. The
proof of the following lemma is long and complicated. For the convenience of
the reader, we put it in the last section.
###### Lemma 2.1.
The coefficients $\\{\Gamma_{IJ}^{K}\\}$ defined in the equality (2.1) give
the Levi-Civita connection on $\mathbb{H}_{g}$ associated to the invariant
metric $ds^{2}=\mbox{\rm Tr}(Y^{-1}dZ\,Y^{-1}d\bar{Z})$, and hence give a
modular connection, which we denote by $D$.
### 2.2. Expression of differential forms under $D$
We first compute $D(dZ_{K})$.
###### Lemma 2.2.
Let $K=(r,s)\in\Omega$. We have
$D(dZ_{K})=-\sqrt{-1}(dZ_{s1},dZ_{s2},\cdots,dZ_{sg})Y^{-1}\cdot(dZ_{r1},dZ_{r2},\cdots,dZ_{rg})^{t}.$
###### Proof.
The notations are as above. If $r=s$, then $\Omega_{r}=\Omega_{s}$. By Lemma
2.1, we have
$\Gamma_{IJ}^{K}=\Gamma_{JI}^{K}=\left\\{\begin{array}[]{ll}\sqrt{-1}R_{ij}&\mbox{
if }I\text{ and }J\in\Omega_{r}\\\ 0&\text{ if }I\text{ or
}J\not\in\Omega_{r},\end{array}\right.$
Hence,
$\displaystyle D(dZ_{K})$ $\displaystyle=$
$\displaystyle-\sum_{I,J\in\Omega}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-\sum_{I,J\in\Omega_{r}}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-\sum_{i,j=1}^{g}\sqrt{-1}R_{ij}dZ_{si}dZ_{rj}$
$\displaystyle=$
$\displaystyle-\sqrt{-1}(dZ_{s1},dZ_{s2},\cdots,dZ_{sg})Y^{-1}\cdot(dZ_{r1},dZ_{r2},\cdots,dZ_{rg})^{t}.$
If $r\not=s$, we assume $r<s$. Then $\Omega_{r}\bigcap\Omega_{s}=\\{(r,s)\\}$
and
$(\Omega_{r}\times\Omega_{s})\bigcap(\Omega_{s}\times\Omega_{r})=\\{(r,s)\times(r,s)\\}$.
Put $A=(\Omega_{r}\times\Omega_{s})\bigcup(\Omega_{s}\times\Omega_{r})$ and
$B=(\Omega_{r}\times\Omega_{s})\bigcap(\Omega_{s}\times\Omega_{r})$. We have
again by Lemma 2.1,
$\Gamma_{IJ}^{K}=\Gamma_{JI}^{K}=\left\\{\begin{array}[]{lll}\frac{\sqrt{-1}R_{ij}}{2^{1-\delta(a_{i},b_{j})}}=\frac{\sqrt{-1}R_{ij}}{2}&\mbox{
if }I\times J\in A\setminus B\\\ \
\frac{\sqrt{-1}R_{ij}}{2^{1-\delta(a_{i},b_{j})}}=\sqrt{-1}R_{ij}&\mbox{ if
}I\times J\in B.\\\ \ 0&\text{ if }I\times J\not\in A\end{array}\right.$
Hence,
$\displaystyle
D(dZ_{K})=-\sum_{I,J\in\Omega}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-\sum_{I\times J\in
A}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}$ $\displaystyle=$
$\displaystyle-2\sum_{I\in\Omega_{r}\atop
J\in\Omega_{s}}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}+\sum_{I\in\Omega_{r}\bigcap\Omega_{s}\atop
J\in\Omega_{r}\bigcap\Omega_{s}}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-2\sum_{I\times
J\in\Omega_{r}\times\Omega_{s}-B}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}-\sum_{I\times
J\in B}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}$ $\displaystyle=$
$\displaystyle-\sum_{i,j=1}^{g}\sqrt{-1}R_{ij}dZ_{si}dZ_{rj}=-\sqrt{-1}(dZ_{s1},dZ_{s2},\cdots,dZ_{sg})Y^{-1}\cdot(dZ_{r1},dZ_{r2},\cdots,dZ_{rg})^{t}.$
The case $s<r$ is similar. ∎
###### Proposition 2.3.
$D(\det(dZ))=-\sqrt{-1}\mbox{\rm Tr}(Y^{-1}dZ)\det(dZ)$.
###### Proof.
Put $\alpha_{i}=(dZ_{i1},dZ_{i2},\cdots,dZ_{ig})$ and
$\beta_{j}=(dZ_{1j},dZ_{2j},\cdots,dZ_{gj})^{t}$. By Lemma 2.2,
$D(d(Z_{ij}))=-\sqrt{-1}\alpha_{i}Y^{-1}\beta_{j}$ and
$D(\alpha_{i})=-\sqrt{-1}\alpha_{i}Y^{-1}dZ$. Thus
$D(\det(dZ))=-\sqrt{-1}\left\\{\det\left(\begin{array}[]{c}\alpha_{1}Y^{-1}dZ\\\
\alpha_{2}\\\ \vdots\\\
\alpha_{g}\end{array}\right)+\det\left(\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}Y^{-1}dZ\\\ \vdots\\\
\alpha_{g}\end{array}\right)+\cdots+\det\left(\begin{array}[]{c}\alpha_{1}\\\
\vdots\\\ \alpha_{g-1}\\\ \alpha_{g}Y^{-1}dZ\end{array}\right)\right\\}.$
Let $A=Y^{-1}dZ=(A_{ij})$ and $dZ[i,j]$ be the algebraic cofactor of $dZ$ at
the position $(i,j)$. By the formula above, we have
$\displaystyle\sqrt{-1}D(\det(dZ))=\sum_{k,j,i=1}^{n}dZ_{ki}\cdot A_{ij}\cdot
dZ[k,j]=\sum_{j,i=1}^{n}A_{ij}\sum_{k=1}^{n}dZ_{ik}\cdot dZ[k,j]$
$\displaystyle=$
$\displaystyle\sum_{j,i=1}^{n}A_{ij}\delta(i,j)\det(dZ)=\mbox{\rm
Tr}(A)\det(dZ)=\mbox{\rm Tr}(Y^{-1}dZ)\det(dZ).$
∎
Put
$\frac{\partial}{\partial Z}=(\partial_{ij})_{g\times
g},\qquad\partial_{ij}=\frac{1+\delta(i,j)}{2}\cdot\frac{\partial}{\partial
Z_{ij}}$
as in the introduction. Then for any $\mathbb{C}^{\infty}$-function $f$ on
$\mathbb{H}_{g}$,
$df=\sum_{1\leq i\leq j\leq g}\frac{\partial f}{\partial
Z_{ij}}dZ_{ij}=\sum_{i=1}^{g}\sum_{j=1}^{g}\partial_{ij}fdZ_{ij}=\mbox{\rm
Tr}\left(\frac{\partial}{\partial Z}f\cdot dZ\right)$
###### Proposition 2.4.
For any $\mathbb{C}^{\infty}$-function $f$ on $\mathbb{H}_{g}$, we have
$D\left(f\det(dZ)^{k}\right)=\mbox{\rm Tr}\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)\det(dZ)^{k}.$
###### Proof.
Since $df=\mbox{\rm Tr}(\frac{\partial}{\partial Z}f\cdot dZ)$, we have, by
Proposition 2.3,
$\displaystyle D\left(f\det(dZ)^{k}\right)$ $\displaystyle=$ $\displaystyle
df\cdot\det(dZ)^{k}+f\cdot D((\det(dZ)^{k})=(df-\sqrt{-1}kf\mbox{\rm
Tr}(Y^{-1}dZ))\det(dZ)^{k}$ $\displaystyle=$ $\displaystyle\mbox{\rm
Tr}\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)\det(dZ)^{k}.$
∎
In the following, $\Omega^{i}_{\mathbb{H}_{g}}$ is the sheaf of holomorphic
$i$-forms on $\mathbb{H}_{g}$. Recall that a section of
$\Omega^{1}_{\mathbb{H}_{g}}$ can be written as $\mbox{\rm Tr}(GdZ)$, where
$G$ is a symmetric matrix of holomorphic functions on $\mathbb{H}_{g}$.
###### Proposition 2.5.
For any section $\mbox{\rm Tr}(GdZ)\in\Omega_{\mathbb{H}_{g}}^{1}$, where $G$
is a symmetric matrix of holomorphic functions on $\mathbb{H}_{g}$, we have
$D(\mbox{\rm Tr}(GdZ))=\mbox{\rm
Tr}\left\\{\left[\left(\frac{\partial}{\partial Z}\right)^{t}\otimes
G\right]\cdot[dZ\otimes dZ]\right\\}-\sqrt{-1}\mbox{\rm Tr}\left(GdZ\cdot
Y^{-1}dZ\right),$
where $\otimes$ is the Kronecker product of matrices.
To show this proposition, we need the following lemma.
###### Lemma 2.6.
We have
1. (1)
Let $A=(a_{ij})_{n\times n},B=(b_{ij})_{n\times n},C=(c_{ij})_{n\times
n},D=(d_{ij})_{n\times n}$. Then
$\mbox{\rm Tr}((A\otimes B)(C\otimes
D))=\sum_{i,j,k,l=1}^{n}a_{ij}b_{kl}c_{lk}d_{ji}=\mbox{\rm Tr}((A\otimes
C)(B\otimes D)),$
2. (2)
$d(\mbox{\rm Tr}(GdZ))=\mbox{\rm Tr}\left(\left(\left(\frac{\partial}{\partial
Z}\right)^{t}\otimes G\right)\cdot(dZ\otimes dZ)\right)$ for a symmetric
matrix $G=(G_{ij})$ of functions.
###### Proof.
The proof of (1) is easy. We only show (2). By (1),
$\displaystyle d(\mbox{\rm Tr}(GdZ))$ $\displaystyle=$ $\displaystyle
d\left(\sum_{i,j=1}^{g}G_{ij}dZ_{ij}\right)=\sum_{i,j=1}^{g}\sum_{k,l=1}^{g}\partial_{kl}G_{ij}\cdot
dZ_{kl}dZ_{ji}$ $\displaystyle=$ $\displaystyle\mbox{\rm
Tr}\bigg{(}\big{(}(\partial_{kl})^{t}\otimes(G_{ij})\big{)}\cdot(dZ\otimes
dZ)\bigg{)}.$
∎
###### Proof of Proposition 2.5.
As
$dZ=(\alpha_{1}^{t},\alpha_{2}^{t},\cdots,\alpha_{g}^{t})^{t}=(\beta_{1},\beta_{2},\cdots,\beta_{g})$,
we have
$\displaystyle D(\mbox{\rm Tr}(GdZ))-\mbox{\rm
Tr}\left\\{\left[\left(\frac{\partial}{\partial Z}\right)^{t}\otimes
G\right]\cdot[dZ\otimes dZ]\right\\}=\mbox{\rm Tr}\left(GD(dZ)\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{g}G_{ij}D(dZ_{ij})=-\sqrt{-1}\sum_{i,j=1}^{g}G_{ij}d(\alpha_{i})Y^{-1}d(\beta_{j})=-\sqrt{-1}\mbox{\rm
Tr}\left(GdZY^{-1}dZ\right).$
∎
### 2.3. Derivative operators
If $0\neq f\in M_{2k}(\Gamma_{g})$, we have, by Proposition 2.4 and Lemma 1.5,
$\mbox{\rm Tr}\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)\det(dZ)^{k}$
is invariant under the action of $\Gamma_{g}$. Let
$\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\in\Gamma_{g}$.
Since $\gamma(\det(dZ))^{k}=\frac{\det(dZ)^{k}}{\det(CZ+D)^{2k}}$ and
$\gamma(f)=\det(CZ+D)^{2k}f$, we have
$\mbox{\rm Tr}\left(\frac{1}{f}\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)$
is invariant under $\Gamma_{g}$. Put
$h:=\frac{1}{f}\left(\frac{\partial}{\partial Z}-\sqrt{-1}kY^{-1}\right)f$.
Then $h$ is a symmetric matrix of functions on $\mathbb{H}_{g}$, and
$\mbox{\rm Tr}(hdZ)$ is invariant under the action of $\Gamma_{g}$. Since for
any $\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\\\
\end{array}\right)\in\Gamma_{g}$, $d(\gamma Z)=(ZC^{t}+D^{t})^{-1}\cdot
dZ\cdot(CZ+D)^{-1}$, we have (see ([3], P210))
$h(\gamma Z)=(CZ+D)h(Z)(ZC^{t}+D^{t}).$
Thus $\det(h)=\det\left(\frac{1}{f}\left(\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right)f\right)$ is a non-holomorphic Siegel modular form
of weight 2. Let $\widetilde{M}_{k}(\Gamma_{g})$ be the
$\mathbb{C}^{\infty}$-Siegel modular forms of weight $k$ as in the
introduction. Finally we get
###### Theorem 2.7.
If $f\in M_{2k}(\Gamma_{g})$, then $\det\left(\left(\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right)f\right)\in\widetilde{M}_{2kg+2}(\Gamma_{g})$.
Let $f\in M_{2r}(\Gamma_{g})$ and $h\in M_{2s}(\Gamma_{g})$. We have
$\displaystyle D(f\det(dZ)^{r})\cdot h\det(dZ)^{s}-f\det(dZ)^{r}\cdot
D(h\det(dZ)^{s})$ $\displaystyle=$ $\displaystyle\mbox{\rm
Tr}\left(\left[h\frac{\partial}{\partial Z}f-f\frac{\partial}{\partial
Z}h\right]dZ\right)\det(dZ)^{r+s}$
is invariant under $\Gamma_{g}$. The same consideration as above gives us
$\det\left(h\frac{\partial}{\partial Z}f-f\frac{\partial}{\partial
Z}h\right)\in M_{2(r+s)g+2}.$
We can continue this construction to find combinations of higher derivatives
of $f$ and $h$ which are modular. By setting $[f,h]_{0}:=fh$,
$[f,h]_{1}:=\det(h\frac{\partial}{\partial Z}f-f\frac{\partial}{\partial
Z}h)$, and so on, one would get the Rankin-Cohen brackets.
### 2.4. The unique theorem
We first show a lemma.
###### Lemma 2.8.
Let $\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\in\mbox{\rm
Sp}(2g,\mathbb{Z})$. Then
$(CZ+D)^{-1}\sqrt{-1}(\text{Im}\gamma(Z))^{-1}=\sqrt{-1}(\text{Im}(Z))^{-1}\cdot(CZ+D)^{t}+2C^{t}.$
###### Proof.
Since Im$(\gamma(Z))=((C\bar{Z}+D)^{t})^{-1}Y(CZ+D)^{-1}$ (see [1]) and
$Y^{t}=Y$, we have
$\displaystyle(\text{Im}\gamma(Z))^{-1}$ $\displaystyle=$
$\displaystyle(CZ+D)Y^{-1}\cdot(C\bar{Z}+D)^{t}=(CZ+D)Y^{-1}\cdot(CZ+D-2\sqrt{-1}CY)^{t}$
$\displaystyle=$
$\displaystyle(CZ+D)Y^{-1}\cdot(CZ+D)^{t}-(CZ+D)2\sqrt{-1}C^{t},$
and thus the result. ∎
###### Theorem 2.9.
For any symmetric $g\times g$ matrix $G=(G_{ij})$ consisting of
$\mathbb{C}^{\infty}$ (or holomorphic) functions on $\mathbb{H}_{g}$ which
satisfies the transformation formula
$(CZ+D)^{-1}\gamma(G)=G\cdot(CZ+D)^{t}+2C^{t},$
there exists a unique modular connection $\mathbb{D}$ such that
$\mathbb{D}(dZ_{rs})=-\sum_{i,j=1}^{g}G_{ij}dZ_{si}dZ_{rj}\quad{and}\quad\mathbb{D}(f(\det(dZ)^{k})=\mbox{\rm
Tr}\left(\left[\frac{\partial}{\partial Z}-kG\right]fdZ\right)(\det(dZ))^{k},$
and thus $G$ gives a derivative operator
$M_{2k}\rightarrow\widetilde{M}_{2kg+2}$ by
$f\mapsto\det\left(\left[\frac{\partial}{\partial Z}-kG\right]f\right)$.
Furthermore, there exists at most one holomorphic symmetric matrix $G$ to
satisfy the transformation formula. If such a $G$ exists, the operator
corresponding to $G$ is holomorphic.
###### Proof.
One notes that in the definition of modular connection coefficients
$\Gamma_{IJ}^{K}$, we only need the transformation law
$\gamma(\omega)=-S^{-1}\cdot dS+S^{-1}\cdot\omega\cdot S$ for
$\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$. If $G$ has the same transformation
law as $\sqrt{-1}(\text{Im}(Z))^{-1}$, then we can use the same method in
Lemma 2.1 to construct $\\{\Gamma_{IJ}^{K}\\}$ and to calculate the
expressions of the differential forms under $\mathbb{D}$. The same discussion
as in subsection 2.3 tells us $\det\left(\left[\frac{\partial}{\partial
Z}-kG\right]f\right)\in\widetilde{M}_{2kg+2}$.
On the uniqueness, let $G$ and $\tilde{G}$ be two holomorphic matrices to
satisfy the transformation formula. Then
$(CZ+D)^{-1}(G(\gamma Z)-\tilde{G}(\gamma Z))=(G(Z)-\tilde{G}(Z))(CZ+D)^{t}.$
So $\mbox{\rm
Tr}\\{(G-\tilde{G})dZ\\}\in(\Omega_{\mathbb{H}_{g}}^{1})^{\Gamma_{g}}$. In
[11] and [12] R. Weissauer proved that if $v$ is not of the form
$[u]:=ug-\frac{1}{2}u(u-1)$, then
$(\Omega_{\mathbb{H}_{g}}^{v})^{\Gamma_{g}}=0$. If $g\geq 2$, one gets
$(\Omega_{\mathbb{H}_{g}}^{1})^{\Gamma_{g}}=0$, and thus $0=\mbox{\rm
Tr}\\{(G-\tilde{G})dZ\\}=2\sum\limits_{i<j}(G_{ij}-\tilde{G}_{ij})dZ_{ji}+\sum\limits_{i=1}^{g}(G_{ii}-\tilde{G}_{ii})dZ_{ii}$.
Hence $G_{ij}=\tilde{G}_{ij}$ for all $1\leq i,j\leq g$. The case $g=1$ has
been proved in Lemma 1.9. ∎
###### Question 2.10.
Does there exist such a $G$? If so, how to construct it?
## 3\. Explicit Construction of the Levi-Civita Connection
In this section we give the proof of Lemma 2.1. We denote $I,J,K,L,\cdots$ the
elements in $\Omega$, $\ i,j,k,l,r,s,\cdots$ the elements in
$\\{1,2,\cdots,g\\}$ and $\alpha,\beta,\gamma,\delta,\epsilon$ the elements in
$\\{1,2,\cdots,g(g+1)/2\\}$.
### 3.1. Riemannian metric
Let $R:=(R_{ij})_{g\times g}=Y^{-1}$. Then
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\mbox{\rm Tr}(R\cdot
dZ\cdot R\cdot d\overline{Z})=\sum_{i\leq j}\sum_{r\leq
s}2^{2-\delta(i,j)-\delta(r,s)}\times\frac{R_{ir}R_{js}+R_{jr}R_{is}}{2}dZ_{ij}d\overline{Z}_{rs},$
and thus the Riemannian metric matrix associated to $ds^{2}=\mbox{\rm
Tr}(Y^{-1}dZ\cdot Y^{-1}d\overline{Z})$ is given by
$G=\left(\begin{array}[]{cc}0&W\\\ W&0\\\ \end{array}\right)$
where
$W=(W_{IJ})_{I,J\in\Omega},\quad
W_{IJ}=\frac{R_{ir}R_{js}+R_{jr}R_{is}}{2^{\delta(i,j)+\delta(r,s)}},~{}~{}\text{
if }I=(i,j)\text{ and }J=(r,s).$
###### Lemma 3.1.
The inverse $W^{-1}$ of $W$ is given by
$M:=(M_{IJ})_{I,J\in\Omega},\quad
M_{IJ}=Y_{ir}Y_{js}+Y_{jr}Y_{is},~{}~{}\text{ if }I=(i,j)\text{ and }J=(r,s).$
###### Proof.
We need to show that for any $I=(i,j)\in\Omega$ and $K=(p,q)\in\Omega$,
$\sum_{J\in\Omega}M_{IJ}W_{JK}=\delta(I,K).$
By direct computations, we have
$\displaystyle\sum_{1\leq r\leq s\leq
g}M_{(i,j),(r,s)}W_{(r,s),(p,q)}=\sum_{1\leq r\leq s\leq
g}(Y_{ir}Y_{js}+Y_{jr}Y_{is})\frac{R_{rp}R_{sq}+R_{sp}R_{rq}}{2^{\delta(r,s)+\delta(p,q)}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq r\leq
g}2^{1-\delta(p,q)}Y_{ir}Y_{jr}R_{rp}R_{rq}+\sum_{1\leq r<s\leq
g}(Y_{ir}Y_{js}+Y_{jr}Y_{is})\frac{R_{rp}R_{sq}+R_{sp}R_{rq}}{2^{\delta(p,q)}}$
$\displaystyle=$ $\displaystyle 2^{-\delta(p,q)}\sum_{1\leq r\leq
g}\sum_{1\leq s\leq g}(Y_{ir}Y_{js}R_{rp}R_{sq}+Y_{jr}Y_{is}R_{rp}R_{sq})$
$\displaystyle=$ $\displaystyle 2^{-\delta(p,q)}\sum_{1\leq r\leq
g}(Y_{ir}R_{rp}\delta(j,q)+Y_{jr}R_{rp}\delta(i,q))$ $\displaystyle=$
$\displaystyle
2^{-\delta(p,q)}\\{\delta(i,p)\delta(j,q)+\delta(j,p)\delta(i,q)\\}=\delta(I,K).$
Notice that, in the last three steps, we have used the equality $\sum_{1\leq
r\leq g}Y_{ir}R_{rp}=\delta(r,p)$, which comes from $R=Y^{-1}$. ∎
### 3.2. Connection Coefficients
As before, we put $u^{N(i,j)}=Z_{ij}$, $u^{N(g,g)+N(i,j)}=\overline{Z}_{ij}$,
$G=\left(\begin{array}[]{cc}0&W\\\ W&0\\\
\end{array}\right)\quad\text{and}\quad\widehat{G}:=G^{-1}=\left(\begin{array}[]{cc}0&M\\\
M&0\\\ \end{array}\right).$
By the Equality (1.1), we have
$\Gamma_{\alpha,\beta}^{\gamma}=\sum_{1\leq\rho\leq
g(g+1)}\frac{1}{2}\widehat{G}_{\gamma,\rho}\left(\frac{\partial
G_{\alpha,\rho}}{\partial u^{\beta}}+\frac{\partial G_{\beta,\rho}}{\partial
u^{\alpha}}-\frac{\partial G_{\alpha,\beta}}{\partial u^{\rho}}\right).$
Assume $0\leq\alpha,\beta,\gamma\leq\frac{g(g+1)}{2}$, then
$G_{\alpha,\beta}=0$. We have:
$\Gamma_{\alpha,\beta}^{\gamma}=\sum_{1\leq\rho\leq
g(g+1)}\frac{1}{2}\widehat{G}_{\gamma,\rho}\left(\frac{\partial
G_{\alpha,\rho}}{\partial u^{\beta}}+\frac{\partial G_{\beta,\rho}}{\partial
u^{\alpha}}\right).$
Hence for $I,J,K\in\Omega$
$\Gamma_{I,J}^{K}=\sum_{L\in\Omega}\frac{1}{2}M_{K,L}\left(\frac{\partial
W_{I,L}}{\partial Z_{J}}+\frac{\partial W_{J,L}}{\partial Z_{I}}\right).$
Again, notice that
$\sum_{L\in\Omega}M_{K,L}W_{I,L}=\delta(K,I)\ \text{ and }\
\sum_{L\in\Omega}M_{K,L}W_{J,L}=\delta(K,J).$
Do partial derivatives on both sides with respect to $Z_{J}$ and $Z_{I}$
respectively, we have
$\sum_{L\in\Omega}M_{K,L}\frac{\partial W_{I,L}}{\partial
Z_{J}}+\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{J}}W_{I,L}=0,$
and
$\sum_{L\in\Omega}M_{K,L}\frac{\partial W_{J,L}}{\partial
Z_{I}}+\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{I}}W_{J,L}=0.$
Finally we get
$\Gamma_{I,J}^{K}=-\frac{1}{2}\left(\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{J}}W_{I,L}+\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{I}}W_{J,L}\right).$
If $I=(i,j),J=(r,s),K=(p,q)$ and $L=(a,b)\in\Omega$, then
$M_{I,J}=Y_{ir}Y_{js}+Y_{jr}Y_{is}$ and
$\displaystyle\frac{\partial M_{K,L}}{\partial
Z_{J}}=\frac{\partial(Y_{pa}Y_{qb}+Y_{qa}Y_{pb})}{\partial Z_{J}}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2}\left\\{\sigma_{(p,a),(r,s)}Y_{qb}+\sigma_{(q,b),(r,s)}Y_{pa}+\sigma_{(q,a),(r,s)}Y_{pb}+\sigma_{(p,b),(r,s)}Y_{qa}\right\\}$
Here we define:
$\sigma_{(p,a),(r,s)}=\left\\{\begin{array}[]{ll}1,&\mbox{ if
}Z_{pa}=Z_{rs},\\\ 0,&\mbox{ if }Z_{pa}\neq Z_{rs}.\end{array}\right.$
One should notice the difference of the notation above with the notation
$\delta_{(p,a),(r,s)}:=\delta((p,a),(r,s))=\delta(p,r)\delta(a,s)$. These two
notations have the following relations:
$\sigma_{(p,a),(r,s)}=\delta_{(p,a),(r,s)}+\delta_{(p,a),(s,r)}-\delta_{(p,a),(r,s)}\cdot\delta_{(p,a),(s,r)}.$
Using the equality
$W_{IJ}=\frac{R_{ir}R_{js}+R_{jr}R_{is}}{2^{\delta(i,j)+\delta(r,s)}}$ and
others above, we have
$\displaystyle\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial
Z_{J}}W_{I,L}=-\frac{\sqrt{-1}}{2}\sum_{L=(a,b)\in\Omega}\\{\sigma_{(p,a),(r,s)}Y_{qb}+\sigma_{(q,b),(r,s)}Y_{pa}$
$\displaystyle+\sigma_{(q,a),(r,s)}Y_{pb}+\sigma_{(p,b),(r,s)}Y_{qa}\\}W_{I,L}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2^{1+\delta(i,j)}}\Bigg{\\{}\sum_{1\leq a\leq
g}\sigma_{(p,a),(r,s)}\delta(q,j)R_{ia}+\sum_{1\leq b\leq
g}\sigma_{(p,b),(r,s)}\delta(q,i)R_{jb}$ $\displaystyle+\sum_{1\leq a\leq
g}\sigma_{(q,a),(r,s)}\delta(p,i)R_{ja}+\sum_{1\leq b\leq
g}\sigma_{(q,b),(r,s)}\delta(p,j)R_{ib}\Bigg{\\}}.$
While
$\displaystyle\sum_{1\leq a\leq g}\sigma_{(p,a),(r,s)}\delta(q,j)R_{ia}$
$\displaystyle=$ $\displaystyle\sum_{1\leq a\leq
g}\\{\delta(p,r)\delta(a,s)+\delta(p,s)\delta(a,r)$
$\displaystyle-\delta(p,r)\delta(a,s)\delta(p,s)\delta(a,r)\\}\delta(q,j)R_{ia}$
$\displaystyle=$
$\displaystyle\delta(q,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{is}\\},$
$\displaystyle\sum_{1\leq b\leq g}\sigma_{(p,b),(r,s)}\delta(q,i)R_{jb}$
$\displaystyle=$
$\displaystyle\delta(q,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{js}\\},$
$\displaystyle\sum_{1\leq a\leq g}\sigma_{(q,a),(r,s)}\delta(p,i)R_{ja}$
$\displaystyle=$
$\displaystyle\delta(p,i)\\{\delta(q,r)R_{js}+\delta(q,s)R_{jr}-\delta(q,r)\delta(q,s)R_{js}\\},$
$\displaystyle\sum_{1\leq b\leq g}\sigma_{(q,b),(r,s)}\delta(p,j)R_{ib}$
$\displaystyle=$
$\displaystyle\delta(p,j)\\{\delta(q,r)R_{is}+\delta(q,s)R_{ir}-\delta(q,r)\delta(q,s)R_{is}\\}.$
Combining these equalities together, we have
$\displaystyle\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{J}}W_{I,L}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2^{1+\delta(i,j)}}\\{\delta(q,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{is}\\}$
$\displaystyle+\delta(q,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{js}\\}$
$\displaystyle+\delta(p,i)\\{\delta(q,r)R_{js}+\delta(q,s)R_{jr}-\delta(q,r)\delta(q,s)R_{js}\\}$
$\displaystyle+\delta(p,j)\\{\delta(q,r)R_{is}+\delta(q,s)R_{ir}-\delta(q,r)\delta(q,s)R_{is}\\}\\}$
Similarly, we have
$\displaystyle\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{I}}W_{J,L}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2^{1+\delta(r,s)}}\\{\delta(q,s)\\{\delta(p,i)R_{rj}+\delta(p,j)R_{ri}-\delta(p,i)\delta(p,j)R_{rj}\\}$
$\displaystyle+\delta(q,r)\\{\delta(p,i)R_{sj}+\delta(p,j)R_{si}-\delta(p,i)\delta(p,j)R_{sj}\\}$
$\displaystyle+\delta(p,r)\\{\delta(q,i)R_{sj}+\delta(q,j)R_{si}-\delta(q,i)\delta(q,j)R_{sj}\\}$
$\displaystyle+\delta(p,s)\\{\delta(q,i)R_{rj}+\delta(q,j)R_{ir}-\delta(q,i)\delta(q,j)R_{rj}\\}\\}$
Finally we get
###### Lemma 3.2.
$\displaystyle\Gamma_{I,J}^{K}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{J}}W_{I,L}+\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{I}}W_{J,L}\right)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2^{2+\delta(i,j)}}\\{\delta(q,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{is}\\}$
$\displaystyle+\delta(q,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{js}\\}$
$\displaystyle+\delta(p,i)\\{\delta(q,r)R_{js}+\delta(q,s)R_{jr}-\delta(q,r)\delta(q,s)R_{js}\\}$
$\displaystyle+\delta(p,j)\\{\delta(q,r)R_{is}+\delta(q,s)R_{ir}-\delta(q,r)\delta(q,s)R_{is}\\}\\}$
$\displaystyle+\frac{\sqrt{-1}}{2^{2+\delta(r,s)}}\\{\delta(q,s)\\{\delta(p,i)R_{rj}+\delta(p,j)R_{ri}-\delta(p,i)\delta(p,j)R_{rj}\\}$
$\displaystyle+\delta(q,r)\\{\delta(p,i)R_{sj}+\delta(p,j)R_{si}-\delta(p,i)\delta(p,j)R_{sj}\\}$
$\displaystyle+\delta(p,r)\\{\delta(q,i)R_{sj}+\delta(q,j)R_{si}-\delta(q,i)\delta(q,j)R_{sj}\\}$
$\displaystyle+\delta(p,s)\\{\delta(q,i)R_{rj}+\delta(q,j)R_{ir}-\delta(q,i)\delta(q,j)R_{rj}\\}\\}$
Using the lemma 3.2 above, one can easily show that in the case
$K=(p,q)=(1,1),I=(i,j)=(1,j),J=(r,s)=(1,s)$, we have
* •
if $j=s=1$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{1,1}$;
* •
if $j=1,s\neq 1$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{1,s}$;
* •
if $j\neq 1,s\neq 1$, then
$\Gamma_{I,J}^{K}=\sqrt{-1}R_{j,s}=\sqrt{-1}R_{s,j}$.
In general case, if $K=(p,q),I=(i,j),J=(r,s)$, and if both $(Z_{ij},Z_{rs})$
and $(Z_{rs},Z_{ij})$ do not belong to
$\\{Z_{1p},Z_{2p},\cdots,Z_{gp}\\}\times\\{Z_{1q},Z_{2q},\cdots,Z_{gq}\\}$,
then all terms in the last equality of lemma 3.2 are zero, hence
$\Gamma_{I,J}^{K}=\Gamma_{J,I}^{K}=0$.
If $p=q$, then
$\displaystyle\Gamma_{I,J}^{(p,p)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2^{1+\delta(i,j)}}\\{\delta(p,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{jr}\\}$
$\displaystyle+\delta(p,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{ir}\\}\\}$
$\displaystyle+\frac{\sqrt{-1}}{2^{1+\delta(r,s)}}\\{\delta(p,r)\\{\delta(p,i)R_{sj}+\delta(p,j)R_{si}-\delta(p,i)\delta(p,j)R_{sj}\\}$
$\displaystyle+\delta(p,s)\\{\delta(p,i)R_{rj}+\delta(p,j)R_{ri}-\delta(p,i)\delta(p,j)R_{rj}\\}\\}.$
If $Z_{ij}=Z_{ji}$ and $Z_{rs}=Z_{sr}$ belong to the same row or column with
$Z_{pp}$, then $i=r=p$, or $i=s=p$, or $j=r=p$, or $j=s=p$.
* •
If $i=r=p$, one can use the formula above to show that
$\Gamma_{I,J}^{K}=\sqrt{-1}R_{js}$
* •
If $i=s=p$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{jr}$.
* •
If $j=r=p$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{is}$.
* •
If $j=s=p$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{ir}$.
If $p<q$, we may assume that $Z_{ij}(i\leq j)$ belong to the same row with
$Z_{pq}$ and $Z_{rs}(r\leq s)$ belongs to the same column with $Z_{pq}$ (Other
cases can be proved in the same way). Then $i=p\leq j$, $r\leq s=q$ and
$\displaystyle\Gamma_{I,J}^{K}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2^{2+\delta(i,j)}}(\delta(q,j)\delta(p,r)R_{is}+R_{jr}+\delta(p,j)R_{ir})$
$\displaystyle+\frac{\sqrt{-1}}{2^{2+\delta(r,s)}}(\delta(p,r)\delta(q,j)R_{si}+R_{rj}+\delta(q,r)R_{sj}).$
* •
If $i=p=j\leq q$ and $r<s=q$, then
$\Gamma_{I,J}^{K}=\frac{\sqrt{-1}}{8}(R_{jr}+R_{ir})+\frac{\sqrt{-1}}{4}R_{rj}=\frac{\sqrt{-1}}{2}R_{jr}.$
* •
If $i=p=j$ and $r=s=q$, then $\Gamma_{I,J}^{K}=\frac{\sqrt{-1}}{2}R_{jr}$.
* •
If $i=p<j$ and $r<s=q$, then
$\displaystyle\Gamma_{I,J}^{K}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{4}(\delta(q,j)\delta(p,r)R_{is}+R_{jr})+\frac{\sqrt{-1}}{4}(\delta(p,r)\delta(q,j)R_{is}+R_{rj})$
$\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2}(\delta(q,j)\delta(p,r)R_{pq}+R_{jr})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\sqrt{-1}R_{pq}&\mbox{ if }j=q\text{
and }r=p,\\\ \frac{\sqrt{-1}}{2}R_{jr}&\text{ otherwise}.\end{array}\right.$
* •
If $i=p<j$ and $r=s=q$, then $\Gamma_{I,J}^{K}=\frac{\sqrt{-1}}{2}R_{jr}$.
At last, we complete the proof of Lemma 2.1.
## References
* [1] A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Springer 2008.
* [2] S. S. Chern, W. H. Chen, K. S. Lam, Lectures on Differential Geometry, World Scientific, 2000.
* [3] G. van der Geer, Siegel Modular Forms and Their Applications, In ”The 1-2-3 of Modular Forms : Lectures at a Summer School in Nordfjordeid, Norway” (K. Ranestad, ed.), Universitext, Springer-Verlag, Berlin, 2008, 181-245.
* [4] J. Jost, Riemannian Geometry and Geometric Analysis, Springer(Universitext), Berlin, 1988.
* [5] N. Koblitz, Introduction to Elliptic curves and Modular Forms, Springer-Verlag, New York, 1984.
* [6] H. Maass, Lectures on Siegel’s Modular Functions, Tate Institute of Fundamental Research, Bombay, 1955.
* [7] H. Maass, Siegel’s Modular Forms and Dirichlet Series, LNM 216, Berlin-Heidelberg New York, 1971.
* [8] G. Shimura, Arithmetic of Differential Operators on Symmetric Domains, Duke Math. J. 48, 1981, 813-843.
* [9] J.-P.Serre,Congruences Et Formes Modulaires, Seminaire Bourbaki, 24e annee,1971/72,n416.
* [10] Paul-Emile Paradan, Symmetric Spaces of the Non-compact Type: Lie Groups, http://math.univ-lyon1.fr/~ remy/smf_ sec_ 18_ 02.pdf
* [11] R. Weissauer, Vektorwertige Siegelsche Modulformen Kleinen Gewichtes, J. Reine Angew. Math. 343, 1983 , 184-202.
* [12] R. Weissauer, Divisors of the Siegel Modular Variety, LNM 1240, Berlin-Heidelberg New York, 1987, 304-324.
* [13] D. Zagier, Elliptic Modular Forms and Their Applications, In ”The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway” (K. Ranestad, ed.), Universitext, Springer-Verlag, Berlin, 2008, 1-103.
|
arxiv-papers
| 2012-07-07T03:16:49 |
2024-09-04T02:49:32.765267
|
{
"license": "Public Domain",
"authors": "Enlin Yang and Linsheng Yin",
"submitter": "Yin Linsheng",
"url": "https://arxiv.org/abs/1207.1756"
}
|
1207.1834
|
# On the Dirichlet’s type of Eulerian polynomials
Serkan Araci University of Gaziantep, Faculty of Science and Arts, Department
of Mathematics, 27310 Gaziantep, TURKEY mtsrkn@hotmail.com , Mehmet Acikgoz
University of Gaziantep, Faculty of Science and Arts, Department of
Mathematics, 27310 Gaziantep, TURKEY acikgoz@gantep.edu.tr and Deyao Gao
Yuren Lab., No. 8 Tongsheng Road, Changsha, P. R. China 13607433711@163.com
###### Abstract.
In the present paper, we introduce Eulerian polynomials attached to $\chi$ by
using $p$-adic $q$-integral on $\mathbb{Z}_{p}$. Also, we give new interesting
identities via the generating functions of Dirichlet’s type of Eulerian
polynomials. After, by applying Mellin transformation to this generating
function of Dirichlet’s type of Eulerian polynomials, we derive $L$-function
for Eulerian polynomials which interpolates of Dirichlet’s type of Eulerian
polynomials at negative integers.
2010 Mathematics Subject Classification. 11S80, 11B68.
Keywords and phrases. Eulerian polynomials, $p$-adic $q$-integral on
$\mathbb{Z}_{p}$, Mellin transformation, $L$-function.
## 1\. Introduction
Recently, Kim $et$ $al.$ have studied to Eulerian polynomials. They gave not
only Witt’s formula for Eulerian polynomials but also relations between
Genocchi, Tangent and Euler numbers (for more details, see [1]). In arithmetic
works of T. Kim have introduced many different generating functions of
families of Bernoulli, Euler, Genocchi numbers and polynomials by using
$p$-adic $q$-integral on $\mathbb{Z}_{p}$ (see [1-5]). After, many
mathematicians are motivated from his papers and introduced new generating
function for special functions (for more information about this subject, see
[16-28]). Y. Simsek also gave new $q$-twisted Euler numbers and polynomials
and ($h,q$)-Bernoulli numbers and polynomials by using Kim’s $p$-adic
$q$-integral on $\mathbb{Z}_{p}$. He also derived some interesting properties
in his works [26], [27], [28].
The $p$-adic $q$-integral on $\mathbb{Z}_{p}$ was originally defined by Kim.
He also investigated that $p$-adic $q$-integral on $\mathbb{Z}_{p}$ is related
to non-Archimedean combinatorial analysis in mathematical physics. That is,
the functional equation of the $q$-zeta function, the $q$-Stirling numbers and
$q$-Mahler theory and so on (for details, see[5], [6]).
We firstly list some properties of familiar Eulerian polynomials for sequel of
this paper as follows:
As it is well-known, the Eulerian polynomials, $\mathcal{A}_{n}\left(x\right)$
are given by means of the following generating function:
(1)
$e^{\mathcal{A}\left(x\right)t}=\sum_{n=0}^{\infty}\mathcal{A}_{n}\left(x\right)\frac{t^{n}}{n!}=\frac{1-x}{e^{t\left(1-x\right)}-x}$
where $\mathcal{A}^{n}\left(x\right):=\mathcal{A}_{n}\left(x\right)$ as
symbolic. To find Eulerian polynomials, it has the following recurrence
relation:
(2)
$\left(\mathcal{A}\left(t\right)+\left(t-1\right)\right)^{n}-t\mathcal{A}_{n}\left(t\right)=\left\\{\begin{array}[]{cc}1-t&\text{if
}n=0\\\ 0&\text{if }n\neq 0,\end{array}\right.$
(for details, see [1]).
Suppose that $p$ be a fixed odd prime number. Throughout this paper, we use
the following notations. By $\mathbb{Z}_{p}$, we denote the ring of $p$-adic
rational integers, $\mathbb{Q}$ denotes the field of rational numbers,
$\mathbb{Q}_{p}$ denotes the field of $p$-adic rational numbers, and
$\mathbb{C}_{p}$ denotes the completion of algebraic closure of
$\mathbb{Q}_{p}$. Let $\mathbb{N}$ be the set of natural numbers and
$\mathbb{N}^{\ast}=\mathbb{N}\cup\left\\{0\right\\}$.
The $p$-adic absolute value is defined by
$\left|p\right|_{p}=\frac{1}{p}\text{.}$
In this paper we assume $\left|q-1\right|_{p}<1$ as an indeterminate. Let
$UD\left(\mathbb{Z}_{p}\right)$ be the space of uniformly differentiable
functions on $\mathbb{Z}_{p}$. For a positive integer $d$ with
$\left(d,p\right)=1$, set
$\displaystyle X$ $\displaystyle=$ $\displaystyle
X_{m}=\lim_{\overleftarrow{m}}\mathbb{Z}/dp^{m}\mathbb{Z}\text{,}$
$\displaystyle X^{\ast}$ $\displaystyle=$
$\displaystyle\underset{\underset{\left(a,p\right)=1}{0<a<dp}}{\cup}a+dp\mathbb{Z}_{p}$
and
$a+dp^{m}\mathbb{Z}_{p}=\left\\{x\in X\mid x\equiv
a\left(\mathop{\mathrm{m}od}dp^{m}\right)\right\\}\text{,}$
where $a\in\mathbb{Z}$ satisfies the condition $0\leq a<dp^{m}$.
Firstly, for introducing fermionic $p$-adic $q$-integral, we need some basic
information which we state here. A measure on $\mathbb{Z}_{p}$ with values in
a $p$-adic Banach space $B$ is a continuous linear map
$f\mapsto\int f(x)\mu=\int_{\mathbb{Z}_{p}}f(x)\mu(x)$
from $C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$, (continuous function on
$\mathbb{Z}_{p}$ ) to $B$. We know that the set of locally constant functions
from $\mathbb{Z}_{p}$ to $\mathbb{Q}_{p}$ is dense in
$C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$ so.
Explicitly, for all $f\in C^{0}(\mathbb{Z}_{p},\mathbb{C}_{p})$, the locally
constant functions
$f_{n}=\sum_{i=0}^{p^{m}-1}f(i)1_{i+p^{m}\mathbb{Z}_{p}}\rightarrow f\text{ in
}C^{0}$
Now, set
$\mu(i+p^{m}\mathbb{Z}_{p})=\int_{\mathbb{Z}_{p}}1_{i+p^{m}\mathbb{Z}_{p}}\mu$.
Then $\int_{\mathbb{Z}_{p}}f\mu$, is given by the following Riemannian sum
$\int_{\mathbb{Z}_{p}}f\mu=\lim_{m\rightarrow\infty}\sum_{i=0}^{p^{m}-1}f(i)\mu{(i+p^{m}\mathbb{Z}_{p})}$
The following $q$-Haar measure is defined by Kim in [3] and [5]:
$\mu_{q}(a+p^{m}\mathbb{Z}_{p})=\frac{q^{a}}{[p^{m}]_{q}}$
So, for $f\in UD\left(\mathbb{Z}_{p}\right)$, the $p$-adic $q$-integral on
$\mathbb{Z}_{p}$ is defined by Kim as follows:
(3)
$I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(\eta\right)d\mu_{q}\left(\eta\right)=\lim_{n\rightarrow\infty}\frac{1}{\left[p^{n}\right]_{q}}\sum_{\eta=0}^{p^{n}-1}q^{\eta}f\left(\eta\right)\text{.}$
The bosonic integral is considered as the bosonic limit $q\rightarrow 1,$
$I_{1}\left(f\right)=\lim_{q\rightarrow 1}I_{q}\left(f\right)$. In [10], [11]
and [12], similarly, the $p$-adic fermionic integration on $\mathbb{Z}_{p}$
defined by Kim as follows:
(4)
$I_{-q}\left(f\right)=\lim_{q\rightarrow-q}I_{q}\left(f\right)=\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q}\left(x\right)\text{.}$
By (4), we have the following well-known integral eguation:
(5)
$q^{n}I_{-q}\left(f_{n}\right)+\left(-1\right)^{n-1}I_{-q}\left(f\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{n-1-l}q^{l}f\left(l\right)$
Here $f_{n}\left(x\right):=f\left(x+n\right)$. By (5), we have the following
equalities:
If $n$ odd, then
(6)
$q^{n}I_{-q}\left(f_{n}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{l}q^{l}f\left(l\right)\text{.}$
If $n$ even, then we have
(7)
$I_{-q}\left(f\right)-q^{n}I_{-q}\left(f_{n}\right)=\left[2\right]_{q}\sum_{l=0}^{n-1}\left(-1\right)^{l}q^{l}f\left(l\right)\text{.}$
Substituting $n=1$ into (6), we readily see the following
(8)
$qI_{-q}\left(f_{1}\right)+I_{-q}\left(f\right)=\left[2\right]_{q}f\left(0\right)\text{.}$
Replacing $q$ by $q^{-1}$ in (8), we easily derive the following
(9)
$I_{-q^{-1}}\left(f_{1}\right)+qI_{-q^{-1}}\left(f\right)=\left[2\right]_{q}f\left(0\right)\text{.}$
In [1], Kim $et$ $al.$ is considered $f(x)=e^{-x\left(1+q\right)t}$ in (9),
then they gave Witt’s formula of Eulerian polynomials as follows:
For $n\in\mathbb{N}^{\ast}$,
(10)
$I_{-q^{-1}}\left(x^{n}\right)=\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n}\left(-q\right)\text{.}$
Now also, we consider $I_{-q^{-1}}\left(\chi\left(x\right)x^{n}\right)$ in the
next section. We shall call as Dirichlet’s type of Eulerian polynomials. After
we shall give arithmetic properties for Dirichlet’s type of Eulerian
polynomials.
## 2\. On the Dirichlet’s type of Eulerian polynomials
Firstly, we consider the following equality by using (6): For $d$ odd natural
numbers,
(11)
$\displaystyle\int_{\mathbb{Z}_{p}}f\left(x+d\right)d\mu_{-q^{-1}}\left(x\right)+q^{d}\int_{\mathbb{Z}_{p}}f\left(x\right)d\mu_{-q^{-1}}\left(x\right)$
$\displaystyle=\left[2\right]_{q}\sum_{0\leq l\leq
d-1}\left(-1\right)^{l}q^{d-l+1}f\left(l\right)\text{.}$
Let $\chi$ be a Dirichlet’s character of conductor $d,$ which is any multiple
of $p$ (=$odd$). Then, substituting
$f(x)=\chi\left(x\right)e^{-x\left(1+q\right)t}$ in (11), then we compute as
follows:
$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(x+d\right)e^{-\left(x+d\right)\left(1+q\right)t}d\mu_{-q^{-1}}\left(x\right)+q^{d}\int_{\mathbb{Z}_{p}}\chi\left(x\right)e^{-x\left(1+q\right)t}d\mu_{-q^{-1}}\left(x\right)$
$\displaystyle=\left[2\right]_{q}\sum_{0\leq l\leq
d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)e^{-l\left(1+q\right)t}$
After some applications, we discover the following
(12)
$\int_{\mathbb{Z}_{p}}\chi\left(x\right)e^{-x\left(1+q\right)t}d\mu_{-q^{-1}}\left(x\right)=\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)\frac{e^{-l\left(1+q\right)t}}{e^{-d\left(1+q\right)t}+q^{d}}\text{.}$
Let
$\mathcal{F}_{q}\left(t\mid\chi\right)=\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}$.
Then, we introduce the following definition of generating function of
Dirichlet’s type of Eulerian polynomials.
###### Definition 1.
For $n\in\mathbb{N}^{\ast}$, then we define the following:
(13)
$\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}=\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)\frac{e^{-l\left(1+q\right)t}}{e^{-d\left(1+q\right)t}+q^{d}}\text{.}$
By (12) and (13), we state the following theorem which is the Witt’s formula
for Dirichlet’s type of Eulerian polynomials.
###### Theorem 2.1.
The following identity holds true:
(14)
$I_{-q^{-1}}\left(\chi\left(x\right)x^{n}\right)=\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)\text{.}$
By using (13), we discover the following applications:
$\displaystyle\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{d-l+1}\chi\left(l\right)\frac{e^{-l\left(1+q\right)t}}{e^{-d\left(1+q\right)t}+q^{d}}$
$\displaystyle=$
$\displaystyle\left[2\right]_{q}\sum_{l=0}^{d-1}\left(-1\right)^{l}q^{-l+1}\chi\left(l\right)e^{-l\left(1+q\right)t}\sum_{m=0}^{\infty}\left(-1\right)^{m}q^{-md}e^{-md\left(1+q\right)t}$
$\displaystyle=$ $\displaystyle
q\left[2\right]_{q}\sum_{m=0}^{\infty}\sum_{l=0}^{d-1}\left(-1\right)^{l+md}\chi\left(l+md\right)q^{-\left(l+md\right)}e^{-\left(l+md\right)\left(1+q\right)t}$
$\displaystyle=$ $\displaystyle
q\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}\chi\left(m\right)q^{-m}e^{-m\left(1+q\right)t}\text{.}$
Thus, we get the following theorem.
###### Theorem 2.2.
The following
(15)
$\mathcal{F}_{q}\left(t\mid\chi\right)=\sum_{n=0}^{\infty}\mathcal{A}_{n,\chi}\left(-q\right)\frac{t^{n}}{n!}=\left[2\right]_{q}\sum_{m=0}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)q^{-m}e^{-m\left(1+q\right)t}}{q^{m-1}}$
is true.
By considering Taylor expansion of $e^{-m\left(1+q\right)t}$ in (15), we
procure the following theorem.
###### Theorem 2.3.
For $n\in\mathbb{N}$, then we have
(16)
$\frac{\left(-1\right)^{n}}{q\left(1+q\right)^{n+1}}\mathcal{A}_{n,\chi}\left(-q\right)=\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)m^{n}}{q^{m}}\text{.}$
From (14) and (16), we easily derive the following corollary:
###### Corollary 2.4.
For $n\in\mathbb{N}$, then we procure the following
$\lim_{m\rightarrow\infty}\sum_{x=1}^{p^{m}-1}\frac{\left(-1\right)^{x}\chi\left(x\right)x^{n}}{q^{x}}=2\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)m^{n}}{q^{m-2}}\text{.}$
Now, we give distribution formula for Dirichlet’s type of Eulerian polynomials
by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$, as follows:
$\displaystyle\int_{\mathbb{Z}_{p}}\chi\left(x\right)x^{n}d\mu_{-q^{-1}}\left(x\right)$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{\left[dp^{m}\right]_{-q^{-1}}}\sum_{x=0}^{dp^{m}-1}\left(-1\right)^{x}\chi\left(x\right)x^{n}q^{-x}$
$\displaystyle=$
$\displaystyle\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\left(\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q^{-d}}}\sum_{x=0}^{p^{m}-1}\left(-1\right)^{x}\left(\frac{a}{d}+x\right)^{n}q^{-dx}\right)$
$\displaystyle=$
$\displaystyle\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\int_{\mathbb{Z}_{p}}\left(\frac{a}{d}+x\right)^{n}d\mu_{-q^{-d}}\left(x\right).$
Thus, we state the following theorem.
###### Theorem 2.5.
The following identity holds true:
(17)
$\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)=\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\int_{\mathbb{Z}_{p}}\left(\frac{a}{d}+x\right)^{n}d\mu_{-q^{-d}}\left(x\right)\text{.}$
From this, we notice that the above equation is related to $q$-Genocchi
polynomials with weight zero, $\widetilde{G}_{n,q}\left(x\right)$, and
$q$-Euler polynomials with weight zero, $\widetilde{E}_{n,q}\left(x\right)$,
which is defined by Araci $et$ $al.$ and Kim and Choi in [25] and [15] as
follows:
(18)
$\frac{\widetilde{G}_{n+1,q}\left(x\right)}{n+1}=\lim_{m\rightarrow\infty}\frac{1}{\left[p^{m}\right]_{-q}}\sum_{y=0}^{p^{m}-1}\left(-1\right)^{y}\left(x+y\right)^{n}q^{y}$
and
(19)
$\widetilde{E}_{n,q}\left(x\right)=\int_{\mathbb{Z}_{p}}\left(x+y\right)^{n}d\mu_{-q}\left(y\right).$
By expressions of (17), (18) and (19), we easily discover the following
corollary.
###### Corollary 2.6.
For $n\in\mathbb{N}^{\ast}$, then we have
$\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)=\frac{d^{n}}{\left(n+1\right)\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\widetilde{G}_{n+1,q^{-d}}\left(\frac{a}{d}\right)$
Moreover,
$\frac{\left(-1\right)^{n}}{\left(1+q\right)^{n}}\mathcal{A}_{n,\chi}\left(-q\right)=\frac{d^{n}}{\left[d\right]_{-q^{-1}}}\sum_{a=0}^{d-1}\left(-1\right)^{a}\chi\left(a\right)q^{-a}\widetilde{E}_{n,q^{-d}}\left(\frac{a}{d}\right)\text{.}$
## 3\. On the Eulerian-$L$ function
Our goal in this section is to introduce Eulerian-$L$ function by applying
Mellin transformation to the generating function of Dirichlet’s type of
Eulerian polynomials. By (15), for $s\in\mathbb{C}$, we define the following
$L_{E}\left(s\mid\chi\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}\mathcal{F}_{q}\left(t\mid\chi\right)dt$
where $\Gamma\left(s\right)$ is the Euler Gamma function. It becomes as
follows:
$\displaystyle L_{E}\left(s\mid\chi\right)$ $\displaystyle=$ $\displaystyle
q\left[2\right]_{q}\sum_{m=0}^{\infty}\left(-1\right)^{m}\chi\left(m\right)q^{-m}\left\\{\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}t^{s-1}e^{-m\left(1+q\right)t}dt\right\\}$
$\displaystyle=$
$\displaystyle\frac{q}{\left(1+q\right)^{s-1}}\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)}{q^{m}m^{s}}$
So, we give definition of Eulerian $L$-function as follows:
###### Definition 2.
For $s\in\mathbb{C}$, then we have
(20)
$L_{E}\left(s\mid\chi\right)=\frac{q}{\left(1+q\right)^{s-1}}\sum_{m=1}^{\infty}\frac{\left(-1\right)^{m}\chi\left(m\right)}{q^{m}m^{s}}\text{.}$
Substituting $s=-n$ into (16), then, relation between Eulerian $L$-function
and Dirichlet’s type of Eulerian polynomials are given by the following
theorem.
###### Theorem 3.1.
The following equality holds true:
$L_{E}\left(-n\mid\chi\right)=\left\\{\begin{array}[]{cc}-\mathcal{A}_{n,\chi}\left(-q\right)&\text{if
}n\text{ odd,}\\\ \mathcal{A}_{n,\chi}\left(-q\right)&\text{if }n\text{
even.}\end{array}\right.$
## References
* [1] D. S. Kim, T. Kim, W. J. Kim and D. V. Dolgy, A note on Eulerian polynomials, Abstract and Applied Analysis (Article in press).
* [2] T. Kim, On explicit formulas of $p$-adic $q$-$L$-functions, Kyushu. J. Math. 48 (1994), 73-86.
* [3] T. Kim, On a $q$-analogue of the $p$-adic log gamma functions and related integrals, J. Number Theory 76 (1999), 320-329.
* [4] T. Kim and S. H. Rim, A note on $p$-adic Carlitz’s $q$-Bernoulli numbers, Bull. Austral. Math. Soc. Vol. 62 (2000), 227-234.
* [5] T. Kim, $q$-Volkenborn integration, Russ. J. Math Phys., 19 (2002), 288-299.
* [6] T. Kim, Non-archimedean $q$-integrals associated with multiple Changhee $q$-Bernoulli Polynomials, Russ. J. Math Phys., 10 (2003), 91-98.
* [7] T. Kim, $p$-adic $q$-integrals associated with the Changhee-Barnes’ $q$-Bernoulli Polynomials, Integral Transform. Spec. Funct., 15 (2004), 415-420.
* [8] T. Kim, Analytic continuation of multiple $q$-zeta functions and their values at negative integers, Russ. J. Math Phys., 11 (2004), 71-76.
* [9] T. Kim, $q$-Generalized Euler Numbers and polynomials, Russian Journal of Mathematical Physics, Vol. 13, No. 3, 2006, pp. 293-308.
* [10] T. Kim, Some identities on the $q$-Euler polynomials of higher order and $q$-stirling numbers by the fermionic $p$-adic integral on $\mathbb{Z}_{p}$, Russian J. Math. Phys. 16 (2009), 484–491.
* [11] T. Kim, On the $q$-extension of Euler and Genocchi numbers, J. Math. Anal. Appl. 326 (2007) 1458–1465.
* [12] T. Kim, On the analogs of Euler numbers and polynomials associated with $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=1$, J. Math. Anal. Appl. 331 (2007) 779–792.
* [13] T. Kim, Some Identities on the integral representation of the product of several $q$-Bernstein-type polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 634675, 11 pages.
* [14] T. Kim, S. H. Lee, H. H. Han and C. S. Ryoo, On the values of the weighted $q$-Zeta and $L$-functions, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 476381, 7 pp.
* [15] T. Kim and J. Choi, On the $q$-Euler numbers and polynomials with weight 0, Abstract and Applied Analysis, vol. 2012, Article ID 795304, 7 pages, 2012.
* [16] L. C. Jang, The $q$-analogue of twisted Lerch type Euler Zeta functions, Bull. Korean Math. Soc. 47 (2010), No. 6, pp. 1181-1188.
* [17] L. C. Jang, V. Kurt, Y. Simsek, and S. H. Rim, $q$-analogue of the $p$-adic twisted $l$-function, Journal of Concrete and Applicable Mathematics, vol. 6, no. 2, pp. 169–176, 2008.
* [18] H. Ozden, I. N. Cangul, Y. Simsek, Multivariate interpolation functions of higher order $q$-Euler numbers and their applications, Abstract and Applied Analysis 2008 (2008), Article ID 390857, 16 pages.
* [19] I. N. Cangul, H. Ozden, Y. Simsek, A new approach to $q$-Genocchi numbers and their interpolation functions, Nonlinear Analysis 71 (2009), pp. 793-799.
* [20] E. Cetin, M. Acikgoz, I. N. Cangul and S. Araci, A note on the ($h$,$q$)-Zeta type function with weight $\alpha$, http://arxiv.org/pdf/1206.5299.pdf.
* [21] S. Araci, M. Acikgoz, K. H. Park and H. Jolany, On the unification of two families of multiple twisted type polynomials by using $p$-adic $q$-integral on $\mathbb{Z}_{p}$ at $q=-1$, Bulletin of the Malaysian Mathematical Sciences and Society (Article in press).
* [22] S. Araci, M. Acikgoz and K. H. Park, A note on the $q$-analogue of Kim’s $p$-adic $\log$ gamma type functions associated with $q$-extension of Genocchi and Euler numbers with weight $\alpha$, Bulletin of the Korean Mathematical Society (accepted for publication).
* [23] S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Volume 2011, Article ID 649248, 10 pages.
* [24] S. Araci, J. J. Seo and D. Erdal, New construction weighted $\left(h,q\right)$-Genocchi numbers and polynomials related to Zeta type function, Discrete Dynamics in Nature and Society, Volume 2011, Article ID 487490, 7 pages, doi:10.1155/2011/487490.
* [25] S. Araci, M. Acikgoz and Feng Qi, On the $q$-Genocchi numbers and polynomials with weight zero and their applications, http://arxiv.org/abs/1202.2643.
* [26] Y. Simsek, $q$-analogue of twisted $l$-series and $q$-twisted Euler numbers, Journal of Number Theory, vol. 110, no. 2, pp. 267–278, 2005.
* [27] Y. Simsek, $q$-Dedekind type sums related to $q$-zeta function and basic $L$-series, Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 333–351, 2006.
* [28] Y. Simsek, Twisted $\left(h,q\right)$-Bernoulli numbers and polynomials related to twisted $\left(h,q\right)$-zeta function and $L$-function, Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.
* [29] M. Acikgoz and Y. Simsek, On Multiple Interpolation functions oof the Nörlund-Type $q$-Euler polynomials, Abstract and Applied Analysis, Volume 2009, Article ID 382574, 14 pages.
|
arxiv-papers
| 2012-07-08T01:29:21 |
2024-09-04T02:49:32.775726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Serkan Araci, Mehmet Acikgoz and Deyao Gao",
"submitter": "Serkan Araci",
"url": "https://arxiv.org/abs/1207.1834"
}
|
1207.1856
|
# Neutron penumbral imaging simulation and reconstruction for Inertial
Confinement Fusion Experiments
Xian-You Wang wangphysics@126.com Department of Physics, Chongqing
University, Chongqing 400044, P.R. China Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049, P.R. China Zhen-yun Fang
Department of Physics, Chongqing University, Chongqing 400044, P.R. China
Yun-qing Tang tangyq@126.com Department of Physics, Chongqing University,
Chongqing 400044, P.R. China Institute of theoretical Physics, Chinese
Academy of Sciences, Beijing 100049,P.R. China Zhi-Cheng Tang Hong Xiao
Ming Xu Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, P.R. China
###### Abstract
Neutron penumbral imaging technique has been successfully used as the
diagnosis method in Inertial Confined Fusion. To help the design of the
imaging systems in the future in CHINA. We construct the Monte carlo imaging
system by Geant4. Use the point spread function from the simulation and decode
algorithm (Lucy-Rechardson algorithm) we got the recovery image.
Inertial confinement fusion (ICF) has attracted much attention from all over
the world. The National Ignition Facility (NIF) programs in America and Laser
MegaJoule (LMJ) in France aim at reaching the inertial confinement fusion of a
deuterium-tritium (DT) filled cryogenic target. China’s ICF program are also
in progress, this paper’s aim is just study the neutron diagnosis system which
will be used in the future ICF program in China.
The process of ICF can be divided into four steps as Fig 1 : First step
generate radiation to drive implosion. There are two approaches to drive
implosion, one is laser direct driven, another is laser energy injected inside
the cavity generates X-rays that heat the DT fuel. Under X-ray or laser
compression, the target density as well as the mean ions’ temperature
increases. Fusion reactions will occur within a hot spot generating 14 MeV
neutrons and alpha particlesICFSim ; ICFSim1 . If the density is sufficient to
trap the energetic alphas or other radiation, these energy will be deposited
on the surrounding cold DT medium. Then the chained thermonuclear reactions
will lead to a combustion in the compressed target. For a well compressed
target, combustion will last long enough to generate more energy than the
injected energy. If ever, drive symmetry is poor or pulse shaping is
imprecise, burning will be low. So we can use the neutrons escaped from the
compressed cores of target to study the compressed process to make sure the
drive be symmetrical.
There are several neutron imaging techniques for inertial confinement fusion
(ICF) experiment to detect the size, shape, and uniformity of the compressed
cores. The pinhole imaging is the easiest way, but the low neutron yield has
an effect on the sensitivity. In NIF experiment a pinhole array is designed to
increase the sensitivitypinhole1 ; pinhole2 ; pinhole3 ; pinhole4 ; pinhole5 .
The another technique is the neutron penumbral imaging (NPI), which have been
widely studied by LMJpenum1 ; penum2 ; penum3 ; penum4 ; penum5 ; penum6 and
other groups penum_other1 ; penum_other2 ; penum_other3 ; penum_other4 ;
penum_other5 ; penum_other6 . Compared with other neutron imaging techniques,
the sensitivity of neutron penumbral imaging is higher. Annular imaging is
nearly the same as penumbral imaging, the only difference of the imaging
process is aperture.
Figure 1: ICF physics process (Radiation compressed, Implosion, Hot spot
creation, Combustion)
In this paper we just based on the simulation model in the referencecqu1 ;
cqu2 ; cqu3 ; cqu4 to design a software package NGPINEX which can be used to
simulate the neutron imaging process and recovery the coded image. The imaging
process simulation have been described in Geant4, and then the recover
algorithm we used Lucy-Richardson methodLR1 ; LR2 , which have been
successfully applied on decoding x-ray ring code imagexrayLR . The whole
program is written in C++, which can be run under the linux system with multi-
core server.
In the program we simulated neutron penumbral imaging and Annular imaging, the
pinhole imaging block is also included in the program. Based on the NIF and
LMJ experiment we design an imaging system aimming to achieve $20\mu m$
resolution. At last we use the Lucy-Richardson method successfully got the
recovered image with $20\mu m$ resolution
## I The imaging system simulation
The structure of neutron penumbral imaging system is shown in Fig.2. The
source , aperture and detector composed the whole imaging system. The fast
neutrons emitting from a burning target are scattered by the aperture
materials, and form an penumbral image on the detector arrays. In our
simulation the coded aperture is placed at $55mm$ from the source, the
detector is placed at $7m$ from the aperture. The aperture is designed as a
$50mm(h)$ thick $10mm$ diameter(tungsten) cylinder to achieve an effective
absorption of the $14MeV$ neutrons in the opaque parts of the aperture. The
pattern of the aperture is a disk (penumbra) which consists in a biconical
hole, the diameters $(D_{1},D_{2},D_{3})$ of the biconical hole are
$0.3mm,0.38mm,0.535mm$. An enlarged coded image is projected on the neutron
detector which is placed on the target bay floor, at $7895mm$ from the
aperture. The image obtained from the detector is a convolution of the source
spatial distribution with the aperture transfer function.
Figure 2: Penumbral imaging
In the Fig.2 only the red annular of the coded image contains the effective
information of the source shape changing. To achieve the high SNR we can place
a plug inside the hole of an aperture of the same dimensions as the
penumbra(Fig.2) to define a annular, which will be study in another paper.
This pattern can shield a lot of background but also face the alignment
tolerance problem and not enough neutron can be detected.
The neutron detector is composed of $255\times 255$ array of scintillating
fibers Fig.3. Each scintillating fiber can be realized as a sub-detector, the
structure was designed as the left figure of Fig.3. The core part is a tubes
of $460\mu m$ diameter and the material is polystyrene BCF-10 (molecular
formula C6H5CH=CH2, $1.05g/cm^{3}$). 14 MeV neutrons mainly interact by
elastic scattering on hydrogen nuclei inside the fibers. As they lose kinetic
energy, the recoil protons will produce light, a part of the light is guided
through these fibers. Outside the core part is a layer of the clad material,
which is used to avoid the scintillating light interaction in the array of
scintillating fibers. The material of the envelope is Polyvinyl
alcohol(molecular formula CH2=CHOH, $1.26g/cm^{3}$) and the thickness is
$20\mu m$. An optical relay then casts the scintillating light from the end of
the array to an image intensifier tube (IIT) which gates the 14 MeV neutrons
at their arrival time on the detector. The image is then reduced with a
fibered optical taper onto a CCD.
Figure 3: Fiber array
Overall spatial resolution of the system for both the annular and the
penumbral apertures is the same, which is determined by the point spread
functions (PSF) of the coded aperture and detector resolution ($\Delta
s_{detector}$) scaled down the source plane as equation 1. In the paraxial
approximation, the broadening of the apertures point spread function is
identical, determined by the FOV and the aperture to target distance
($L_{0}$).
$\Delta s=\sqrt{\frac{ln(2)\times FOV}{2\times L_{0}\times\mu}+\Delta
s_{detector}^{2}\times\left(\frac{L_{0}}{L_{1}}\right)^{2}}$ (1)
where $\mu$ is the attenuation of the neutrons in tungsten and $L_{1}$ is the
detector to the target distance. With this design, the overall spatial
resolution of the system is about 20 $\mu m$ in source plane.
In the GEANT4 simulation we used the physics list QGSP_BERT_HP which contains
the high precision neutron package (NeutronHP) to simulate the process of
neutrons’ transportation with the energy below 20 MeV down to thermal
energies. The physical process neutron capture, fission, elastic and inelastic
scattering (including absorption) are treated by referring to the ENDF-B VI
cross-section data. Besides, in the refMCNP has demonstrated the validity of
GEANT4 calculations in neutron generation and transportation which is the same
as MCNP. Moreover, object-oriented programming and highly sophisticated
processing of both electromagnetic interactions and ionization processes
enhance GEANT4 capabilities for the study of mixed fields and complicated
geometries.
In the ICF experiment the neutrons generated by hot spot spread in whole
space. The idealized aperture only allow the neutrons in the solid angle as
Fig.2 to deposit the energy on the detector. But actually because the neutrons
have the high ability to penetrate the start part of the aperture, and also
the neutrons will scatter with the wall of the aperture, these neutrons which
hit the detector will become the background of the imaging. To simulate the
complete imaging process and also save the calculation time we ask the program
only generate random distribution neutron in the cone with $R_{i}$ radius as
Fig.4. As equation 2 $R_{i}$ is determined by $R_{0}$ which is the width of
the detector project on the front face of the aperture.
With this design there are about one half of the events be scattered by the
penumbral aperture.To get the high statistic sample we simulated $1\times
10^{10}$ effective events of penumbral imaging.
Figure 4: The neutron radiated region in the simulation
$\rm R_{i}=\left\\{\begin{array}[]{lll}D_{0}/2,&R_{0}>D_{0}/2\\\
1.3R_{0},&D_{3}/2\leq R_{0}\leq D_{0}/2\\\
D_{3}/2,&R_{0}<D_{3}/2\end{array}\right.$ (2)
## II Reconstruction of the coded image
At present, there are a lot of reconstruction methods have been developed.
Among them the simplest and quickest method is Wiener filter method, which is
a linear reconstruction method. As we cannot get prior knowledge of the
signal-noise ratio, wiener filter method cannot get good results. Therefore we
can use this method as the reference or use the result as the input of some
nonlinear reconstruction method. In these years there are some nonlinear
reconstruct methods have been introduced to reconstruct penumbral image such
as genetic algorithm and heuristic algorithm proposed by Chen et
al.penum_other1 , and the classical molecular dynamics reconstruction method
proposed by Liu et al.penum_other2 ; penum_other3 ,
In the experiment L. Disdier et al. had successfully used the filtered
autocorrelation techniquefilauto1 to get $20\mu m$ resolution reconstruction
image from the penumbral imaging system with $22.3\mu m$ theoritical
resolution. Filtered autocorrelation technique is different from other method
which don’t need to rely on the simulation PSF in the reconstruction.
Another nonlinear method Lucy-Rechardson algorithmLR1 ; LR2 has been
successfully used on X-ray ring coded imaging in experimentxrayLR . And this
algorithm has been most widely used for restoring images with noise from
counting statistics and the data approximate Poisson statistics. In this
program we first try to use the Lucy-Rechardson method to get reconstructed
image. The process of the imaging can be described as a spatial intensity
distribution $g(x,y)$ convolved by a neutron source spatial intensity
distribution $f(x,y)$ with a point spread function (PSF) $h(x,y)$. Add the
noise $n(x,y)$ the detector image can be expressed as
$g(x,y)=f(x,y)\otimes h(x,y)+n(x,y)$ (3)
Lucy-Rechardson algorithm is an iterative method. Based on the referenceLR1 ;
LR2 , the decode formulation can be expressed as 4. Where PSF(h) and image on
detector(g) come from the simulation, the reconstructed image(f) can be
extracted after k times iterate.
$f_{k+1}=f_{k}(h\odot\frac{g}{h\otimes f_{k}})$ (4)
The PSF for penumbral and annular imaging are presented in Fig. 5
respectively. In Fig. 6 we presented the raw images with a ‘F’ shape neutron
source and the unfolded results with Lucy-Rechardson method. The width of the
‘F’ shape source is $20\mu m$.
Figure 5: Distribution of point spread function(PSF).
Figure 6: ‘F’ source imaging for penumbral aperture(left), unfolded result
(right).
From the Fig. 6 we can see with this design the imaging system can achieve the
aim to reconstruct the neutron source. With Lucy-Rechardson method the
unfolded image of Penumbral aperture is clear. To get the real resolution of
the system we simulate two point source with distance $20\mu m$ imaging on the
detector. In the Fig. 7 present the raw image and unfolded image.
Figure 7: Two point source imaging for penumbral aperture(left), unfolded
result(right).
For the annular imaging reconstruction and detail of the unfolding method we
will present in further study.
## III Conclusion
We have successfully construct one software NGPINEX which integrated the
imaging system simulation and the reconstruction method. The simulation and
reconstruction result show we can use this imaging system with these
parameters to get $20\mu m$ resolution image of DT implore. To attend the aim
of $5\mu m$ resolution in the future we can use this software to optimize the
system parameter.
Acknowledgments: The authors would like to thank Ming Jiang and Bing-quan Hu
for the help on the simulation model construction. This work was supported in
part by the Fundamental Research Funds for the Central Universities under
Grant No.CDJXS1102209 and the Program for New Century Excellent Talents in
Univresity under Grant No. NCET-10-0882, and by Natural Science Foundation of
China under Grant No.10805082 and No.11075225.
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|
arxiv-papers
| 2012-07-08T09:08:43 |
2024-09-04T02:49:32.782234
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xian-You Wang, Zhen-yun Fang, Yun-qing Tang, Zhi-Cheng Tang, Hong\n Xiao, and Ming Xu",
"submitter": "Xianyou Wang",
"url": "https://arxiv.org/abs/1207.1856"
}
|
1207.1871
|
# Long-range adiabatic quantum state transfer through tight-binding chain as
quantum data bus
Bing Chen1, Wei Fan1, Yan Xu1,3, Zhao-yang Chen2, Xun-li Feng3 and C. H. Oh3
1School of Science, Shandong University of Science and Technology, Qingdao
266510, China 2 Department of Mechanical Engineering, University of
California, Berkeley, CA 94720, USA 3 Center for Quantum Technologies and
Physics Department, Faculty of Science, National University of Singapore, 2
Science Drive 3, Singapore 117542
###### Abstract
We introduce a scheme based on adiabatic passage which allows for long-range
quantum communication through tight-binding chain with _always-on_
interaction. By adiabatically varying the external gate voltage applied on the
system, the electron can be transported from the sender’s dot to the aim one.
We numerically solve the schrödinger equation for a system with given number
of quantum dots. It is shown that this scheme is a simple and efficient
protocol to coherently manipulate the population transfer under suitable gate
pulses. The dependence of the energy gap and the transfer time on system
parameters is analyzed and shown numerically. Our method provides a guidance
for future realization of adiabatic quantum state transfer in experiments.
###### pacs:
03.67.Hk, 03.65.-w, 73.23.Hk
## I Introduction
Quantum state transfer (QST), as the name suggests, refers to the transfer of
an arbitrary quantum state from one qubit to another, which is a central task
in quantum information science. For the solid-state based quantum computing at
the large scale, it is very crucial to have a solid system serving as such
quantum data bus, which can provide us with a quantum channel for quantum
communication. During the last years many efforts have been made in different
fields to design a feasible proposal for perfect QST. One kind of proper QST
proposals is based on solid-state system with _always-on_ interaction Bose1 ;
Song ; Christandle1 . The communication is achieved by simply placing a
quantum state at one end of the chain and waiting for an optimized time to let
this state propagate to the other end with a high fidelity. The other kinds of
proposals have paid much attention to adiabatic passage for coherent QST in
time-evolving quantum systems, which is a powerful tool for manipulating a
quantum system from an initial state to a target state. This way of population
transfer has the important property of being robust against small variations
of the Hamiltonian and the transport time, which is crucial in experiment
since the system parameters are often hard to control. The typical scheme for
coherently spatial population transfer has been independently proposed for
neutral atoms in optical traps Eckert and for electrons in quantum dot (QD)
systems CTAP via a dark state of the system, which is termed coherent
tunneling via adiabatic passage (CTAP) following Ref. CTAP . In such a scheme,
the tunneling interaction between adjacent quantum units is dynamically tuned
by changing either the distance or the height of the neighboring potential
wells following a counterintuitive scheme which is a solid-state analog of the
well-known stimulated Raman adiabatic passage (STIRAP) protocol STIRAP of
quantum optics. Since then, the CTAP technique has been proposed in a variety
of physical systems for transporting single atoms atom1 ; atom2 , spin states
spin , electrons electron1 ; electron2 and Bose-Einstein condensates BEC1 ;
BEC2 ; BEC3 . It has also been considered as a crucial element in the scale up
to large quantum processors LR1 ; LR2 .
Recently, Ref. chen1 presented a scheme to adiabatically transfer an electron
from the left end to the right end of a three dot chain using the ground state
of the system. This technique is a copy of the frequency chirping method CF1 ;
CF2 , which is used in quantum optics to transfer the population of a three-
level atom of the Lambda configuration. The scheme chen1 is presented as an
alternative to a well known transfer scheme (CTAP) CTAP . However, different
from CTAP process, the protocol in Ref. chen1 considers a three QD array with
_always-on_ interaction which can be manipulated by the external gate voltage
applied on the two external dots (sender and receiver). Through maintaining
the system in the ground state, it shows that it is a high-fidelity process
for a proper choice of system parameters and also robust against experimental
parameter variations. The obvious extension of this work is to consider the
passage through more than one intervening dot chen2 . In this paper we will
consider a quasi-one-dimensional chain of QDs, which is schematically
illustrated in Fig. 1. The central tight-binding chain serves as the quantum
channel and two external QDs are attached to the media chain. The sender
(Alice) and the receiver (Bob) control one external QD each and Alice
transmits information via a qubit using the chain to Bob by adiabatically
changing the gate voltages. Different from previously discussed schemes, we
will consider a fixed $N$-site coupled QDs media chain and QST can be realized
in required transfer distance by modulating the positions where QD A and B are
connected to the chain. In particular, the nearest-neighbor hopping amplitudes
are set to be uniform. We first theoretically elaborate the adiabatic QST in
this scheme. Taking a 50-dot structure as an example, we show that the
electron can be robustly transported from Alice to Bob through the media
chain, by slowly varying the gate voltages.
The paper is organized as follows. In Sec. II the model is setup and we
describe the adiabatic transfer of an electron between QDs. In Sec. III we
show numerical results that substantiate the analytical results. The last
section is the summary and discussion of the paper.
## II Model Setup
Figure 1: (a) Schematic illustrations of adiabatic QST in multi-dot array. The
system is controlled by external gates voltage $\mu_{i}(t)$ $(i=A,$ $B)$. By
adiabatically varying the gates voltage, one can achieve long-range QST from
the QD $A$ to QD $B$ of the chain. (b) Gate voltages as a function of time (in
units of $\tau$). $\mu_{A}(t)$ is the solid line and $\mu_{B}(t)$ is the dash
line.
Consider a quasi-one-dimensional chain of QDs, realized by the empty or singly
occupied states of a positional eigenstate, see Fig. 1. The whole quantum
system consists of two sites ($A$ and $B$) and a simple tight-binding $N$-site
chain. The sender (Alice) and the receiver (Bob) can only control the external
gates voltage $\mu_{\alpha}(t)$ $(\alpha=A,$ $B)$. The total Hamiltonian
$\mathcal{H}(t)=\mathcal{H}_{M}+\mathcal{H}_{I}+\mathcal{H}_{C},$contains
three parts, the medium Hamiltonian
$\mathcal{H}_{M}=-J\sum_{j=1}^{N-1}\left|j\right\rangle\left\langle
j+1\right|+\text{h.c.}$ (1)
describing the tight-binding chain with uniform nearest neighbor hopping
integral $-J$ $(J>0),$ the coupling Hamiltonian
$\mathcal{H}_{I}=-J_{0}\left(\left|A\right\rangle\left\langle
l\right|+\left|B\right\rangle\left\langle
l^{\prime}\right|\right)+\text{h.c.},$ (2)
describing the connections between QDs $A$, $B$, and the chain with hopping
integral $-J_{0}$ $(J_{0}>0)$, and the operating Hamiltonian
$\mathcal{H}_{C}=\mu_{A}(t)\left|A\right\rangle\left\langle
A\right|+\mu_{B}(t)\left|B\right\rangle\left\langle B\right|$ (3)
describing the adiabatic manipulation of the Hamiltonian parameters. In
$\mathcal{H}(t)$, $\left|j\right\rangle$ represents the Wannier state
localized in the $j$-th quantum site for $j=A,$ $1,$ $2,...,$ $N,$ $B$. In
$\mathcal{H}_{I}$, $l$ and $l^{\prime}$ denote the sites of medium connecting
to the QDs $A$ and $B$. The distance between $A$ and $B$ is
$D=l^{\prime}+1-l$. In this proposal, we just consider one connection way:
$l^{\prime}=N+1-l$, that is the quantum state is transferred between site $A$
and its _mirror-conjugate_ site $B$. In term $\mathcal{H}_{C}$, $\mu_{A}(t)$
and $\mu_{B}(t)$ are site energies (externally controlled), which are
modulated by a Gaussian pulses (shown in Fig. 1(b))
$\displaystyle\mu_{A}(t)$ $\displaystyle=$
$\displaystyle-\mu_{0}\exp\left[-\frac{1}{2}\alpha^{2}t^{2}\right],$
$\displaystyle\mu_{B}(t)$ $\displaystyle=$
$\displaystyle-\mu_{0}\exp\left[-\frac{1}{2}\alpha^{2}\left(t-\tau\right)^{2}\right],$
(4)
where $\mu_{0}$ is the peak voltage of the pulse; $\tau$ and $\alpha$ the
total adiabatic evolution time and standard deviation of the control pulse. To
realize high fidelity transfer in this scheme, the peak voltage $\mu_{0}$ must
be much larger than hopping integral, i.e. $\mu_{0}\gg J,J_{0}$. The reason is
that small peak values improve adiabaticity, but lead to a low fidelity
because the final instantaneous eigenstate is not the desired one. According
to Ref. chen2 , the transfer is optimized when we choose $\alpha=8/\tau$.
Throughout this paper, all energies ($J_{0}$ and $\mu_{0}$) are scaled in
units of $J$, and evolution time $\tau$ is in units of $1/J$.
In this proposal, we focus our study on the ground state
$\left|\psi_{g}(t)\right\rangle$ of Hamiltonian $\mathcal{H}(t)$ to induce
population transfer from state $\left|A\right\rangle$ to
$\left|B\right\rangle$. For single electron transfer, a state in the single
particle Hilbert space is assumed as
$\left|\psi_{k}\right\rangle=f_{A}^{k}\left|A\right\rangle+\sum_{j=1}^{N}f_{j}^{k}\left|j\right\rangle+f_{B}^{k}\left|B\right\rangle$,
where $k$ denotes the momentum. Duo to the translational symmetry of the
present system, the instantaneous Hamiltonian’s eigen equation for
$f_{j}^{k},$ $j\in\left[1,N\right]$ is easily shown to be
$\displaystyle-J\left[f_{j-1}^{k}+f_{j+1}^{k}\right]$ $\displaystyle=$
$\displaystyle\left[\varepsilon_{k}-V_{A}\left(\varepsilon_{k}\right)\delta_{j,l}-V_{B}\left(\varepsilon_{k}\right)\delta_{j,N+1-l}\right]f_{j}^{k},$
$\displaystyle J_{0}f_{A}^{k}$ $\displaystyle=$ $\displaystyle
V_{A}\left(\varepsilon_{k}\right)f_{l}^{k},$ (5) $\displaystyle
J_{0}f_{B}^{k}$ $\displaystyle=$ $\displaystyle
V_{B}\left(\varepsilon_{k}\right)f_{N+1-l}^{k},$
here $\varepsilon_{k}$ is the eigenenergy and the term
$V_{i}\left(\varepsilon_{k}\right)=\frac{J_{0}^{2}}{\varepsilon_{k}+\mu_{i}},i\in\left[A,B\right]$
(6)
on the right hand side is contributed by the interactions between the sites
$A$, $B$, and medium chain which is dependent on the eigenenergy
$\varepsilon_{k}$. The $\delta$-type potential forms a confining barrier to
the transportation of single electron in the chain and forms a bounded state
of single electron, similar to those proposed in Ref. Sun . In this work, we
focus our attention on the bound state, which is the ground state of the total
system to realize the long-range QST.
Starting from $t=0$, we have $\mu_{A}(0)=-\mu_{0}$ and $\mu_{B}(0)\approx 0$.
The solution to Eq. (5) is
$\left|\psi_{g}(0)\right\rangle=\mathcal{N}^{-1/2}\left[\sum_{j=1}^{N}e^{-\kappa\left|j-l\right|}\left|j\right\rangle+\frac{\lambda_{+}}{J_{0}}\left|A\right\rangle+\frac{\lambda_{-}}{J_{0}}e^{-\kappa(N+1-2l)}\left|B\right\rangle\right],$
(7)
where $\lambda_{\pm}=\left(\sqrt{\mu_{0}^{2}+4J_{0}^{2}}\pm\mu_{0}\right)/2$
and $\kappa=\ln\left(\lambda_{+}/J\right)$;
$\mathcal{N}=\sum_{j=1}^{N}e^{-2\kappa\left|j-l\right|}+\left(\mu_{0}^{2}+2J_{0}^{2}\right)/J_{0}^{2}$
is the normalization factor. By choosing a sufficiently large value of
$\mu_{0}$, the ground state $\left|\psi_{g}(0)\right\rangle$ can be reduced to
$\left|\psi_{g}(0)\right\rangle\approx\left|A\right\rangle$.
With the same results, in the time limit $t=\tau$, the parameter $\mu_{A}(t)$
goes to zero and $\mu_{B}(t)$ goes to $-\mu_{0}$. Duo to the reflection
symmetry (relabeling sites from right to left) of the system, for
$j=A,1,2,...,N,B,$
$\langle
j\left|\psi_{g}(\tau)\right\rangle=\langle\bar{j}\left|\psi_{g}(0)\right\rangle.$
(8)
We have used $\bar{j}=N+1-j$ to indicate the _mirror-conjugate_ site of $j$.
This leads to the final ground state
$\left|\psi_{g}(\tau)\right\rangle\approx\left|B\right\rangle$. To illustrate
with an example, the probability of $\left|B\right\rangle$ in
$\left|\psi_{g}(\tau)\right\rangle$ can achieve 99.7% when the parameters are
set to be $J_{0}/J=1$ and $\mu_{0}/J=20$. Preparing the system in state
$\left|\Psi\left(t=0\right)\right\rangle=\left|A\right\rangle\,$and
adiabatially changing $\mu_{A}(t)$ and $\mu_{B}(t)$, one can see that the
system will end up in $\left|B\right\rangle$,
$\left|\Psi\left(t=0\right)\right\rangle=\left|A\right\rangle\rightarrow\left|\Psi\left(t=\tau\right)\right\rangle=\left|B\right\rangle,$
(9)
hence we can see that a high fidelity transfer may still be possible, even
with imperfect controls.
In the absence of hopping term between the two external QDs $A$, $B$ and the
medium chain ($J_{0}=0$), the Hamiltonian $\mathcal{H}(t)$ can be diagonalized
as
$\mathcal{H}(t)=\sum_{k}-2J\cos k\left|k\right\rangle\left\langle
k\right|+\mu_{A}(t)\left|A\right\rangle\left\langle
A\right|+\mu_{B}(t)\left|B\right\rangle\left\langle B\right|$
with
$\left|k\right\rangle=\sqrt{2/\left(N+1\right)}\sin(kj)\left|j\right\rangle$,
where $k=n\pi/(N+1),n=1,2,...,N$. One can see that at $t=\tau/2$, the
eigenstates $\left|A\right\rangle$ and $\left|B\right\rangle$ are degenerate
which leads to a breakdown of adiabaticity. The presence of the hopping term
$J_{0}$ will open up energy gap at the crossing. To evaluate instantaneous
eigenvalues of the Hamiltonian is generally only possible numerically. In Fig.
2(a) we present the results showing the eigenenergy gap between the
instantaneous first-excited state and ground state undergoing evolution due to
modulation of the gate voltages according to pulse Eq. (4). The eigenvalues
shown in this figure exhibit pronounced avoided crossing and approach nonzero
minimum
$\Delta=\varepsilon_{1}\left(\tau/2\right)-\varepsilon_{g}\left(\tau/2\right)$.
This minimum energy gap plays a significant role in the transfer, because the
total evolution time should be larger enough compared to $\Delta$. In this
scheme, the energy gap $\Delta$ depends both on the transfer distance $D$ and
the coupling constance $J_{0}$. To study the relationship between the total
evolution time and system parameters is one of the important contributions of
this paper.
As an example, we show in Fig. 2 the effect two factors has on the energy gap
$\Delta$ for a system with $N=48$ QDs and coupling strength $J=1.0$. The
pulse’s parameters we choose are $\alpha=8/\tau$, and $\mu_{0}=20$. In Fig.
2(b), we plot the energy gap $\Delta$ as a function of transfer distance $D$
for $J_{0}=0.5J$, and $J_{0}=0.7J$. The horizontal line $\delta\approx
3J\pi^{2}/N^{2}$ indicates the minimum gap of medium chain $\mathcal{H}_{M}$.
One can see that the logarithmic scales chosen suggest that as transfer
distance $D$ increasing there is an exponential disappearance of the gap
$\Delta$. The smaller hopping constant $J_{0}$, the slower the decay of
$\Delta$. The other thing is that $\Delta$ is also determined by the coupling
strength $J_{0}$. Fig. 2(c) shows the numerically computed behavior of
$\Delta$ as a function of $J_{0}$ for $D=8$, $16$, and $24$. As $J_{0}$
increases, the gap $\Delta$ increase for short-distance transfer ($D<N/3$) and
decreases for long-distance transfer ($D>N/3$). The results shown in Fig. 2(c)
also suggest that decreasing coupling strength $J_{0}$ can obtain relatively
large $\Delta$ for long-range transfer. But it does not mean the weaker the
coupling $J_{0}$, the better the result of QST will be. The reason is that the
energy gap is not the sufficient and necessary condition for adiabatic
process. In this proposal, the negative effects of distance on the gap can be
partially compensated by $J_{0}$.
Figure 2: (a) The instantaneous eigenenergy (in units of $J$) of the lowest
two states $\psi_{g}$ and $\psi_{1}$ through the gate pulse shown in Fig.
1(b). The gap is minimum at $t=\tau/2$,
$\Delta=\varepsilon_{1}\left(\tau/2\right)-\varepsilon_{g}\left(\tau/2\right)$.
(b) Plot of gap $\Delta$ (in units of $J$) as a function of transfer distance
$D$ for $J_{0}=$0.5$J$, and 0.7$J$. The energy gap $\Delta$ is exponentially
decreasing as transfer distance $D$ increase for a given $J_{0}$. (c) The gap
$\Delta$ (in units of $J$) as a function of coupling constant $J_{0}$ for the
transfer distance $D=$8, 16, and 24. The horizontal line $\delta$ is the
minimum gap of the medium chain $\mathcal{H}_{M}$.
## III Numerical Simulations
In this section let us firstly review the transfer process of this protocol.
At $t=0$ we initialize the device so that the electron occupies site-$1$,
i.e., the total initial state is $\left|A\right\rangle$. Provided we transform
the gate pulses adiabatically, then the adiabatic theorem states that the
system will stay in the same eigenstate. Therefore, the quantum state starting
in $\left|A\right\rangle$ will end up in $\left|B\right\rangle$.
The analysis above is based on the assumption that the adiabaticity is
satisfied. The adiabaticity parameter defined for this scheme is
$\mathcal{A}\left(t\right)=\frac{\left|\left\langle\psi_{g}(t)\right|\partial\mathcal{H}/\partial
t\left|\psi_{1}(t\right\rangle\right|}{\left|\varepsilon_{g}(t)-\varepsilon_{1}(t)\right|^{2}},$
(10)
where $\left|\psi_{g}(t\right\rangle$ ($\left|\psi_{1}(t\right\rangle$) is the
instantaneous ground state (first-excited state) of the Hamiltionian
$\mathcal{H}(t)$ and $\varepsilon_{g}(t)$ ($\varepsilon_{1}(t)$) is the
corresponding instantaneous eigenvalue of the state
$\left|\psi_{g}(t\right\rangle$ ($\left|\psi_{1}(t\right\rangle$). For
adiabatic evolution of the system we require $\mathcal{A}\left(t\right)\ll 1$
for all time, which greatly suppresses the quantum transition from the ground
state $\left|\psi_{g}(t)\right\rangle$ to the first-excited state
$\left|\psi_{1}(t)\right\rangle$. Fig. 3(a) and (b) show the numerical result
of energy difference $\varepsilon_{1}(t)-\varepsilon_{g}(t)$ and
$\mathcal{A}\left(t\right)\tau$ as a function of pulse time. Note that the
appearance time of maxima of $\mathcal{A}\left(t\right)$ is not at the middle
of the pulse sequence, but the energy difference
$\varepsilon_{1}(t)-\varepsilon_{g}(t)$ at this time is slightly larger than
the minimum gap $\Delta$. So we can use minimum gap $\Delta$ to estimate the
minimum pulse time required for high-fidelity transfer. Fig. 3(c) shows the
maximum adiabaticity $\max\\{\mathcal{A}\left(t\right)\tau\\}$ through the
protocol as a function of $J_{0}$ for $D=10$, $16$, and $20$. One sees that
there is an optimal value of $J_{0}$ which ensures the shortest time for
realizing perfect QST and the optimal value decreases as transfer distance $D$
increases. To sum up, the adiabatic regime necessary to obtain transport with
high fidelity can be concluded to the condition $\tau\gg
1/\Delta^{\ast}\left(D\right)$, where $\Delta^{\ast}\left(D\right)$ is the
minimum gap of the system when the coupling strength $J_{0}$ takes the optimal
value corresponding to transfer distance $D$.
Figure 3: (a) The energy difference $E_{1}(t)-E_{g}(t)$ (in units of $J$) and
(b) adiabaticity $\mathcal{A}\left(t\right)\tau$ (in units of $10^{3}/J$) as a
function of time in the time interval $t\in[0.3\tau,0.7\tau]$. Note that the
energy gap approximately equals $\Delta$ in a wide time range and the maxima
of adiabaticity is not at the middle of the pulse sequence. (c) Maximum
adiabaticity $\max\\{\mathcal{A}\left(t\right)\tau\\}$ through the protocol as
a function of $J_{0}$ for three different transfer distances.
$\max\\{\mathcal{A}\left(t\right)\tau\\}$ varies with $J_{0}$, and reaches
minimum value when the $J_{0}$ takes specific value.
The consequent time evolution of the state is given by the Schrödinger
equation (assuming $\hbar=1$)
$i\frac{d}{dt}\left|\Psi\left(t\right)\right\rangle=\mathcal{H}(t)\left|\Psi\left(t\right)\right\rangle.$
(11)
The time evolution creates a coherent superposition:
$\left|\Psi\left(t\right)\right\rangle=c_{A}(t)\left|A\right\rangle+\sum_{j=1}^{N}c_{j}(t)\left|j\right\rangle+c_{B}(t)\left|B\right\rangle,$
(12)
where $c_{j}(t)$ denotes the time-dependent probability amplitude for the
electron to be in $j$-th QD. We define the probability of finding the electron
on the medium chain as
$\left|c_{M}(t)\right|^{2}=\sum_{j=1}^{N}\left|c_{j}(t)\right|^{2}$ that obeys
the normalization
condition$\,\left|c_{A}(t)\right|^{2}+\left|c_{M}(t)\right|^{2}+\left|c_{B}(t)\right|^{2}=1$.
At time $\tau$ the fidelity of initial state transferring to the dot-$B$ is
defined as $F=\left|\langle
B\left|\Psi(\tau)\right\rangle\right|^{2}=\left|c_{B}(\tau)\right|^{2}$.
Figure 4: (Color online) QST from QD $A$ to $B$ attached to a tight-binding
chain of length with $N=48$ for two different transfer distance: $D=11$
(a)-(d) and $D=21$ (e)-(h). (a) The transfer fidelity as a function of
coupling strength $J_{0}$ for transfer distance $D=11$. The calculation was
done for various values of $J_{0}$ at fixed total evolution time $\tau=480/J$.
Note that there is an optimal value of $J_{0}$ to achieve high fidelity QST.
Population transfer as a function of time obtained for three coupling strength
$J_{0}$:(b) $J_{0}=0.8J$; (c) $J_{0}=0.91J$; (d) $J_{0}=1.0J$. Initially the
population is on QD $A$ (dashed red line) and finally mainly on QD $B$ (solid
blue line). The population on the media chain is shown as a dotted black line.
(e) The same as in (a), but for transfer distance $D=21$ and $\tau=1200/J$.
(f)-(h) The population behavior under the influence of $J_{0}$. The parameters
are like that in (b)-(d).
In order to proceed, we used standard numerical methods to integrate the
Schrödinger equation for probability amplitudes. Because the scheme relies on
maintaining adiabatic conditions, we examine the effect of system parameters
on the target state population. In Fig. 4, we consider the system with $N=48$
and show QST from QD $A$ to QD $B$ for two different transfer distance: $D=11$
and $D=21$. Firstly, we examine the effect of coupling strength $J_{0}$ on
transfer fidelity. It is seen that there is an optimal value of $J_{0}$ to
achieve high-fidelity transfer.
To illustrate the process of QST for $D=11$, we exhibit in Fig. 4(b)-(d) the
exact evolution of the probabilities of finding electron in QD $A$ (red dashed
line), $B$ (blue solid line), and media chain (black dotted line) as a
function of time for three different values of coupling strength $J_{0}$ but
the same remaining parameters ($\mu_{0}=20J$, $\tau=480/J$ and
$\alpha=8/\tau$). We get good results for the transfer if we choose
$J_{0}=0.89J$ as shown in Fig. 4(b). The populations on the QD $A$ and QD $B$
are exchanged in the expected adiabatic manner. If we choose parameters
deviated from the optimum value, $J_{0}=0.8J$ and $1.0J$, we find the results
in Fig. 4(c) and Fig. 4(d). We can see that a slight deviations from the
values will breaks adiabaticity and lead to major deteriorations of the
quality of transfer. When the transfer distance becomes large, we should
enlarge evolution time $\tau$ to enhance the adiabaticity. In Fig. 4(e),
transfer fidelity as a function of $J_{0}$ for transfer distance $D=21$ and
evolution time $\tau=1200/J$. The time evolution of the probabilities the same
as Fig. 4(b)-(d) for $D=21$ and three different $J_{0}$ are illustrated in
Fig. 4(f)-(g). We can still see from Fig. 2 that the optimum value of $J_{0}$
to achieve high-fidelity transfer decreases as the transfer distance increase
which is consistent with the results shown before.
Figure 5: (Color online) For a given chain length $N=48$ and a chosen fidelity
$F=0.995$, (a) the optimum value of coupling strength $J_{0}$ (in units of
$J$) and (b) the minimum time $\tau$ (in units of $1/J$) required for state
transfer varies as a function of transfer distance $D$. The other system
parameters we choose is $J=1$, $\mu_{0}=20$ and $\alpha=8/\tau$. It shows that
optimum value of $J_{0}$ decreases as $D$ increasing and time $\tau$ scales
linearly with transfer distance $D$.
In order to provide the most economical choice of parameters for reaching high
transfer efficiency, we perform numerical analysis, as shown in Fig. 5.
Specifically, we depict the optimum coupling $J_{0}$ (shown in Fig. 5(a)) and
the corresponding minimum state transfer time $\tau$ (shown in Fig. 5(b)) for
a given chain length $N=48$ and a given tolerable transfer fidelity
$F=99.5\%$, the minimum time varies as a function of transfer distance $D$.
One can see that the time $\tau$ required for high-fidelity transfer scales
linearly with transfer distance $D$.
## IV Summary and Discussion
An efficient QST scheme should not only admit a state transfer of any quantum
state in a fixed period of time of the state evolution with high fidelity, but
also the transfer time can not grow fast as communication distance increases.
In this paper, we have introduced a long-range transport mechanism for quantum
information around a quasi-one-dimensional QDs network, based on adiabatic
passage. This scheme is realized by modulation of gate voltages applied on the
two external QDs which is connected to the tight-binding chain. Under suitable
system parameters, the electron can be transported from the sender QD to the
receiver one with high efficiency, carrying along with it the quantum
information encoded in its spin. Different from the CTAPn Scheme CTAPn , our
method is to induce population transfer through tight-binding chain by
maintaining the system in its ground state and this is more operable in
experiments. We have studied the adiabatic QST through the system by
theoretical analysis and numerical simulations of the ground state evolution
of tight-binding model. The result demonstrates that it is an efficient high-
fidelity process (99.5%) for an economical choice of system parameters.
Increasing the transfer distance, we found that the efficiency of QST is
inversely proportional to the distance of the two QDs.
In a real system, decoherence is the main obstacle to the experimental
implementation of quantum information Ivanov ; Kam . There are two sources of
quantum decoherence in QDs, one is due to charge dephasing brought by lead-QD
coupling and the other is due to the hyperfine interaction. For the former,
the coherence time of quantum dot is $\sim$1 ns, which plays a role in this
case. On the other hand, the maximum time in $N=48$ QD system needed for the
appearance of the better fidelity is roughly proportional to $10^{4}/J$. As a
simple estimate of the effects of decoherence, we compare this time with the
dephasing time, which leads to a limit of coupling strength of $J$ of $\sim$10
THz. The probability of realization of this idea in experiment can be
maximized by more precise manipulation technology and by cooling the system.
Furthermore, the development of cold atom physics provides us with an
alternative realization of our systems in experiment, because decoherence in
cold-atom system is much less destructive.
## Acknowledgments
We acknowledge the support of the NSF of China (Grant No.10847150 and
No.11105086), the National Research Foundation and Ministry of Education,
Singapore (Grant No. WBS: R-710-000-008-271), the Shandong Provincial Natural
Science Foundation (Grant No. ZR2009AM026 and BS2011DX029), and the basic
scientific research project of Qingdao (Grant No.11-2-4-4-(6)-jch). Y. X. also
thanks the Basic Scientific Research Business Expenses of the Central
University and Open Project of Key Laboratory for Magnetism and Magnetic
Materials of the Ministry of Education, Lanzhou University (Grant No.
LZUMMM2011001) for financial support.
## References
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* (21) D. Z. Xu, H. Lan, T. Shi, H. Dong, and C. P. Sun, Sci China Ser G-Phys Mech Astron, 53(7): 1234-1238 (2010).
* (22) L. C. L. Hollenberg, A. D. Greentree, A. G. Fowler, and C. J. Wellard, Phys. Rev. B 74, 045311 (2006).
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|
arxiv-papers
| 2012-07-08T13:37:08 |
2024-09-04T02:49:32.789298
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Bing Chen, Wei Fan, Yan Xu, Zhao-yang Chen, Xun-li Feng, and C. H. Oh",
"submitter": "Bing Chen",
"url": "https://arxiv.org/abs/1207.1871"
}
|
1207.1969
|
# On the existence of $3$–way $k$–homogeneous Latin trades
Behrooz Bagheri Gh.a,b, Diane Donovanc, and E. S. Mahmoodianb
###### Abstract
A $\mu$-way Latin trade of volume $s$ is a collection of $\mu$ partial Latin
squares $T_{1},T_{2},\ldots,T_{\mu}$, containing exactly the same $s$ filled
cells, such that if cell $(i,j)$ is filled, it contains a different entry in
each of the $\mu$ partial Latin squares, and such that row $i$ in each of the
$\mu$ partial Latin squares contains, set-wise, the same symbols and column
$j$, likewise. It is called $\mu$–way $k$–homogeneous Latin trade, if in each
row and each column $T_{r}$, for $1\leq r\leq\mu,$ contains exactly $k$
elements, and each element appears in $T_{r}$ exactly $k$ times. It is also
denoted by $(\mu,k,m)$ Latin trade, where $m$ is the size of partial Latin
squares.
We introduce some general constructions for $\mu$–way $k$–homogeneous Latin
trades and specifically show that for all $k\leq m$, $6\leq k\leq 13$ and
$k=15$, and for all $k\leq m$, $k=4,\ 5$ (except for four specific values), a
$3$–way $k$–homogeneous Latin trade of volume $km$ exists. We also show that
there are no $(3,4,6)$ Latin trade and $(3,4,7)$ Latin trade. Finally we
present general results on the existence of $3$–way $k$–homogeneous Latin
trades for some modulo classes of $m$.
$a$ Department of Mathematical Sciences
Isfahan University of Technology
84156-83111, Isfahan, I. R. Iran
$b$ Department of Mathematical Sciences
Sharif University of Technology
P. O. Box 11155–9415, Tehran, I. R. Iran
$c$ Department of Mathematics
The University of Queensland
Brisbane 4072
Australia
AMS Subject Classification: 05B15
Keywords: Latin square; Latin trade; $\mu$–way Latin trade; $\mu$–way
$k$–homogeneous Latin trade.
## 1 Introduction
A Latin square $L$ of order $n$ is an $n\times n$ array usually on the set
$N=\\{1,\ldots,n\\}$ where each element of $N$ appears exactly once in each
row and exactly once in each column. We can represent each Latin square as a
subset of $N\times N\times N$,
$L=\\{(i,j;k)\mid\mbox{ element $k$ is located in position }(i,j)\\}.$
A partial Latin square $P$ of order $n$ is an $n\times n$ array of elements
from the set $N$, where each element of $N$ appears at most once in each row
and at most once in each column. The set $S_{P}=\\{(i,j)\mid(i,j;k)\in P\\}$
of the partial Latin square $P$ is called the shape of $P$ and $|S_{P}|$ is
called the volume of $P$. By ${\cal R}_{P}^{i}$ and ${\cal C}_{P}^{j}$ we mean
the set of entries in row $i$ and column $j$, respectively of $P$. A $\mu$-way
Latin trade, $(T_{1},T_{2},\ldots,T_{\mu})$, of volume $s$ is a collection of
$\mu$ partial Latin squares $T_{1},T_{2},\ldots,T_{\mu}$, containing exactly
the same $s$ filled cells, such that if cell $(i,j)$ is filled, it contains a
different entry in each of the $\mu$ partial Latin squares, and such that row
$i$ in each of the $\mu$ partial Latin squares contains, set-wise, the same
symbols and column $j$, likewise. If $\mu=2$, $(T_{1},T_{2})$ is called a
Latin bitrade. The study of Latin trades and combinatorial trades in general,
has generated much interest in recent years. For a survey on the topic see
[3], [9], and [6].
A $\mu$–way Latin trade which is obtained from another one by deleting its
empty rows and empty columns, is called a $\mu$–way $k$–homogeneous Latin
trade $(\mu\leq k)$ or briefly a $(\mu,k,m)\mbox{ \sf Latin trade}$, if it has
$m$ rows and in each row and each column $T_{r}$, for $1\leq r\leq\mu,$
contains exactly $k$ elements, and each element appears in $T_{r}$ exactly $k$
times.
In Figure 1($a$) a $(3,5,7)\mbox{ Latin trade}$ is demonstrated. The elements
of $T_{2}$ and $T_{3}$ are written as subscripts in the same array as $T_{1}$.
($\bullet$ means the cell is empty.)
${1_{{\bf\displaystyle 2}_{\displaystyle 3}}}$ | ${3_{{\bf\displaystyle 5}_{\displaystyle 2}}}$ | ${5_{{\bf\displaystyle 3}_{\displaystyle 7}}}$ | ${7_{{\bf\displaystyle 1}_{\displaystyle 5}}}$ | ${2_{{\bf\displaystyle 7}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$
---|---|---|---|---|---|---
${{}_{{}_{\displaystyle\bullet}}}$ | ${2_{{\bf\displaystyle 3}_{\displaystyle 4}}}$ | ${4_{{\bf\displaystyle 6}_{\displaystyle 3}}}$ | ${6_{{\bf\displaystyle 4}_{\displaystyle 1}}}$ | ${1_{{\bf\displaystyle 2}_{\displaystyle 6}}}$ | ${3_{{\bf\displaystyle 1}_{\displaystyle 2}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$
${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${3_{{\bf\displaystyle 4}_{\displaystyle 5}}}$ | ${5_{{\bf\displaystyle 7}_{\displaystyle 4}}}$ | ${7_{{\bf\displaystyle 5}_{\displaystyle 2}}}$ | ${2_{{\bf\displaystyle 3}_{\displaystyle 7}}}$ | ${4_{{\bf\displaystyle 2}_{\displaystyle 3}}}$
${5_{{\bf\displaystyle 3}_{\displaystyle 4}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${4_{{\bf\displaystyle 5}_{\displaystyle 6}}}$ | ${6_{{\bf\displaystyle 1}_{\displaystyle 5}}}$ | ${1_{{\bf\displaystyle 6}_{\displaystyle 3}}}$ | ${3_{{\bf\displaystyle 4}_{\displaystyle 1}}}$
${4_{{\bf\displaystyle 5}_{\displaystyle 2}}}$ | ${6_{{\bf\displaystyle 4}_{\displaystyle 5}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${5_{{\bf\displaystyle 6}_{\displaystyle 7}}}$ | ${7_{{\bf\displaystyle 2}_{\displaystyle 6}}}$ | ${2_{{\bf\displaystyle 7}_{\displaystyle 4}}}$
${3_{{\bf\displaystyle 1}_{\displaystyle 5}}}$ | ${5_{{\bf\displaystyle 6}_{\displaystyle 3}}}$ | ${7_{{\bf\displaystyle 5}_{\displaystyle 6}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${6_{{\bf\displaystyle 7}_{\displaystyle 1}}}$ | ${1_{{\bf\displaystyle 3}_{\displaystyle 7}}}$
${2_{{\bf\displaystyle 4}_{\displaystyle 1}}}$ | ${4_{{\bf\displaystyle 2}_{\displaystyle 6}}}$ | ${6_{{\bf\displaystyle 7}_{\displaystyle 4}}}$ | ${1_{{\bf\displaystyle 6}_{\displaystyle 7}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${7_{{\bf\displaystyle 1}_{\displaystyle 2}}}$
${1_{{\bf\displaystyle 2}_{\displaystyle 3}}}$ | ${3_{{\bf\displaystyle 5}_{\displaystyle 2}}}$ | ${5_{{\bf\displaystyle 3}_{\displaystyle 7}}}$ | ${7_{{\bf\displaystyle 1}_{\displaystyle 5}}}$ | ${2_{{\bf\displaystyle 7}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$
---|---|---|---|---|---|---
${{}_{{}_{\displaystyle\bullet}}}$ | $\searrow$ | $\searrow$ | $\searrow$ | $\searrow$ | $\searrow$ | ${{}_{{}_{\displaystyle\bullet}}}$
${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | | | | |
| ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | | | |
| | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | | |
| | | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | |
| | | | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ |
($a$) ($b$)
Figure 1: A $(3,5,7)\mbox{ Latin trade}$ and its base row
A $(\mu,k,m)\mbox{ Latin trade}$ $(T_{1},T_{2},\ldots,T_{\mu})$ is called
circulant, if it can be obtained from the elements of its first row, called
the base row and denoted by $\mu$–$B_{m}^{k}$, by permuting the coordinates
cyclically along the diagonals. For example in Figure 1($b$), a
$3$–$B_{7}^{5}$ base row,
$\\{(1,2,3)_{1},(3,5,2)_{2},(5,3,7)_{3},(7,1,5)_{4},(2,7,1)_{5}\\}$, is shown.
Actually if a base row $B=\\{(a_{1},a_{2},\ldots,a_{\mu})_{c_{l}}\mid 1\leq
l\leq k\\}$, where $a_{r}$ and $c_{l}$ $\in\\{1,2,\ldots,m\\},$ is given, we
construct a set of $\mu$ partial Latin squares as in the following manner:
$1\leq r\leq\mu,\ T_{r}=\\{(1+i,c_{l}+i;a_{r}+i)({\rm mod}\ m)|0\leq i\leq
m-1,1\leq l\leq k\\}.$
###### Algorithm 1
To check that $B=\\{(a_{1},a_{2},\ldots,a_{\mu})_{c_{l}}\mid 1\leq l\leq
k\\}$, where $a_{r}$ and $c_{l}$ $\in\\{1,2,\ldots,m\\},$ is a base row of a
$(\mu,k,m)\mbox{ Latin trade}$:
we note that for each $r$, $1\leq r\leq\mu$, ${\cal
R}_{T_{r}}^{1}=\\{a_{r}\mid(a_{1},a_{2},\ldots,a_{\mu})_{c_{l}}\in B\ {\rm
and}\ 1\leq l\leq k\\}$ and ${\cal C}_{T_{r}}^{m}=\\{a_{r}+m-c_{l}\ \equiv
a_{r}-c_{l}({\rm mod}\ m)\mid(a_{1},a_{2},\ldots,a_{\mu})_{c_{l}}\in B\ {\rm
and}\ 1\leq l\leq k\\}$. Now if $B$ satisfies the following conditions, then
it will suffice to be a base row of a $(\mu,k,m)\mbox{ Latin trade}$.
(i) $a_{r}$’s are distinct, for each $(a_{1},a_{2},\ldots,a_{\mu})_{c_{l}}\in
B$.
(ii) $c_{l}$’s are distinct.
(iii) ${\cal R}_{T_{1}}^{1}={\cal R}_{T_{2}}^{1}=\cdots={\cal
R}_{T_{\mu}}^{1}$.
(iv) ${\cal C}_{T_{1}}^{m}={\cal C}_{T_{2}}^{m}=\cdots={\cal
C}_{T_{\mu}}^{m}$.
###### Lemma 1
For each $k\geq\mu$, a $(\mu,k,k)\mbox{ Latin trade}$ exists.
###### Proof.
By taking a Latin square of order $k$ and permuting its rows, cyclically,
$\mu$ times we obtain the desired Latin trade.
A $(\mu,\mu,\mu)$ Latin trade is called a $\mu$–intercalate.
${1_{{\bf\displaystyle 3}_{\displaystyle 2}}}$ | ${2_{{\bf\displaystyle 1}_{\displaystyle 3}}}$ | ${3_{{\bf\displaystyle 2}_{\displaystyle 1}}}$
---|---|---
${3_{{\bf\displaystyle 2}_{\displaystyle 1}}}$ | ${1_{{\bf\displaystyle 3}_{\displaystyle 2}}}$ | ${2_{{\bf\displaystyle 1}_{\displaystyle 3}}}$
${2_{{\bf\displaystyle 1}_{\displaystyle 3}}}$ | ${3_{{\bf\displaystyle 2}_{\displaystyle 1}}}$ | ${1_{{\bf\displaystyle 3}_{\displaystyle 2}}}$
Figure 2: A $3$–intercalate
The following question is of interest.
###### Question 1
For given $m$ and $k$, $m\geq k\geq\mu$, does there exist a $(\mu,k,m)\mbox{
Latin trade}$?
For Latin bitrades, Question 1 is discussed and is answered completely in [4],
[5], [2], [1], and [7]. In this paper applying earlier results we introduce
some general constructions for $(\mu,k,m)\mbox{ Latin trade}$s and
specifically concentrate on the case of $\mu=3$. Our main result is stated in
the following theorem.
###### Theorem 1
All $(3,k,m)\mbox{ Latin trade}$s $(m\geq k\geq 3)$ exist, for
* •
$k=4$, except for $m=6$ and $7$ and possibly for $m=11$,
* •
$k=5$, except possibly for $m=6$,
* •
$6\leq k\leq 13$,
* •
$k=15$,
* •
$k\geq 4$ and $m\geq k^{2}$,
* •
$m$ a multiple of $5$, except possibly for $m=30$,
* •
$m$ a multiple of $7$, except possibly for $m=42$ and $(3,4,7)\mbox{ Latin
trade}$.
## 2 General constructions
###### Theorem 2
If $l\neq 2,6$ and for each $k\in\\{k_{1},\ldots,k_{l}\\}$ there exists a
$(\mu,k,p)$ Latin trade, then a $(\mu,k_{1}+\cdots+k_{l},lp)\mbox{ Latin
trade}$ exists. (Some $k_{i}$s can possibly be zero.)
###### Proof.
Since $l\neq 2,6$, there exist two $l\times l$ orthogonal Latin squares.
Denote these Latin squares by $L_{1}$ and $L_{2}$, with elements chosen from
the sets $\\{e_{1},e_{2},\ldots,e_{l}\\}$ and
$\\{f_{1},f_{2},\ldots,f_{l}\\}$, respectively. Assume that $L^{*}$ is a
square that is formed by superposing $L_{1}$ and $L_{2}$. We replace each
$(e_{i},f_{j})$ in $L^{*}$ with a $(\mu,k_{j},p)\mbox{ Latin trade}$ whose
elements are from the set $\\{(i-1)p+1,(i-1)p+2,\ldots,ip\\}$. As a result we
obtain a $(\mu,k_{1}+\cdots+k_{l},lp)\mbox{ Latin trade}$.
###### Theorem 3
If the number of mutually orthogonal Latin squares of order $k+1$, ${\rm
MOLS}(k+1)$, is greater than or equal to $\mu+1$, then there exists a
$(\mu,k,k+1)$ Latin trade.
###### Proof.
By Exercise $5.2.11$ of [10] page $103$, there are $\mu$ idempotent ${\rm
MOLS}(k+1).$ If in each of those ${\rm MOLS}$ we delete the main diagonals, we
obtain a $(\mu,k,k+1)$ Latin trade.
Actually by applying results of existence of idempotent ${\rm MOLS}(n)$ ([8],
Section $3.6$, Table $3.83$), we can improve Theorem 3 for the case $\mu=3$ as
follows.
###### Theorem 4
If $k\geq 11$, then there exists a $(3,k,k+1)\mbox{ Latin trade}$.
###### Theorem 5
Any $(\mu,\mu,m)\mbox{ Latin trade}$, ${\bf T}=(T_{1},T_{2},\ldots,T_{\mu})$,
can be partitioned into disjoint $\mu$–intercalates.
###### Proof.
We prove this result by induction. Without loss of generality, let $(1,1;r)\in
T_{r}$ for each $1\leq r\leq\mu$. Therefore $\\{1,2,\ldots,\mu\\}\subset{\cal
R}_{T_{r}}^{i}\cap{\cal C}_{T_{r}}^{i}$ for each $1\leq i,r\leq\mu$. Since
$|{\cal R}_{T_{r}}^{i}|=|{\cal C}_{T_{r}}^{i}|=\mu$ for each $1\leq
i,r\leq\mu$, ${\cal R}_{T_{r}}^{i}={\cal C}_{T_{r}}^{i}=\\{1,2,\ldots,\mu\\}$
for each $1\leq i,r\leq\mu$. Again without loss of generality, let $(i,1;i)\in
T_{1}$ and $(1,j;j)\in T_{1}$ for $1\leq i,j\leq\mu$. This implies that
$\\{(i,j)\mid 1\leq i,j\leq\mu\\}$ is a subset of shape of $T_{1}$. Therefore
subarray $\\{(i,j)\mid 1\leq i,j\leq\mu\\}$ with elements
$\\{1,2,\ldots,\mu\\}$ is a $\mu$–intercalate. We can apply the same argument
to the $(m-\mu)\times(m-\mu)$ subsquare obtained by removing rows
$1,2,\ldots,\mu$ and columns $1,2,\ldots,\mu$. This completes the proof.
###### Corollary 1
For every $m\geq 1$, there exists a $(\mu,k,m)\mbox{ Latin trade}$ with
$k=\mu$, if and only if $k|m$.
###### Theorem 6
Assume that $m_{i}\geq k_{i}$, for $i=1,2$. If there exists a
$(\mu_{i},k_{i},m_{i})$ Latin trade for $i=1,2$, then there exists a
$(\mu_{1}\mu_{2},k_{1}k_{2},m_{1}m_{2})\mbox{ Latin trade}$.
###### Proof.
We construct a $(\mu_{1}\mu_{2},k_{1}k_{2},m_{1}m_{2})\mbox{ Latin trade}$ in
the following way:
Suppose $(T_{1},T_{2},\ldots,T_{\mu_{1}})$ is a $(\mu_{1},k_{1},m_{1})\mbox{
Latin trade}$ and ${\bf U}=(U_{1},U_{2},\ldots,U_{\mu_{2}})$ is a
$(\mu_{2},k_{2},m_{2})\mbox{ Latin trade}$. For each entry $i$ in
$T_{1},T_{2},\ldots,T_{\mu_{1}}$, we replace $i$ with a copy of ${\bf U}$
where elements are chosen from the set
$\\{(i-1)m_{2}+1,(i-1)m_{2}+2,\ldots,im_{2}\\}$; replace the empty cells in
$T_{1},T_{2},\ldots,T_{\mu_{1}}$ with an empty $m_{2}\times m_{2}$ array. As a
result we obtain a $(\mu_{1}\mu_{2},k_{1}k_{2},m_{1}m_{2})\mbox{ Latin
trade}$.
###### Corollary 2
Suppose $k=k_{1}k_{2}$ and $m=m_{1}m_{2}$ where $m_{i}\geq k_{i}\geq 2$, for
$i=1,2$. Then there exists a $(4,k,m)\mbox{ Latin trade}$, provided that if
$k_{j}=2$, for some $j$, then $m_{j}$ must be assumed to be even.
###### Proof.
It is shown that Latin homogeneous bitrades (i.e $(2,k,m)\mbox{ Latin trade}$)
exist for all $m\geq k\geq 3$ and for all even $m$, when $k=2$. (See [4], [5],
[2], [1], and [7].)
###### Theorem 7
For every $k$, if there exists a $(\mu,k,m)\mbox{ Latin trade}$ and a
$(\mu,k,n)$ Latin trade, then there exists a $(\mu,k,m+n)\mbox{ Latin trade}$.
###### Proof.
Let ${\bf T_{1}}$ be a $(\mu,k,m)\mbox{ Latin trade}$ and ${\bf T_{2}}$ be a
$(\mu,k,n)\mbox{ Latin trade}$ such that the elements of ${\bf T_{1}}$ are in
the set $\\{1,\ldots,m\\}$ and the elements of ${\bf T_{2}}$ are chosen from
the set $\\{m+1,\ldots,m+n\\}$. Therefore, the following Latin trade is a
$(\mu,k,m+n)\mbox{ Latin trade}$.
$\begin{array}[]{|@{\hspace{2pt}}c@{\hspace{2pt}}@{\hspace{1pt}}c@{\hspace{1pt}}
@{\hspace{1pt}}c@{\hspace{1pt}}@{\hspace{1pt}}c@{\hspace{1pt}}
|}\hline\cr\hskip 2.0pt\lx@intercol\hfil{\bf T_{1}}\hfil\hskip 2.0pt\hskip
1.0&\hfil\hskip 1.0pt\hskip 1.0&\hfil\hskip 1.0pt\hskip 1.0&\hfil\hskip 1.0\\\
\hskip 2.0pt\lx@intercol\hfil\hfil\hskip 2.0pt\hskip 1.0&{\bf
T_{2}}\hfil\hskip 1.0pt\hskip 1.0&\hfil\hskip 1.0pt\hskip 1.0&\hfil\hskip
1.0\\\ \hline\cr\end{array}$
###### Corollary 3
If the number of ${\rm MOLS}(k+1)\geq\mu+1$, then for each $m$ where $m\geq
k^{2}$, there exists a $(\mu,k,m)\mbox{ Latin trade}$.
###### Proof.
If $m\geq k^{2}$, then we can write $m$ as $m=rk+s(k+1)$, where $r,s\geq 0$.
Theorem 7 and Theorem 3 lead us to a conclusion.
By Theorems 7 and 4 we have:
###### Corollary 4
If $k\geq 11$, then for each $m$ where $m\geq k^{2}$, there exists a $(3,k,m)$
Latin trade.
###### Theorem 8
Consider an arbitrary natural number $k$. If for every ${k+1}\leq l\leq 2k-1$
there exists a $(\mu,k,l)\mbox{ Latin trade}$, then for any $m\geq k$ there
exists a $(\mu,k,m)$ Latin trade.
###### Proof.
For every $m\geq 2k$, we can write $m=rk+sl$, where $r,s\geq 0$ and ${k+1}\leq
l\leq 2k-1$. Since there exist a $(\mu,k,k)\mbox{ Latin trade}$ and a
$(\mu,k,l)\mbox{ Latin trade}$, by Theorem 7 we conclude that there exists a
$(\mu,k,m)$ Latin trade.
## 3 $\mu=3$
In this section we apply the above constructions to establish the existence of
$3$–way $k$–homogeneous Latin trades for specific values of $k$, and when $m$
is a multiple of $5$ or $7$. We also show that there is no $(3,4,6)$ Latin
trade.
### 3.1 Small even $k$
###### Proposition 1
There exists a $(3,4,m)\mbox{ Latin trade}$ for every $m\geq 4$, except
possibly for $m=6,7$ and $11$.
###### Proof.
By Lemma 1 and Theorem 3 there exist a $(3,4,4)\mbox{ Latin trade}$ and a
$(3,4,5)$ Latin trade, respectively. Since $8=2\times 4,\ 9=4+5,\ 10=2\times
5,\ 12=3\times 4,\ 13=2\times 4+5,\ 14=4+2\times 5,$ and $15=3\times 5$;
Theorem 7 results that there exist $(3,4,m)\mbox{ Latin trade}$s for
$m=8,9,10,12,13,14$, and $15$. Since the number ${\rm MOLS}(5)=4$, then by
Corollary 3 there exists a $(3,4,m)$ Latin trade, for every $m\geq 16$.
###### Proposition 2
There is no $(3,4,6)$ Latin trade.
###### Proof.
By contradiction. Suppose $T=(T_{1},T_{2},T_{3})$ is a $(3,4,6)\mbox{ Latin
trade}$. By applying some permutations on rows and columns, if necessary, we
may assume that all cells containing the element 1 form a $4\times 4$ array
minus a transversal $\tau$, which will be labeled $L$. For example in Figure 3
one of the possible positions of 1 is shown. Note that there are 12 cells in
$L$ each of which has a 1 in one of the $T_{i}$’s. In what follows the
argument is based only on the assumption that in each of those cells there
exists one 1 from one of the $T_{i}$’s. ($\bullet$ means the cell is empty.)
${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | |
---|---|---|---|---|---
${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | |
${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | |
${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | |
| | | | |
| | | | |
Figure 3: Positions of 1 in $T=(T_{1},T_{2},T_{3})$
In the first stage we show that the cells of $\tau$ in $T$ are empty. Suppose
without loss of generality the cell $T_{14}$ in $\tau$ is not empty. Then
$T_{54}$ and $T_{64}$ must be empty. Thus at least 4 cells of
$\\{T_{51},T_{52},T_{53},T_{61},T_{62},T_{63}\\}$ must be filled. Then by
pigeonhole principal there exists a column in $T$ with at least 5 filled
cells, a contradiction. So all cells of $\tau$ are empty. Therefore exactly 4
cells of $\\{T_{51},T_{52},T_{53},T_{54},T_{61},T_{62},T_{63},T_{64}\\}$ are
filled, and from $T$ being $4$-homogeneous all the cells:
$\\{T_{55},T_{56},T_{65},T_{66}\\}$ are filled.
In the second stage we show that no element, other than 1, appears more than
two times in any row or in any column of $L$. For example let us denote by
$\\{1,x,y,z\\}$, the elements which appear in the first row and without loss
of generality $T_{15}$ is another filled cell of that row. In contrary, assume
that $x$ appears three times in the first row of $L$, i.e. in the cells
$T_{11},T_{12},$ and $T_{13}$. This leaves only two elements $y$ and $z$ to
appear in $T_{15}$, which is a contradiction for $T$ being a $3$-way Latin
trade. So each of the elements other than 1, either does not appear in a row
of $L$ or it appears exactly two times in a row of $L$. Now each element other
than 1 if it appears in $L$, it occupies 4, 6, or 8 cells.
In the third stage we show that no element occupies 6 or 8 cells of $L$. If an
element, say $u\neq 1$ appears 8 times in $L$, then since $u$ appears 2 times
in each row and in each column of $L$, so it appears once in each row of the
$[1,\ldots,4]\times[5,6]$ block. This means that $u$ appears at least 16 times
in $T$, which is a contradiction. If $u\neq 1$ appears 6 times in $L$ then
three rows and three columns of $L$ each contains $u$ twice. So without loss
of generality one of the following cases happens.
${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | u | ${{}_{{}_{\displaystyle\bullet}}}$
---|---|---|---|---|---
${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | u | ${{}_{{}_{\displaystyle\bullet}}}$
${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | u | ${{}_{{}_{\displaystyle\bullet}}}$
${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | |
| | | | |
| | | | |
${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | u | ${{}_{{}_{\displaystyle\bullet}}}$
---|---|---|---|---|---
${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | u | ${{}_{{}_{\displaystyle\bullet}}}$
${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | u
${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | |
| | | | |
| | | | |
($a$) ($b$)
Figure 4: Positions of u in the fifth and sixth columns of $T$
In case ($a$) the fifth column has at least 5 filled cells which is a
contradiction. In case ($b$) there are five columns of $T$ which have $u$ and
since each column containing an $u$ will contain 3 of them, so there are at
least 15 cells containing $u$ in $T$, which is a contradiction.
Now we have shown that each $u\neq 1$ if it appears in $L$, it appears exactly
4 times. The array $L$ has exactly $36-12=24$ places for elements different
from 1 to occupy while the 5 other elements can fill at most $5\times 4=20$
places, which is a contradiction.
###### Proposition 3
There is no $(3,4,7)$ Latin trade.
###### Proof.
By contradiction. Suppose $T=(T_{1},T_{2},T_{3})$ is a $(3,4,7)\mbox{ Latin
trade}$. By applying some permutations on rows and columns, if necessary, we
may assume that all cells containing the element 1 form a $4\times 4$ array
minus a transversal $\tau$, which will be labeled $L$. For example in Figure 5
one of the possible positions of 1 is shown. Note that there are 12 cells in
$L$ each of which has a 1 in one of the $T_{i}$’s. In what follows the
argument is based only on the assumption that in each of those cells there
exists one 1 from one of the $T_{i}$’s. ($\bullet$ means the cell is empty.)
${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | | |
---|---|---|---|---|---|---
${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | | |
${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | | |
${{}_{{}_{\displaystyle\bullet}}}$ | ${1_{{\bf\displaystyle.}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle 1}_{\displaystyle.}}}$ | ${._{{\bf\displaystyle.}_{\displaystyle 1}}}$ | | |
| | | | | |
| | | | | |
| | | | | |
Figure 5: Positions of 1 in $T=(T_{1},T_{2},T_{3})$
If we focus on the placement of the remaining filled cells in $T$, we see that
rows $1$ to $4$ of $T$ each have one additional filled cell in one of columns
$5,6$ or $7$. Likewise for columns $1$ to $4$ of rows $5,6$ or $7$. Further,
the subsquare defined by the intersection of rows $5,6,$ and $7$ with columns
$5,6,$ and $7$, can have at most three filled cells in any row or column.
Hence it follows that without loss of generality columns $5$ has two filled
cell in rows $1$ to $4$ (similarly row $5$ has two filled cells in columns $1$
to $4$) and columns $6$ and $7$ have one filled cell in rows $1$ to $4$
(similarly rows $6$ and $7$ have one filled cell in columns $1$ to $4$). Thus
we may assume cell $(5,5)$ is empty and one possible distribution of empty
cells (one out of $36$) is:
| | | ${{}_{{}_{\displaystyle\bullet}}}$ | | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$
---|---|---|---|---|---|---
| | ${{}_{{}_{\displaystyle\bullet}}}$ | | | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$
| ${{}_{{}_{\displaystyle\bullet}}}$ | | | ${{}_{{}_{\displaystyle\bullet}}}$ | | ${{}_{{}_{\displaystyle\bullet}}}$
${{}_{{}_{\displaystyle\bullet}}}$ | | | | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ |
| | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | |
${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | | | |
${{}_{{}_{\displaystyle\bullet}}}$ | ${{}_{{}_{\displaystyle\bullet}}}$ | | ${{}_{{}_{\displaystyle\bullet}}}$ | | |
We can assume that the cell $T_{15}$ contains symbols $2,3,4$. Then the first
row must contain only symbols $1,2,3,4,$ and these are distributed among the
four filled cells (in the first row) according to one of three possible ways:
123, 124, 134, 234 (or 123, 134, 124, 234)
124, 134, 123, 234 (or 124, 123, 134, 234)
134, 124, 123, 234 (or 134, 123, 124, 234)
The idea is to label the filled columns with one of these configurations, to
label the first row 1234, and then attempt to complete the labeling of the
rows and columns as follows:
$\bullet$ each row and column is labeled by 4 elements from
$\\{1,\ldots,7\\}$,
$\bullet$ the first 4 rows and first 4 columns contain 1 in its label,
$\bullet$ first row is labeled $\\{1,2,3,4\\}$,
$\bullet$ columns with filled cells in the first row are filled as above,
$\bullet$ for any $i$, the number $i$ appears in precisely 4 row labels and in
precisely 4 column labels,
$\bullet$ if the cell $T_{ij}$ is filled, $A$ is the label of row $i$ and $B$
is the label of row $j$, then $|A\cup B|\leq 5$ (because the cell $T_{ij}$
contains three elements of $A\cap B$).
By applying a depth-first search, we found no solutions (Indeed, we tried all
36 distributions of filled cells and all three configurations in the first
row). The search takes a minute with no optimization. So, it is already
impossible to distribute elements in rows and columns according to the
restrictions of the $(3,4,7)\mbox{ Latin trade}$ disregarding how the cell
symbols are distributed among the three components of the purported Latin
trade. Therefore, there is no $(3,4,7)\mbox{ Latin trade}$.
At this point we will show the existence of some $(3,k,m)\mbox{ Latin
trade}$s. For this purpose we will need some small cases. We have found base
rows of those Latin trades computationally, sometimes by trial and errors. But
we have checked all of them by Algorithm 1.
###### Theorem 9
If $k=6,8,10$ and $12$ then there exists a $(3,k,m)\mbox{ Latin trade}$ for
every $m\geq k$.
###### Proof.
We will show for the given $k$, there exist $(\mu,k,l)\mbox{ Latin trade}$s
for $l$, where ${k+1}\leq l\leq 2k-1$. Then by Theorem 8, we will get all
$m\geq k$ where $k=6,8,10$, and $12$.
* •
$k=6$.
If $8\leq m=2l\leq 10$, by Corollary 2 a $(3,6,m)\mbox{ Latin trade}$ exists.
And the following are the base rows of a $(3,6,m)\mbox{ Latin trade}$ for
$m=7,9,11$:
$3$–$B_{7}^{6}=\\{(1,5,4)_{1},(3,4,2)_{2},(5,3,1)_{3},(7,2,5)_{4},(2,1,7)_{5},(4,7,3)_{6}\\}$,
$3$–$B_{9}^{6}=\\{(1,8,3)_{1},(3,2,1)_{2},(2,5,6)_{3},(6,3,2)_{4},(8,6,5)_{5},(5,1,8)_{7}\\}$,
$3$–$B_{11}^{6}=\\{(1,6,3)_{1},(3,2,7)_{2},(6,4,1)_{3},(2,7,4)_{4},(7,3,6)_{5},(4,1,2)_{10}\\}.$
* •
$k=8$.
If $10\leq m=2l\leq 14$, by Corollary 2 a $(3,8,m)\mbox{ Latin trade}$ exists.
And the following are the base rows of a $(3,8,m)\mbox{ Latin trade}$ for
$m=9,11,13,15$:
$3$–$B_{9}^{8}=\\{(1,8,7)_{1},(3,2,9)_{2},(2,4,3)_{3},(7,1,6)_{4},(9,7,4)_{5},(8,9,1)_{6},(4,6,8)_{7},$
$(6,3,2)_{8}\\}$,
$3$–$B_{11}^{8}=\\{(1,5,4)_{1},(3,2,11)_{2},(2,4,5)_{3},(6,1,3)_{4},(8,3,2)_{5},(4,8,6)_{6},(11,6,8)_{7},$
$(5,11,1)_{8}\\}$,
$3$–$B_{13}^{8}=\\{(1,5,3)_{1},(3,1,5)_{2},(2,6,11)_{3},(6,4,2)_{4},(8,3,4)_{5},(4,8,6)_{6},(11,2,8)_{7},$
$(5,11,1)_{10}\\}$,
$3$–$B_{15}^{8}=\\{(1,11,4)_{1},(3,2,6)_{2},(2,4,3)_{3},(6,7,2)_{4},(8,3,7)_{5},(4,8,1)_{6},(11,6,8)_{7},$
$(7,1,11)_{12}\\}.$
* •
$k=10$.
If $12\leq m=2l\leq 18$, by Corollary 2 a $(3,10,m)\mbox{ Latin trade}$
exists.
And the following are the base rows of a $(3,10,m)\mbox{ Latin trade}$ for
$m=13,15,17,19$:
$3$–$B_{13}^{10}=\\{(1,11,6)_{1},(3,2,13)_{2},(2,4,3)_{3},(6,8,7)_{4},(8,7,4)_{5},(4,5,2)_{6},(11,3,8)_{7},$
$(13,6,5)_{8},(5,1,11)_{9},(7,13,1)_{10}\\}$,
$3$–$B_{15}^{10}=\\{(1,6,5)_{1},(3,2,4)_{2},(2,4,14)_{3},(6,8,3)_{4},(8,1,2)_{5},(4,3,6)_{6},(11,5,8)_{7},$
$(5,7,11)_{8},(14,11,7)_{9},(7,14,1)_{11}\\}$,
$3$–$B_{17}^{10}=\\{(1,6,4)_{1},(3,2,6)_{2},(2,7,14)_{3},(6,1,2)_{4},(8,4,5)_{5},(4,8,3)_{6},(11,5,8)_{7},$
$(5,11,7)_{8},(14,3,11)_{9},(7,14,1)_{13}\\}$,
$3$–$B_{19}^{10}=\\{(1,6,2)_{1},(3,2,6)_{2},(2,4,14)_{3},(6,8,7)_{4},(8,7,3)_{5},(4,3,5)_{6},(11,5,4)_{7},$
$(5,11,8)_{8},(14,1,11)_{9},(7,14,1)_{15}\\}.$
* •
$k=12$.
If $14\leq m=2l\leq 22$ or $m=15,21$, by Corollary 2 a $(3,12,m)\mbox{ Latin
trade}$ exists.
And the following are the base rows of a $(3,12,m)\mbox{ Latin trade}$ for
$m=17,19,23$:
$3$–$B_{17}^{12}=\\{(1,16,4)_{1},(3,7,2)_{2},(2,4,14)_{3},(6,8,3)_{4},(8,5,11)_{5},(4,3,10)_{6},(11,1,8)_{7}$,
$(5,14,6)_{8},(14,11,5)_{9},(16,6,7)_{10},(7,10,16)_{11},(10,2,1)_{16}\\}$,
$3$–$B_{19}^{12}=\\{(1,16,7)_{1},(3,2,6)_{2},(2,4,3)_{3},(6,9,1)_{4},(8,7,4)_{5},(4,3,11)_{6},(11,5,2)_{7},$
$(5,11,9)_{8},(14,8,5)_{9},(16,14,8)_{10},(7,6,14)_{11},(9,1,16)_{14}\\}$,
$3$–$B_{23}^{12}=\\{(1,7,5)_{1},(3,2,8)_{2},(2,4,1)_{3},(6,9,3)_{4},(8,1,7)_{5},(4,3,9)_{6},(11,5,2)_{7},$
$(5,11,4)_{8},(14,8,6)_{9},(16,14,11)_{10},(7,6,16)_{11},(9,16,14)_{14}\\}.$
### 3.2 Small odd $k$
###### Proposition 4
There exists a $(3,5,m)\mbox{ Latin trade}$ for every $m\geq 5$, except
possibly $m=6$.
###### Proof.
By Lemma 1 there exists a $(3,5,5)\mbox{ Latin trade}$. The following are the
base rows of a $(3,5,m)\mbox{ Latin trade}$ for $m=7,8,9,11$:
$3$–$B_{7}^{5}=\\{(1,3,2)_{1},(3,2,5)_{2},(5,7,3)_{3},(7,5,1)_{4},(2,1,7)_{5}\\}$,
$3$–$B_{8}^{5}=\\{(1,6,2)_{1},(3,2,4)_{2},(2,4,3)_{3},(6,3,1)_{4},(4,1,6)_{7}\\}$,
$3$–$B_{9}^{5}=\\{(1,4,3)_{1},(4,3,8)_{2},(7,1,4)_{4},(3,8,7)_{6},(8,7,1)_{7}\\}$,
$3$–$B_{11}^{5}=\\{(1,6,9)_{1},(9,2,11)_{5},(11,1,6)_{6},(2,11,1)_{7},(6,9,2)_{9}\\}.$
By Theorem 7, a $(3,5,10)\mbox{ Latin trade}$ exists. So a $(3,5,m)\mbox{
Latin trade}$ exists for 5 consecutive values $m\in\\{7,8,\ldots,11\\}$. Thus
a $(3,5,m)\mbox{ Latin trade}$ exists for all $m\geq 7$ by Theorem 7.
###### Theorem 10
If $k=7,9,11$ and $13$ then there exists a $(3,k,m)\mbox{ Latin trade}$ for
every $m\geq k$.
###### Proof.
We introduce the following base rows:
* •
$k=7$.
* $m\geq 8$:
$3$–$B_{m}^{7}=\\{(1,4,2)_{1},(3,1,4)_{2},(2,3,6)_{3},(6,5,1)_{4},(8,2,3)_{5},(4,8,5)_{6},(5,6,8)_{8}\\}.$
* •
$k=9$.
* $m=10$:
$3$–$B_{10}^{9}=\\{(1,7,9)_{1},(3,2,8)_{2},(2,4,5)_{3},(7,6,4)_{4},(9,3,2)_{5},(8,9,7)_{6},(4,1,6)_{7},$
$(6,5,1)_{8},(5,8,3)_{9}\\}.$
* $m\geq 11$:
$3$–$B_{m}^{9}=\\{(1,5,4)_{1},(3,4,6)_{2},(2,3,1)_{3},(6,2,5)_{4},(8,1,2)_{5},(4,7,8)_{6},(11,6,3)_{7},$
$(5,11,7)_{8},(7,8,11)_{11}\\}.$
* •
$k=11$.
* $m\geq 11$:
$3$–$B_{m}^{11}=\\{(6,1,2)_{1},(1,7,4)_{2},(7,2,1)_{3},(2,8,7)_{4},(8,3,10)_{5},(3,9,5)_{6},(9,4,11)_{7},$
$(4,10,3)_{8},(10,5,9)_{9},(5,11,6)_{10},(11,6,8)_{11}\\}.$
* •
$k=13$.
* $m\geq 13$:
$3$–$B_{m}^{13}=\\{(7,1,2)_{1},(1,8,4)_{2},(8,2,1)_{3},(2,9,3)_{4},(9,3,8)_{5},(3,10,11)_{6},(10,4,13)_{7},$
$(4,11,12)_{8},(11,5,6)_{9},(5,12,10)_{10},(12,6,5)_{11},(6,13,7)_{12},(13,7,9)_{13}\\}.$
###### Theorem 11
If $k=15$ and $m\geq 15$ then there exists a $(3,15,m)\mbox{ Latin trade}$.
###### Proof.
By Lemma 1, Theorem 3 and Corollary 2, we have a $(3,15,m)\mbox{ Latin trade}$
for $m=15,16,18$ and $20$. The following is a base row of a $(3,15,m)\mbox{
Latin trade}$ for $m\geq 21$:
$3$–$B_{m}^{15}=\\{(1,5,4)_{1},(3,1,2)_{2},(2,9,11)_{3},(6,11,3)_{4},(8,6,7)_{5},(4,14,10)_{6},(11,4,8)_{7},$
$(5,3,6)_{8},(14,7,5)_{9},(16,10,1)_{10},(7,2,16)_{11},(19,8,9)_{12},(21,16,19)_{13},$
$(9,19,21)_{14},(10,21,14)_{19}\\}.$
The following are the base rows of a $(3,15,m)\mbox{ Latin trade}$ for
$m=17,19$:
$3$–$B_{17}^{15}=\\{(5,2,12)_{3},(7,15,11)_{4},(9,17,4)_{5},(11,13,14)_{6},(13,16,5)_{7},(15,11,13)_{8},$
$(17,14,6)_{9},(2,12,16)_{10},(4,9,7)_{11},(6,8,15)_{12},(8,10,17)_{13},$
$(10,7,8)_{14},(12,6,10)_{15},(14,5,9)_{16},(16,4,2)_{17}\\},$
$3$–$B_{19}^{15}=\\{(1,2,11)_{1},(3,4,2)_{2},(5,17,4)_{3},(7,10,9)_{4},(9,15,14)_{5},(11,9,13)_{6},$
$(13,19,10)_{7},(15,13,5)_{8},(17,6,1)_{9},(19,14,3)_{10},(2,11,15)_{11},$
$(4,1,7)_{12},(6,3,19)_{13},(10,7,17)_{15},(14,5,6)_{17}\\}.$
### 3.3 General cases
###### Theorem 12
Let $m\equiv 1({\rm mod\ 6})$ and $m\geq 7$. Then there exists a $(3,m-2,m)$
Latin trade.
###### Proof.
The following is a base row of a $(2,m-2,m)\mbox{ Latin trade}$:
$2$–$B_{m}^{m-2}=\bigcup_{i=0}^{(m-13)/6}\\{(6i+2,6i+3)_{3i+1},(6i+4,6i+2)_{3i+2},(6i+3,6i+4)_{3i+3},$
$(6i+5,6i+6)_{(m+3)/2+3i+1},(6i+7,6i+5)_{(m+3)/2+3i+2},(6i+6,6i+7)_{(m+3)/2+3i+3}\\}$
$\bigcup\\{(m-5,m-4)_{(m-7)/2+1},(m-2,m-5)_{(m-7)/2+2},(m-4,m)_{(m-7)/2+3},$
$(1,m-2)_{(m-7)/2+4},(m,1)_{(m-7)/2+5}\\}.$
Now, for $1\leq i\leq m-2$ we put $2i-1({\rm mod\ m})$ in $i$-th cell of
$2$–$B_{m}^{m-2}$, as a result we obtain a base row of a $(3,m-2,m)\mbox{
Latin trade}$.
###### Example 1
As an example of the previous theorem, the following is a base row of a
$(3,11,13)\mbox{ Latin trade}$:
$3$–$B_{13}^{11}=\\{(1,2,3)_{1},(3,4,2)_{2},(5,3,4)_{3},(7,8,9)_{4},(9,11,8)_{5},(11,9,13)_{6},(13,1,11)_{7},$
$(2,13,1)_{8},(4,5,6)_{9},(6,7,5)_{10},(8,6,7)_{11}\\}.$
###### Theorem 13
For every $m=5l$ and $4\leq k\leq m$, $l\neq 6$, there exists a $(3,k,m)$
Latin trade.
###### Proof.
The theorem trivially holds for $l=1$. If $l=2$, then by Theorem 3, Theorem 7,
Theorem 10 and Theorem 9, we can construct a $(3,k,10)\mbox{ Latin trade}$ for
every $4\leq k\leq 10$. By Theorems 9 and 10 there exists a $(3,k,m)\mbox{
Latin trade}$ for $k=6,7$ and $11$, so suppose that $k\neq 6,7$ and $11$.
We may also assume that $m>k$.
We have the following cases to consider, each case follows from Theorem 2:
* •
$k=5l^{{}^{\prime}}$.
We set $k_{i}=5$ for $1\leq i\leq l^{{}^{\prime}}$ and $k_{i}=0$ for
$l^{{}^{\prime}}+1\leq i\leq l$ and $p=5$.
* •
$k=5l^{{}^{\prime}}+1$.
We set $k_{i}=5$ for $1\leq i\leq l^{{}^{\prime}}-3$ and $k_{i}=4$ for
$l^{{}^{\prime}}-2\leq i\leq l^{{}^{\prime}}+1$ and $k_{i}=0$ for
$l^{{}^{\prime}}+2\leq i\leq l$ and $p=5$.
* •
$k=5l^{{}^{\prime}}+2$.
We set $k_{i}=5$ for $1\leq i\leq l^{{}^{\prime}}-2$ and $k_{i}=4$ for
$l^{{}^{\prime}}-1\leq i\leq l^{{}^{\prime}}+1$ and $k_{i}=0$ for
$l^{{}^{\prime}}+2\leq i\leq l$ and $p=5$.
* •
$k=5l^{{}^{\prime}}+3$.
We set $k_{i}=5$ for $1\leq i\leq l^{{}^{\prime}}-1$,
$k_{l^{{}^{\prime}}}=k_{l^{{}^{\prime}}+1}=4$, and $k_{i}=0$ for
$l^{{}^{\prime}}+2\leq i\leq l$, and $p=5$.
* •
$k=5l^{{}^{\prime}}+4$.
We set $k_{i}=5$ for $1\leq i\leq l^{{}^{\prime}}$, $k_{l^{{}^{\prime}}+1}=4$,
and $k_{i}=0$ for $l^{{}^{\prime}}+2\leq i\leq l$, and $p=5$.
###### Theorem 14
For every $m=7l$ and $5\leq k\leq m$, $l\neq 6$, there exists a $(3,k,m)$
Latin trade.
###### Proof.
The theorem trivially holds for $l=1$. If $l=2$, then by Theorem 7, Theorem 10
and Theorem 9, we can construct a $(3,k,14)\mbox{ Latin trade}$ for every
$5\leq k\leq 14$. For $l\neq 2,6$ by Theorems 9 and 10 there exists a
$(3,k,m)\mbox{ Latin trade}$ for $k=8,9$, so suppose that $k\neq 8,9$.
We may also assume that $m>k$.
We have the following cases to consider, each case follows from Theorem 2:
* •
$k=7l^{{}^{\prime}}$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}$ and $k_{i}=0$ for
$l^{{}^{\prime}}+1\leq i\leq l$ and $p=7$.
* •
$k=7l^{{}^{\prime}}+1$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}-2$ and $k_{i}=5$ for
$l^{{}^{\prime}}-1\leq i\leq l^{{}^{\prime}}+1$ and $k_{i}=0$ for
$l^{{}^{\prime}}+2\leq i\leq l$ and $p=7$.
* •
$k=7l^{{}^{\prime}}+2$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}-2$ and
$k_{l^{{}^{\prime}}-1}=k_{l^{{}^{\prime}}}=5$, $k_{l^{{}^{\prime}}+1}=6$ and
$k_{i}=0$ for $l^{{}^{\prime}}+2\leq i\leq l$ and $p=7$.
* •
$k=7l^{{}^{\prime}}+3$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}-1$,
$k_{l^{{}^{\prime}}}=k_{l^{{}^{\prime}}+1}=5$, and $k_{i}=0$ for
$l^{{}^{\prime}}+2\leq i\leq l$, and $p=7$.
* •
$k=7l^{{}^{\prime}}+4$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}-1$, $k_{l^{{}^{\prime}}}=6$,
$k_{l^{{}^{\prime}}+1}=5$ and $k_{i}=0$ for $l^{{}^{\prime}}+2\leq i\leq l$,
and $p=7$.
* •
$k=7l^{{}^{\prime}}+5$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}$, $k_{l^{{}^{\prime}}+1}=5$,
and $k_{i}=0$ for $l^{{}^{\prime}}+2\leq i\leq l$, and $p=7$.
* •
$k=7l^{{}^{\prime}}+6$.
We set $k_{i}=7$ for $1\leq i\leq l^{{}^{\prime}}$, $k_{l^{{}^{\prime}}+1}=6$,
and $k_{i}=0$ for $l^{{}^{\prime}}+2\leq i\leq l$, and $p=7.$
Now by the results given above we have proved Theorem 1, given at the end of
the Introduction.
Acknowledgement. We appreciate very useful comments of anonymous referee, who
also added the proof of Proposition 3. Also we thank to Amir Hooshang
Hosseinpoor and Mayssam Mohammadi Nevisi for their computer programming.
## References
* [1] Behrooz Bagheri Gh. and E. S. Mahmoodian. On the existence of $k$–homogeneous Latin bitrades. Util. Math., 85:333–345, 2011.
* [2] Richard Bean, Hoda Bidkhori, Maryam Khosravi, and E. S. Mahmoodian. $k$-homogeneous Latin trades. Bayreuth. Math. Schr., 74:7–18, 2005.
* [3] Elizabeth J. Billington. Combinatorial trades: a survey of recent results. In Designs, 2002, volume 563 of Math. Appl., pages 47–67. Kluwer Acad. Publ., Boston, MA, 2003.
* [4] Nicholas Cavenagh, Diane Donovan, and Aleš Drápal. 3-homogeneous Latin trades. Discrete Math., 300(1-3):57–70, 2005.
* [5] Nicholas Cavenagh, Diane Donovan, and Aleš Drápal. 4-homogeneous Latin trades. Australas. J. Combin., 32:285–303, 2005.
* [6] Nicholas J. Cavenagh. The theory and application of Latin bitrades: a survey. Math. Slovaca, 58(6):691–718, 2008.
* [7] Nicholas J. Cavenagh and Ian M. Wanless. On the number of transversals in Cayley tables of cyclic groups. Discrete Appl. Math., 158(2):136–146, 2010.
* [8] Charles J. Colbourn and Jeffrey H. Dinitz, editors. Handbook of combinatorial designs. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, second edition, 2007.
* [9] A. D. Keedwell. Critical sets in Latin squares and related matters: an update. Util. Math., 65:97–131, 2004.
* [10] C. C. Lindner and C. A. Rodger. Design theory. CRC Press LLC, 1997.
|
arxiv-papers
| 2012-07-09T07:21:14 |
2024-09-04T02:49:32.799134
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Behrooz Bagheri Gh., Diane Donovan, and E. S. Mahmoodian",
"submitter": "Behrooz Bagheri Ghavam Abadi",
"url": "https://arxiv.org/abs/1207.1969"
}
|
1207.1978
|
# Can contribution from magnetic-penguin operator with real photon to
$B_{s}\to\ell^{+}\ell^{-}\gamma$ in the standard model be neglected?
Wenyu Wang, Zhao-Hua Xiong and Si-Hong Zhou Institute of Theoretical Physics,
College of Applied Science, Beijing University of Technology, Beijing 100124,
China
###### Abstract
Using the $B_{s}$ meson wave function extracted from non-leptonic $B_{s}$
decays, we reevaluate the rare decays
$B_{s}\to\ell^{+}\ell^{-}~{}\gamma,~{}(\ell=e,\mu)$ in the standard model,
including two kinds of contributions from magnetic-penguin operator with
virtual and real photon. We find that the contributions from magnetic-penguin
operator $b\to s\gamma$ with real photon to the exclusive decays, which is
regarded as to be negligible in previous literatures, are large, and the
branchings of $B_{s}\to\ell^{+}\ell^{-}\gamma$ are nearly enhanced by a factor
2. With the predicted branching ratios at order of $10^{-8}$, it is expected
that the radiative dileptonic decays will be detected in the LHC-b and B
factories in near future.
###### pacs:
12.15.Ji, 13.26.He
## I Introduction
The standard model (SM) of electroweak interaction has been remarkably
successful in describing physics below the Fermi scale and is in good
agreement with the most experiment data. Thanks to the efforts of the B
factories and LHC, the exploration of quark-flavor mixing is now entering a
new interesting era. Measurements of rare B mesons decays such as $B\to
X_{s}\gamma$, $B\to X_{s}\ell^{+}\ell^{-}(\ell=e,\mu)$ and
$B_{s}\to\mu^{+}\mu^{-}$, $B_{s}\to\ell^{+}\ell^{-}\gamma$ are likely to
provide sensitive test of the SM. In fact, these decays, induced by the flavor
changing neutral currents which occur in the SM only at loop level, play an
important role in testing higher order effects of SM and in searching for the
physics beyond the SM Aliev97 ; Xiong01 . Nevertheless, these processes are
also important in determining the parameters of the SM and some hadronic
parameters in QCD, such as the CKM matrix elements, the meson decay constant
$f_{B_{s}}$, providing information on heavy meson wave functions li .
The rare B inclusive radiative decays $B\to X_{s}\gamma$ and $B\to
X_{s}\ell^{+}\ell^{-}(\ell=e,\mu)$ as well as the exclusive decays
$B_{s}\to\mu^{+}\mu^{-}$ have been studied extremely at the leading logarithm
order BLOSM and high order in the SM BHOSM and various new physics models.
In previous works, prediction for the exclusive decays
$B_{s}\to\ell^{+}\ell^{-}\gamma$ have been carried out by using the light cone
sum rule Aliev97 ; Xiong01 , the simple constituent quark model Eilam97 , and
the B meson distribution amplitude extracted from non-leptonic B decays LU06 .
long distance QCD effects describing the neutral vector-meson resonances
$\phi$ and $J/\Psi$ family have received special attention in Melikhov04 ;
Kruger03 ; Nikitin11 . At parton level, $B_{s}\to\ell^{+}\ell^{-}\gamma$
decays have been thought to be obtained from decay $b\to
s\ell^{+}\ell^{-}\gamma$, and further, from $b\to s\ell^{+}\ell^{-}$ directly.
To achieve this, a necessary work is attaching real photon to any charged
internal and external lines in the Feynman diagrams of $b\to
s\ell^{+}\ell^{-}$ with two statements: i) Contributions from the attachment
of photon to any charged internal propagator are regraded as to be strongly
suppressed and can be neglected safely Aliev97 ; Xiong01 ; LU06 ; Eilam97 ;
ii) Contributions from the attachment of real photon with magnetic-penguin
vertex to any charged external lines are always neglected Aliev97 ; Xiong01
or stated to be negligibly small LU06 . Here we would like to address that the
conclusion of the first statement is correct, but the explanation is not as
what it is described Dong . The second statement seems to be questionable, for
that the pole of propagator of the charged line attached by photon may enhance
the decay rate greatly which make some diagrams can not be neglected in the
calculation. Since the weak radiative B-meson decay is well known to be a
sensitive probe of new physics, it is essential to calculate the Standard
Model value of its branching ratio as precisely as possible. Although the
second contribution has been calculated in Ref. Melikhov04 , it mainly
concentrated on the long distance effects of the meson resonances, whereas the
short distance contribution which was incompletely analyzed.
In this letter, we will concentrate on the short distance contribution to
$B_{s}\to\ell^{+}\ell^{-}\gamma$ and check whether the contribution from
magnetic-penguin operator with real photon to $B_{s}\to\ell^{+}\ell^{-}$ is
negligible or not, and give some remarks including a comparison with other
works. The paper is organized as follows. In sec. II, we present the detailed
calculation of exclusive decays $B_{s}\to\ell^{+}\ell^{-}\gamma$, including
full contribution from magnetic-penguin operator with real photon. Sec. III
contains the numerical results and comparison with previous works, and the
conclusions are given in sec. IV.
## II The calculation
In order to simplify the decay amplitude for $B_{s}\to\ell^{+}\ell^{-}\gamma$,
we have to utilize the $B_{s}$ meson wave function, which is not known from
the first principal. Fortunately, many studies on non-leptonic $B$ bdecay ;
cdepjc24121 and $B_{s}$ decays bs have constrained the wave function
strictly. It was found that the wave function has form
$\Phi_{B_{s}}=(\not\\!p_{B_{s}}+m_{B_{s}})\gamma_{5}~{}\phi_{B_{s}}({x}),$ (1)
where the distribution amplitude $\phi_{B_{s}}(x)$ can be expressed as form :
$\phi_{B_{s}}(x)=N_{B_{s}}x^{2}(1-x)^{2}\exp\left(-\frac{m_{B}^{2}\
x^{2}}{2\omega_{b_{s}}^{2}}\right)$ (2)
with $x$ being the momentum fractions shared by $s$ quark in $B_{s}$ meson.
The normalization constant $N_{B_{s}}$ can be determined by comparing
$\displaystyle\langle
0\left|\bar{s}\gamma^{\mu}\gamma_{5}b\right|B_{s}\rangle=i\int_{0}^{1}\phi_{B_{s}}(x)dx{\rm
Tr}\left[\gamma^{\mu}\gamma_{5}(\not\\!p_{B_{s}}+m_{B_{s}})\gamma_{5}\right]dx=-4ip_{B_{s}}^{\mu}\int_{0}^{1}\phi_{B_{s}}(x)dx$
(3)
with
$\displaystyle\langle
0\left|\bar{s}\gamma^{\mu}\gamma_{5}b\right|B_{s}\rangle=-if_{B_{s}}p_{B_{s}}^{\mu},$
(4)
the $B$ meson decay constant $f_{B_{s}}$ is thus determined by the condition
$\displaystyle\int_{0}^{1}\phi_{B_{s}}(x)dx=\frac{1}{4}f_{B_{s}}.$ (5)
Let us start with the quark level processes $b\to s\ell^{+}\ell^{-}$. They are
subject to the QCD corrected effective weak Hamiltonian, obtained by
integrating out heavy particles, i.e., top quark, higgs, and $W^{\pm},\ Z$
bosons:
$\displaystyle{\cal H}_{eff}(b\to s\ell^{+}\ell^{-})$ $\displaystyle=$
$\displaystyle-\frac{\alpha_{em}{G_{F}}}{\sqrt{2}\pi}V_{tb}V^{*}_{ts}\left\\{\left[-\frac{2{C^{eff}_{7}}{m_{b}}}{q^{2}}\bar{s}i\sigma^{\mu\nu}q_{\nu}P_{R}b+C^{eff}_{9}\bar{s}\gamma^{\mu}{P_{L}}b\right]\bar{\ell}\gamma_{\mu}{\ell}\right.$
(6)
$\displaystyle\left.+C_{10}(\bar{s}\gamma^{\mu}{P_{L}}b)~{}\bar{\ell}\gamma_{\mu}\gamma_{5}{\ell}\right\\},$
where $P_{L,R}=(1\mp\gamma_{5})/{2}$, $q^{2}$ is the dilepton invariant mass
squared. The QCD corrected Wilson coefficients $C_{7}^{eff}$, $C^{eff}_{9}$
and $C_{10}$ at $\mu=m_{b}$ scale can be found in Ref. Misiak93 .
If an additional photon line is attached to any of the charged lines in
diagrams contributing to the Hamiltonian above, we will have the radiative
leptonic decays $b\to s\ell^{+}\ell^{-}\gamma$. Therefore, there are two kinds
of diagrams: photon connecting to the internal propagators, and photon
connecting to the external line. As addressed in the introduction, the
contribution from the first kind of diagrams is neglected safely. Now we will
only consider the second category of diagrams which are displayed in Fig. 1.
Figure 1: Feynman diagrams for $B_{q}\to\ell^{+}\ell^{-}\gamma~{}(q=d,s)$ in
the SM. The black dot in (a), (c) and (e), (f) stands for the magnetic-penguin
operator $O_{7}$ with virtual and real photon, respectively, and the black dot
in (b),(d),(g) and (f) denotes operators $O_{9}$ and $O_{10}$.
At first, we recalculate the diagrams (a)-(d) in Fig. 1 with photon emitted
from the external quark lines $b$ or $s$ by using the $B_{s}$ meson wave
function extracted from non-leptonic $B_{s}$ decays. Note that these diagrams
are already studied in previous literatures and considered as giving the
dominant contribution to $B_{s}\to\ell^{+}\ell^{-}\gamma$. At parton level,
the amplitudes for transition $b\to s\ell^{+}\ell^{-}\gamma$ can be calculated
directly using from the Hamiltonian of $b\to s\ell^{+}\ell^{-}$ in (6). For
example, contribution from the magnetic-penguin operator with virtual photon
shown in Fig. 1 (a) reads:
$\displaystyle{\cal A}_{a}$ $\displaystyle=$ $\displaystyle-
iGee_{d}\frac{m_{b}}{q^{2}}C_{7}^{eff}\left(\bar{s}\left[\frac{2p_{s}\cdot\epsilon+\not\\!\epsilon\not\\!k}{p_{s}\cdot
k}\gamma^{\mu}\not\\!qP_{R}\right]b\right)\left[\bar{\ell}\gamma_{\mu}\ell\right],$
(7)
where $p_{b,s},k$ denotes the momentum of quarks and photon respectively,
$\epsilon$ is the vector polarization of photon and
$G=\alpha_{em}G_{F}V_{tb}V_{ts}^{*}/(\sqrt{2}\pi)$, $e_{d}=-1/3$ is the number
of electrical charge of the external quarks. In deriving above equation, we
have used motion equation for quarks and
$q^{\mu}\bar{\ell}\gamma_{\mu}\ell=0$. Using Eqs. (1) and (7), we write the
amplitude of $B_{s}\to\ell^{+}\ell^{-}\gamma$ at meson level as:
$\displaystyle A_{a}$ $\displaystyle=$ $\displaystyle
2iGee_{d}\frac{m_{b}m_{B_{s}}}{q^{2}}C_{7}^{eff}\frac{1}{p_{B_{s}}\cdot
k}\int_{0}^{1}\frac{\phi_{B_{s}}(x)}{x}dx\left[k_{\mu}q\cdot\epsilon-\epsilon_{\mu}k\cdot
q-i\epsilon_{\mu\nu\alpha\beta}\epsilon^{\nu}k^{\alpha}q^{\beta}\right]\left[\bar{\ell}\gamma^{\mu}\ell\right],$
(8)
where $x$, $y=1-x$ are the momentum fractions shared by $s$, $b$ quark in
$B_{s}$. By doing the similar calculation to diagrams (b)-(d), the decay
amplitude is then obtained as:
$\displaystyle A_{a+b+c+d}$ $\displaystyle=$ $\displaystyle
iGee_{d}\frac{1}{p_{B_{s}}\cdot
k}\biggl{\\{}\left[C_{1}i\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{B_{s}}\varepsilon^{\beta}{k^{\nu}}+C_{2}p_{B_{s}}^{\nu}(\varepsilon_{\mu}k_{\nu}-k_{\mu}\varepsilon_{\nu})\right]\bar{\ell}\gamma^{\mu}{\ell}$
(9)
$\displaystyle+C_{10}\left[C_{+}i\epsilon_{\alpha\beta\mu\nu}p^{\alpha}_{B_{s}}\varepsilon^{\beta}{k^{\nu}}+C_{-}p_{B_{s}}^{\nu}(\varepsilon_{\mu}k_{\nu}-k_{\mu}\varepsilon_{\nu})\right]\bar{\ell}\gamma^{\mu}\gamma_{5}\ell\biggl{\\}}.$
The form factors in Eq. (9) are found to be:
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
C_{+}\left(C_{9}^{eff}-2\frac{m_{b}m_{B_{s}}}{q^{2}}C_{7}^{eff}\right),$
$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle
C_{9}^{eff}C_{-}-2\frac{m_{b}m_{B_{s}}}{q^{2}}C_{7}^{eff}C_{+},$ (10)
where
$\displaystyle
C_{\pm}=\int_{0}^{1}\left(\frac{1}{x}\pm\frac{1}{y}\right)\phi_{B_{s}}(x)dx.$
(11)
The expression in (9) can be compared with Ref. LU06 .
Now we will focus attention on calculating the diagrams (e) and (f) in Fig. 1
which are always neglected in other works. In these two diagrams, photon of
the magnetic-penguin operator is real, thus its contribution to
$B_{s}\to\ell^{+}\ell^{-}\gamma$ is different from that of magnetic-penguin
operator with virtual photon in diagram (a) and (c). We get the amplitude:
$\displaystyle
A_{e+f}=i2Gee_{d}C_{7}^{eff}\frac{m_{b}m_{B_{s}}}{q^{2}}\frac{1}{p_{B_{s}}\cdot
q}\overline{C}_{+}\left[k_{\mu}q\cdot\epsilon-\epsilon_{\mu}k\cdot
q-i\epsilon_{\mu\nu\alpha\beta}\epsilon^{\nu}k^{\alpha}q^{\beta}\right]\left[\bar{\ell}\gamma^{\mu}\ell\right],$
(12)
with coefficients $\overline{C_{+}}$ obtained by a replacement:
$\displaystyle\overline{C}_{+}$ $\displaystyle=$ $\displaystyle
C_{+}(x\to\bar{x}=x-z-i\epsilon;~{}y\to\bar{y}=y-z-i\epsilon)$ (13)
$\displaystyle=$ $\displaystyle
N_{B}\int_{0}^{1}dx({\frac{1}{x-z-i\epsilon}}+{\frac{1}{1-x-z-i\epsilon}})x^{2}(1-x)^{2}\exp\left[-\frac{m_{B_{s}}^{2}}{2\omega_{B_{s}}^{2}}x^{2}\right],$
where $z=\frac{q^{2}}{2p_{B_{s}}\cdot q}$. Note that pole in
$\overline{C_{+}}$ corresponds to the pole of the quark propagator when it is
connected by the off-shell photon propagator. Thus the $\overline{C_{+}}$ term
may enhance the decay rate of $B_{s}\to\ell^{+}\ell^{-}\gamma$ and its
analytic expression reads
$\displaystyle\overline{C}_{+}$ $\displaystyle=$ $\displaystyle 2N_{B_{s}}\pi
iz^{2}(1-z)^{2}\exp\left[-\frac{m_{B_{s}}^{2}}{2\omega_{B_{s}}^{2}}z^{2}\right]$
(14) $\displaystyle+$ $\displaystyle
N_{B_{s}}\int_{0}^{1}dx({\frac{1}{x+z}}-{\frac{1}{1+x-z}})x^{2}(1+x)^{2}\exp\left[-\frac{m_{B_{s}}^{2}}{2\omega_{B_{s}}^{2}}x^{2}\right]$
$\displaystyle-$ $\displaystyle
N_{B_{s}}\int_{-1}^{1}\left(\frac{1}{\frac{1}{x}-z}+\frac{1}{1-\frac{1}{x}-z}\right)\frac{dx}{x^{4}}(1-\frac{1}{x})^{2}\exp\left[-\frac{m_{B_{s}}^{2}}{2\omega_{B_{s}}^{2}}\frac{1}{x^{2}}\right].$
As the contribution from the Fig.1 (g) and (h) with photon attached to
external lepton lines, considering the fact that (i) being a pseudoscalar
meson, $B_{s}$ meson can only decay through axial current, so the magnetic
penguin operator $O_{7}$ ’s contribution vanishes; (ii) the contribution from
operators $O_{9},\ O_{10}$ has the helicity suppression factor
$m_{\ell}/m_{B_{s}}$, so for light lepton electron and muon, we can neglect
their contribution safely.
The total matrix element for the decay $B_{s}\to\ell^{+}\ell^{-}\gamma$ is
obtained a sum of the $A_{a+b+c+d}$ and $A_{e+f}$. After summing over the
spins of leptons and polarization of the photon, and then performing the phase
space integration over one of the two Dalitz variables, we get the
differential decay width versus the photon energy $E_{\gamma}$,
$\displaystyle\frac{d\Gamma}{dE_{\gamma}}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{3}{G^{2}_{F}}}{108\pi^{4}}|V_{tb}V^{*}_{ts}|^{2}(m_{B_{s}}-2E_{\gamma})E_{\gamma}\left[|\overline{C}_{1}|^{2}+|\overline{C_{2}}|^{2}+C_{10}^{2}(|C_{+}|^{2}+|C_{-}|^{2})\right].$
(15)
The coefficients $\overline{C}_{i}\ (i=1,2)$ can be obtained by a shift:
$\displaystyle\overline{C}_{i}=C_{i}-\frac{2m_{b}m_{B_{s}}}{q^{2}}\frac{p_{B_{s}}\cdot
k}{p_{B_{s}}\cdot q}\overline{C_{+}}C_{7}^{eff}.$ (16)
## III Results and discussions
The decay branching ratios can be easily obtained by integrating over photon
energy. In numerical calculations, we use the following parameters PDG2012 :
$\alpha_{em}=\frac{1}{137},~{}G_{F}=1.166\times 10^{-5}{\rm
GeV}^{-2},~{}m_{b}=4.2{\rm GeV},$
$|V_{tb}|=0.88,~{}|V_{ts}|=0.0387,~{}|V_{td}|=0.0084$ $m_{B_{s}}=5.37{\rm
GeV},~{}\omega_{B_{s}}=0.5,~{}f_{B_{s}}=0.24{\rm
GeV},~{}\tau_{B_{s}}=1.47\times 10^{-12}s.$ $m_{B_{d}^{0}}=5.28{\rm
GeV},~{}\omega_{B_{d}}=0.4,~{}f_{B_{d}}=0.19{\rm
GeV},~{}\tau_{B_{d}}=1.53\times 10^{-12}s.$
The ratios of $B_{s}\to\ell^{+}\ell^{-}\gamma$ with and without the
contribution from magnetic-penguin operator with real photon are shown in
Table 1 together with results of $B_{d,s}\to\ell^{+}\ell^{-}\gamma$ from this
work and other models for comparison. The errors shown in the Table 1 comes
from the heavy meson wave function, by varying the parameter
$\omega_{B_{d}}=0.4\pm 0.1$, and $\omega_{B_{s}}=0.5\pm 0.1$ LU06 . Note that,
the predicted branching ratios receive errors from many parameters, such as
meson decay constant, meson and quark masses etc.
Table 1: Comparison of branching ratios with other model calculations Branching Ratios ($\times 10^{-9}$) | Our Results | Quark Model | light cone
---|---|---|---
| Excluded Fig.1(e),(f) | Included Fig.1(e),(f) | Ref.LU06 | Ref.Eilam97 | Ref.Aliev97
$B_{s}\to\ell^{+}\ell^{-}\gamma$ | $3.74_{-1.00}^{+1.76}$ | $7.45_{-1.82}^{+2.98}$ | 1.90 | 6.20 | 2.35
$B_{d}^{0}\to\ell^{+}\ell^{-}\gamma$ | $0.16_{-0.05}^{+0.11}$ | $0.31_{-0.10}^{+0.20}$ | 0.08 | 0.82 | 0.15
A couple of remarks on the $B_{s}$ rare exclusive radiative decays are
follows:
* 1.
As pointed out in Ref. li ; LU06 , the branching ratios are proportional to
the heavy meson wave function squared, the radiative leptonic decays are very
sensitive probes in extracting the heavy meson wave functions;
* 2.
The contributions from magnetic-penguin operator with real photon to the
exclusive decay is large, and the branching of
$B_{s}\to\ell^{+}\ell^{-}\gamma$ is enhanced nearly by a factor 2 compared
with that only contribution from magnetic-penguin operator with virtual photon
and nearly up to $10^{-8}$, implying the search of
$B_{s}\to\ell^{+}\ell^{-}\gamma$ can be achieved in near future.
* 3.
Due to the large contributions from magnetic-penguin operator with real
photon, the form factors for matrix elements
$\langle\gamma|\bar{s}\gamma^{\mu}(1\pm\gamma_{5})b|B_{s}\rangle$ and
$\langle\gamma|\bar{s}\sigma_{\mu\nu}(1\pm\gamma_{5})q^{\nu}b|B_{s}\rangle$ as
a function of dilepton mass squared $q^{2}$ are not as simple as
$1/(q^{2}-q_{0}^{2})^{2}$ where $q_{0}^{2}$ is constant Eilam95 . The
$B_{s}\to\gamma$ transition form factors predicted in this works have also
some differences from those in Ref. Melikhov04 ; Kruger03 ; Nikitin11 . For
instance, Ref. Kruger03 predicted the form factors $F_{TV}(q^{2},0)$,
$F_{TA}(q^{2},0)$ induced by tensor and pseudotensor currents with emission of
the virtual photon, as shown in diagrams (a) and (c) of FIG. 1, are only equal
at maximum photon energy, whereas the corresponding formula in this work have
the same expression as $-\frac{e_{d}m_{B_{s}}}{p_{B_{s}}\cdot k}C_{+}\propto
1/(q^{2}-q_{0}^{2})$ in Eq. (9). The research of
$B_{s}\to\ell^{+}\ell^{-}\gamma$ may give some hints on these form factors.
At this stage, we think it is necessary to present a few more comments about
the calculation of Ref. Melikhov04 . In order to estimate the contribution of
emission of the real photon from the magnetic-penguin operator, the authors of
Ref. Melikhov04 calculated the form factors $F_{TA,TV}(0,q^{2})$ by including
the short distance contribution in $q^{2}\to 0$ limit and additional long
distance contribution from the resonances of vector mesons such as $\rho^{0}$,
$\omega$ for $B_{d}$ decay and $\phi$ for $B_{s}$ decay. Obviously, this means
the pole mass enhancement of the valence quark were not appropriately taken
into account. Moreover, if $F_{TA,TV}(0,q^{2})=F_{TA,TV}(0,0)$ stands for the
short distance contribution, it seems double counting since in this case
photons which emits from magnetic-penguin vertex and quark lines directly are
not able to be distinguished.
## IV Conclusion
We evaluated the rare decays $B_{s}\to\gamma\ell^{+}\ell^{-}$ in the SM,
including two kinds of contributions from magnetic-penguin operator with
virtual and real photon. In contrast to the previous works which treated
contribution from magnetic-penguins operators with real photon to the decays
as negligible small, we found that the contributions is large, leading to the
branching of $B_{s}\to\ell^{+}\ell^{-}\gamma$ being nearly enhanced by a
factor 2. In the current early phase of the LHC era, the exclusive modes with
muons in the final states are among the most promising decays. The decay
$B_{s}\to\mu^{+}\mu^{-}$ is likely to be confirmed before the end of 2012
Talk11 . Although there are some theoretical challenges in calculation of the
hadronic form factors and non-factorable corrections, with the predicted
branching ratios at order of $10^{-8}$, $B_{s}\to\ell^{+}\ell^{-}\gamma$ can
be expected as the next goal once $B_{s}\to\mu^{+}\mu^{-}$ measurement is
finished since the final states can be identified easily and branching ratios
are large. Our predictions for such processes can be tested in the LHC-b and B
factories in near future.
###### Acknowledgements.
This work was supported in part by the NSFC No. 11005006, 11172008 and Doctor
Foundation of BJUT No. X0006015201102.
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|
arxiv-papers
| 2012-07-09T08:06:46 |
2024-09-04T02:49:32.806811
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wenyu Wang, Zhao-Hua Xiong and Si-Hong Zhou",
"submitter": "Wenyu Wang",
"url": "https://arxiv.org/abs/1207.1978"
}
|
1207.2002
|
2012 Vol. 12 No. 8, 1081–1106
11institutetext: Department of Astronomy and Key Laboratory of Modern
Astronomy and Astrophysics in Ministry of Education, Nanjing
University,Nanjing 210093, China; zhoujl@nju.edu.cn
Received 2012.7.9; accepted 2012.7.17
# Forming Different Planetary Systems
Ji-Lin Zhou Ji-Wei Xie Hui-Gen Liu Hui Zhang Yi-Sui Sun
###### Abstract
With the increasing number of detected exoplanet samples, the statistical
properties of planetary systems have become much clearer. In this review, we
summarize the major statistical results that have been revealed mainly by
radial velocity and transiting observations, and try to interpret them within
the scope of the classical core-accretion scenario of planet formation,
especially in the formation of different orbital architectures for planetary
systems around main sequence stars. Based on the different possible formation
routes for different planet systems, we tentatively classify them into three
major catalogs: hot Jupiter systems, standard systems and distant giant planet
systems. The standard system can be further categorized into three sub-types
under different circumstances: solar-like systems, hot Super-Earth systems,
sub-giant planet systems. We also review the planet detection and formation in
binary systems as well as planets in star clusters.
###### keywords:
Planetary systems: dynamical evolution and stability—formation—planet-disk
interactions—stars: binary: general—clusters: general
INVITED REVIEWS
## 1 Introduction
Since the discovery of a giant planet by radial velocity measurements in 51
Peg by Mayor & Queloz (1995), as well as the planets around 47 UMa and 70 Vir
(Butler & Marcy 1996; Marcy & Butler 1996), the new era of detecting
exoplanets around main sequence stars was opened at the end of last century.
Now exoplanet detection has become one of the most rapidly developed area.
With the development of measurement techniques, the precision of radial
velocity(RV) can be better than 0.5 $ms^{-1}$ with the HARPS spectrograph at
La Silla Observatory (Mayor et al. 2003), making possible to detect Earth-
sized planets in close orbits around bright stars. The detection of transiting
signals when exoplanets pass in front of their host stars has become another
powerful method in searching for planet candidates, especially after the
successful launch of CoRoT and Kepler . The unprecedented high precision of
photometric observation ($\sim 10$ppm) and long-duration continuous
observation (up to years) achieved by space missions make transits an ideal
tool to detected near-Earth-sized planets around solar type stars.
To date, around 780 exoplanets have been detected mainly by RV measurements,
with more than 100 multiple planet systems 111http://exoplanet.eu/,and
hereafter for all RV based statistics in the paper.. The first 16 months’
observation of the Kepler mission revealed more than 2321 transit candidates,
with 48 candidates in the habitable zone of their host
stars222http://kepler.nasa.gov/,and hereafter in the paper (Borucki et al.
2011; Batalha et al. 2012; Fabrycky et al. 2012).
The study of planet formation can be traced back to the 18th century, when E.
Swedenborg, I. Kant, and P.-S. Laplace developed the nebular hypothesis for
the formation of the solar system. At that time the solar system was the only
sample of a planetary system. The architecture of the solar system implied
that it was formed in a Keplerian disk of gas and dust (for reviews, see
Wetherill 1990; Lissauer 1995; Lin & Papaloizou 1996). With the discovery of
more exoplanet systems, planet formation theory has developed dramatically.
For example, the discovery of hot Jupiters(HJs) stimulated the classical
migration theory of planets embedded in the proto-stellar disk (Lin &
Papaloizou 1986, 1996), the moderate eccentricities of exoplanet orbits remind
us the planet-planet scattering(hereafter, PPS) theory (Rasio & Ford 1996),
and the observation of some HJs in retrograde orbits extends the classical
Lidov-Kozai mechanism to eccentric cases (Kozai 1962; Lidov 1962; Naoz et al.
2011b). Through population synthesis, N-body and hydrodynamical simulations,
the planet formation processes have been well revealed but are still far from
fully understood.
In this paper, we focus on recent processes in the theory of detection and
formation of solar type stars, either around single stars (§2), binary
stars(§3), or in star clusters (§4).
## 2 Planets around single stars
### 2.1 Overview of Observations
#### 2.1.1 Planet Occurrence Rate
The occurrence rate of gas-giant exoplanets around solar-type stars has been
relatively well studied. Tabachnik & Tremaine (2002) fitted 72 planets from
different Doppler surveys, and found a mass ($M$)-period ($P$) distribution
with the form of a double power law,
$dN\propto M^{\alpha}P^{\beta}dlnMdlnP,$ (1)
after accounting for selection effects. They obtained $\alpha=-0.11\pm 0.1$,
and $\beta=0.27\pm 0.06$. Recently, Cumming et al. (2008) analyzed eight years
of precise RV measurements from the Keck Planet Search, including a sample of
585 chromospherically quiet stars with spectrum types from F to M. Such a
power-law fit in equation (1) for planet masses $>0.3M_{J}$ (Jupiter Mass) and
periods $<2000$ days was re-derived with $\alpha=-0.31\pm 0.2$, and
$\beta=0.26\pm 0.1$. They concluded $10.5\%$ of solar type stars have a planet
with mass $>0.3M_{J}$ and period $2-2000$ days, with an extrapolation of
$17-20\%$ of stars having gas giant planets within 20 AU.
Based on the 8-year high precision RV survey at the La Silla Observatory with
the HARPS spectrograph, Mayor et al. (2011) concluded that 50% of solar-type
stars harbor at least one planet of any mass and with a period up to 100 days.
About 14% of solar-type stars have a planetary companion more massive than
$50M_{\oplus}$ in an orbit with a period shorter than 10 years. The mass
distribution of Super-Earths and Neptune-mass planets is strongly increasing
between 30 and $15M_{\oplus}$, indicating small mass planets are more frequent
around solar type stars.
Howard et al. (2010) calculated the occurrence rate of close planets (with
$P<50$ days), based on precise Doppler measurements of 166 Sun-like stars.
They fitted the measures as a power-law mass distribution,
$df=0.39M^{-0.48}d\log M,$ (2)
indicating an increasing planet occurrence with decreasing planet mass. It
also predicted that 23% of stars harbor a close Earth-mass planet, ranging
from 0.5 to 2.0 $M_{\oplus}$ (Earth mass).
With 12 years of RV data with long-term instrumental precision better than
$3ms^{-1}$, the Anglo-Australian Planet Search targets 254 stars, and
estimates an occupance rate of 3.3% of stars hosting Jupiter analogs, and no
more than 37% of stars hosting a giant planet between 3 and 6 AU(Wittenmyer et
al. 2011a). Their results also reveal that the planet occurrence rate
increases sharply with decreasing planetary mass. In periods shorter than 50
days, they found that 3.0% of stars host a giant ($m\sin i>100M_{\oplus}$)
planet, and that 17.4% of stars host a planet with ($m\sin i<10M_{\oplus}$)
(Wittenmyer et al. 2011b). Moreover, the lack of massive planets $(>8M_{\rm
Jup})$ beyond 4 AU was reported in Boisse et al. (2012), although with less
than 20 RV detected candidates at the moment. Such a distribution agrees with
population synthesis (Mordasini et al. 20012) , where they showed that a
decrease in frequency of massive giant planets at large distance ($\geq$ 5AU)
is a solid prediction of the core-accretion theory.
Transit observations from the Kepler spacecraft give qualitatively similar
results. Howard & Marcy (2011) checked the distribution of planets in close
orbits. For $P<50$ days, the distribution of planet radii ($R$) is given by a
power law,
$df=k(R/R_{\oplus})^{\alpha}d\log R$ (3)
with $k=2.9^{+0.5}_{-0.4}$ and $\alpha=-1.92\pm 0.11$, and $R_{\oplus}$ is the
Earth radius. They find the occurrence of $0.130\pm 0.008$, $0.023\pm 0.003$,
and $0.013\pm 0.002$ planets per star for planets with radii $2-4$, $4-8$, and
$8-32R_{\oplus}$ respectively.
The rapid increase of planet occurrence with decreasing planet size indicates
the presence of Super-Earth and Neptune size cases are quite common. Although
this agrees with the prediction of the conventional core-accretion scenario,
it conflicts with the results predicted by the population synthesis models
that a paucity of extrasolar planets with mass in the range $10-100M_{\oplus}$
and semi-major axis less than 3 AU are expected, the so called ’planet desert’
(Ida & Lin 2004).
#### 2.1.2 Stellar masses of planet hosting stars
Kepler results also revealed a correlation between the planet’s occurrence
with the effective temperature ($T_{\rm eff}$) of host stars (Howard & Marcy
2011). The occurrence rate $f$ is inversely correlated with $T_{\rm eff}$ for
small planets with $R_{p}=2-4R_{\oplus}$, i.e.,
$f=f_{0}+k(T_{\rm eff}-5100K)/1000K,$ (4)
with $f_{0}=0.165\pm 0.011$ and $k=-0.081\pm 0.011$. The relation is valid
over $T_{\rm eff}=3600-7100K$, corresponding to stellar spectral typed from M
($\leq 0.45M_{\odot}$, solar mass)to F($1.4M_{\odot}$). This implies that
stars with smaller masses tends to have small size planets.
However, the occurrence of planets with radii larger than $4R_{\oplus}$ from
Kepler does not appear to correlate with $T_{\rm eff}$ (Howard & Marcy 2011).
This is in contrast with the observation by RV measurements. In fact, a much
lower incidence of Jupiter-mass planets is found around M dwarfs (Butler et
al. 2006; Michael et al. 2006; Johnson et al. 2007), and higher mass stars are
more likely to host giant planets than lower-mass ones (e.g., Lovis & Mayor
2007; Johnson et al. 2007, 2010; Borucki et al. 2011). The result is
compatible with the prediction in the core accretion scenario for planet
formation (Laughlin et al. 2004; Ida & Lin 2005; Kennedy & Kenyon 2008).
#### 2.1.3 Metallicity Dependence
It has been well-established that more metal-rich stars have a higher
probability of harboring a giant planet than their lower metallicity
counterparts (Gonzalez 1997; Santos et al. 2001, 2004; Fischer & Valenti 2005;
Udry & Santos 2007; Sozzetti et al. 2009; Sousa et al. 2011). The occurrence
rate increases dramatically with increasing metallicity. Based on the CORALIE
and HARPS samples, around 25% of the stars with twice the metal content of our
Sun are orbited by a giant planet. This number decreases to $\sim 5\%$ for
solar-metallicity cases (Sousa et al. 2011; Mayor et al. 2011). Recently,
Mortier et al. (2012) showed the frequency of giant planets is
$(f=4.48^{+4.04}_{-1.38}\%)$ around stars with $[Fe/H]>-0.7$, as compared with
$f\leq 2.36\%$ around stars with $[Fe/H]\leq-0.7$. Curiously, no such
correlation between planet host rate and stellar metallicity is observed for
the lower-mass RV planets, and the stars hosting Neptune-mass planets seem to
have a rather flat metallicity distribution (Udry et al. 2006; Sousa et al.
2008, 2011; Mayor et al. 2011).
By re-evaluating the metallicity, Johnson et al. (2009) find that M dwarfs
with planets appear to be systematically metal rich, a result that is
consistent with the metallicity distribution of FGK dwarfs with planets.
Schlaufman & Laughlin (2011) find that star hosting Kepler exoplanet
candidates are preferentially metal-rich, including the low-mass stars that
host candidates with small-radius , which confirms the correlation between the
metallicity of low-mass stars and the presence of low-mass and small-radius
exoplanets.
Figure 1: Distribution of exoplanets detected by RV measure(left,
http://exoplanet.eu/), and candidates by the Kepler mission (right,
http://kepler.nasa.gov/).
#### 2.1.4 Mass and Period Distributions
Figure 1 shows the distribution of planetary orbital periods for different
mass regimes. 705 planets detected by RV measurement and 2320 candidates
revealed by the Kepler mission are included. Several features of the mass-
period distribution have been well known and widely discussed in the
literatures. However, it seems that the distribution features from RV detected
exoplanets are slightly different from those of Kepler candidates.
* •
All kinds of RV planets show a ”pile-up” at orbital periods 2-7 days (e.g.,
Gaudi et al. 2005; Borucki et al. 2011), while Kepler results show that
Jupiter-size ($>6R_{\oplus}$) candidates are more or less flat up to orbits
with $>100$ days; Neptune-size ($2M_{\oplus}<R_{p}<6R_{\oplus}$) and super-
Earth candidates ($1.25M_{\oplus}<R_{p}<2R_{\oplus}$) peak at $10-20$days.
Both of them are more abundanct relative to Jupiter-size candidates in the
period range from one week to one month(Borucki et al. 2011). Extrapolating
the distribution by considering the $(R_{p}/a)$ probability of a transiting
exoplanet could extend these peaks to a bit more distant orbits. Earth size or
smaller candidates($<1.25M_{\oplus}$) show a peak of $\sim 3$ days .
* •
RV planets show a paucity of massive planets ($M>1M_{J}$) in close orbits
(Udry et al. 2002, 2003; Zucker & Mazeh 2002; Zucker& Mazeh 2003), and a
deficit of planets at intermediate orbital periods of $10-100$ days (Jones et
al. 2003; Udry et al. 2003; Burkert & Ida 2007). However, this is not obvious
as Kepler results show at least several tens of candidates with the radius
lying in the $10-20R_{\oplus}$ line within 10-day orbits, and the distribution
extending to 100 days is rather flat. (Fig.1). The lack of all types of
planets with orbital periods $\sim 10-1000$ days observed by RV is clear, but
from Kepler results, the lack of planets with period $>100$ days is also
shown, possibly due to the observational bias (Fig.1). RV observations from
the Anglo-Australian Planet Search indicate that, such a lack of giant planets
($M>100M_{\oplus}$) with periods between 10 and 100 days is indeed real.
However, for planets in the mass range $10-100M_{\oplus}$, the results suggest
that the deficit of such planets may be a result of selection effects
(Wittenmyer et al. 2010).
### 2.2 Hot Jupiter Systems
The HJ class is referred to a class of extrasolar planets and has mass close
to or exceeding that of Jupiter ($M_{p}\geq 0.1M_{J}$, or radius $\geq
8R_{\oplus}$ for densities of $1.4{\rm g/cm^{3}}$, a typical value of gas
giants with small rocky cores), with orbital peroids $\leq 10$ days (or
$a<0.1$ AU) to their parent stars (Cumming et al. 2008; Howard & Marcy 2011;
Wright et al. 2012). According to this definition, the RV exoplanets have 202
HJs, while Kepler candidates have 89 HJs. HJs are notable since they are easy
to detecte either by RV or by transit measurements. For example, the first
exoplanet discovered around 51 Peg was such a close-in giant planet(Mayor &
Queloz 1995). Transiting HJs also give us information about their radii, which
is crucial for understanding their compositions(e.g., Fortney et al. 2003;
Sara & Drake 2010 ). However, with the increase of RV precision and the number
of detected exoplanets, HJs are found to be in fact rare objects (e.g.,
Cumming et al. 2008; Wright et al. 2012). More interestingly, some HJs were
observed in orbits that are retrograde with respect to the spin direction of
their host stars(e.g., Winn et al. 2010), indicating that their formation
process might have been quite different with that of our solar system.
#### 2.2.1 Occurrence rate and distributions
Marcy et al. (2005a) analyzed 1330 stars from the Lick, Keck, and Anglo-
Australian Planet Searches, and the rate of HJs among FGK dwarfs surveyed by
RV was estimated to be $1.2\pm 0.1\%$. Mayor et al. (2011) used the HARPS and
CORALIE RV planet surveys and found the occurrence rate for planets with
$M\sin i>50M_{\oplus}$ and $P\leq 11$ days is $0.89\pm 0.36\%$. Recently,
Wright et al. (2012) used the California Planet Survey from the Lick and Keck
planet searches, and found the rate to be $1.2\pm 0.38\%$. These numbers are
more than double the rate reported by Howard & Marcy (2011) for Kepler stars
($0.5\pm 0.1\%$) and the rate of Gould et al. (2006) from the OGLE-III transit
search. The difference might be, as pointed out by Wright et al. (2012), that
transit surveys like OGLE and Kepler (centered at galactic latitude
$b=+13.3^{o}$) probe a lower-metallicity population, on average, than RV
surveys.
Previous RV measurements show that, there is a sharp inner cutoff in the three
day pileup of HJs. They appear to avoid the region inward of twice the Roche
radius (Ford& Rasio 2006), where the Roche radius is the distance within which
a planet would be tidally shredded. However, recent RV detected exoplanets and
Kepler candidates indicate the presence of more than 200 exoplanet and
candidates within 3-day orbits, with the inner most orbital peroid being 0.24
days for system KOI-55, corresponding to a location close to its Roche radius
(Fig.1).
Also, RV detected HJs appear to be less massive than more distant planets
(Pätzold& Rauer 2002; Zucker & Mazeh 2002). For planets discovered with the RV
method, close planets have projected masses ($M\sin i$) less than twice
Jupiter’s mass. But numerous planets farther out have $M\sin i>2M_{J}$ ( Udry
& Santos 2007).
#### 2.2.2 Spin-Orbit misalignment
One of the most fascinating features for HJs is that, some HJs have orbits
that are misaligned with respect to the spin of their host stars(Winn et al.
2010; Triaud et al. 2010). The sky-projected angle between the stellar spin
and the planet’s orbital motion can be probed with the Rossiter-McLaughlin
(RM) effect (Rossiter 1924; McLaughlin 1924). To date, the RM effect has now
been measured for at least 47 transiting exoplanets (see Winn et al. 2010,
Table 1 for a list of 28 planets, and Brown et al. (2012), Table 5 for a list
of 19 additional planets, and references therein). Only 7 (HAT-P-6b,HAT-P-7b,
HATP-164b, WASP-8b,WASP-15b,WASP-17b,WASP-33) of the 47 samples have projected
angles above $90^{o}$, indicating a ratio of $\sim 15\%$ that are in
retrograde motion. It is still not clear what type of stars could host HJs in
retrograde orbits. Winn et al. 2010 showed that the stars hosting HJs with
retrograde orbits might have high effective temperatures ($>6250$ K). The
underlying physics remans further study.
#### 2.2.3 Lack of close companions
Few companion planets have been found in HJ systems within several AU (Wright
et al. 2009; Hébrard etal. 2010). To date, only six RV detected planetary
system have multiple planets with the inner one being HJs (HIP14810, ups And,
HAT-P-13,HD187123, HD21707, HIP 11952 ). Compared to the total number of 89 RV
detected HJs, the ratio is less than $7\%$. Interestingly, all these planetary
companions are in orbits with periods $>140$ days. This relative deficit also
shows up in the transit samples, where most attempts at detecting transit
timing variations caused by close companions have been unsuccessful (Holman&
Murray 2005; Agol et al. 2005; Rabus etal. 2009; Csizmadia etal. 2010;
Hrudková et al. 2010; Haghighipour & Kirste 2011; Steffen et al. 2012)).
Kepler data also revealed the lack of a close companion in HJ systems. Steffen
et al. (2012) present the results of a search for planetary companions
orbiting near HJ candidates in the Kepler data. Special emphasis is given to
companions between the 2:1 interior and exterior mean-motion resonances(MMRS).
A photometric transit search excludes the existence of companions with sizes
ranging from roughly $2/3$ to 5 times the size of Earth.
### 2.3 Multiple Planet systems
Figure 2: The distribution of Kepler candidates in single and multiple planet
systems for different mass regimes. Data are from http://kepler.nasa.gov.
With the increasing number of exoplanets being detected, the number of
multiple planet systems is also steadily increasing. The first 16 months of
Kepler data show that, among the 2321 candidates, 896 ones are in multiple
planet systems, so that 20% of the stars cataloged have multiple
candidates(Borucki et al. 2011; Batalha et al. 2012). Considering the present
observation bias towards large mass planets, as well as the increasing
occurrence rate of small mass planets, we have a good reason to believe that
multiple planets are very common and might occur at a much higher rate. The
systems that have been revealed with the most numerous exoplanets are HD 10180
(Lovis et al. 2011, up to seven planets) and Kepler 11 (Lissauer et al. 2011,
with six planets). All of them are mainly composed of small mass planets
(Super Earth or Neptune mass). Several important signatures have been revealed
by the Kepler mission:
* •
Multiple planets have on average smaller masses than single planet systems.
Fig.2 shows the paucity of giant planets at short orbital periods in multiple
planet systems, and the ratio of giant planets (with radius $>6R_{\oplus}$) in
single and multiple planet systems is roughly $5.7:1$, with orbital periods of
up to $\sim 500$ days’ orbits.
* •
Many planet pairs are near MMRs. The presence of MMR is a type of strong
evidence for the migration history of the planet pairs (e.g., Lee & Peale
2002; Zhou et al. 2005). Wright et al. 2011 summarized the data from RV
detected planets, and found 20 planetary systems are apparently in MMRs,
indicating one-third of the well-characteried RV multiple planet systems have
planet pairs in apparent MMRs. Fabrycky et al. 2012 found the Kepler multiple
transiting planet systems show some pile-up for planets pairs near lower order
MMRs (especially 3:2 and 2:1 MMRs).
* •
Multiple planet systems are nearly coplanar. Checking the Kepler multiple-
transiting system indicates that these planets are typically coplanar to
within a few degrees (Batalha et al. 2012). Also the comparison between the
Kepler and RV surveys shows that the mean inclination of multi-planet systems
is less than $5^{o}$ (Tremaine & Dong 2012). Figueira et al. (2012)
demonstrated that, in order to match the ratio of single planet systems to the
2-planet ones observed in HARPS and Kepler surveys, the distribution of mutual
inclinations of multi-planet systems has to be of the order of $1^{o}$.
### 2.4 Planet Formation Theory
Now it is widely accepted that planets were formed in the protoplanetary disk
during the early stage of star formation (e.g.,Wetherill 1990; Lissauer 1995;
Lin & Papaloizou 1996; Tutukov & Fedorva 2012). According to the conventional
core accretion model, planets are formed through the following processes
(e.g., Lissauer 1993; Armitage 2007):
(1) grain condensation in the mid-plane of the gas disk, forming kilometer-
sized planetesimals ($10^{18}-10^{22}$ g) on timescales on the order of
$10^{4}$ yrs, from sticking collisions of dust (Weidenschilling & Cuzzi 1993;
Weidenschilling 1997) , with gravitational fragmentation of a dense particle
sub-disk near the midplane of the protoplanetary disk (Goldreich & Ward 1973;
Youdin & Shu 2002). Further growth of planetesimals can be helped by
procedures such as the onset of streaming instability (Johansen et al. 2007)
or vortices in turbulence (Cuzzi et al. 2008), or the sweeping of dust with
the ”snowball” model (Xie et al. 2010b; Ormel & Kobayashi 2012; Windmark et
al. 2012).
(2) accretion of planetesimals into planetary embryos ($10^{26}-10^{27}$ g,
Mercury to Mars size) through a phase of ”runaway” and ”oligarchic” growth on
a timescale of $\sim 10^{4}-10^{5}$ yrs (Greenberg et al. 1978; Wetherill &
Stewart 1989; Aarseth et al. 1993; Kokubo & Ida 1996, 2000; Rafikov 2003,
2004).
(3) gas accretion onto solid embryos with mass bigger than a critical mass
($\sim 10M_{\oplus}$) after a $\sim$Myrs long quasi-equilibrium stage before
gas depletion (Mizuno 1980; Bodenheimer & Pollack 1986; Pollack et al. 1996;
Ikoma et al. 2000).
(4) giant impacts between embryos, producing full-sized $(10^{27}-10^{28}$ g)
terrestrial planets in about $10^{7}-10^{8}$ yrs (Chambers & Wetherill 1998;
Levison & Agnor 2003; Kokubo et al. 2006). Thus the presence of big solid
embryos and the lifetime of the gas disk are crucial for the presence of giant
plants, while the presence of enough heavy element determines the mass of
solid embryos and terrestrial planets.
According to the above scenario, the correlation between stars that host giant
planets and stellar metallicity can be understood. By cosmological assumption,
a high stellar metallicity implies a protoplanety disk with more heavy
elements, thus a metal-rich protoplanetary disk enable the rapid formation of
Earth-mass embryos necessary to form the cores of giant planets before the
gaseous disk is dissipated. That correlation might also indicate a lower limit
on the amount of solid material necessary to form giant planets. Johnson & Li
(2012) estimated a lower limit of the critical abundance for planet formation
of $[Fe/H]_{\rm crit}\sim-1.5+\log({\rm r}/1{\rm AU})$, where $r$ is the
distance to the star. Another key point may be the correlation between
metallicity and the lifetime of the gas disk. There is observational evidence
that the lifetime of circumstellar disks is short at lower metallicity, likely
due to the great susceptibility to photoevaporation(Yasui et al. 2009).
Although the above procedures for single planet formation are relatively
clear, there are some bottleneck questions (see previous listed reviews).
Next, we focus on the formation of orbital architectures for different planet
systems.
#### 2.4.1 Formation of Hot Jupiter systems
Due to the high temperature that might hinder the accretion of gas in forming
giant planets, the HJs were assumed to be formed in distance orbits rather
than formed in situ. There are mainly three theories that were proposed to
explain the formation of HJ systems with the observed configurations.
Disk migration model. The earliest model for the formation of HJ systems is
the planet migration theory embed in protostellar disks (Lin & Papaloizou
1986; Lin et al. 1996). Giant planets formed in distant orbits, then migrated
inward under the planet-disk interactions and angular momentum
exchanges(Goldreich & Tremaine 1980; Lin & Papaloizou 1986). The so called
type II migration will be stalled at the inner disk edge truncated by the
stellar magnetic field. The maximum distance of disk truncation is estimated
to be $\sim 9$ stellar radii (Königl 1991). Considering the radius of the
protostar is generally 2-3 times larger than their counterpart in the main
sequence, the inner disk truncation would occur at $\sim 0.1$AU. This might
naturally explain the pileup of orbits with periods of $3-10$ days’ for HJs.
However, as type II migration is effective only in the plane of the disk, and
disk’s tidal forces try to dampen the inclination of planets (Goldreich &
Tremaine 1980), this procedure can not explain the formation of HJs in orbits
with high inclination, as well as the lack of planetary companions in close
orbits. Recently, Lai et al. (2011) proposed that stellar-disk interaction
may gradually shift the stellar spin axis away from the disk plane, on a time
scale up to Gyrs.
Planetary scattering model. Another mechanism that might account for the
formation of HJ systems is the PPS model. Close-encounters among planets can
excite their orbital eccentricities ($e$). In the extreme case that $e$ is
near unity, the orbital periastron will be small enough so that star-planet
tidal interactions might be effective and circularize the orbits to become HJs
(Rasio & Ford 1996; Ford et al. 2001; Papaloizou & Terquem 2001; Ford &
Rasio 2008). The planetary scattering model can reproduce the observed
eccentricity distribution of moderately eccentric $(e\sim 0.1-0.3)$, non-HJ
extra-solar planets (Zhou at al. 2007; Chatterjee et al. 2008; Juric &
Tremaine 2008). However, the required high eccentricity and the long
timescale required for tidal damping be effective might be not easy to achieve
unless some secular effects (e.g. the Lidov-Kozai mechanism) are excited
(e.g., Nagasawa et al. 2008).
Secular models. The third class of models relate to the Lidov-Kozai effect
(Lidov 1962; Kozai 1962) in the presence of a third body. To account for the
high inclination of HJs, Wu & Murray (2003) proposed that a companion star
which is a third body in a high inclination orbit can induce Kozai
oscillations on the planet’s evolution, gradually exciting the planet’s orbit
to an eccentricity near unity so that it can reach a proximity close to the
central star, until tidal dissipation circularizes the orbit into a HJ.
Fabrycky & Tremaine (2007) found such resulting HJs should be double-peaked
with orbital inclinations of $\sim 40^{o}$ and $140^{o}$. Such an idea has
been extended to brown dwarf companion by Naoz et al. (2011a).
However, because the population studies have established that only $10\%$ of
HJs can be explained by Kozai migration due to binary companions (Wu et al.
2007; Fabrycky & Tremaine 2007), but studies show that most of the HJ systems
do not have stellar or substellar companions. Whether this mechanism can
account for the formation of most HJs is not known. Another question is that,
in the stellar companion case ($m_{c}$, a star or a brown dwarf), the orbital
angular momentum (AM) of $m_{c}$ dominates that of the system and determines
an invariant plane, thus the $z-$component of AM (perpendicular to the
invariant plane) of the planet ($m_{p}$) is conserved when $m_{c}$ is in a
distant orbit. Thus $m_{p}$ can be in an apparent retrograde orbit relative to
the spin axis of the main star only when $m_{c}$ has a relatively large
inclination with respect to the equator of the main star(Wu et al. 2007;
Fabrycky & Tremaine 2007), and this retrograde motion is not with respect to
the invariant plane determined by the total AM.
To avoid relying on the effects of stars or brown dwarf companions, and also
to find the occurrence of retrograde motion relative to the invariant plane,
one resorts to the conditions under which the Lidov-Kozai mechanism works for
planet mass companions ($m_{c}$). Naoz et al. (2011b) study the mechanism with
a general three-body model. Denote $a,a_{c}$ as the semimajor axes of inner
planet ($m_{p}$) and companion, respectively, with $e_{c}$ being the
eccentricity of $m_{c}$; they find that as long as
$(a/a_{c})e_{c}/(1-e_{c}^{2})$ is not negligible, the octuple-level of the
three-body Hamiltonian would be effective, so that the z-component of $m_{p}$
in AM is no longer conserved, allowing the occurrence of retrograde motion
relative to the invariant plane. Thus , to make a retrograde HJ, a companion
in a close and eccentric orbit is required, but the mass of the companion is
not important.
However, newly-born planets are assumed to be in near circular and coplanar
orbits. To generate the required eccentricity, Nagasawa et al. 2008, 2011
introduced planet scattering into the above pictures. Starting from a
relatively compact system ($\sim 3.6R_{H}$, Hill’s radius) with three Jupiter-
mass planets, the planets scatter one another on a timescale of $\sim 10^{3}$
years. They found $\sim 30\%$ of the simulations can result in a planet with
eccentricity high enough, that Kozai excitations from outer planets can become
effective, so that it can be either in a close orbit with non-negligible
eccentricity, or in a highly inclined (even a retrograde orbit) with
relatively small eccentricity over a timescale of $10^{9}$ years. However, it
is unclear whether the initial condition of a compact and highly unstable
planetary system can exist, as required by this theory (Matsumura et al.
2010). Also the scatted planets can be observed to test the theory.
Another route to generate eccentricities other than through violent PPS is the
diffusive chaos arising from a multiple planet system after it forms. The
generation of eccentricity in a multiple planet system is a slow, random walk
diffusion in the velocity dispersion space, and the timescale increases with
the logarithm of the initial orbital separations (Zhou at al. 2007).
Recently, Wu & Lithwick 2011 proposed that secular chaos may be excited in an
orderly space system, and it may lead to natural excitation of the
eccentricity and inclination of the inner system, resulting in observed HJ
systems. They inferred that such a theory can also explain the eccentricities
and inclinations for distant giant planets. However, to what extent such a
mechanism could be effective within the age of planetary systems remains for
further study.
To summarize, the Lidov-Kozai mechanism seems to be the most promising
mechanism for the formation of HJs. Provided that initial eccentricities of
the planet’s companion can reach high enough value, inter planetary Kozai
oscillations can bring the inner planets into HJ orbits with sufficiently high
inclinations.
#### 2.4.2 Formation of multiple planet architectures and a system of
classification
What should a ’standard’ planet system be like? Before answering this
question, let us first check the possible outcome of a planet system after the
formation of individuals by the procedure listed at the beginning of section
2.4.
According to the core accretion scenario, by depleting all the heavy elements
in a nearby region (called the feeding zone, roughly 10 Hill radii), an embryo
without any migration will be stalled from growing, which is a case called an
isolation mass(Ida & Lin 2004). In a disk with metallicity $f_{d}$ times the
minimum mass solar nebula(MNSN) (Hayashi 1981), the isolation mass can be
estimated as (Ida & Lin 2004, Eq.19)
$M_{\rm iso}\approx 0.16\eta_{\rm ice}^{3/2}f_{d}^{3/2}(\frac{a}{\rm
1AU})^{3/4}(\frac{M_{*}}{M_{\odot}})^{-1/2}M_{\oplus},$ (5)
where $\eta_{\rm ice}$ is the enhancement factor, with a value of $1$ and
$\approx 4.2$ respectively inside and outside the snow line ( location with
temperature 170K beyond which water is in the form of ice, $\sim 2.7$ AU in
the solar system). The time required for the core to accrete nearby materials
and become isolated is on the order of (Ida & Lin 2004. Eq.18)
$\tau\approx 1.2\times 10^{5}\eta_{\rm
ice}^{-1}f_{d}^{-1}f_{g}^{-2/5}(\frac{a}{1\rm AU})^{27/10}(\frac{M_{\rm
iso}}{M_{\oplus}})^{1/3}(\frac{M_{*}}{M_{\odot}})^{-1/6}~{}{\rm yr},$ (6)
where $f_{g}$ is the enhancement factor of gas disk over MMSN. So for a
typical disk with 2 times the MNSN ($f_{d}=f_{g}=2)$, isolation embryos inside
the snow line are small ($<1M_{\oplus}$), and they can not develop. Embryos
beyond the snow line can grow to $\sim 10M_{\oplus}$ so that they can accrete
gas to form gas giants. However, the growth time of embryos with mass
$10M_{\oplus}$ in distant orbits ($>20AU$) is long ($\sim 10Myr$ at 10AU and
$\sim 70Myr$ at 20 AU). Within a disk with a moderate lifetime of $\sim 3$ Myr
for classical T-Tauri stars (Haisch et al. 2001) , embryos in distant orbits
do not have enough time to accrete gas, thus they will stall their growth at
the mass of a sub-giant mass, like Uranus and Neptune in the solar system.
As the gas disk is depleted, the induced secular resonance sweeps through the
inner region of the planetary systems, causing further mergers of cores
(Nagasawa et al. 2003). Terrestrial planet formed after the gas disk was
depleted at $\sim 200$Myrs(Chambers 2001) . After depletion of the gas disk ,
a debris disk with leftover cores interacted with giant plants, causing small
scale migration, such as in the Nice model(Gomes et al. 2005; Morbidelli et
al. 2005; Tsiganis et al. 2005). Thus, assuming no giant migrations occurred,
the solar system is the basic ”standard” multiple planet system. As all
planetary embryos were formed in near mid-plane of the gas disk, without
perturbations in the vertical direction, such a standard planet system is
nearly coplanar, like many multiple planet systems observed by the Kepler
mission.
However, several procedures make the above picture more complicated. One of
the most difficult task is to understand the migration of embryos or planets
embedded in the gas disk before depletion. For a sub-Earth protoplanet , the
exchanges of angular momentum between it and the nearby gas disk will cause a
net momentum lose on it, which results in a so-called type I migration over a
timescale on the order of $<0.1Myr$ (Goldreich & Tremaine 1979; Ward 1986,
1997; Tanaka et al. 2002). If the protoplanet can avoid such disastrous inward
migration, and successfully grow massive enough to accrete gas and become a
gas giant, the viscous evolution of the disk may cause the giant planet to
undergo a type II migration, with a timescale of Myrs(Lin & Papaloizou 1986).
Recent studies infer that, under more realistic conditions, the migration
speeds of both types can be reduced or even with their direction-being
reversed, leading to an even rarer outcome Kley & Nelson 2012).
The evidence for planet migration is the observed systems in MMR. Since 2:1
MMR has the widest resonance width, especially for planetesimals in nearly
circular orbits (Murray & Dermott 1999), many planet pair are expected to show
2:1 MMRs if they had a history of convergent migration (e.g. Rivera et al.
2010; Zhou et al. 2010; Wang et al. 2012; Gerlach & Haghighipour 2012).
However, Kepler planets give many planets conditions near but not in MMR. This
can be understood by the phenomenon that later stage planetesimal and planet
interactions may cause further migrations but with smaller extensions, causing
strict commensurability to be lost (Terquem & Papaloizou, 2007). Giant
planets in MMR might be strong enough and survive under such perturbations,
like the GJ 876 system(Lee & Peale 2002; Rivera et al. 2010; Gerlach &
Haghighipour 2012). Hydrodynamical simulations show that different disk
geometries might lead the planet pair to either convergent migration (thus
possibly the trap of different MMRs), or sometimes to divergent
migrations(Zhang & Zhou 2010a, b). However planet pairs may not necessarily
lead to MMRs configurations for some dynamical configurations (Batygin &
Morbidelli 2012), e.g. the resonant repulsion of planet pairs is discussed by
Lithwick & Wu (2011).
The orbital configurations of multiple planet systems incorporating planetary
migration have been studied extensively by population syntheses (e.g., Ida &
Lin 2008, 2010,Mordasini et al. 2009a, b) and N-body simulations(e.g.,Thommes
et al. 2008, Liu et al. 2011). Thommes et al. (2008) found that for giant
planet formation, two timescales are crucial: the lifetime of the gas disk
$\tau_{\rm disk}$ and the time to form the first gas giant $\tau_{\rm giant}$.
In cases with $\tau_{\rm giant}>\tau_{\rm disk}$, the gas is removed before
any gas giant has a chance to form, leaving behind systems consisting solely
of rocky-icy bodies. In cases with $\tau_{\rm giant}<\tau_{\rm disk}$, such
systems generally produced a number of gas giants that migrated inward a
considerable distance. Liu et al. (2011) also showed that $\tau_{\rm disk}$ is
crucial for forming planet systems, as large $\tau_{disk}$ tends to form more
giant planets in close and nearly circular orbits, while small $\tau_{\rm
disk}$ favors forming planets with small masses in distant and eccentric
orbits.
According to the above theories as well as the currently available
observations, the planet systems around solar type stars can be roughly
classified into the following categories. A detailed classification will be
presented later( Zhou et al. , in preparation).
* •
Class I: Hot Jupiter systems. These might be formed through some secular
mechanisms such as Lidov-Kozai cycling , as discussed previously. Typical
example: 51 Peg b.
* •
Class II: Standard systems . They are formed either through processes similar
to our solar system, or by undergoing some large scale migrations, as
mentioned perviously. According to scenarios, they undergo migration, due to
the deficit of heavy elements in the gas disk, or due to the short lifetime of
disk, They can further be classified as,
* –
Subclass II-1: Solar-like systems. These have planetary configurations similar
to the solar system: terrestrial planets in the inner part, 2-3 gas giant
planets in middle orbits, and Neptune-size sub-giants in outer orbits, due to
insufficient gas accretion. Typical example: Mu Ara, ups And, and HD125612
systems.
* –
Subclass II-2: Hot super-Earth systems. With the migration of giant planets,
the sweeping of inward MMRs or secular resonances will trap the isolated
masses ($0.1-1M_{\oplus}$), and excite their eccentricities , causing further
mergers, which result in the formation of hot super Earth, like GJ 876d (Zhou
et al. 2005; Raymond et al. 2006, 2008). Other formation scenarios, see a
review (Haghighipour 2011). Typical example: GJ 876, and Kepler 9 systems.
* –
Subclass II-3: Sub-giant planet systems. Due to the low disk mass or low
metallicity, planet embryos around some stars (especially M dwarf) might not
grow massive enough to accrete sufficient gas to become a gas giant, thus
planets in these systems are generally sub-giants, like most of the systems
discovered in Mayor et al. (2011). Typical example: the Kepler 11 system.
* •
Class III. Distant giant systems. Through direct imaging, a type of system was
detected with many massive companions (up to several times the mass of of
Jupiter) in distant orbits, such as Fomalhaut b (Kalas et al. 2008) , the
HR8799 system(Marois et al. 2008), and beta Pic b (Lagrange et al. 2009).
Interestingly, all these stars have short ages $(\sim 100-300$ Myrs). Whether
the planets were formed in situ through gravitational instability(Boss 1997),
or formed through outward migration or scattering, is still not clear. Typical
examples: Fomalhaut, HR 8799, and beta Pic systems.
## 3 Planets in Binary Star Systems
### 3.1 Overview of Observation
Planets in binaries are of particular interest as most stars are believed to
be born not alone but in a group, e.g., binaries and multiple stellars
systems. Currently, the multiplicity rate of solar like stars is $\sim
44-46\%$, including $\sim 34-38\%$ for only binaries (Duquennoy & Mayor 1991;
Raghavan et al. 2010). Different resulting values of the multiplicity rate of
planet-bearing stars (compared to all the planet hosts) were found to be
$23\%$ (Raghavan et al. 2006) and $\sim 17\%$ (Mugrauer & Neuhäuser 2009), and
most recently $\sim 12\%$ (Roell et al. 2012). The decreasing multiplicity
rate is mainly because of the quickly increasing number of transiting planets
discovered in recent years. For example, Kepler has discovered more than 60
planets since 2010, however, followup multiplicity studies on such planet
hosts are usually postponed or even considered impracticable. In any case, the
multiplicity rate of a planet host is significantly less than the multiplicity
rate of stars. This may be because of selection biases in planet-detection
against binary systems and/or because of impacts of binarity on planet
formation and evolution (Eggenberger et al. 2011).
Depending on the orbital configuration, planets in binaries are usually
divided into two categories (Haghighipour et al. 2010; Haghighipour 2010), S
type for planets orbiting around one of the stellar binary components,
i.e.,the circumprimary case, and P type for those orbiting around both the
stellar binary components, i.e., the circumbinary case. Currently, most of
them are S type, and only a few are found in P type, including NN Ser
(Beuermann et al. 2010), HW Vir (Lee at al. 2009), DP Leo (Qian et al. 2010),
HU Aqr (Qian et al. 2010; Hinse et al. 2012), UZ For (Dai et al. 2010; Potter
et al. 2011), Kepler-16 (AB)b, Kepler-34 (AB)b, and Kepler-35 (AB)b (Doyle et
al. 2011; Welsh et al. 2012). In the following, we will focus more on the
former, and a binary system, hereafter, refers to S type unless explicitly
noted otherwise.
According to the most recent summary of observations (Roell et al. 2012),
there are 57 S type planet-bearing binary systems 333In fact, 10 of them are
triple stellar systems, but with the third star being very far away and thus
exerting less effects on the binaries with planets, which, as a subsample of
extra-solar planetary systems, may provide some significant statistics. Here
we summarize several points worth noting.
1. 1.
_Binary separation (or orbital semimajor axis, $a_{B}$)._ Most S type systems
have a $a_{B}$ larger than 100 AU. However, there seems to be a pileup at
$a_{B}\sim 20$ AU with 4 systems: $\gamma$ Cephei (Hatzes et al. 2003), Gl 86
(Queloz et al. 2000b), HD 41004 (Zucker et al. 2004), and HD196885 (Correia et
al. 2008; Chauvin et al. 2011). Planets are slightly less frequent in
binaries with $a_{B}$ between 35 and 100 AU (Eggenberger et al. 2011). No
planet has been found in binaries with $a_{B}<10$ AU (excluding P type).
2. 2.
_Planetary mass_. Planets in wide binaries ($a_{B}>100$ AU) has a mass range
($0.01-10M_{J}$) that is close to those in single star systems but much more
extended than those ($0.1-10M_{J}$) in close binaries ($a_{B}<100$ AU) (Roell
et al. 2012).
3. 3.
_Planetary multiplicity_. Planets in close binaries ($a_{B}>100$ AU) are all
singleton, while those in wide binaries are diverse (Fig.3 of Roell et al.
(2012)). The occurrence rate of multiple planets in wide binaries is close to
that in single star systems (Desidera & Barbieri 2007).
4. 4.
_Planetary orbit_. Most extremely eccentric planets are found in wider
binaries (e.g., $e=0.935$ for HD 80606 b and $e=0.925$ for HD 20782 b). The
distribution od planetary eccentricity in binaries also seems to be different
compared to those in single star systems (Kaib et al. 2012). Planetary orbital
periods are slightly smaller in close binaries as compared to those in wide
binaries and single star systems (Desidera & Barbieri 2007).
How are these planets formed with double suns? Are they behaving in a similar
way as our solar system or other single star systems? In the following, we
review some important effects on planet formation and evolution in a binary
system as compared to those in a single star system, which may provide some
clues to answer these questions.
### 3.2 Binary Effects on a Protoplanetary Disk
#### 3.2.1 Disk Truncation
Planets are considered to be born in a protoplanetary disk. Such a disk, in
the solar system, could be extended to the location of the Kuiper belt, e.g.,
30-50 AU from the Sun. But in a binary system, the disk could be severely
truncated by the companion star. For the S type case, the typical radial size
of a truncated disk is about $20-40\%$ of the binary’s separation, depending
on the mass ratio and orbital eccentricity of the binary. For the P type case,
the binary truncates the circumbinary disk by opening a gap in the inner
region. The typical radial size of the gap is about 2-5 times the binary’s
separation distance. Various empirical formulas for estimating the boundary of
the truncated disk are given by Artymowicz & Lubow (1994); Holman & Wiegert
(1999); Pichardo et al. (2005)444The boundaries given by Holman & Wiegert
(1999) and Pichardo et al. (2005) are actually the boundaries of stable orbits
of a test particle. . The size range of the truncated disk puts the first
strict constraint on planet formation, determining where planets are allowed
to reside and how much material is available for their formation. The reason
why no S-type planet has been found in binaries with $a_{B}<10$ AU could be
that the truncated protoplanetary disk was too small to have enough material
for formation of a giant planet (Jang-Condell 2007).
#### 3.2.2 Disk Distortion
After the violent truncation process, the left-over, truncated disk, is still
subject to strong perturbations from the companion star, and thus it is not as
dynamically quiet as disks around single stars. First, a binary in an
eccentric orbit can also cause the disk to be eccentric (Paardekooper et al.
2008; Kley & Nelson 2008; Müller & Kley 2012). Second, if the binary orbital
plane is misaligned with respect to the disk plane, then binary perturbations
can cause the disk to become warped, twisted or even disrupted (Larwood et al.
1996; Fragner & Nelson 2010). Third, the eccentric, warped disk is precessing.
All the above effects cause planet formation in binary systems to be more
complicated than that in single star systems.
#### 3.2.3 Disk Lifetime
Estimating the lifetime of the protoplanetary disk is crucial as it provides a
strong constraint on the timescale of planet formation. Observations of disks
around single stars show that the typical disk life time is in the range 1-10
Myr (Haisch et al. 2001). Although disks around wide binaries show a similar
lifetime, those in close binaries ($a_{B}<40$ AU) show evidence of shorter
lifetime, i.e., $\sim 0.1-1$ Myr (Cieza et al. 2009). Such results are not
unexpected as disks in close binaries are truncated to a much smaller size and
thus have much smaller timescales of viscous evolution. In any case, such a
short disk lifetime requires that planets in close binaries (such as $\gamma$
Cephei) should form quickly, probably on a timescale less than 1 Myr.
### 3.3 Binary Effects on Planet Formation
We consider planet formation based on the core accretion scenario (Lissauer
1993; Chambers 2004) 555Gravitational instability is another candidate
scenario for planet formation in binaries (see Mayer et al. (2010) for a
review), starting from planetesimals (usually having a radius on the scale of
kilometers) embedded in a protoplanetary disk. This is the standard way that
people consider planet formation in single star systems, though planetesimal
formation itself is still unclear (Blum & Wurm 2008; Chiang & Youdin 2010).
Nevertheless, some observational indications imply that the first stages of
planet formation, i.e., dust settling and growing to planetesimals, could
proceed in binaries as well as in single star systems (Pascucci et al. 2008).
#### 3.3.1 Growing Planetesimals
One straightforward way for growing planetesimals is via mutual collisions and
mergers, as long as the collisional velocity $V_{col}$ is low enough. For a
protoplanetary disk around a single star system, if the disk turbulence is
weak, e.g., in a dead zone, growth by mutual collisions could be efficient,
and it is thought that planetesimals have undergone a runaway and oligarchic
phase of growth to become planetary embryos or protoplanets (Kokubo & Ida
1996, 1998). However, the situation becomes less clear in binary systems. On
one hand, the outcome of planetesimal-planetesimal collision is highly
sensitive to $V_{col}$ (Benz & Asphaug 1999; Stewart & Leinhardt 2009). On the
other hand, perturbations from a close binary companion can excite
planetesimal orbits and increase their mutual impact velocities, $V_{col}$, to
values that might exceed their escape velocities or even the critical
velocities for the onset of eroding collisions (Heppenheimer 1978; Whitmire et
al. 1998). This is a thorny problem for those binaries with separation of only
$\sim 20$ AU, such as $\gamma$ Cephei and HD 196885. Recently, many studies
have been performed to address this issue.
An earlier investigation by Marzari & Scholl (2000) found that the combination
of binary perturbations and local gas damping could force a strong orbital
alignment between planetesimal orbits, which significantly reduced $V_{col}$
despite relatively high planetesimal eccentricities. This mechanism was
thought to solve the problem of planetesimal growth until Thébault et al.
(2006) found the orbital alignment is size-dependent. Planetesimals of
different sizes align their orbits to different orientations, thus $V_{col}$
values between different sized planetesimals are still high enough to inhibit
planetesimal growth (Thébault et al. (2008, 2009) for S-type, and Meschiari
(2012) for P type). Moreover, the situation would become much more complicated
(probably unfavorable) for planetesimal growth if the eccentricity,
inclination and precession of the gas disk are also considered (Ciecielä G et
al. 2007; Paardekooper et al. 2008; Marzari et al. 2009; Beaugé et al. 2010;
Xie et al. 2011; Fragner et al. 2011; Batygin et al. 2011; Zhao et al. 2012).
Nevertheless, the problem could be somewhat simplified if the effects of a
dissipating gas disk are taken into account (Xie & Zhou 2008) and/or a smaller
inclination ($i_{B}<10^{\circ}$) between the binary orbit and the plane of the
protoplanetary disk is considered (Xie & Zhou 2009). Optimistically,
planetesimals may undergo a delayed runaway growth mode (called Type II
runaway) towards planets (Kortenkamp et al. 2001). In any case, however, it
seems that planetesimal-planetesimal collision is not an efficient way for
growing planetesimals in close binary systems.
An alternative way of growing planetesimals could be via accretion of dust
that they pass through in the disk. Both analytical studies and simulations
(Xie et al. 2010b; Paardekooper & Leinhardt 2010; Windmark et al. 2012) have
shown this could be promising to solve the problem of growing planetesimals
not only in binaries but also in single star systems (e.g., the well known
“meter-barrier” puzzle). For an efficient dust accretion to occur, one needs,
first, a source of dust, which could be either from the primordial
protoplanetary disk or from fragmentation of planetesimal-planetesimal
collisions, and second, weak disk turbulence to maintain a high volume density
of dust (Johansen et al. 2008).
#### 3.3.2 Formation of Terrestrial and Gaseous Planets
Once planetesimals grew to 100-1000 km in radius (usually called planetary
embryos or protoplanets), they are no longer as fragile as before. Their own
gravity is strong enough to prevent them from fragmenting by mutual
collisions. In such a case, most collisions lead to mergers and thus growth of
planetesimals. Hence, one way to speed up growth is by increasing $V_{col}$,
which is readily available in a binary star system. For close binaries, such
as $\alpha$ Centauri AB, simulations (Barbieri et al. 2002; Quintana 2004;
Quintana & Lissauer 2006; Quintana et al. 2007; Haghighipour & Raymond 2007;
Guedes et al. 2008) have shown that Earth-like planets could be formed in
10-100 Myr.
If a protoplanet reaches several Earth masses, the critical mass for
triggering a runway gas accretion, before the gas disk is depleted, then it
could accrete the surrounding gas to become a gaseous planet. Generally,
planets would stop gas accretion after they have cleared all the surrounding
gas and opened a gap. However, because of the binary perturbation, gas could
be pushed inward to refill the gap and finally accreted by the planet (Kley
2001), leading to a higher gas accretion rate and more massive gaseous
planets. Such an effect could partially explain one of the observed facts:
gaseous planets in close binaries are slightly more massive than those in
single star systems.
### 3.4 Binary Effects on Planetary Dynamical Evolution
#### 3.4.1 With a Gas Disk
Due to the complication of the problem itself, the studies of this aspect
mainly rely on numerical simulations. Kley & Nelson (2008) considered the
evolution of a low-mass planet (30 Earth masses) embedded in a gas disk of the
$\gamma$ Cephei system (S type). They found that the planet would rapid
migrate inward and accrete a large fraction of the disk’s gas to become a gas
giant planet, which is similar to the observed planet. For the circumbinary
case, i.e., P type, simulations (Pierens & Nelson 2007, 2008a, 2008b) showed
the results were sensitive to planet mass. Low mass planets (tens of Earth
masses) would successively migrate inward to the inner edge of the gas disk
and subsequently merge, scatter, and/or lock into a MMR. A high mass planet
($>$ Jupiter mass) would enter a 4:1 resonance with the binary, which pumped
up the eccentricity of the planet and probably led to instability. The model
favoring the low mass planet from the simulation is consistent with the recent
observation: the masses of the three confirmed circumbinary planets
(Kepler-16,-34,-35) are all $\leq$ Saturn’s mass.
#### 3.4.2 Without a Gas Disk
As the lifetime of the disk, typically $\leq$ 10 Myr, is only less than 1% of
that of a planet (typically on the order of Gyr), the subsequent gas free
phase could dominate the evolution of planets after they have formed. In fact,
several mechanisms are found to play an important role in shaping the final
structure of planetary systems in binaries.
* •
_Planet-planet scattering_. Multiple planets could form in a protoplanetary
disk, and because of damping from the gas disk , planets could maintain their
near circular orbits and thus avoid close encounters. Once the gas disk
dissipated, planet-planet interaction would excite the eccentricities of
planets, leading to close encounters and finally instability of the systems;
e.g., merger and/or ejection. Such a mechanism (usually called PPS) is thought
to explain the eccentricity distribution of observed giant planets (Ford &
Rasio 2008; Nagasawa et al. 2008; Chatterjee et al. 2008). In a binary system,
PPS would be more violent because of the additional perturbation from the
binary stars (Malmberg et al. 2007a). In S type binaries (especially those
with close separations and highly inclined and/or eccentric orbits),
simulations (Marzari et al. (2005), Xie et al. in prep.) have shown that PPS
often causes the system to finally have only a single planet, and the
remaining planet is usually the most massive one. Such results may explain one
observed fact: planets in close binaries are single and massive. In P type
binaries, PPS again favors a single planet. In addition, it predicts a
positive correlation between the planet’s orbital semimajor axis and
eccentricity (Gong et al. in prep.), which currently fit well to the three
confirmed circumbinary planets (Kepler-16,-34,-35). More P type planets
detected in the future will further test this correlation.
* •
_Lidov-Kozai Effect_. In an S type binary, if a planet is on a highly inclined
orbit666This could be either primordial or induced by planet-planet
scattering., then it could undergo the Lidov-Kozai effect (Kozai 1962; Lidov
1962). One of the most striking features of the effect is that the planet’s
eccentricity could be pumped to a very high value and oscillate with its
inclination out of phase. Recently, it has also been found that the planet
could flip its orbit back and forth when its eccentricity approaches unity
(Lithwick & Naoz 2011; Naoz et al. 2011a) if the binary orbit is eccentric,
hence exhibiting the so called eccentric Lidov-Kozai effect. One application
of this effect is that it could produce a HJ; when the planet oscillates to
very high eccentricity, with a very small periastron, tides from the central
star kick in and dampen its orbits to form a close planet (Wu & Murray 2003;
Fabrycky & Tremaine 2007). Recently, there have been examples of such
candidates showing evidence that they are on the way to being HJs via the
Lidov-Kozai effect (Socrates et al. 2012; Dawson & Johnson 2012). In addition,
as the planet could flip during the Lidov-Kozai evolution, there are
significant chances to form an HJ in a retrograde orbit (Naoz et al. 2011a),
which has been observed in some extrasolar systems (Triaud et al. 2010).
Nevertheless, depending on specific conditions, e.g, if general relativistic
effects and/or perturbation by another planet is relevant, the Lidov-Kozai
effect can be suppressed (Takeda et al. 2008; Saleh & Rasio 2009).
### 3.5 Non-Primordial Scenario
There is another possibility that a currently observed planet-bearing binary
was not the original one when the planet was born, namely the non-primordial
scenario. Various mechanisms can lead to such a result, and we briefly
summarize these two kinds as follows.
* •
_Encounters with other stars and/or planets._ A binary star system has a
larger collisional cross section than a single star and thus a larger chance
to have a close encounter with other stars, during which they could have their
planets lost and/or exchanged (Pfahl 2005; Marti & Beauge 2012). In the end,
the binaries probably dramatically changed their orbits, and the surviving
planets were probably excited to highly eccentric and/or inclined
orbits(Malmberg et al. 2007b; Spurzem et al. 2009; Malmberg et al. 2011). In
addition, free floating planets(FFPs) could be recaptured by flyby binary
stellar systems (Perets & Kouwenhoven 2012).
* •
_Steller Evolution._ If one of the binary component star evolves away from the
main sequence, it could induce instabilities in the planetary system in the
binary. Planets could bounce back and be forced between the space around the
two component binary stars (Kratter & Perets 2012). If a close binary star
evolve to some phase to have mass transfer, the mass lost from the donor star
could form a circumbinary disk, which could potentially harbor new planets
(Perets 2010).
## 4 Planet in star Clusters
Almost all the planets found now are around field stars. However the normal
theory of star formation predicts that most field stars are formed from a
molecular cloud, having the same initial mass function(IMF) as
stars($<3M_{\odot}$) in an open cluster indicating that these field stars
initially formed in clusters, e.g. our solar system’s, initial birth
environment was reviewed by Adams (2010). According to the chemical
composition of our solar system, our Sun may have formed in an environment
with thousands of stars, i.e. a star cluster or association. Thus scientists
are very interested in planet detection in clusters which would be more
effective than that around field stars due to many more objects existing in
the same size of a telescope’s field of view.
To survey planets around stars in a cluster, we have some advantages in
obtaining more effective and credible results. Some correlations between
planets occurrences as well as their properties and characteristics of their
host stars are not very clear due to the bias of measurements for these field
stars, such as age, mass, [Fe/H] etc. Large differences among these field
stars, especially the type of environment in the early stage, is a problem for
surveying the correlations. However, in one cluster, most of its members have
homogeneous physical parameters, i.e. age and [Fe/H]. The comparative study of
planets in clusters will provide more valid, credible correlations.
Unfortunately, except for some FFPs, few planets are found to be bounded
around members of either globular clusters(GCs) or open clusters(OCs). The
following sections will introduce the observational results and theoretical
works in both GCs and OCs respectively.
### 4.1 Planets in Globular Clusters
Because of the fruitful observation results of GCs and the huge number of
stars in GCs, especially main sequence stars(MSSs), people naturally expect to
find planets around these MSSs in GCs. As these stars are, on average about 50
times denser than field stars near the Sun, GCs have advantages for planet
searching. For example, in the two brightest GCs: $\omega$ Cen and 47 Tuc,
there are more than 60000 MSSs, approximately half of the total number of
Kepler targets. However, the extreme star density near the center of GCs
($10^{5}$ stars within a few arc min), requires an extremely high precision of
photometry. Until now, it has been hard to individually distinguish two nearby
stars in the core of GCs. The stars in the outer region of GCs are more widely
separated from each other, therefore they are more suitable for planet
searches.
The first planet system in GC was found in the nearest GC: M4, PSR B1620-26 b
(Backer et al. 1993), a $2.5M_{\rm J}$ planet around a binary radio pulsar
composed with a $1.35M_{\odot}$ pulsar and a $0.6M_{\odot}$ white dwarf.
However, if we focus on sun-like stars in GCs, no planets have been confirmed
until now.
To search for bounded planets around MSSs, some efforts have been made by
several groups. As the brightest GCs in the sky, 47 Tucanae and $\omega$
Centauri are good targets for planet searching by transiting. Using HST to
find planets in the core of 47 Tucanae, Gilliland et al. (2000) provided a
null result. In the outer halo, the same result was obtained by Weldrake et
al. (2005). Furthermore, Weldrake et al. (2007) found no bounded planets by
transiting in both of the two clusters, under the precision of $P<7$day,
$1.3-1.6R_{\rm J}$. The most recent works to find planets in the nearby
globular cluster NGC 6397 is contributed by Nascimbeni et al. (2012), butstill
no highly-significant planetary candidates have been detected for early-M type
cluster members.
Do the null results in GCs indicate the low occurrence of planets? For some
dense star environments, the stability of planets is crucial. Although planets
at $1$AU in the core of 47 Tuc can only survive around $10^{8}$yrs in such a
violent dynamical environment(Davies & Sigurdsson, 2001), planets at $10$AU
in the uncrowded halo of GCs can be preserved for several Gyrs (Bonnell et
al., 2001). Therefore HJs with periods around a few days can survive much
longer in the halo. If HJs formed near these cluster members, they would have
a chance to be detected in GCs (Fregeau et al., 2006).
The null results are mainly attributed to the low metallicity of these GCs.
Fischer & Valenti (2005) surveyed planet systems not far from the Sun, and
pointed out that the occurrence of gas giants depends on the metallicity of
their host stars. The most recent work by Mortier et al. (2012) found a
frequency of HJs $<1\%$ around metal-poor stars, while the frequency of gas
giants is $<2.36\%$ around stars with $[Fe/H]<-0.7$. Both 47 Tuc and $\omega$
Cen have a low [Fe/H](respectively -0.78 and $<-1$, from data collected by
Harris (1996)). Hence, these two GCs contains few HJs. Higher frequencies of
giant planet are expected in GCs with higher [Fe/H].
Additionally, different properties of a circumstellar gas disk, especially its
structure, might influence the final architecture of planet systems, e.g. if
the gas disk in GCs is depleted much faster due to Extreme-Ultraviolet(EUV)
and Far-Ultraviolet(FUV) evaporation from nearby massive stars (Matsuyama et
al., 2003), formation of gas giants may be unlikely, as well as the formation
of a hot planet. The different structure of a gas disk might not force the
planet to migrate inward enough to form hot planets, and naturally they are
hard to detect using transit.
In these old GCs, mass segregation is obvious due to energy equipartition,
i.e. massive objects concentrate in the center of the cluster while small
objects are easily ejected outside. Energy equipartition results in FFPs,
which might be ejected to become unbound by some mechanisms (Parker & Quanz
2012; Veras & Raymond, 2012) and have a lower-mass than stars. It is hard for
FFPs to stay in old GCs.
### 4.2 Planets in Open Clusters and Associations
None of the planets around solar-like stars are found in GCs,because of the
reasons mentioned before. OCs and associations which still contain lots of
MSSs, are also useful for planet searching. The main dissimilarities between
OCs and GCs are:
1. 1.
_Cluster ages_. OCs and associations are much younger than GCs, and have a
much larger [Fe/H], probably leading to more planets being formed around the
cluster members.
2. 2.
_The dynamical environments_. The dynamical environment in OCs and
associations is still less violent than that in GCs because of lower star
density, which can preserve the two-body systems more easily than in GCs.
3. 3.
_Binary fraction_. The much larger fraction of binary systems in OCs than GCs
is a good way to understand the formation of planet systems in binary stars.
Additionally, many more OCs and associations($\sim 1200$) are observed than
GCs ( $\sim 160$ ) in our galaxy. Due to these dissimilarities, a higher
probability of planet detection is expected.
As for the different properties of OCs, surveyed planets in OCs have their own
values. Some OCs are only a few Myr old, e.g. NGC6611(Bonatto et al., 2006)
and NGC 2244 (Bonatto & Bica, 2009). Their ages are comparable with the
timescale of planet formation. Surveying planets and circumstellar disks in
these very young clusters will provide valuable samples to check and enhance
the current theories of planet formation, particularly the influences via
different environments in clusters during the early stages of planet
formation.
#### 4.2.1 Bounded Planets and Debris Disks
Many groups have made efforts to search for planets by transits in OCs: e.g.
Bruntt et al. (2003) in NGC 6791, Bramich et al. (2005) in NGC 7789, Rosvick &
Robb (2006) in NGC 7086, Mochejska et al. (2006) in NGC 2158, etc. Only few
candidates were found but none were confirmed. The most significant progresses
were made in 2007. In NGC 2423, a gas giant with a minimum mass of $10.6M_{\rm
J}$ around a $2.4M_{\odot}$ red giant was found by Lovis & Mayor (2007) using
RV measurement. Another planet was soon found soon afterward by RV around the
giant star $\epsilon~{}$Tauri (Sato et al., (2007)) in the Hyades, the
nearest OC, with a minimum mass of $\sim 7.6M_{\rm J}$ and a period of $\sim
595$ days. Using transit, some smaller candidates have also been found without
RV confirmation, e.g. a single transit of a candidate $\sim 1.81M_{\rm J}$ in
NGC 7789 found by Bramich et al. (2005), which may indicate another exoplanet
with a long period. Most recent work by Quinn et al. (2012) claims that they
found two HJs by RV: Pr0201b and Pr0211b in Praesepe, these planet are the
first known HJs in OCs. Parameters describing these planets are listed in
Table 1 as well as properties of their host cluster.
Table 1: Parameters of the planets and their host stars in OCs, as well as
their host cluster. Data are from Lovis & Mayor (2007); Sato et al., (2007);
Harris (1996); Quinn et al. (2012)
. $M_{\rm p}\sin i$ Period semi-major ecc $M_{\rm star}$ host cluster age
[Fe/H] dist $(M_{\rm J})$ (day) axis (AU) $M_{\odot}$ (Gyr) (pc) $10.6$
$714.3$ $2.1$ $0.21$ $2.4$ $NGC2423$ $0.74$ $0.14$ $766$ $7.6$ $594.9$ $1.93$
$0.15$ $2.7$ $Hyades$ $0.6$ $0.19$ $47$
Compared with the null results in GCs, the encuraging results of planet
searching in OCs confirm the formation and survival ability of planets in
cluster environment, especially observations of the circumstellar disk in
young OCs, which is related to the occurrence of planet formation.
Haisch et al. (2001) showed the fraction of disks in OCs decayed with their
ages. Some recent results verify this correlation: $30-35\%$ of T-Tauri stars
have a disk in the $\sigma$ Ori cluster with ages $\sim 3$ Myr(Hernández et
al., 2007). Using the Chandra X-ray Observatory, Wang et al. (2011) found a
K-excess disk frequency of $3.8\pm 0.7\%$ in the $5\sim 10r$My old cluster:
Trumpler 15.
Although the disk structures around cluster members are not well known, the
large fraction of gas disk in very young OCs makes the formation of planets
possible, especially for gas giants. The two confirmed planets found were not
HJs, but another two planets found most recently are HJs. However, lack of
more samples is a big problem in making a credible conclusion and surveying
the statistical characteristics of planet formation and evolution in OCs.
#### 4.2.2 Free-Floating Planets
Ages, metallicity and star density are the main dissimilarities between OCs
and GCs. The formation of planets in OCs is thought to be common, but few
planets bound around stars have been observed. However a population of FFPs
has been found in OCs. In 2000, Lucas & Roche (2000) found a population of
FFPs in Orion. Bihain et al. (2009) also found three additional FFPs with
$4-6M_{\rm J}$ in the $\sim 3$ Myr old OC: $\sigma$ Orions. A huge number
(nearly twice the number around MSSs) of unbound planets have been found in
the direction of the Galactic Bulge (Sumi et al., 2011).
These planets have multiple origins. They may also form around some cluster
members, but were ejected out of the original systems and cruise into clusters
(Sumi et al., 2011). Because of their young ages, energy equipartition in OCs
is less effective than that in GCs. The dissolution timescale for objects to
escape from a cluster is $t_{\rm dis}\sim 2{\rm
Myr}\times\frac{N}{\ln(0.4N)}\times\frac{R_{\rm G}}{\rm kpc}$ (Baumgardt &
Makino, 2003). For a typical OC, with $N=1000$ stars at distance $R_{\rm
G}=1$kpc, $t_{\rm dis}\sim 0.1$Gyr and therefore FFPs can still stay in their
host clusters for most young OCs. It is hard to find the original host stars
of these FFPs, but surveying them is still useful for evaluating the frequency
of planet formation in OCs and GCs.
### 4.3 Planetary Systems in Clusters: theoretic works
The planet occurrent rate including formation rate and stability related with
the cluster environment is very important for predicting the rate of further
observations. From their respective dynamics, the large distinctions between
OCs and GCs generally predict more planets in OCs and HJs in halo of GCs.
Dynamical works focus on the stability and orbital architecture of planetary
systems in a cluster. Considering a fly-by event, the previous works show the
stability of planet systems depends on the bounded energy of planetary
systems, fly-by parameters as well as the star density of the environment,
which decides the occurrent rate of a fly-by event ($t_{\rm enc}\propto 1/n$,
Binney & Tremaine, 1987). Spurzem et al. (2009) used a strict N-body
simulation as well as a Monte Carlo method to survey the dynamical
characteristics, especially the effective cross section of planetary systems
with different orbital elements in a cluster’s gravity field. Adding
substructure of a young OC by Parker & Quanz (2012), the fraction of liberated
planet depends on the initial semi-major axis and virial parameter. The planet
systems in binary systems were also been surveyed by Malmberg et al. (2007a,
b, 2009). They considered encounters between a binary system and a single
star. After obvious changes of the inclination, a fraction of planets will
suffer the Kozai effect after encounters and consequently show instabilities.
The stability and orbital architecture of multi-planet systems in clusters
still need to be surveyed in further works, because planet-planet interactions
play an important role in deciding the final configuration of a planet system
after fly-bys. The dynamical evolution in clusters is much more complex than
in a single fly-by. In some very open clusters, the tidal effect can also
disrupt planet systems in the outer region. The effect of interstellar gas in
very young OCs is still uncertain. The fine structure of the circumstellar
disks still needs to be investigated during the formation of planet systems.
Planet formation in star clusters must have a strong dependence on the
physical and dynamical environments of their host stars. The environments in
clusters are very different from that around field stars, or binary pairs,
e.g. the different properties of the circumstellar disk, dynamical
instabilities in different stages during planet formation, as well as the
stability of a planetary system after the planets are formed. The protoplanet
gas disk plays a very important role in the formation of gas giant planets. A
comparison between the timescale of gas disk dispersion and that of gas giant
formation is a crucial clue to judge the formation rate of giant planets. On
the other hand, the observation of circumstellar disks and giant planets
(including FFPs) in some very young OCs, can also give a limit on the rate
that a planetary gas disk is preserved, which is related to the planet
formation rate in a cluster environment. The distinctions in the different
environments for OCs and small bounded planet samples in OCs have limited our
knowledge about the formation of planets in clusters.
## 5 Conclusions
With the increasing data of observed exoplanets, the study of orbital
architectures for multiple planet systems becomes timely. Unlike the
relatively mature theory for formation of a single planet (except for some
bottleneck problems), the properties of planet’s architecture is relatively
far from clear. Dynamical factors, such as interactions among planets, tidal
interactions with the host star and a protostellar disk, or in some cases
perturbations from a third companion (a star or brown dwarf), etc, tend to
sculpt the orbital evolution and sculpt the final architectures of the planet
systems.
According to our present knowledge, we tentatively classify the planet systems
around single stars into three major catalogs: HJ systems, standard systems
and distanct giant planet systems. The standard systems can be further
categorized into three sub-types under different circumstances: solar-like
systems, hot super-Earth systems and sub-giant planet systems. The
classification is based on the major process that occurred in their history.
It may help to predict unseen planets, as well as to understand the possible
composition of planets, since through the history of their evolution, we can
judge whether large orbital mixing has occurred.
Due to the presence of a third companion, planet formation in a binary
environment has raised some more challenging problems, especially for the
stage of planetesimal formation. Anyway, the observed exoplanets around binary
stars, especially the circumbinary exoplanets like Kepler 34b and 35b,
indicate that planet formation is a robust procedure around solar type stars.
Planets in clusters will provide a useful clue for understanding the formation
of planets in a cluster environment. Although only very limited observational
results have been obtained, theories can still predict some properties of
exoplanets in clusters. Planet samples in some young OCs might be especially
interesting for revealing the difference between planet formation around field
stars and members of clusters.
###### Acknowledgements.
We would like to thank Drs. G.Marcy, S.Udry, N.Haghighipour,M.Nagasawa, Y.Q.
Wu, A.V.Tutukov and R.Wittenmyer, I.Boisse for their helpful suggestions. The
work is supported by the National Natural Science Foundations of China
(Nos.10833001,10925313,11078001 and 11003010), Fundamental Research Funds for
the Central Universities(No.1112020102), the Research Fund for the Doctoral
Program of Higher Education of China(No.20090091110002 and 20090091120025).
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|
arxiv-papers
| 2012-07-09T10:31:51 |
2024-09-04T02:49:32.814971
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "J.-L. Zhou, J.-W. Xie, H.-G. Liu, H. Zhang, Y.-S. Sun",
"submitter": "Hui-gen Liu",
"url": "https://arxiv.org/abs/1207.2002"
}
|
1207.2187
|
# Dynamics of Feshbach Molecules in an Ultracold Three-Component Mixture
Alexander Y. Khramov Anders H. Hansen Alan O. Jamison William H. Dowd
Subhadeep Gupta Department of Physics, University of Washington, Seattle WA
98195
(August 27, 2024)
###### Abstract
We present investigations of the formation rate and collisional stability of
lithium Feshbach molecules in an ultracold three-component mixture composed of
two resonantly interacting fermionic 6Li spin states and bosonic 174Yb. We
observe long molecule lifetimes ($>\,100\,$ms) even in the presence of a large
ytterbium bath and extract reaction rate coefficients of the system. We find
good collisional stability of the mixture in the unitary regime, opening new
possibilities for studies and probes of strongly interacting quantum gases in
contact with a bath species.
Magnetic Feshbach resonances allow precise control of collisional properties,
making them a key tool in ultracold atom systems. They have been extensively
used to study ultracold molecules, as well as few- and many-body physics Chin
et al. (2010). Two-component Fermi gases near a Feshbach resonance provide
excellent opportunities to study strongly interacting quantum systems Giorgini
et al. (2008). This is possible due to the remarkable collisional stability of
the atom-molecule mixture on the positive scattering length side of the
resonance Jochim et al. (2003); Cubizolles et al. (2012), attributed largely
to Fermi statistics Esry et al. (2001); Petrov et al. (2004). Extending the
system to three-component mixtures in which only two are resonantly
interacting Spiegelhalder et al. (2009) offers the exciting possibility of
modifying or probing pairing dynamics by selective control of the third
component. A third-component may also be used as a coolant bath for exothermic
molecule-formation processes, provided that inelastic processes with the bath
are negligible. In the context of many-body physics, a third non-resonant
component can be useful as a microscopic probe of superfluid properties
Spiegelhalder et al. (2009); Targonska and Sacha (2010), as a stable bath for
studies of non-equilibrium phenomena Robertson and Galitski (2009), or as an
accurate thermometer of deeply degenerate fermions Nascimbène et al. (2011).
Collisional stability of Feshbach molecules in the absence of Fermi statistics
becomes a crucial question for multi-component mixtures D’Incao and Esry
(2008); Zirbel et al. (2008); Spiegelhalder et al. (2009). A recent
theoretical analysis of such mixtures suggests a possibility for enhanced
molecule formation rates with good collisional stability D’Incao and Esry
(2008). Enhanced atom loss has been observed near a 6Li _p_ -wave resonance in
the presence of a 87Rb bath Deh et al. (2008), while a small sample of the
probe species 40K has been found to be stable within a larger strongly
interacting 6Li sample Spiegelhalder et al. (2009).
In this paper we investigate a mixture composed of two resonantly interacting
spin states of fermionic 6Li immersed in a large sample of bosonic 174Yb
atoms. While the Li interstate interactions are arbitrarily tunable by means
of an $s$-wave Feshbach resonance at $834\,$G Dieckmann et al. (2002), the
interspecies interactions between Li and Yb are constant and small Ivanov et
al. (2011). We present the first observations of formation and evolution of
Feshbach molecules in a bath of a second atomic species. In the unitary
regime, we observe good collisional stability of the mixture with elastic
interactions dominating over inelastic losses. We extract the reaction rate
constants from a classical rate equations model of the system.
Our experimental procedure has been described in earlier work Hansen et al.
(2011). Briefly $3\\!\times\\!10^{6}$ atoms of 174Yb in the ${}^{1}S_{0}$
state and up to $4\times 10^{4}$ atoms of 6Li , distributed equally between
the two ${}^{2}S_{1/2}$, $F\\!=\\!\frac{1}{2}$ states (denoted Li$|1\rangle$,
Li$|2\rangle$), are loaded from magneto-optical traps into a crossed-beam
optical dipole trap. We then perform forced evaporative cooling on Yb to the
final trap depth $U_{{\rm Yb(Li)}}=15(55)\,\mu$K, with mean trap frequency
$\bar{\omega}_{{\rm Yb(Li)}}=2\pi\times 0.30(2.4)\,$kHz odt (2012), during
which Li is cooled sympathetically by Yb. Following evaporation, the mixture
is held at constant trap depth to allow inter-species thermalization. With a
time constant of $1\,$s the system acquires a common temperature $T_{{\rm
Yb}}\,=\,T_{{\rm Li}}\,=\,2\,\mu$K with atom number $N_{{\rm Yb(Li)}}=2\times
10^{5}(3\times 10^{4})$. This corresponds to $T_{{\rm Li}}/T_{F}\simeq 0.4$,
and $T_{{\rm Yb}}/T_{C}\simeq 2.5$, where $T_{F}$ is the Li Fermi temperature
and $T_{C}$ is the Yb Bose-Einstein condensation temperature bec (2012).
Figure 1: Li atom loss spectroscopy in the presence (filled circles) and
absence (open squares) of an Yb bath near the 6Li $834\,$G Feshbach resonance
(inset). We plot the number of Li atoms after $500\,$ms of evolution
normalized to that at $10\,$ms. The thick dashed line indicates the resonance
center and the thin dashed line indicates the magnetic field at which
$\epsilon_{B}=k_{B}T_{\rm Li}$ for the initial conditions.
After this initial preparation, we ramp up the magnetic field to a desired
value and observe the system after a variable hold time. For fields in the
vicinity of the Feshbach resonance, there is a field-dependent number loss and
heating for the Li cloud during the $20\,$ms ramp time, resulting in $T_{{\rm
Li}}$ rising to as high as $4.5\,\mu$K. At this point, the density-weighted
average density $\langle n_{{\rm Yb(Li)}}\rangle$ is $2.6(0.35)\times
10^{13}\,$cm-3. For interrogation in the absence of the bath, Yb is removed
from the trap with a $1\,$ms light pulse resonant with the
${}^{1}S_{0}\rightarrow{{}^{1}P_{1}}$ transition ybb (2012). Atom number and
temperature are monitored using absorption imaging for both species after
switching off the magnetic field.
We first present our results on atom loss spectroscopy near the Feshbach
resonance (see Fig. 1). The atom-loss maximum obtained in the absence of Yb
has been observed previously Dieckmann et al. (2002) and can be explained as a
result of the formation and subsequent decay of shallow lithium Feshbach
dimers Jochim et al. (2003); Cubizolles et al. (2012); Chin and Grimm (2004);
Kokkelmans et al. (2004); Zhang and Ho (2011) which form only on the positive
$a$ side of the resonance. Here $a$ denotes the Li$|1\rangle$-Li$|2\rangle$
scattering length. In the presence of the Yb bath, the loss feature is shifted
and broadened. We interpret the behavior of the mixture in terms of five
chemical processes:
$\displaystyle\mathrm{Li|1\rangle+Li|2\rangle+Li}\hskip 11.38109pt$
$\displaystyle\rightleftharpoons$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Li\hskip
17.64069pt(+\epsilon_{B})}$ (I)
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Li}\hskip 11.38109pt$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{d}}+Li\hskip
17.64069pt(+\epsilon_{D})}$ (II)
$\displaystyle\mathrm{Li|1\rangle+Li|2\rangle+Yb}\hskip 11.38109pt$
$\displaystyle\rightleftharpoons$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Yb\hskip
14.22636pt(+\epsilon_{B})}$ (III)
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Yb}\hskip 11.38109pt$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{d}}+Yb\hskip
14.22636pt(+\epsilon_{D})}$ (IV)
$\displaystyle\mathrm{Li|1\rangle+Li|2\rangle+Yb}\hskip 11.38109pt$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{d}}+Yb\hskip
14.22636pt(+\epsilon_{D})}$ (V)
Forward process (I) corresponds to a three-body collision event which produces
a shallow Feshbach dimer (denoted $\mathrm{Li_{2}^{\mathnormal{s}}}$)
accompanied by the release of the dimer binding energy
$\epsilon_{B}=\frac{\hbar^{2}}{2m_{{\rm Li}}a^{2}}$. Li denotes a 6Li atom in
either of the two spin states. Process (II) corresponds to two-body loss to a
deeply bound dimer (denoted $\mathrm{Li_{2}^{\mathnormal{d}}}$) with binding
energy $\epsilon_{D}$. Processes (III) and (IV) are similar to (I) and (II)
with the spectator atom being Yb rather than Li processiv (2012). Process (V)
corresponds to direct three-body loss to a deeply-bound molecule. Processes
(II, IV, V) always result in particle loss from the trap since
$\epsilon_{D}\gg U_{\rm Li}$. Vibrational relaxation due to collisions between
${\rm Li}_{2}^{\mathnormal{s}}$ Feshbach molecules may contribute at the
lowest fields, but has a negligible rate for the fields at which we perform
our analysis Jochim et al. (2003); Zhang and Ho (2011). We have experimentally
checked that direct three-body loss processes to deeply-bound states involving
three Li atoms as well as those involving one Li atom and two Yb atoms are
negligible for this work threebody (2012). Three-body losses involving Yb
atoms alone have a small effect Takasu et al. (2003) and are taken into
account in our analysis.
Figure 2: Evolution of Li Feshbach molecule number at 709 G without (a) and
with (b) an Yb bath. The numbers are obtained by comparing Li atom numbers
(insets) ramped across resonance (diamonds) or not (open squares) as described
in the text. Lower inset also shows Yb number (filled squares). The curves are
fits with a rate equations-based model.
In the absence of Yb, only processes (I) and (II) contribute. If we neglect
loss process (II), the atom-molecule mixture approaches an equilibrium,
characterized by an equality of the forward and reverse rates and an
equilibrium molecule fraction $\frac{2N_{m}}{N_{\rm
Li}+2N_{m}}=\left(1+\frac{e^{-\epsilon_{B}/k_{B}T}}{\phi_{\mathrm{Li}}}\right)^{-1},$
where $N_{m}$ is the molecule number and $\phi_{{\rm Li}}$ is the phase space
density for each spin component in the ground state of the trap Chin and Grimm
(2004); Kokkelmans et al. (2004); Zhang and Ho (2011). The timescale for
achieving equilibrium depends on the three-body rate constant $L_{3}$ for
process (I), which scales with the scattering length as $a^{6}$, whereas rate
constant $L_{2}$ for process (II) scales as $a^{-3.3}$ Petrov (2003); D’Incao
and Esry (2005). The shape of the loss spectrum can thus be qualitatively
explained by noting that the dimer formation rate increases with magnetic
field while equilibrium dimer fraction and molecule decay rate decrease. The
large rate for process (I) at high fields close to resonance ensures
equilibrium molecule fraction at all times. Broadly speaking, the rate-
limiting step determining the system evolution is the molecule formation rate
at low fields and decay rate at high fields. The trap depth also affects the
loss spectrum shape, since it determines the magnetic field range over which
the formed shallow dimers remain trapped.
In the presence of Yb, the additional dimer formation (III), dimer decay (IV),
and 3-body loss (V) processes contribute. The observed loss spectrum is
broadened on the higher field side, suggesting that for our parameters,
processes (IV) and/or (V) play an important role while process (III) does not.
The rate constants $L^{\prime}_{3}$, $L^{\prime}_{2}$, and $L_{3}^{d}$, for
processes (III), (IV) and (V), have theoretical scalings $a^{4}$, $a^{-1}$,
and $a^{2}$, respectively D’Incao and Esry (2008); Petrov (2003). Overall, we
see two regimes of behavior - a lossy one where molecule formation is
energetically favored ($\epsilon_{B}>k_{B}T_{\rm Li}$) and a stable one closer
to resonance ($\epsilon_{B}<k_{B}T_{\rm Li}$). The criterion
$\epsilon_{B}=k_{B}T$ separating these two regimes is equivalent to $ka=1$
where $\frac{\hbar^{2}k^{2}}{2m_{\rm Li}}=k_{B}T_{\rm Li}$, i.e., the unitary
criterion.
In order to expand upon this qualitative picture, we study the time evolution
of the three-component mixture at representative magnetic fields in the above
two regimes. We are then able to extract quantitative information for the
above processes from a rate-equations model of the system.
Fig. 2 shows the Li atom and molecule number evolution at $709\,$G
($\epsilon_{B}=k_{B}\times 8.3\,\mu$K) a field value where modifications due
to the Yb bath are apparent in Fig. 1. The number of Feshbach molecules at a
particular field is determined by using a procedure similar to earlier works
Jochim et al. (2003); Cubizolles et al. (2012). After variable evolution time,
we ramp the magnetic field with a speed of $40\,$G/ms either up to $950\,$G,
which dissociates the molecules back into atoms that remain in the trap, or to
$506\,$G, which does not. We then rapidly switch off the magnetic field and
image the atomic cloud. The molecule number is obtained from the number
difference in the two images (see insets in Fig. 2).
We see that the presence of Yb alters the molecule decay rate while the
formation rate is unchanged. The Feshbach molecules appear to coexist for a
long time ($>100\,$ms) with the Yb bath, even in the absence of Pauli blocking
D’Incao and Esry (2008). We adapt the recent rate-equations analysis of
Feshbach losses in a Fermi-Fermi mixture Zhang and Ho (2011) to incorporate a
third component, temperature evolution, and trap inhomogeneity. $T_{{\rm
Li}}/T_{F}>0.5$ is satisfied throughout the measurement range, allowing a
classical treatment of the Li cloud. We model the density evolutions due to
processes (I-V) using:
$\displaystyle\dot{n}_{m}$ $\displaystyle=$ $\displaystyle
R_{m}+R^{\prime}_{m}-L_{2}n_{m}n_{{\rm Li}}-L^{\prime}_{2}n_{m}n_{{\rm Yb}}$
(1) $\displaystyle\dot{n}_{{\rm Li}}$ $\displaystyle=$
$\displaystyle-2R_{m}-2R^{\prime}_{m}-L_{2}n_{m}n_{{\rm Li}}-2L_{3}^{d}n_{{\rm
Li}}^{2}n_{{\rm Yb}}$ (2) $\displaystyle\dot{n}_{{\rm Yb}}$ $\displaystyle=$
$\displaystyle-L^{\prime}_{2}n_{m}n_{{\rm Yb}}-L_{3}^{d}n_{{\rm
Li}}^{2}n_{{\rm Yb}}.$ (3)
Here $n_{m}$, $n_{{\rm Li}}$ and $n_{{\rm Yb}}$ are the densities of shallow
dimers $\mathrm{Li_{2}^{\mathnormal{s}}}$, Li atoms and Yb atoms,
respectively. $R_{m}(R^{\prime}_{m})=\frac{3}{4}L_{3}(L^{\prime}_{3})n_{{\rm
Li}}^{2}n_{{\rm Li(Yb)}}-qL_{3}(L^{\prime}_{3})n_{m}n_{{\rm Li(Yb)}}$ is the
net-rate for molecule production via process (I)((III)). We determine $q$
through the constraints on the molecule fraction at equilibrium
($R_{m}(R^{\prime}_{m})=0$). We obtain an upper bound for $L_{3}^{d}$ by
observations at large negative $a$ (described below) which indicates a
negligible effect for the data in Fig. 2, allowing us to set $L_{3}^{d}=0$ for
the analysis at $709\,$G.
Figure 3: The evolution of temperature and number at $810\,$G for Li atomic
cloud with Yb (filled circles) and without (empty circles) and also for Yb in
the presence of Li (filled squares). The curves are fits with a rate
equations-based model.
The time evolution of $T_{{\rm Li}}$ and $T_{{\rm Yb}}$ are modeled
considering the energy deposition from processes (I) and (III) as well as
heating from the density-dependent loss processes (II), (IV) and (V) Weber et
al. (2003). In addition, our model also takes into account the effects of
evaporative cooling O’Hara and Thomas (2001), inter-species thermalization
Ivanov et al. (2011), one-body losses from background gas collisions, and Yb
three-body losses Takasu et al. (2003); Weber et al. (2003). The Li scattering
length at 709 G is $a=1860\,a_{0}$, ensuring rapid thermalization ($<1\,$ms)
in the lithium atom-Feshbach molecule mixture Petrov (2003). This allows the
assumption of equal temperature $T_{{\rm Li}}$ for lithium atoms and Feshbach
molecules. The heating from molecule-formation at $709\,$G dominates over
inter-species thermalization, maintaining $T_{\rm Li}\simeq 4.5\,\mu$K and
$T_{\rm Yb}\simeq 2\,\mu$K, as observed in both experiment and model.
The best-fit rate coefficients extracted from the atom data (shown in the
insets) are $L_{3}=(1.4\pm 0.3)\times 10^{-24}\,{\rm cm}^{6}/$s,
$L_{2}=(1.3\pm 0.3)\times 10^{-13}\,{\rm cm}^{3}/$s, and
$L^{\prime}_{2}=(2.3\pm 0.2)\times 10^{-13}\,{\rm cm}^{3}/$s. $L^{\prime}_{3}$
is consistent with 0. All reported uncertainties are statistical. The $L_{3}$
value is consistent with that obtained in Jochim et al. (2003), after
accounting for the slight differences in experimental parameters. Using
$L^{\prime}_{2}\langle n_{{\rm Yb}}\rangle$ as a measure of the dimer decay
rate, we get $170\,$ms as the lifetime of a Li Feshbach molecule in the Yb
bath.
We now turn to the unitary regime, where we choose 810 G ($ka=+6$,
$\epsilon_{B}=k_{B}\times 0.11\,\mu$K) as our representative field to study
the mixture properties. It is difficult to reliably observe the molecule
number using our earlier method in this regime, so we only monitor the atoms
(see Fig. 3). Starting with an inter-species temperature differential as
before, we observe a fast drop in $T_{{\rm Li}}$ in the presence of Yb and
clear evidence of inter-species thermalization. The Li number in the three-
component mixture exhibits a long $1/e$ lifetime of $2\,$s, far larger than at
$709\,$G. However this is still an order of magnitude shorter than that
obtained in the absence of Yb. The interpretation of the decay is not
straightforward as both two-body (process (IV)) and three-body (process (V))
inelastic loss can contribute Du et al. (2009); Spiegelhalder et al. (2009).
The large rate for process (I) in this regime ensures equilibrium molecule
fraction at all times. By fitting to data taken at $935\,$G where $ka=-2$ and
process (V) is expected to dominate inelastic loss, we obtain
$L_{3}^{d}=(4.3\pm 0.3)\times 10^{-28}\,{\rm cm}^{6}/$s. This sets a lower
bound for $L_{3}^{d}$ at $810\,$G. We fit the first $2.5\,$s of data in Fig. 3
after fixing $L^{\prime}_{2}$ to its value scaled from $709\,$G and find
$L_{3}^{d}=(9.5\pm 0.5)\times 10^{-28}\,{\rm cm}^{6}/$s at $810\,$G. The
slight disagreement in Li atom number at long times may be due to a small
($<10\%$) inequality in our spin mixture composition, which the model does not
take into account.
The qualitative features of both spectra in Fig. 1 can be theoretically
reproduced by using field-dependent reaction coefficients scaled from our
measured values at $709$ and $810\,$G. However, a full quantitative comparison
will need to take into account the theoretical deviations from scaling
behavior in the unitary regime as well as experimental variations in the
initial temperature, and is open to future investigation.
By extending the forced evaporative cooling step, lower temperature mixtures
can be produced where bosonic 174Yb shrinks to a size smaller than the Fermi
diameter of the 6Li cloud. Such experiments at $834\,$G yield $T_{{\rm
Li}}/T_{F}\simeq 0.25$ with $N_{{\rm Yb}}=N_{{\rm Li}}=2.5\times 10^{4}$.
Here, the estimated volume of the Yb sample is $\simeq 0.3$ of the Li sample
volume, compared to 3.3 in the classical regime. The mixture is thus also
capable of achieving the opposite regime of a second species being immersed
inside a strongly interacting quantum degenerate Fermi gas, similar to earlier
studies in the K-Li mixture Spiegelhalder et al. (2009).
Our experiments with the Yb-Li mixture near a Feshbach resonance demonstrate
effects of an additional species on chemical reaction rates in the microKelvin
regime. We observe a long lifetime for Feshbach molecules, even in the absence
of Pauli blocking. Our demonstrated stability of the mixture near the unitary
regime of the resonance opens various possibilities of studying strongly
interacting fermions immersed in a bath species or being interrogated by a
small probe species. Future experimental opportunities include realizations of
non-equilibrium states, and studies of superfluid properties, for instance by
controlled relative motion between the two species. Finally, our results
constitute an advance in the manipulation of ultracold mixtures of alkali and
alkaline-earth-like atoms Hara et al. (2011); Hansen et al. (2011); Baumer et
al. (2011). These mixtures may be used to produce quantum gases of
paramagnetic polar molecules which are promising for diverse applications such
as quantum simulation of lattice spin models Micheli et al. (2006), tests of
fundamental symmetries Hudson et al. (2011), and probes of time variations in
fundamental constants Kajita et al. (2011).
We thank Ben Plotkin-Swing for experimental assistance, and J.P. D’Incao,
T.-L. Ho and M.W. Zwierlein for helpful discussions. We gratefully acknowledge
support from the National Science Foundation and the Air Force Office of
Scientific Research. A.K. thanks the NSERC.
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* Du et al. (2009) X. Du, Y. Zhang, and J. E. Thomas, Phys. Rev. Lett. 102, 250402 (2009).
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|
arxiv-papers
| 2012-07-09T21:25:13 |
2024-09-04T02:49:32.837180
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Y. Khramov, A. H. Hansen, A. O. Jamison, W. H. Dowd, S. Gupta",
"submitter": "Alexander Khramov",
"url": "https://arxiv.org/abs/1207.2187"
}
|
1207.2195
|
# BRIEF REVIEW OF CHARM PHYSICS
Marco Gersabeck CERN, 1211 Geneva, Switzerland
marco.gersabeck@cern.ch
###### Abstract
Charm physics has attracted increased attention after first evidence for charm
mixing was observed in 2007. The level of attention has risen sharply after
LHCb reported first evidence for $C\\!P$ violation in the charm sector.
Neither mixing nor $C\\!P$ violation have been established by a single
unambiguous measurement to date. This review covers the status of mixing and
$C\\!P$ violation measurements and comments on the challenges on the road
ahead, both on the experimental and theoretical side, and on ways to tackle
them.
###### keywords:
Charm physics; Meson mixing; CP violation.
Received (Day Month Year)Revised (Day Month Year)
PACS Nos.: 13.20.Fc, 13.25.Ft, 14.40.Lb
## 1 Introduction
Charm physics covers the studies of a range of composite particles containing
charm quarks which provide unique opportunities for probing the strong and
weak interactions in the standard model and beyond. The charm quark, being the
up-type quark of the second of the three generations, is the third-heaviest of
the six quarks. Charm particles can exist as so-called open charm mesons or
baryons, containing one or several (for baryons) charm quarks, or as
charmonium states which are bound states of charm-anticharm quark pairs.
The uniqueness of charm particles lies in their decays. The charm quark can
only decay via annihilation with an anti-charm quark in the case of charmonium
states or as a weak decay, mediated by a $W^{\pm}$-boson, into a strange or
down quark. Thus, open charm particles are the only ones allowing the study of
weak decays of an up-type quark in a bound state.
In 2009, Ikaros Bigi asked whether charm’s third time could be the real charm
[1]. Charm’s first time was the discovery of the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ [2, 3], which followed
three years after the possible first observation of an open charm decay in
cosmic ray showers [4]. This discovery confirmed the existence of a fourth
quark as expected by the GIM mechanism [5] motivated by the non-existence of
flavour-changing neutral currents [6, 7] in conjunction with the observation
of the mixing of neutral kaons [8, 9, 10]. The second time charm attracted
considerable attention was caused by the observation of $D_{\mathrm{s}J}$
states [11, 12, 13, 14] which could not be accommodated by QCD [15, 16, 17,
18, 19]. Until today, excited charmonium and open charm particles provide an
excellent laboratory for studying QCD, however, this topic is beyond the scope
of this review.
Charm’s third time started with the first evidence for the mixing of neutral
charm mesons reported by BaBar [20] and Belle [21] in 2007. Since then a lot
of work went into more precise measurements of the mixing phenomenon as well
as into searches for charge-parity ($C\\!P$) symmetry violation in the charm
sector. At the same time theoretical calculations were improved even though
precise standard model predictions are still a major challenge. This paper
reviews the current situation of studies of processes, which are mediated by
the weak interaction at leading order, using open charm particles. Particular
focus is given to mixing and $C\\!P$ violation, followed by comments on rare
charm decays at the end of this review.
### 1.1 Charm production
Charm physics has been and is being performed at a range of different
accelerators. These come with different production mechanisms and thus with
largely varying production cross-sections. At $\mathrm{e}^{+}\mathrm{e}^{-}$
colliders two different running conditions are of interest to charm physics.
Tuning the centre-of-mass energy to resonantly produce $\psi(3770)$ states
leads to the production of quantum-correlated $D^{0}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ or $D^{+}D^{-}$ pairs. This is the
case for the CLEO-c experiment at the CESR-c collider as well as for BESIII at
BEPCII. The most commonly used alternative is running at a higher centre-of-
mass energy to resonantly produce $\Upsilon(4S)$ which decay into quantum-
correlated $B^{0}\kern 1.99997pt\overline{\kern-1.99997ptB}{}^{0}$ or
$B^{+}B^{-}$ pairs. This is used by the BaBar and Belle experiments which are
located at the PEP-II and KEKB colliders, respectively. Both PEP-II and KEKB
are asymmetric colliders, thus having a collision system that is boosted with
respect to the laboratory frame. This allows measurements with decay-time
resolutions about a factor two to four below the $D^{0}$ lifetime and
therefore decay-time dependent studies.
The production cross-section for producing $D\kern
1.99997pt\overline{\kern-1.99997ptD}{}$ pairs at the $\psi(3770)$ resonance is
approximately $8\rm\,nb$ [22]. When running at the $\Upsilon(4S)$ resonance,
the cross-section for producing $\mathrm{c}\overline{\mathrm{c}}$ pairs is
$1.3\rm\,nb$ [23]. The latter scenario gives access to all species of charm
particles while the $\psi(3770)$ only decays into $D^{0}\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ or $D^{+}D^{-}$ pairs. The BaBar
and Belle experiments have collected integrated luminosities of about
$500\mbox{\,fb}^{-1}$ and $1000\mbox{\,fb}^{-1}$, respectively. CLEO-c has
collected $0.5\mbox{\,fb}^{-1}$ at the $\psi(3770)$ resonance as well as
around $0.3\mbox{\,fb}^{-1}$ above the threshold for
$D^{+}_{\mathrm{s}}D^{-}_{\mathrm{s}}$ production. BESIII has so far collected
nearly $3\mbox{\,fb}^{-1}$ in their 2010 and 2011 runs.
At hadron colliders the production cross-sections are significantly higher.
The cross-section for producing $\mathrm{c}\overline{\mathrm{c}}$ pairs in
proton-proton collisions at the LHC with a centre-of-mass energy
$7\mathrm{\,Te\kern-1.00006ptV}$ is about $6\rm\,mb$ [24], i.e. more than six
orders of magnitude higher compared to operating an
$\mathrm{e}^{+}\mathrm{e}^{-}$ collider at the $\Upsilon(4S)$ resonance. This
corresponds to a cross-section of about $1.5\rm\,mb$ for producing $D^{0}$ in
the LHCb acceptance111The LHCb acceptance is given as a range in momentum
transverse to the beam direction and rapidity as $p_{\rm
T}<8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},2<y<4.5$.. This number may be
compared to its equivalent at CDF which has been measured to $13\rm\,\mu b$
inside the detector acceptance222The CDF acceptance is defined as $p_{\rm
T}>5.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c},|y|<1$. for proton-antiproton
collisions at the Tevatron with $\sqrt{s}=1.96\mathrm{\,Te\kern-1.00006ptV}$
[25]. CDF has collected a total of about $10\mbox{\,fb}^{-1}$ while LHCb has
collected about $1.8\mbox{\,fb}^{-1}$ by the time of writing this review,
corresponding to $1.3\times 10^{11}$ and $2.7\times 10^{12}$ $D^{0}$ mesons
produced in the respective detector acceptances.
The production of charm quarks in hadron collisions occurs predominantly in
very asymmetric collisions which result in heavily boosted quarks with high
rapidities. Therefore, LHCb is ideally suited for performing decay-time
dependent studies of charm decays. At the same time, the
$\mathrm{c}\overline{\mathrm{c}}$ cross-section at the LHC is about $10\%$ of
the total inelastic cross-section which allows to have reasonably low
background levels for a hadronic environment. The coverage of nearly the full
solid angle of the $\mathrm{e}^{+}\mathrm{e}^{-}$-collider experiments
mentioned here makes them very powerful instruments for analysing decays
involving neutral particles that may remain undetected or for inclusive
studies.
### 1.2 Mixing and $C\\!P$ violation in neutral mesons
For neutral mesons, the mass eigenstates, i.e. the physical particles,
generally do not coincide with the flavour eigenstates, i.e. those governing
the interactions. The mass eigenstates of neutral mesons, $|M_{1,2}\rangle$,
with masses $m_{1,2}$ and widths $\Gamma_{1,2}$, are linear combinations of
the flavour eigenstates, $|M^{0}\rangle$ and $|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$, as
$|M_{1,2}\rangle=p|M^{0}\rangle\pm{}q|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$ with complex coefficients
satisfying $|p|^{2}+|q|^{2}=1$. This allows the definition of the averages
$m\equiv(m_{1}+m_{2})/2$ and $\Gamma\equiv(\Gamma_{1}+\Gamma_{2})/2$. The
phase convention of $p$ and $q$ is chosen such that, in the limit of no
$C\\!P$ violation, $C\\!P|M^{0}\rangle=-|\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}\rangle$.
Mixing, i.e. the periodical transformation of mesons into their anti-mesons
and back, occurs if there is a non-zero difference in the masses or widths of
the two mass eigenstates of a meson. This is quantified in the differences
$\Delta{}m\equiv{}m_{2}-m_{1}$ and $\Delta\Gamma\equiv\Gamma_{2}-\Gamma_{1}$.
Furthermore, the mixing parameters are defined as $x\equiv\Delta{}m/\Gamma$
and $y\equiv\Delta\Gamma/(2\Gamma)$.
It is these mixing parameters $x$, and $y$ which define the characteristic
behaviour of the four neutral meson systems, which are subject to mixing,
kaons ($K$), charm ($D$), $B^{0}_{\mathrm{d}}$, and $B^{0}_{\mathrm{s}}$
mesons. To appreciate the different mixing behaviour it is instructive to
consider the time evolution of the meson and anti-meson states. The
probability of observing a neutral meson state $M^{0}$ or $\kern
1.99997pt\overline{\kern-1.99997ptM}{}^{0}$ after a time $t$ has passed since
the observation of an initial state $M^{0}$ is
$\displaystyle P(M^{0}\to M^{0},t)$
$\displaystyle=\frac{1}{2}e^{-\Gamma{}t}(\cosh(y\Gamma{}t)+\cos(x\Gamma{}t)),$
$\displaystyle P(M^{0}\to\kern 1.99997pt\overline{\kern-1.99997ptM}{}^{0},t)$
$\displaystyle=\frac{1}{2}\left|\frac{q}{p}\right|^{2}e^{-\Gamma{}t}(\cosh(y\Gamma{}t)-\cos(x\Gamma{}t)),$
(1)
where the oscillatory behaviour is governed by the mixing parameter $x$.
Figure 1: The widths and mass differences of the physical states of the
flavoured neutral mesons. The width corresponds to the inverse lifetime while
the mass difference determines the oscillation frequency.
For charm mesons the mixing parameters are drastically different compared to
those of kaons or $B$ mesons. Figure 1 shows the widths and mass differences
of the four neutral meson systems. The kaon system is the only one to have
$y\approx 1$, resulting in two mass eigenstates with vastly different
lifetimes, hence their names $K$-short ($K^{0}_{\rm\scriptstyle S}$) and
$K$-long ($K^{0}_{\rm\scriptstyle L}$). Furthermore, also $x\approx 1$ which
results in a sizeable sinusoidal oscillation frequency as shown in Eq. (1.2).
The two $B$-meson systems have reasonably small width splitting, however, they
have sizeable values for $x$. Particularly for the $B^{0}_{\mathrm{s}}$ system
this leads to fast oscillations which require high experimental accuracy to be
resolved. The charm meson system is the only one where both $x$ and $y$ are
significantly less than $1$, hence the nearly overlapping curves in Fig. 1.
Experimentally, the different mixing parameters lead to rather different
challenges for measurements in the various meson systems. The vast lifetime
difference in the kaon system leads to the possibility of studying nearly
clean samples of just one of the two mass eigenstates by either measuring
decays close to a production target where $K^{0}_{\rm\scriptstyle S}$ decays
dominate, or far away where most $K^{0}_{\rm\scriptstyle S}$ have decayed
before entering the detection region. In the $B$ systems the oscillation
frequency puts a challenge to the decay-time resolution, particularly for
$B^{0}_{\mathrm{s}}$ mesons as mentioned before. The smallness of $y$
requires, to first order, large data samples to acquire the necessary
statistical precision for measuring such a small quantity. The latter is
particularly true for the charm system, where both $x$ and $y$ are small. This
is the reason why it was only in 2007 when first evidence for charm mixing was
observed.
The symmetry under $C\\!P$ transformation is violated for a deviation from
unity of the quantity $\lambda_{f}$, defined as
$\lambda_{f}\equiv\frac{q\bar{A}_{\bar{f}}}{pA_{f}}=-\eta_{C\\!P}\left|\frac{q}{p}\right|\left|\frac{\bar{A}_{f}}{A_{f}}\right|e^{i\phi},$
(2)
where the right-hand expression is valid for a $C\\!P$ eigenstate $f$ with
eigenvalue $\eta_{C\\!P}$ and $\phi$ is the $C\\!P$ violating relative phase
between $q/p$ and $\bar{A}_{f}/A_{f}$. Besides the mixing parameters
introduced above this expression contains the decay amplitudes $A_{f}$ and
$\bar{A}_{f}$ for decays into a final state $f$.
$C\\!P$ violation can have different origins: the case $|q/p|\neq 1$ is called
$C\\!P$ violation in mixing, $|\bar{A}_{f}/A_{f}|\neq 1$ is $C\\!P$ violation
in the decay, and a non-zero phase $\phi$ between $q/p$ and
$\bar{A}_{f}/A_{f}$ causes $C\\!P$ violation in the interference between
mixing and decay. Mixing is common to all decay modes and hence $C\\!P$
violation originating in this process is universal which is called indirect
$C\\!P$ violation. Decay-specific $C\\!P$ violation is called direct $C\\!P$
violation. An excellent discussion on the different types of $C\\!P$ violation
can be found in section 7.2.1 of Ref. Sozzi:2008zza. As opposed to the strange
and the beauty system, $C\\!P$ violation has not yet been discovered in the
charm system, though the LHCb collaboration has recently found first evidence
for $C\\!P$ violation in two-body $D^{0}$ decays [27], arguably the most
surprising result from the LHC in 2011.
## 2 Charm mixing
The studies of charm mesons have gained in momentum with the measurements of
first evidence for meson anti-meson mixing in neutral charm mesons in 2007
[20, 21]. Mixing of $D^{0}$ mesons is the only mixing process where down-type
quarks contribute to the box diagram. Unlike $B$-meson mixing, where the top-
quark contribution dominates, the third generation quark is of similar mass to
the other down-type quarks. This leads to a combination of GIM cancellation
[5] and CKM suppression [28, 29], which results in a strongly suppressed
mixing process [30, 31, 32, 33, 34].
There are two approaches for theoretical calculations of charm mixing. The
“inclusive” approach is based an operator product expansion (OPE) in
$\Lambda/m_{\mathrm{c}}$ [35, 36, 37, 38, 39, 40, 34]. Due to the
cancellations mentioned above it is higher order operators that give the
largest contributions to the mixing parameters. Furthermore, it is not yet
clear whether the expansion series really converges. Calculations of the charm
meson lifetimes are being performed to test whether the OPE approach can
properly reproduce the large difference between the $D^{0}$ and the $D^{+}$
lifetimes. In the $B^{0}_{\mathrm{s}}$ system, the OPE approach successfully
predicted the width splitting of the two $B^{0}_{\mathrm{s}}$ mass eigenstates
[41] which has recently been confirmed by an LHCb measurement [42].
The “exclusive” approach sums over intermediate hadronic states, taking input
from models or experimental data [43, 44, 45, 46, 47, 48]. Also in this
approach, different modes of the same $SU(3)$ multiplet lead to cancellations
which is why their individual contributions have to be known to high
precision. Due to the considerable mass of the $D^{0}$ meson, many different
modes need to be taken into account simultaneously. Of these, only phase space
differences can be evaluated at the moment. Estimates indicate that mixing in
the experimentally observed range is conceivable when taking into account
$SU(3)$-breaking effects. However, neither the inclusive nor the exclusive
approach have thus far permitted a precise theoretical calculation of charm
mixing.
It was discussed whether the measured size of the mixing parameters could be
interpreted as a hint for physics beyond the standard model [49, 50, 51, 52,
53, 54, 55]. The biggest problem in answering this question is the non-
existence of a precise standard model calculation. Effects of physics beyond
the standard model were also searched for in numerous $C\\!P$ violation
measurements and searches for rare decays both of which are covered in the
remainder of this review.
Mixing of $D^{0}$ mesons can be measured in several different modes. Most
require identifying the flavour of the $D^{0}$ at production as well as at the
time of the decay. Tagging the flavour at production usually exploits the
strong decay $D^{*+}\\!\to D^{0}\pi^{+}$ (and charge conjugate333Charge
conjugate decays are implicitly included henceforth.) where the charge of the
pion determines the flavour of the $D^{0}$. The small amount of free energy in
this decay leads to the difference in the reconstructed invariant mass of the
$D^{*+}$ and the $D^{0}$, $\delta m\equiv{}m_{D^{*+}}-m_{D^{0}}$, exhibiting a
sharply peaking structure over a threshold function as background. An
alternative to using this decay mode is tagging the $D^{0}$ flavour by
reconstructing a flavour-specific decay of a $B$ meson. This method has not
yet been used in a measurement as it did not yet yield competitive quantities
of tagged $D^{0}$ mesons. At LHCb this approach may be of interest due to
differences in trigger efficiencies compensating for lower production rates.
Another option available particularly at $\mathrm{e}^{+}\mathrm{e}^{-}$
colliders is the reconstruction of the opposite side charm meson in a flavour
specific decay.
Theoretically, the most straight-forward mixing measurement is that of the
rate of the forbidden decay $D^{0}\\!\to K^{+}\mu^{-}\overline{\nu}_{\mu}$
which is only accessible through $D^{0}$-$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing. The ratio of the time-
integrated rate of these forbidden decays to their allowed counterparts,
$D^{0}\\!\to K^{-}\mu^{+}\nu_{\mu}$, determines $R_{\rm
m}\equiv(x^{2}+y^{2})/2$. As this requires very large samples of $D^{0}$
mesons no measurement has thus far reached sufficient sensitivity to see
evidence for $D^{0}$ mixing. The most sensitive measurement to date has been
made by the Belle collaboration [56] to $R_{\rm m}=(1.3\pm 2.2\pm 2.0)\times
10^{-4}$, where the first uncertainty is of statistical and the second is of
systematic nature444This notation is applied to all results where two
uncertainties are quoted..
Related to the semileptonic decay is the suppressed decay $D^{0}\\!\to
K^{+}\pi^{-}$, called wrong-sign (WS) decay. For this decay, a doubly Cabibbo-
suppressed (DCS) amplitude interferes with the decay through a mixing process
followed by the Cabibbo-favoured (CF) decay $D^{0}\\!\to K^{-}\pi^{+}$. The
time-dependent decay rate of the WS decay is, in the limit of $C\\!P$
conservation, proportional to
$\frac{\Gamma(D^{0}(t)\to K^{+}\pi^{-})}{e^{-\Gamma t}}\propto\left(R_{\rm
D}+\sqrt{R_{\rm D}}y^{\prime}\Gamma{}t+R^{2}_{\rm m}(\Gamma{}t)^{2}\right),$
(3)
where the mixing parameters are rotated by the strong phase between the DCS
and the CF amplitude, leading to the observable
$y^{\prime}=y\cos\delta_{K\pi}-x\sin\delta_{K\pi}$ [57]. The parameter $R_{\rm
D}$ is the ratio of the DCS to the CF rate. Measurements with sufficient
sensitivity to unveil evidence for $D^{0}$ mixing have been performed by the
BaBar and CDF collaborations, leading to
| $x^{\prime 2}$ in $10^{-3}$ | $y^{\prime}$ in $10^{-3}$
---|---|---
BaBar [20] | $-0.22\pm 0.30\pm 0.20$ | $9.7\pm 4.4\pm 3.1$
CDF [58] | $-0.12\pm 0.35$ | $8.5\pm 7.6$.
Similarly, the CF and DCS amplitudes can also lead to excited states of the
same quark content. The decay $D^{0}\\!\to K^{-}\pi^{+}\pi^{0}$ is the final
state of several such resonances. Thus, by studying the decay-time dependence
of the various resonances a mixing measurement can be obtained. The BaBar
collaboration achieved a measurement showing evidence for $D^{0}$ mixing [59]
with central values of $x^{\prime\prime}=(26.1^{+5.7}_{-6.8}\pm 3.9)\times
10^{-3}$ and $y^{\prime\prime}=(-0.6^{+5.5}_{-6.4}\pm 3.4)\times 10^{-3}$,
where the rotation between the observables and the system of mixing parameters
is given by a strong phase as
$\displaystyle x^{\prime\prime}$
$\displaystyle=x\cos\delta_{K^{-}\pi^{+}\pi^{0}}+y\sin\delta_{K^{-}\pi^{+}\pi^{0}}$
(4) $\displaystyle y^{\prime\prime}$
$\displaystyle=y\cos\delta_{K^{-}\pi^{+}\pi^{0}}-x\sin\delta_{K^{-}\pi^{+}\pi^{0}}.$
(5)
The significant advantage of this analysis over that using two-body final
states is that both mixing parameters are measured at first order rather than
one at first and one at second order.
The strong phases are not accessible in these measurements but have to come
from measurements performed using quantum-correlated $D^{0}$-$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ pairs produced at threshold. Such
measurements are available from CLEO [60, 61, 62, 63] and can be further
improved at BESIII.
The comparison of effective inverse lifetimes in decays of $D^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) mesons into final states which
are $C\\!P$ eigenstates, $\hat{\Gamma}$ ($\hat{\bar{\Gamma}}$), to that of a
Cabibbo-favoured flavour eigenstate ($\Gamma$) leads to the observable
$y_{CP}=\frac{\hat{\Gamma}+\hat{\bar{\Gamma}}}{2\Gamma}-1\approx\eta_{C\\!P}\left[\left(1-\frac{A_{m}^{2}}{8}\right)y\cos\phi-\frac{A_{m}}{2}x\sin\phi\right],$
(6)
where $A_{m}$ is the $C\\!P$ violation in mixing defined alongside the direct
$C\\!P$ violation $A_{d}$ by $|\lambda_{f}^{\pm 1}|^{2}\approx(1\pm
A_{m})(1\pm A_{d})$ [64]. In the limit of $C\\!P$ conservation $y_{CP}$ equals
the mixing parameter $y$. As the $C\\!P$-violating contributions $A_{m}$ and
$\phi$ enter only at second order, measurements of $y_{CP}$ are among the most
powerful constraints of the mixing parameter $y$.
Comparing the $C\\!P$ eigenstates $K^{-}K^{+}$ and $\pi^{-}\pi^{+}$ to the
Cabibbo-favoured mode $K^{-}\pi^{+}$, the Belle [21] and BaBar [65]
collaborations have measured $y_{CP}=(13.1\pm 3.2\pm 2.5)\times 10^{-3}$ and
$y_{CP}=(11.6\pm 2.2\pm 1.8)\times 10^{-3}$, respectively. These measurements
have recently been updated by preliminary results based on the full dataset of
flavour-tagged events for both collaborations. The BaBar collaboration has
added the larger sample of untagged events to the analysis, however, with
limited gain in sensitivity due to larger systematic uncertainties for the
untagged sample which has lower purity compared to the $D^{*}$-tagged events.
The updated results are $y_{CP}=(7.2\pm 1.8\pm 1.2)\times 10^{-3}$ and
$y_{CP}=(11.1\pm 2.2\pm 1.1)\times 10^{-3}$, for BaBar [66] and Belle [67],
respectively.
It is worth noting that the central value of the BaBar result is significantly
lower compared the one from 2007. This is thought to be compatible with a
statistical fluctuation due to the added data as well as only partial overlap
in the older dataset following improvements in reconstruction and analysis.
The new BaBar result relaxes the tension that existed between measurements of
$y_{CP}$, which favoured values of about $1\%$, and other mixing measurements,
which tend towards smaller values. Such a tension would be impossible to be
explained by $C\\!P$ violation as that leads to $y_{CP}\leq y$.
Another possibility of measuring $y_{CP}$ is using the decay mode $D^{0}\\!\to
K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$. The Belle collaboration have published a
measurement in which they compare the effective lifetime around the $\phi$
resonance with that measured in sidebands of the $K^{-}K^{+}$ invariant mass
[68]. The effective $C\\!P$ eigenstate content in these regions is determined
with two different models. Their result is $y_{CP}=(1.1\pm 6.1\pm 5.2)\times
10^{-3}$.
The decay $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$ and more so the
decay $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{-}\pi^{+}$ give excellent
access to the mixing parameters $x$ and $y$ individually. At the same time
they allow a measurement of parameters of indirect $C\\!P$ violation as
discussed in the following section. Under the assumption of no $C\\!P$
violation Belle and BaBar have measured
| $x$ in $10^{-3}$ | $y$ in $10^{-3}$
---|---|---
Belle [69] | $8.0\pm 2.9\pm 1.7$ | $3.3\pm 2.4\pm 1.5$
BaBar [70] | $1.6\pm 2.3\pm 1.2\pm 0.8$ | $5.7\pm 2.0\pm 1.3\pm 0.7$,
where the last uncertainty in the BaBar measurement is a model uncertainty.
The Belle result has recently been updated including the full available
dataset based on the final reconstruction [71]. This leads to $x=(0.56\pm
0.19^{+0.03}_{-0.09}\,{}^{+0.06}_{-0.09})\times 10^{-3}$ and $y=(0.30\pm
0.15^{+0.04}_{-0.05}\,{}^{+0.03}_{-0.06})\times 10^{-3}$. Once more, the last
uncertainty is a model uncertainty. Curiously, while the recent BaBar result
on $y_{CP}$ has relaxed the tension with $y$, now there is a nearly $3\sigma$
tension among the latest Belle results for $y_{CP}$ and $y$. Additional
measurements are required to resolve this situation.
The LHCb collaboration has made its first measurement of $y_{CP}$ [72] based
on two-body $D^{0}$ decays recorded in 2010 to $y_{CP}=(5.5\pm 6.3\pm
4.1)\times 10^{-3}$. While this measurement falls short of being at the level
of precision of the B-factory measurements in this mode, LHCb has recorded
over a factor 50 more in integrated luminosity to date. The Heavy Flavor
Averaging Group (HFAG) has combined all measurements of $y_{CP}$ [73] to
$y_{CP}=(8.7\pm 1.6)\times 10^{-3}$.
By the time of this review no single experiment observation of mixing in
$D^{0}$ mesons with a significance exceeding $5\sigma$ has been possible.
However, the combination of the numerous measurements by HFAG excludes the no-
mixing hypothesis by about $10\sigma$ [73]. Under the assumption of no $C\\!P$
violation in mixing or decays, the world average of the mixing parameters is
$x=(6.5^{+1.8}_{-1.9})\times 10^{-3}$ and $y=(7.4\pm 1.2)\times 10^{-3}$.
## 3 Charm $C\\!P$ violation
### 3.1 Indirect $C\\!P$ violation
Indirect $C\\!P$ violation can be measured through the comparison of effective
lifetimes of $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
decays to $C\\!P$ eigenstates. This leads to the observable
$A_{\Gamma}=\frac{\hat{\Gamma}-\hat{\bar{\Gamma}}}{\hat{\Gamma}+\hat{\bar{\Gamma}}}\approx\eta_{C\\!P}\left[\frac{1}{2}\left(A_{m}+A_{d}\right)y\cos\phi-x\sin\phi\right],$
(7)
which has contributions from both direct and indirect $C\\!P$ violation [64,
74]. Currently, there are three measurements of $A_{\Gamma}$ which are all
compatible with zero. The Belle [21], BaBar [75] and LHCb [72] collaborations
have measured $A_{\Gamma}=(0.1\pm 3.0\pm 1.5)\times 10^{-3}$,
$A_{\Gamma}=(2.6\pm 3.6\pm 0.8)\times 10^{-3}$, and $A_{\Gamma}=(-5.9\pm
5.9\pm 2.1)\times 10^{-3}$, respectively. With the LHCb result being based
only on a small fraction of the data recorded so far, significant improvements
in sensitivity may be expected in the near future. In parallel to their
updates of $y_{CP}$, BaBar [66] and Belle [67] have also released preliminary
results for $A_{\Gamma}$ based on their full datasets. They have measured
$A_{\Gamma}=(0.9\pm 2.6\pm 0.6)\times 10^{-3}$ and $A_{\Gamma}=(-0.3\pm 2.0\pm
0.8)\times 10^{-3}$, respectively. The HFAG world average [73] yields
$A_{\Gamma}=(-0.2\pm 1.6)\times 10^{-3}$ in agreement with no $C\\!P$
violation.
Using current experimental bounds values of $A_{\Gamma}$ up to
$\mathcal{O}(10^{-4})$ are expected from theory [74, 76]. It has however been
shown that enhancements up to about one order of magnitude are possible, for
example in the presence of a fourth generation of quarks [34] or in a little
Higgs model with T-parity [76]. This would bring $A_{\Gamma}$ close to the
current experimental limits.
Eventually, the interpretation of $C\\!P$ violation results requires precise
knowledge of both mixing and $C\\!P$ violation parameters. The relative
sensitivity to the $C\\!P$-violating quantities in the observable $A_{\Gamma}$
is limited by the relative uncertainty of the mixing parameters. Therefore, to
establish the nature of a potential non-zero measurement of $A_{\Gamma}$ it is
mandatory to have measured the mixing parameters with a relative precision of
$\approx 10\%$.
The analyses of the decays $D^{0}\\!\to K^{0}_{\rm\scriptstyle
S}\pi^{-}\pi^{+}$ and $D^{0}\\!\to K^{0}_{\rm\scriptstyle S}K^{-}K^{+}$ offer
separate access to the parameters $x$, $y$, $|q/p|$ and $\arg(q/p)$ and are
one of the most promising ways of obtaining precise mixing measurements. These
analyses require the determination of the decay-time dependence of the phase
space structure (Dalitz plot, see Ref. Dalitz:1953cp) of these decays. This
can be obtained in two ways: explicit fits of the time evolution of resonances
based on Dalitz-plot models, or based on a measurement of the strong-phase
difference across the Dalitz-plot carried out by the CLEO collaboration [78].
One measurement made by the Belle collaboration has determined these
parameters based on a Dalitz plot model [69]. Other measurements were
performed by the CLEO [79] and BaBar [70] collaborations assuming $C\\!P$
conservation and thus extracting only $x$ and $y$. With the data samples
available and being recorded at LHCb and those expected at future flavour
factories, these measurements will be very important to understand charm
mixing and $C\\!P$ violation. However, in order to avoid systematic
limitations it will be important to reduce model uncertainties or to improve
model-independent strong-phase difference measurements which are possible at
BESIII.
### 3.2 Direct $C\\!P$ violation
Direct $C\\!P$ violation is searched for in decay-time integrated
measurements. However, for neutral mesons, the decay-time distribution of the
data has to be taken into account to estimate the contribution from indirect
$C\\!P$ violation. Currently, the most striking measurements have been made in
decays of $D^{0}$ mesons into two charged pions or kaons. While early
measurements of BaBar [80] and Belle [81] have not shown significant
deviations from zero, the LHCb collaboration has reported first evidence for
$C\\!P$ violation in the charm sector [27]. They have measured
$\Delta
A_{C\\!P}\equiv{}A_{C\\!P}(K^{-}K^{+})-A_{C\\!P}(\pi^{-}\pi^{+})=(-8.1\pm
2.1\pm 1.1)\times 10^{-3}.$
Meanwhile, the CDF collaboration has released a preliminary measurement of
$\Delta A_{C\\!P}=(-6.2\pm 2.1\pm 1.0)\times 10^{-3}$ which shows a hint of a
deviation from zero [82], in support of the LHCb result. Just before
submission of this review the Belle collaboration has released an update of
their measurement of $\Delta A_{C\\!P}$ based on their full reprocessed
dataset showing a roughly $2\sigma$ deviation from zero [83].
The quantity $\Delta A_{C\\!P}$ exploits first-order cancellation of
systematic uncertainties in the difference of asymmetries. At higher order,
terms that are products of individual asymmetries contribute, e.g. the product
of a production and a $C\\!P$ asymmetry, which no longer cancel. In a
kinematic region with large production asymmetries such a contribution leads
to relative corrections of $\Delta A_{C\\!P}$ of the order of the local
production asymmetry. Hence, these higher order terms need to be taken into
account for a precision measurement of the size of observed $C\\!P$ violation.
The observable $\Delta A_{C\\!P}$ gives access to the difference in direct
$C\\!P$ violation of the two decay modes through
$\Delta A_{C\\!P}=\Delta{}a_{C\\!P}^{\rm
dir}\left(1+y_{CP}\frac{\overline{\langle{}t\rangle}}{\tau}\right)+\overline{A}_{\Gamma}\frac{\Delta\langle{}t\rangle}{\tau},$
(8)
where $\tau$ is the nominal $D^{0}$ lifetime,
$\overline{X}\equiv(X(K^{-}K^{+})+X(\pi^{-}\pi^{+}))/2$, and
$\Delta{}X\equiv{}X(K^{-}K^{+})-X(\pi^{-}\pi^{+})$ for $X=(a_{C\\!P}^{\rm
dir},\langle{}t\rangle)$ [64]. Equation (8) assumes the $C\\!P$-violating
phase $\phi$ to be universal. For a small non-zero difference in this phase
between the two final states, $\Delta\phi_{f}$, an additional term of the form
$x\Delta\phi_{f}\overline{\langle{}t\rangle}/\tau$ arises as pointed out in
Ref. Kagan:2009gb. Given a typical variation of
$\overline{\langle{}t\rangle}/\tau$ between $1$ and $2.5$ for the different
experiments the contribution of $\Delta\phi_{f}$ is suppressed by
$x\overline{\langle{}t\rangle}/\tau\approx 10^{-2}$.
While it was commonly stated in literature that $C\\!P$ violation effects in
these channels were not expected to exceed $10^{-3}$, this statement has been
revisited in numerous recent publications. To date, no clear understanding of
whether [84, 85, 86, 87] or not [76, 88, 89, 90] $C\\!P$ violation of this
level can be accommodated within the standard model has emerged. In parallel
to attempts to better the standard model calculations, many estimates of
potential effects of physics beyond the standard model have been made [76, 85,
88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106,
107, 108, 109, 110, 111].
Within the standard model the central value can only be explained by
significantly enhanced penguin amplitudes. This enhancement is conceivable
when estimating flavour $SU(3)$ or U-spin breaking effects from fits to data
of $D$ decays into two pseudo scalars [85, 86, 95, 102, 112, 113]. However,
attempts of estimating the long distance penguin contractions directly have
failed to yield conclusive results to explain the enhancement.
Lattice QCD has the potential of assessing the penguin enhancement directly.
However, several challenges arise which make these calculations impossible at
the moment [114, 115, 116, 117, 118, 119, 120]. Following promising results on
$K\to\pi\pi$ decays, additional challenges arise in the charm sector as
$\pi\pi$ and $KK$ states mix with $\eta\eta$, $4\pi$, $6\pi$ and other states.
Possible methods have been proposed and first results may be expected within
the next decade.
General considerations on the possibility of interpreting the
$\Delta{}A_{C\\!P}$ in models beyond the standard model have lead to the
conclusion that an enhanced chromomagnetic dipole operator is required. These
operators can be accommodated in minimal supersymmetric models with non-zero
left-right up-type squark mixing contributions or, similarly, in warped extra
dimensional models [91, 92, 97, 121, 122, 123, 124]. Tests of these
interpretations beyond the standard model are in the focus of ongoing
searches. One promising group of channels are radiative charm decays where the
link between the chromomagnetic and the electromagnetic dipole operator leads
to predictions of enhanced $C\\!P$ asymmetries of several percent [125].
Another, complementary, test is to search for contributions beyond the
standard model in $\Delta I=3/2$ amplitudes. This class of amplitudes leads to
several isospin relations which can be tested in a range of decay modes, e.g.
$D\to\pi\pi$, $D\to\rho\pi$, $D\to K\bar{K}$ [85, 126]. Several of these
measurements, such as the Dalitz plot analysis of the decay
$D^{0}\to\pi^{+}\pi^{-}\pi^{0}$, have been performed by BaBar and Belle and
will be possible at LHCb as well as future $\mathrm{e}^{+}\mathrm{e}^{-}$
machines.
Beyond charm physics, the chromomagnetic dipole operators would affect the
neutron and nuclear EDMs, which are expected to be close to the current
experimental bound [97]. Similarly, rare FCNC top decays are expected to be
enhanced, if kinematically allowed. Furthermore, quark compositness can be
related to the $\Delta{}A_{C\\!P}$ measurement and tested in dijet searches.
Current results favour the new physics contribution to be located in the
$D^{0}\to K^{-}K^{+}$ decay as the strange quark compositness scale is less
well constrained [127].
Another group of channels suitable for $C\\!P$ violation searches is that of
decays of $D^{+}$ and $D^{+}_{\mathrm{s}}$ mesons into three charged hadrons,
namely pions or kaons. Here, $C\\!P$ violation can occur in two-body
resonances contributing to these decay amplitudes. Asymmetries in the Dalitz-
plot substructure can be measured using an amplitude model or using model-
independent statistical analyses [128, 129, 130]. The latter allow $C\\!P$
asymmetries to be discovered while eventually a model-dependent analysis is
required to identify its source. The two types of model-independent analyses
differ in being either binned [128, 129] or unbinned [130] in the Dalitz
plane.
The binned approach computes a local per-bin asymmetry and judges the presence
of $C\\!P$ violation by the compatibility of the distribution of local
asymmetries across the Dalitz plane with a normal distribution. This method
obviously relies on the optimal choice of bins. Bins ranging across resonances
can lead to the cancellation of real asymmetries within a bin. Too fine
binning can reach the limit of statistical sensitivity, whereas too coarse
binning can wash out $C\\!P$ violation effects by combining regions of
opposite asymmetry. A model-inspired choice of binning is clearly useful and
this does not create a model-dependence in the way that fitting resonances
directly does. This method does not yield an easy-to-interpret quantitative
result. This issue has been discussed in a recent update of the procedure
[131].
The unbinned asymmetry search calculates a test statistic that allows the
assignment of a $p$-value when comparing to the distribution of the statistic
for many random permutations of the events among the particle and anti-
particle datasets. Moreover, being unbinned, there is no need for a model-
inspired choice of binning. The drawback of this method is its requirement on
computing power. The calculation of the test statistic scales as the square of
the number of events.
Beyond three-body final states similar analyses can be performed in decays
into four hadrons, e.g. decays of $D^{0}$ into four charged hadrons. This too
gives access to interesting resonance structures that may exhibit significant
$C\\!P$ asymmetries. However, rather than having a two-dimensional Dalitz
plane, the phase space for four-body decays is five-dimensional (see e.g. Ref.
Rademacker:2006zx). This poses not only a challenge on the visualisation but
also on any binned approach due to rapidly decreasing sample sizes per bin.
Also, the phase-space substructure can no longer be described only by
interfering amplitudes of pseudo two-body decays as also three-body decays may
contribute. The LHCb collaboration has released a first model-independent
search for $C\\!P$ violation in the decay
$D^{0}\\!\to\pi^{-}\pi^{+}\pi^{+}\pi^{-}$ without finding any hint of $C\\!P$
non-conservation [133].
Neither searches for phase-space integrated asymmetries [134, 135, 136, 137,
138, 139, 140], nor searches for local asymmetries in the Dalitz plot [136,
138, 141, 142, 143] have shown any evidence for $C\\!P$ violation. The largest
signal is the recently reported measurement of $C\\!P$ violation in
$D^{+}\\!\to\phi\pi^{+}$ of $A_{C\\!P}^{\phi\pi^{+}}=(5.1\pm 2.8\pm 0.5)\times
10^{-3}$ by the Belle collaboration [143], which exploits cancellation of
uncertainties through a comparison of asymmetries in the decays of $D^{+}$ and
$D^{+}_{\mathrm{s}}$ mesons into the final state $\phi\pi^{+}$.
Decays of $D^{+}$ and $D^{+}_{\mathrm{s}}$ are into a $K^{0}_{\rm\scriptstyle
S}$ and either a $K^{+}$ or a $\pi^{+}$ are closely related to their $D^{0}$
counterparts. Measurements of time-integrated asymmetries in these decays are
expected to exhibit a contribution from $C\\!P$ violation in the kaon system.
As pointed out recently [144] this contribution depends on the decay-time
acceptance of the $K^{0}_{\rm\scriptstyle S}$. This can lead to different
expected values for different experiments. For Belle [145], the expected level
of asymmetry due to $C\\!P$ violation in the kaon system is $-3.5\times
10^{-3}$. For LHCb on the other hand, there is no significant asymmetry
induced by kaon $C\\!P$ violation [146] as the LHCb acceptance, for
$K^{0}_{\rm\scriptstyle S}$ reconstructed in the vertex detector, corresponds
to about $10\%$ of a $K^{0}_{\rm\scriptstyle S}$ lifetime. $C\\!P$ violation
searches in the decays $D^{+}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{+}$ [65,
139, 147] and $D^{+}_{\mathrm{s}}\\!\to K^{0}_{\rm\scriptstyle S}\pi^{+}$
[139, 147] show significant asymmetries. However, these asymmetries are fully
accommodated in the expected $C\\!P$ violation of the kaon system. These
measurements do not show any hint for an asymmetry in $D$ decay amplitudes.
Future, more precise, measurements will reveal whether or not these remain in
agreement with the expected contribution from the kaon system.
In the light of the recent measurements it is evident that there are four
directions to pursue: more precise measurements of $\Delta A_{C\\!P}$ and the
individual asymmetries are required to establish the effect; further searches
for time-integrated $C\\!P$ violation need to be carried out in a large range
of modes that allow to identify the source of the $C\\!P$ asymmetry; searches
for time-dependent $C\\!P$ asymmetries, particularly via more precise
measurements of $A_{\Gamma}$; and finally a more precise determination of the
mixing parameters is required.
### 3.3 Interplay of mixing, direct and indirect $C\\!P$ violation
Following Eqs. (7) and (8) it is obvious that both $A_{\Gamma}$ and $\Delta
A_{C\\!P}$ share the underlying $C\\!P$-violating parameters. Allowing for a
non-universal $C\\!P$-violating phase $\phi$ one can write
$\displaystyle A_{\Gamma}(f)$ $\displaystyle=-a_{C\\!P}^{\rm
ind}-a_{C\\!P}^{\rm dir}(f)y_{CP}-x\phi_{f},$ (9) $\displaystyle A_{C\\!P}(f)$
$\displaystyle=a_{C\\!P}^{\rm
dir}(f)-A_{\Gamma}(f)\frac{\langle{}t\rangle}{\tau},$ (10)
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle=\Delta{}a_{C\\!P}^{\rm
dir}-\Delta
A_{\Gamma}\frac{\overline{\langle{}t\rangle}}{\tau}-\overline{A}_{\Gamma}\frac{\Delta\langle{}t\rangle}{\tau},$
(11)
where again $\overline{X}\equiv(X(K^{-}K^{+})+X(\pi^{-}\pi^{+}))/2$ and
$\Delta{}X\equiv{}X(K^{-}K^{+})-X(\pi^{-}\pi^{+})$ for $X=(a_{C\\!P}^{\rm
dir},\langle{}t\rangle)$. It is expected that, at least within the standard
model, one has $a_{C\\!P}^{\rm dir}(K^{-}K^{+})=-a_{C\\!P}^{\rm
dir}(\pi^{-}\pi^{+})$ and thus $\overline{A}_{\Gamma}=-a_{C\\!P}^{\rm ind}$.
This set of equations shows that it is essential to measure both time-
dependent ($A_{\Gamma}$) and time-integrated asymmetries ($A_{C\\!P}$)
separately in the decay modes $D^{0}\\!\to K^{-}K^{+}$ and
$D^{0}\\!\to\pi^{-}\pi^{+}$ in order to distinguish the various possible
sources of $C\\!P$ violation. Currently, the experimental precision on
$A_{\Gamma}$ is such that there is no sensitivity to differences in the
contributions from direct $C\\!P$ violation to measurements using
$K^{-}\\!K^{+}$ or $\pi^{-}\\!\pi^{+}$ final states. Hence, the approximation
$A_{\Gamma}\equiv\overline{A}_{\Gamma}\approx A_{\Gamma}(K^{-}K^{+})\approx
A_{\Gamma}(\pi^{-}\pi^{+})$ can be used to obtain
$\displaystyle A_{\Gamma}$ $\displaystyle=-a_{C\\!P}^{\rm ind}$ (12)
$\displaystyle\Delta A_{C\\!P}$ $\displaystyle=\Delta{}a_{C\\!P}^{\rm
dir}\left(1+y_{CP}\frac{\overline{\langle{}t\rangle}}{\tau}\right)-a_{C\\!P}^{\rm
ind}\frac{\Delta\langle{}t\rangle}{\tau}.$ (13)
These equations have been used by HFAG to prepare a fit of the direct and
indirect $C\\!P$ violation contributions [73] as shown in Fig. 2.
Figure 2: Fit of $\Delta{}a_{C\\!P}^{\rm dir}$ and $a_{C\\!P}^{\rm ind}$.
Reproduced from Ref. Amhis:2012hf.
This fit yields a confidence level of about $2\times 10^{-5}$ for the no
$C\\!P$ violation hypothesis with best fit values of $\Delta{}a_{C\\!P}^{\rm
dir}=(-6.78\pm 1.47)\times 10^{-3}$ and $a_{C\\!P}^{\rm ind}=(0.27\pm
1.63)\times 10^{-3}$. This also shows that the most likely source of the large
measured values for $\Delta A_{C\\!P}$ is direct $C\\!P$ violation in one or
both of the relevant decay modes. The fit formalism will have to be refined
using the equations discussed above in the future as more precise measurements
as well as individual asymmetries will become available.
The ultimate goal of mixing and $C\\!P$ violation measurements in the charm
sector is to reach precisions at or below the standard model predictions. In
some cases this requires measurements in several decay modes in order to
distinguish enhanced contributions of higher order standard model diagrams
from effects caused by new particles.
Indirect $C\\!P$ violation measurements at LHCb are mostly constrained by the
observable $A_{\Gamma}$ (see Eq. (9)). The $C\\!P$ violating parameters in
this observable are multiplied by the mixing parameters $x$ and $y$,
respectively. Hence, the relative precision on the $C\\!P$ violating
parameters is limited by the relative precision of the mixing parameters.
Therefore, aiming at a relative precision below $10\%$ for the $C\\!P$
violation quantities and taking into account the current mixing parameter
world averages, the target precision for the mixing parameters is $2-3\times
10^{-4}$. With standard model indirect $C\\!P$ violation expected of the order
of $10^{-4}$, the direct $C\\!P$ violation parameter contributing to
$A_{\Gamma}$ has to be measured to an absolute precision of $10^{-3}$ in order
to distinguish the two types of $C\\!P$ violation in $A_{\Gamma}$.
Direct $C\\!P$ violation is not expected to be as large as the current world
average of $\Delta A_{C\\!P}$ in other decay modes. Therefore a precision of
$5\times 10^{-4}$ or better for asymmetry differences as well as individual
asymmetries is needed for measurements of other singly-Cabibbo-suppressed
charm decays. While measurements of time-integrated raw asymmetries at this
level should be well within reach, the challenge lies in the control of
production and detection asymmetries in order to extract the physics
asymmetries of individual decay modes.
For multibody final states the aim is clearly the understanding of $C\\!P$
asymmetries in the interfering resonances rather than global asymmetries. Of
highest interest are those resonances that are closely related to the two-body
modes used in $\Delta A_{C\\!P}$, for example the vector-pseudoscalar
resonances $K^{*}K$ and $\rho\pi$. The measurement of further suppressed
resonances is of interest as well since those have no contributions from
gluonic penguin diagrams, thus allowing to constrain the source of $C\\!P$
violating effects.
## 4 Rare charm decays
Rare decays provide a wide range of interesting measurements. The list of
decay modes includes flavour-changing neutral currents, radiative, lepton-
flavour violation, lepton-number violation, as well as baryon-number
violation. While a full discussion of rare charm decays would be beyond the
scope of this review a few remarks shall be made here.
There is a direct link between mixing and flavour changing neutral current
decays in several extensions of the standard model [55, 148]. These relate
$\Delta C=1$ annihilation amplitudes to $\Delta C=2$ mixing amplitudes where
the annihilation product creates a new $C\\!P$-conjugated
$\mathrm{c}\overline{\mathrm{u}}$ pair. At tree level one example is a heavy
$Z$-like boson with non-zero flavour-changing couplings.
The current central values of the $D^{0}$-mixing parameters translate into
model-dependent limits for rare decays based on common amplitudes. These rare
decay limits lie significantly below the current experimental limits. As
$D^{0}$ mixing is well established any upper limit from mixing will not change
significantly in the future. However, due to the direct correlation of mixing
and rare decays, any observation above the model-dependent rare decay limits
will rule out the corresponding model. The best limit on flavour-changing
neutral current decays is the recent LHCb limit on the decay
$D^{0}\\!\to\mu^{-}\mu^{+}$ of $1.1\times{}10^{-8}$ at $90\%$ confidence level
[149].
Among lepton-flavour violating decays the most stringent constraint is a Belle
search for $D^{0}\\!\to\mu^{\mp}\mathrm{e}^{\pm}$ achieving a limit of
$2.6\times{}10^{-7}$ at $90\%$ confidence level [150]. Searches for lepton-
flavour violating muon or kaon decays already provide more constraining
limits, however, in scenarious of non-universal couplings charm decays, giving
access to the up-quark sector, are of great interest.
The best limit on lepton-number violating charm decays has been placed by
BaBar on the decay $D^{+}\\!\to K^{-}\mathrm{e}^{+}\mathrm{e}^{+}$ with a
limit of $9.0\times{}10^{-7}$ at $90\%$ confidence level [151]. Only the CLEO
collaboration has carried out searches for baryon-number violating charm
decays. Their best limit on the decay $D^{0}\\!\to p\mathrm{e}^{-}$ is
$10^{-5}$ at $90\%$ confidence level [152]. For a more complete overview of
rare charm decays please refer to Ref. [73].
## 5 Conclusion
The first evidence for $C\\!P$ violation in the charm sector has opened the
door wide for a broad range of measurements. 40 years after the observation of
the first hint of charm particles in cosmic rays and exactly 37 years after
the dicovery of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
mesons, charm measurements may have shown first hints of effects beyond the
standard model at the LHC. The unambiguous observation of $C\\!P$ violation
will be the near term goal. The ultimate task is the interpretation of the
observed effects for which theoretical and experimental communities have to
collaborate closely to overcome the hurdles related to the charm quark mass
and the large cancellations in this system.
With the $B$ factories, CLEO-c and CDF analysing their final datasets, most
new results are expected to come from LHCb and BESIII. These are expected to
explore very interesting territory for charm $C\\!P$ violation. The longer
term future will be shaped by the LHCb upgrade as well as future
$\mathrm{e}^{+}\mathrm{e}^{-}$ collider experiments running both at the beauty
and charm thresholds. Charm’s third time has just begun to yield first fruits
which may well develop into a real charm.
## Acknowledgments
The author would like to thank Yuval Grossman, Claus Grupen, Alex Kagan,
Alexander Lenz, and Alexey Petrov as well as the members of the LHCb
collaboration for very insightful discussions. Special thanks to Alexander
Lenz and Johannes Albrecht who dared to read the article before submission.
The author further acknowledges the support of a Marie Curie Action:
“Cofunding of the CERN Fellowship Programme (COFUND-CERN)” of the European
Community’s Seventh Framework Programme under contract number (PCOFUND-
GA-2008-229600).
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|
arxiv-papers
| 2012-07-09T22:53:31 |
2024-09-04T02:49:32.844751
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Marco Gersabeck",
"submitter": "Marco Gersabeck",
"url": "https://arxiv.org/abs/1207.2195"
}
|
1207.2205
|
arxiv-papers
| 2012-07-10T01:20:46 |
2024-09-04T02:49:32.855931
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ling Li, Shibing Long, and Ming Liu",
"submitter": "Ling Li prof",
"url": "https://arxiv.org/abs/1207.2205"
}
|
|
1207.2341
|
# I/O-Efficient Dynamic Planar Range Skyline Queries
Casper Kejlberg-Rasmussen
MADALGO
Department of Computer Science
Aarhus University, Denmark
ckr@madalgo.au.dk Center for Massive Data Algorithmics - a Center of the
Danish National Research Foundation Konstantinos Tsakalidis
Computer Engineering and
Informatics Department
University of Patras, Greece
tsakalid@ceid.upatras.gr Kostas Tsichlas
Computer Science Department
Aristotle University
of Thessaloniki, Greece
tsichlas@csd.auth.gr
###### Abstract
We present the first fully dynamic worst case I/O-efficient data structures
that support planar orthogonal 3-sided range skyline reporting queries in
$\mathcal{O}(\log_{2B^{\epsilon}}n+\frac{t}{B^{1-\epsilon}})$ I/Os and updates
in $\mathcal{O}(\log_{2B^{\epsilon}}n)$ I/Os, using
$\mathcal{O}(\frac{n}{B^{1-\epsilon}})$ blocks of space, for $n$ input planar
points, $t$ reported points, and parameter $0\leq\epsilon\leq 1$. We obtain
the result by extending Sundar’s priority queues with attrition to support the
operations DeleteMin and CatenateAndAttrite in $\mathcal{O}(1)$ worst case
I/Os, and in $\mathcal{O}(1/B)$ amortized I/Os given that a constant number of
blocks is already loaded in main memory. Finally, we show that any pointer-
based static data structure that supports dominated maxima reporting queries,
namely the difficult special case of 4-sided skyline queries, in
$\mathcal{O}(\log^{\mathcal{O}(1)}n+t)$ worst case time must occupy
$\Omega(n\frac{\log n}{\log\log n})$ space, by adapting a similar lower
bounding argument for planar 4-sided range reporting queries.
## 1 Introduction
We study the problem of maintaining a set of planar points in external memory
subject to insertions and deletions of points in order to support planar
orthogonal 3-sided range skyline reporting queries efficiently in the worst
case. For two points $p,q\in\mathbb{R}^{d}$, we say that $p$ dominates $q$, if
and only if all the coordinates of $p$ are greater than those of $q$. The
skyline of a pointset $P$ consists of the maximal points of $P$, which are the
points in $P$ that are not dominated by any other point in $P$. Planar 3-sided
range skyline reporting queries that report the maximal points among the
points that lie
Skyline computation has been receiving increasing attention in the field of
databases since the introduction of the skyline operator for SQL [3]. Skyline
points correspond to the“interesting” entries of a relational database as they
are optimal simultaneously over all attributes. The considered variant of
planar skyline queries adds the capability of reporting the interesting
entries among those input entries whose attribute values belong to a given
3-sided range. Databases used in practical applications usually process
massive amounts of data in dynamic environments, where the data can be
modified by update operations. Therefore we analyze our algorithms in the I/O
model [1], which is commonly used to capture the complexity of massive data
computation. It assumes that the input data resides in the disk (external
memory) divided in blocks of $B$ consecutive words, and that computation
occurs for free in the internal memory of size $M$ words. An I/O-operation
(I/O) reads a block of data from the disk into the internal memory, or writes
a block of data to the disk. Time complexity is expressed in number of I/Os,
and space complexity in the number of blocks that the input data occupies on
the disk.
#### Previous Results
Different approaches have been proposed for maintaining the $d$-dimensional
skyline in external memory under update operations, assuming for example
offline updates over data streams [19, 13], only online deletions [20], online
average case updates [16], arbitrary online updates [8] and online updates
over moving input points [9]. The efficiency of all previous approaches is
measured experimentally in terms of disk usage over average case data.
However, even for the planar case, no I/O-efficient structure exists that
supports both arbitrary insertions and deletions in sublinear worst case I/Os.
Regarding internal memory, Brodal and Tsakalidis [4] present two linear space
dynamic data structures that support 3-sided range skyline reporting queries
in $\mathcal{O}(\log n+t)$ and $\mathcal{O}(\frac{\log n}{\log\log n}+t)$
worst case time, and updates in $\mathcal{O}(\log n)$ and
$\mathcal{O}(\frac{\log n}{\log\log n})$ worst case time in the pointer
machine and the RAM model, respectively, where $n$ is the input size and $t$
is the output size. They also present an $\mathcal{O}(n\log n)$ space dynamic
pointer-based data structure that supports 4-sided range skyline reporting
queries in $\mathcal{O}(\log^{2}n+t)$ worst case time and updates in
$\mathcal{O}(\log^{2}n)$ worst case time. Adapting these structures to the I/O
model attains $\mathcal{O}(\log^{\mathcal{O}(1)}_{B}n+t)$ query I/Os, which is
undesired since $\mathcal{O}(1)$ I/Os are spent per reported point.
Regarding the static variant of the problem, Sheng and Tao [17] obtain an
I/O-efficient algorithm that computes the skyline of a static $d$-dimensional
pointset in $\mathcal{O}(\frac{n}{B}\log^{d-2}_{\frac{M}{B}}\frac{n}{B})$
worst case I/Os, for $d\geq 3$, by adapting the internal memory algorithms of
[12, 2] to external memory.
$\mathcal{O}(\frac{n}{B}\log_{\frac{M}{B}}\frac{n}{B})$ I/Os can be achieved
for the planar case. There exist two $\mathcal{O}(n\log n)$ and
$\mathcal{O}(n\frac{\log n}{\log\log n})$ space static data structures that
support planar 4-sided range skyline reporting queries in $\mathcal{O}(\log
n+t)$ and $\mathcal{O}(\frac{\log n}{\log\log n}+t)$ worst case time, for the
pointer machine and the RAM, respectively [10, 7].
#### Our Results
In Section 3 we present the basic building block of the structures for dynamic
planar range skyline reporting queries that we present in Section 4. That is
pointer-based I/O-efficient catenable priority queues with attrition
(I/O-CPQAs) that support the operations DeleteMin and CatenateAndAttrite in
$\mathcal{O}(1/B)$ amortized I/Os and in $\mathcal{O}(1)$ worst case I/Os,
using $\mathcal{O}(\frac{n-m}{B})$ disk blocks, after $n$ calls to
CatenateAndAttrite and $m$ calls to DeleteMin. The result is obtained by
modifying appropriately a proposed implementation for priority queues with
attrition of Sundar [18].
In Section 4 we present our main result, namely I/O-efficient dynamic data
structures that support 3-sided range skyline reporting queries in
$\mathcal{O}(\log_{2B^{\epsilon}}n+\frac{t}{B^{1-\epsilon}})$ worst case I/Os
and updates in $\mathcal{O}(\log_{2B^{\epsilon}}n)$ worst case I/Os, using
$\mathcal{O}(\frac{n}{B^{1-\epsilon}})$ blocks, for a parameter
$0\leq\epsilon\leq 1$. These are the first fully dynamic skyline data
structures for external memory that support all operations in polylogarithmic
worst case time. The results are obtained by following the approach of
Overmars and van Leeuwen [14] for planar skyline maintainance and utilizing
confluently persistent I/O-CPQAs (implemented with functional catenable deques
[11]). Applying the same methodology to internal memory pointer-based CPQAs
yields alternative implementations for dynamic 3-sided reporting in the
pointer machine in the same bounds as in [4].
Finally, in Section 5 we prove that any pointer-based static data structure
that supports reporting the maximal points among the points that are dominated
by a given query point in $\mathcal{O}(\log^{\mathcal{O}(1)}n)$ worst case
time must occupy $\Omega(n\frac{\log n}{\log\log n})$ space, by adapting the
similar lower bounding argument of Chazelle [5] for planar 4-sided range
reporting queries to the considered dominated skyline reporting queries. These
queries are termed as dominating minima reporting queries. The symmetric case
of dominated maxima reporting queries is equivalent and comprises a special
case of rectangular visibilty queries [15] and 4-sided range skyline reporting
queries [4, 10]. The result shows that the space usage of the pointer-based
structures in [15, 4, 10] is optimal within a $\mathcal{O}(\log\log n)$
factor, for the attained query time.
## 2 Preliminaries
#### Priority Queues with Attrition
Sundar [18] introduces pointer-based _priority queues with attrition (PQAs)_
that support the following operations in $\mathcal{O}(1)$ worst case time on a
set of elements drawn from a total order: DeleteMin deletes and returns the
minimum element from the PQA, and InsertAndAttrite($e$) inserts element $e$
into the PQA and removes all elements larger than $e$ from the PQA. PQAs use
space linear to the number of inserted elements minus the number of elements
removed by DeleteMin.
#### Functional Catenable Deques
A dynamic data structure is persistent when it maintains its previous versions
as update operations are performed on it. It is fully persistent when it
permits accessing and updating the previous versions. In turn, it is called
confluently persistent when it is fully persistent, and moreover it allows for
two versions to be combined into a new version, by use of an update operation
that merges the two versions. In this case, the versions form a directed
acyclic version graph. A catenable deque is a list that stores a set of
elements from a total order, and supports the operations Push and Inject that
insert an element to the head and tail of the list respectively, Pop and Eject
that remove the element from the head and tail of the list respectively, and
Catenate that concatenates two lists into one. Kaplan and Tarjan [11] present
purely functional catenable deques that are confluently persistent and support
the above operations in $\mathcal{O}(1)$ worst case time.
#### Searching Lower Bound in the Pointer Machine
In the pointer machine model a data structure that stores a data set $S$ and
supports range reporting queries for a query set $\mathcal{Q}$, can be
modelled as a directed graph $G$ of bounded out-degree. In particular, every
node in $G$ may be assigned an element of $S$ or may contain some other useful
information. For a query range $Q_{i}\in\mathcal{Q}$, the algorithm navigates
over the edges of $G$ in order to locate all nodes that contain the answer to
the query. The algorithm may also traverse other nodes. The time complexity of
reporting the output of $Q_{i}$ is at least equal to the number of nodes
accessed in graph $G$ for $Q_{i}$. To prove a lower bound we need to construct
hard instances with particular properties, as discussed by Chazelle and Liu
[5, 6]. In particular, they define the graph $G$ to be
$(\alpha,\omega)$-effective, if a query is supported in $\alpha(t+\omega)$
time, where $t$ is the output size, $\alpha$ is a multiplicative factor for
the output size ($\alpha=\mathcal{O}(1)$ for our purposes) and $\omega$ is the
additive factor. They also define a query set $\mathcal{Q}$ to be
$(m,\omega)$-favorable for a data set $S$, if $|S\cap Q_{i}|\geq\omega,\forall
Q_{i}\in\mathcal{Q}$ and $|S\cap Q_{i_{1}}\cap\cdots\cap
Q_{i_{m}}|=\mathcal{O}(1),\forall i_{1}<i_{2}\cdots<i_{m}$. Intuitively, the
first part of this property requires that the size of the output is large
enough (at least $\omega$) so that it dominates the additive factor of
$\omega$ in the time complexity. The second part requires that the query
outputs have minimum overlap, in order to force $G$ to be large without many
nodes containing the output of many queries. The following lemma exploits
these properties to provide a lower bound on the minimum size of $G$.
###### Lemma 2.1.
[6, Lemma 2.3] For an $(m,\omega)$-favorable graph $G$ for the data set $S$,
and for an $(\alpha,\omega)$-effective set of queries $\mathcal{Q}$, $G$
contains $\Omega(|\mathcal{Q}|\omega/m)$ nodes, for constant $\alpha$ and for
any large enough $\omega$.
## 3 I/O-Efficient Catenable Priority Queues with Attrition
In this Section, we present I/O-efficient catenable priority queues with
attrition (I/O-CPQAs) that store a set of elements from a total order in
external memory, and support the following operations:
FindMin($Q$) returns the minimum element in I/O-CPQA $Q$.
DeleteMin($Q$) removes the minimum element $e$ from I/O-CPQA $Q$ and returns
element $e$ and the new I/O-CPQA $Q^{\prime}=Q\backslash\\{e\\}$.
CatenateAndAttrite($Q_{1},Q_{2}$) concatenates I/O-CPQA $Q_{2}$ to the end of
I/O-CPQA $Q_{1}$, removes all elements in $Q_{1}$ that are larger than the
minimum element in $Q_{2}$, and returns a new I/O-CPQA $Q^{\prime}_{1}=\\{e\in
Q_{1}|e<\min(Q_{2})\\}\cup Q_{2}$. We say that the removed elements have been
_attrited_.
InsertAndAttrite($Q,e$) inserts element $e$ at the end of $Q$ and attrites all
elements in $Q$ that are larger than the value of $e$.
All operations take $\mathcal{O}(1)$ worst case I/Os and $\mathcal{O}(1/b)$
amortized I/Os, given that a constant number of blocks is already loaded into
main memory, for a parameter $1\leq b\leq B$. To achieve the result, we modify
an implementation for the PQAs of Sundar [18].
An I/O-CPQA $Q$ consists of $k_{Q}+2$ deques of records, called the clean
deque $C(Q)$, the buffer deque $B(Q)$ and the dirty deques
$D_{1}(Q),\ldots,D_{k_{Q}}(Q)$, where $k_{Q}\geq 0$. A _record_ $r=(l,p)$
consists of a buffer $l$ of $[b,4b]$ elements of strictly increasing value and
a pointer $p$ to an I/O-CPQA. The ordering of $r$ is; first all elements of
$l$ and then all elements of the I/O-CPQA pointed to by $p$. We define the
queue order of $Q$ to be $C(Q)$, $B(Q)$ and $D_{1}(Q),\ldots,D_{k_{Q}}(Q)$. A
record is _simple_ when its pointer $p$ is _null_. The clean deque and the
buffer deque only contains simple records. See Figure 1 for an overview of the
structure.
$\ldots$$C(Q)$$\ldots$$B(Q)$$\ldots$$D_{1}(Q)$$\ldots$$D_{k_{Q}-1}(Q)$$\ldots$$D_{k_{Q}}(Q)$$\ldots$
Figure 1: A I/O CPQA $Q$ consists of $k_{Q}+2$ deques of records;
$C(Q),B(Q),D_{1}(Q),\ldots,D_{k_{Q}}(Q)$. The records in $C(Q)$ and $B(Q)$ are
simple, the records of $D_{1}(Q),\ldots,D_{k_{Q}}(Q)$ may contain pointers to
other I/O CPQA’s. Gray recordsare always loaded in memory.
Given a record $r=(l,p)$ the minimum and maximum elements in the buffers of
$r$, are denoted by $\min(r)=\min(l)$ and $\max(r)=\max(l)$, respectively.
They appear respectively first and last in the queue order of $l$, since the
buffer of $r$ is sorted by value. Henceforth, we do not distinguish between an
element and its value. Given a deque $q$ the first and the last record is
denoted by $\text{first}(q)$ and $\text{last}(q)$, respectively. Also
$\text{rest}(q)$ denotes all records of the deque $q$ excluding the record
$\text{first}(q)$. Similarly, $\text{front}(q)$ denotes all records for the
deque $q$ excluding the record $\text{last}(q)$. The size $|r|$ of a record
$r$ is defined to be the number of elements in its buffer. The size $|q|$ of a
deque $q$ is defined to be the number of records it contains. The size $|Q|$
of the I/O-CPQA $Q$ is defined to be the number of elements that $Q$ contains.
For an I/O-CPQA $Q$ we denote by $\text{first}(Q)$ and $\text{last}(Q)$, the
first and last of the records in $C(Q),B(Q),D_{1}(Q),\ldots,D_{k_{Q}}(Q)$ that
exists, respectively. By $\text{middle}(Q)$ we denote all records in $Q$ and
the records in the I/O-CPQAs pointed by $Q$, except for records
$\text{first}(Q)$ and $\text{last}(Q)$ and the I/O-CPQAs they point to. We
call an I/O-CPQA $Q$ _large_ if $|Q|\geq b$ and _small_ otherwise. The minimum
value of all elements stored in the I/O-CPQA $Q$ is denote by $\min(Q)$. For
an I/O-CPQA $Q$ we maintain the following invariants:
1. I.1)
For every record $r=(l,p)$ where pointer $p$ points to I/O-CPQA $Q^{\prime}$,
$\max(l)<\min(Q^{\prime})$ holds.
2. I.2)
In all deques of $Q$, where record $r_{1}=(l_{1},p_{1})$ precedes record
$r_{2}=(l_{2},p_{2})$, $\max(l_{1})<\min(l_{2})$ holds.
3. I.3)
For the deques $C(Q),B(Q)$ and $D_{1}(Q)$,
$\max(\text{last}(C(Q)))<\min(\text{first}(B(Q)))<\min(\text{first}(D_{1}(Q)))$
holds.
4. I.4)
Element $\min(\text{first}(D_{1}(Q)))$ has the minimum value among all the
elements in the dirty deques $D_{1}(Q),\ldots,D_{k}(Q)$.
5. I.5)
All records in the deques $C(Q)$ and $B(Q)$ are simple.
6. I.6)
$|C(Q)|\geq\sum_{i=1}^{k_{Q}}{|D_{i}(Q)|}+k_{Q}-1$.
7. I.7)
$|\text{first}(C(Q))|<b$ holds, if and only if $|Q|<b$ holds.
8. I.8)
$|\text{last}(D_{k_{Q}}(Q))|<b$ holds, if and only if record
$\text{last}(D_{k_{Q}}(Q))$ is simple. In this case $|r|\in[b,5b]$ holds.
From Invariants I.2, I.3 and I.4, we have that the minimum element $\min(Q)$
stored in the I/O-CPQA $Q$ is element $\min(\text{first}(C(Q)))$. We say that
an operation improves or aggravates by a parameter $c$ the inequality of
invariant I.6 for I/O-CPQA $Q$, when the operation increases or decreases
$\Delta(Q)=|C(Q)|-\sum_{i=1}^{k_{Q}}{|D_{i}(Q)|}-k_{Q}+1$ by $c$,
respectively. To argue about the $\mathcal{O}(1/b)$ amortized I/O bounds we
define the following potential functions for large and small I/O-CPQAs. In
particular, for large I/O-CPQAs $Q$, the potential $\Phi(Q)$ is defined as
$\Phi(Q)=\Phi_{F}(|\text{first}(Q)|)+|\text{middle}(Q)|+\Phi_{L}(|\text{last}(Q)|),$
where
$\begin{array}[]{ccc}{\Phi_{F}(x)=\left\\{\begin{array}[]{cl}3-\frac{x}{b},&b\leq
x<2b\\\ 1,&2b\leq x<3b\\\ \frac{2x}{b}-5,&3b\leq x\leq 4b\\\
\end{array}\right.}&\text{and}&{\Phi_{L}(x)=\left\\{\begin{array}[]{cl}0,&0\leq
x<4b\\\ \frac{3x}{b}-12,&4b\leq x\leq 5b\\\ \end{array}\right.}\end{array}$
For small I/O-CPQAs $Q$, the potential $\Phi(Q)$ is defined as
$\Phi(Q)=\frac{3|Q|}{b}$
The total potential $\Phi_{T}$ is defined as
$\Phi_{T}=\sum_{Q}{\Phi(Q)}+\sum_{Q|b\leq|Q|}{1},$
where the first sum is over all I/O-CPQAs $Q$ and the second sum is only over
all large I/O-CPQAs $Q$.
### 3.1 Operations
In the following, we describe the algorithms that implement the operations
supported by the I/O-CPQA $Q$. The operations call the auxiliary operation
Bias$(Q)$, which will be described last, that improves the inequality of
invariant I.6 for $Q$ by at least $1$. All operations take $\mathcal{O}(1)$
worst case I/Os. We also show that every operation takes $\mathcal{O}(1/b)$
amortized I/Os, where $1\leq b\leq B$.
#### FindMin($Q$)
returns the value $\min(\text{first}(C(Q)))$.
#### DeleteMin($Q$)
removes element $e=\min(\text{first}(C(Q)))$ from record
$(l,p)=\text{first}(C(Q))$. After the removal, if $|l|<b$ and $|Q|\geq b$
hold, we do the following. If $b\leq|\text{first}(\text{rest}(C(Q)))|\leq 2b$,
then we merge $\text{first}(C(Q))$ with $\text{first}(\text{rest}(C(Q)))$ into
one record which is the new first record. Else if
$2b<|\text{first}(\text{rest}(C(Q)))|\leq 3b$ then we take $b$ elements out of
$\text{first}(\text{rest}(C(Q)))$ and put them into $\text{first}(C(Q))$. Else
we have that $3b<|\text{first}(\text{rest}(C(Q)))|$, and as a result we take
$2b$ elements out of $\text{first}(\text{rest}(C(Q)))$ and put them into
$\text{first}(C(Q))$. If the inequality for $Q$ is aggravated by $1$ we call
Bias($Q$) once. Finally, element $e$ is returned.
Amortization: Only if the size of $\text{first}(C(Q))$ becomes
$|\text{first}(C(Q))|=b-1$ do we incur any I/Os. In this case
$r=\text{first}(Q)$ has a potential of $\Phi_{F}(|r|)=2$, and since we
increase the number of elements in $r$ by $b$ to $2b$ elements, the potential
of $r$ will then only be $\Phi_{F}(|r|)=1$. Thus, the total potential
decreases by $1$, which also pays for any I/Os including those incurred if
Bias$(Q)$ is invoked.
#### CatenateAndAttrite($Q_{1},Q_{2}$)
concatenates $Q_{2}$ to the end of $Q_{1}$ and removes the elements from
$Q_{1}$ with value larger than $\min(Q_{2})$. To do so, it creates a new
I/O-CPQA $Q^{\prime}_{1}$ by modifying $Q_{1}$ and $Q_{2}$, and by calling
Bias($Q^{\prime}_{1}$) and Bias($Q_{2}$).
If $|Q_{1}|<b$, then $Q_{1}$ is only one record $(l_{1},\cdot)$, and so we
prepend it into the first record $(l_{2},\cdot)=\text{first}(Q_{2})$ of
$Q_{2}$. Let $l_{1}^{\prime}$ be the non-attrited elements of $l_{1}$. We
perform the prepend as follows. If $|l_{1}^{\prime}|+|l_{2}|\leq 4b$, then we
prepend $l_{1}^{\prime}$ into $l_{2}$. Else, we take $2b-|l_{1}^{\prime}|$
elements out of $l_{2}$, and make them along with $l_{1}^{\prime}$ the new
first record of $Q_{2}$.
Amortization: If we simply prepend $l_{1}^{\prime}$ into $l_{2}$, then the
potential $\Phi_{S}(|l_{1}|)$ pays for the increase in potential of
$\Phi_{F}(|\text{first}(C(Q_{2}))|)$. Else, we take $2b-|l_{1}^{\prime}|$
elements out of $l_{2}$, and these elements along with $l_{1}^{\prime}$ become
the new first record of $Q_{2}$ of size $2b$. Thus, $\Phi_{F}(2b)=1$ and the
potential drops by $1$, which is enough to pay for the I/Os used to flush the
old first record of $C(Q_{2})$ to disk.
If $|Q_{2}|<b$, then $Q_{2}$ only consists of one record. We have two cases,
depending on how much of $Q_{1}$ is attrited by $Q_{2}$. Let $r_{1}$ be the
second last record for $Q_{1}$ and let $r_{2}=\text{last}(Q_{1})$ be the last
record. If $e$ attrites all of $r_{1}$, then we just pick the appropriate case
among (1–4) below. Else if $e$ attrites partially $r_{1}$, but not all of it,
then we delete $r_{2}$ and we merge $r_{1}$ and $Q_{2}$ into the new last
record of $Q_{1}$, which cannot be larger than $5b$. Otherwise if $e$ attrites
partially $r_{2}$, but not all of it, then we simply append the single record
of $Q_{2}$ into $r_{2}$, which will be the new last record of $Q_{1}$ and it
cannot be larger than $5b$.
Amortization: If $e$ attrites all of $r_{1}$, then we release at least $1$ in
potential, so all costs in any of the cases (1–4) are paid for. If $e$
attrites partially $r_{1}$, then the new record cannot contain more than $5b$
elements, and thus any increase in potential is paid for by the potential of
$Q_{2}$. Thus, the I/O cost is covered by the decrease of $1$ in potential,
caused by $r_{1}$. If $e$ attrites partially $r_{2}$, any increase in
potential is paid for by the potential of $Q_{2}$.
We have now dealt with the case where $Q_{1}$ is a small queue, so in the
following we assume that $Q_{1}$ is large. Let $e=\min(Q_{2})$.
1. 1)
If $e\leq\min(\text{first}(C(Q_{1})))$, we discard I/O-CPQA $Q_{1}$ and set
$Q^{\prime}_{1}=Q_{2}$.
2. 2)
Else if $e\leq\max(\text{last}(C(Q_{1})))$, we remove the simple record
$(l,\cdot)=\text{first}(C(Q_{2}))$ from $C(Q_{2})$, we set
$C(Q^{\prime}_{1})=\emptyset$, $B(Q^{\prime}_{1})=C(Q_{1})$ and
$D_{1}(Q^{\prime}_{1})=(l,p)$, where $p$ points to $Q_{2}$, if it exists. This
aggravates the inequality for $Q_{2}$ by at most $1$, and gives
$\Delta(Q^{\prime}_{1})=-1$. Thus, we call Bias$(Q_{2})$ once and
Bias$(Q^{\prime}_{1})$ once.
3. 3)
Else if $e\leq\min(\text{first}(B(Q_{1})))$ or
$e\leq\min(\text{first}(D_{1}(Q_{1})))$ holds, we remove the simple record
$(l,\cdot)=\text{first}(C(Q_{2}))$ from $C(Q_{2})$, set
$D_{1}(Q^{\prime}_{1})=(l,p)$, and make $p$ point to $Q_{2}$, if it exists. If
$e\leq\min(\text{first}(B(Q_{1})))$, we set $B(Q_{1}^{\prime})=\emptyset$.
This aggravates the inequality for $Q_{2}$ by at most $1$, and aggravates the
inequality for $Q_{1}$ by at most $1$. Thus, we call Bias$(Q_{2})$ once and
Bias$(Q^{\prime}_{1})$ once.
4. 4)
Else, let $(l_{1},\cdot)=\text{last}(D_{k_{Q_{1}}})$. We remove
$(l_{2},\cdot)=\text{first}(C(Q_{2}))$ from $C(Q_{2})$. If $|l_{1}|<b$, then
remove the record $(l_{1},\cdot)$ from $D_{k_{Q_{1}}}$. Let $l_{1}^{\prime}$
be the non-attrited elements under attrition by $e=\min(l_{2})$. If
$|l_{1}^{\prime}|+|l_{2}|\leq 4b$, then we prepend $l_{1}^{\prime}$ into
$l_{2}$ of record $r_{2}=(l_{2},p_{2})$, where $p_{2}$ points to $Q_{2}$.
Otherwise. we make a new simple record $r_{1}$ with $l_{1}^{\prime}$ and $2b$
elements taken out of $r_{2}=(l_{2},p_{2})$. Finally, we put the resulting one
or two records $r_{1}$ and $r_{2}$ into a new deque $D_{k_{Q_{1}}+1}(Q_{1})$.
This aggravates the inequality for $Q_{2}$ by at most $1$, and the inequality
for $Q_{1}$ by at most $2$. Thus, we call Bias$(Q_{2})$ once and
Bias$(Q^{\prime}_{1})$ twice.
Amortization: In all the cases (1–4) both $Q_{1}$ and $Q_{2}$ are large, hence
when we concatenate them we decrease the potential by at least $1$, as the
number of large I/O-CPQA’s decrease by one which is enough to pay for any Bias
operations.
#### InsertAndAttrite($Q$, $e$)
inserts an element $e$ into I/O-CPQA $Q$ and attrites the elements in $Q$ with
value larger than $e$. This is a special case of operation
CatenateAndAttrite($Q_{1}$,$Q_{2}$), where $Q_{1}=Q$ and $Q_{2}$ is an
I/O-CPQA that only contains one record with the single element $e$.
Amortization: Since creating a new I/O-CPQA with only one element and calling
CatenateAndAttrite only costs $\mathcal{O}(1/b)$ I/Os amortized, the operation
InsertAndAttrite also costs $\mathcal{O}(1/b)$ I/Os amortized.
$\ldots$$C(Q)$$\ldots$$D_{1}(Q)$$\ldots$$C(Q^{\prime})$$\ldots$$B(Q^{\prime})$$\ldots$$D_{1}(Q^{\prime})$$\ldots$$D_{k_{Q^{\prime}}}(Q^{\prime})$$\ldots$
Figure 2: In the case of Bias$(Q)$, where $B(Q)=\emptyset$ and $k_{Q}=1$, we
need to follow the pointer $p$ of $(l,p)=\text{first}(D_{1}(Q))$ that may
point to an I/O-CPQA $Q^{\prime}$. If so, we merge it into $Q$, taking into
account attrition of $Q^{\prime}$ by $e=\min(\text{first}(D_{1}(Q)))$.
#### Bias$(Q)$
improves the inequality in I.6 for $Q$ by at least $1$.
Amortization: Since all I/Os incurred by Bias$(Q)$ are already paid for by the
operation that called Bias$(Q)$, we only need to argue that the potential of
$Q$ does not increase due to the changes that Bias$(Q)$ makes to $Q$.
1. 1)
$|B(Q)|>0$: We remove the first record $\text{first}(B(Q))=(l_{1},\cdot)$ from
$B(Q)$ and let $(l_{2},p_{2})=\text{first}(D_{1}(Q))$. Let $l_{1}^{\prime}$ be
the non-attrited elements of $l_{1}$ under attrition from $e=\min(l_{2})$.
1. 1)
$0\leq|l_{1}^{\prime}|<b$: If $|l_{2}|\leq 2b$, then we just prepend
$l_{1}^{\prime}$ onto $l_{2}$. Else, we take $b$ elements out of $l_{2}$ and
append them to $l_{1}^{\prime}$.
2. 2)
$b\leq|l_{1}^{\prime}|<2b$: If $|l_{2}|\leq 2b$, and if furthermore
$|l_{1}^{\prime}|+|l_{2}|\leq 3b$ holds, then we merge $l_{1}^{\prime}$ and
$l_{2}$. Else $|l_{1}^{\prime}|+|l_{2}|>3b$ holds, so we take $2b$ elements
out of $l_{1}^{\prime}$ and $l_{2}$ and put them into $l_{1}^{\prime}$,
leaving the rest in $l_{2}$.
Else $|l_{2}|>2b$ holds, so we take $b$ elements out of $l_{2}$ and put them
into $l_{1}^{\prime}$.
If we did not prepend $l_{1}^{\prime}$ onto $l_{2}$, we insert
$l_{1}^{\prime}$ along with any elements taken out of $l_{2}$ at the end of
$C(Q)$ instead. If $|l_{1}^{\prime}|<|l_{1}|$, we set $B(Q)=\emptyset$. Else,
we did prepend $l_{1}^{\prime}$ onto $l_{2}$, and then we just recursively
call Bias. Since $|B(Q)|=0$ we will not end up in this case again. As a
result, in all cases the inequality of $Q$ is improved by $1$.
Amortization: If $l_{1}=\text{first}(Q)$, then after calling Bias we ensure
that $2b\leq|\text{first}(Q)|\leq 3b$, and so the that potential of $Q$ does
not increase.
2. 2)
$|B(Q)|=0$: When $|B(Q)|=0$ holds, we have two cases depending on the number
of dirty queues, namely cases $k_{Q}>1$ and $k_{Q}=1$.
1. 1)
$k_{Q}>1$: Let $e=\min(\text{first}(D_{k_{Q}}(Q)))$. If
$e\leq\min(\text{last}(D_{k_{Q}-1}(Q)))$ holds, we remove the record
$\text{last}(D_{k_{Q}-1}(Q))$ from $D_{k_{Q}-1}(Q)$. This improves the
inequality of $Q$ by $1$.
Else, if
$\min(\text{last}(D_{k_{Q}-1}(Q)))<e\leq\max(\text{last}(D_{k_{Q}-1}(Q)))$
holds, we remove record $r_{1}=(l_{1},p_{1})=\text{last}(D_{k_{Q}-1}(Q))$ from
$D_{k_{Q}-1}(Q)$ and let $r_{2}=(l_{2},p_{2})=\text{first}(D_{k_{Q}}(Q))$. We
delete any elements in $l_{1}$ that are attrited by $e$, and let
$l_{1}^{\prime}$ denote the non-attrited elements.
1. 1)
$0\leq|l_{1}^{\prime}|<b$: If $|l_{2}|\leq 2b$, then we just prepend
$l_{1}^{\prime}$ onto $l_{2}$. Otherwise, we take $b$ elements out of $l_{2}$
and append them to $l_{1}^{\prime}$.
2. 2)
If $b\leq|l_{1}^{\prime}|<2b$: If $|l_{2}|\leq 2b$ and
$|l_{1}^{\prime}|+|l_{2}|\leq 3b$, then we merge $l_{1}^{\prime}$ and $l_{2}$.
Else, $|l_{1}^{\prime}|+|l_{2}|>3b$ holds, so we take $2b$ elements out of
$l_{1}^{\prime}$ and $l_{2}$ and put them into $l_{1}^{\prime}$, leaving the
rest in $l_{2}$.
Else $|l_{2}|>2b$, so we take $b$ elements out of $l_{2}$ and put them into
$l_{1}^{\prime}$.
If $r_{1}$ still exists, we insert it in the front of $D_{k_{Q}}(Q)$. Finally,
we concatenate $D_{k_{Q}-1}(Q)$ and $D_{k_{Q}}(Q)$ into one deque. This
improves the inequality of $Q$ by at least $1$.
Else $\max(\text{last}(D_{k_{Q}-1}(Q)))<e$ holds, and we just concatenate the
deques $D_{k_{Q}-1}(Q)$ and $D_{k_{Q}}(Q)$, which improves the inequality for
$Q$ by $1$.
Amortization: If not all of $l_{1}$ is attrited then we ensure that its record
$r_{1}$ has size between $2b$ and $3b$. Thus, if $r_{1}=\text{first}(Q)$
holds, we will not have increased the potential of $Q$. In the cases where all
or none of $l_{1}$ is attrited, the potential of $Q$ can only be decreased by
at least $0$.
2. 2)
$k_{Q}=1$: In this case $Q$ contains only deques $C(Q)$ and $D_{1}(Q)$. We
remove the record $r=(l,p)=\text{first}(D_{1}(Q))$ and insert $l$ into a new
record at the end of $C(Q)$. This improves the inequality of $Q$ by at least
$1$. If $r$ is not simple, let $r$’s pointer $p$ point to I/O-CPQA
$Q^{\prime}$. We restore I.5 for $Q$ by merging I/O-CPQAs $Q$ and $Q^{\prime}$
into one I/O-CPQA. See Figure 2 for this case of operation Bias. In
particular, let $e=\min(\text{first}(D_{1}(Q)))$, we now proceed as follows:
If $e\leq\min(Q^{\prime})$, we discard $Q^{\prime}$. The inequality for $Q$
remains unaffected.
Else, if
$\min(\text{first}(C(Q^{\prime})))<e\leq\max(\text{last}(C(Q^{\prime}))$, we
set $B(Q)=C(Q^{\prime})$ and discard the rest of $Q^{\prime}$. The inequality
for $Q$ remains unaffected.
Else if
$\max(\text{last}(C(Q^{\prime}))<e\leq\min(\text{first}(D_{1}(Q^{\prime})))$,
we concatenate the deque $C(Q^{\prime})$ at the end of $C(Q)$. If moreover
$\min(\text{first}(B(Q^{\prime})))<e$ holds, we set $B(Q)=B(Q^{\prime})$.
Finally, we discard the rest of $Q^{\prime}$. This improves the inequality for
$Q$ by $|C(Q^{\prime})|$.
Else $\min(\text{first}(D_{1}(Q^{\prime})))<e$ holds. We concatenate the deque
$C(Q^{\prime})$ at the end of $C(Q)$, we set $B(Q)=B(Q^{\prime})$, we set
$D_{1}(Q^{\prime}),\ldots,D_{k_{Q^{\prime}}}(Q^{\prime})$ as the first
$k_{Q^{\prime}}$ dirty queues of $Q$ and we set $D_{1}(Q)$ as the last dirty
queue of $Q$. This improves the inequality for $Q$ by $\Delta(Q^{\prime})\geq
0$, since $Q^{\prime}$ satisfied I.6 before the operation.
If $r=\text{first}(Q)$ and $|l|\leq 2b$, then we remove $r$ and run Bias
recursively. Let $r^{\prime}=(l^{\prime},p^{\prime})=\text{first}(Q)$. If
$|l|+|l^{\prime}|>3b$, then we take the $2b$ first elements out and make them
the new first record of $C(Q)$. Else we merge $l$ into $l^{\prime}$, so that
$r$ is removed and $r^{\prime}$ is now $\text{first}(Q)$.
Amortization: Since $\text{first}(Q)$ is either untouched or left with $2b$ to
$3b$ elements, in which case its potential is $1$, and since all other changes
decrease the potential by at least $0$, we have that Bias does not increase
the potential of $Q$.
###### Theorem 3.1.
A set of $\ell$ I/O-CPQA’s can be maintained supporting the operations
FindMin, DeleteMin, CatenateAndAttrite and InsertAndAttrite in
$\mathcal{O}(1/b)$ I/Os amortized and $\mathcal{O}(1)$ worst case I/Os per
operation. The space usage is $\mathcal{O}(\frac{n-m}{b})$ blocks after
calling CatenateAndAttrite and InsertAndAttrite $n$ times and DeleteMin $m$
times, respectively. We require that $M\geq\ell b$ for $1\leq b\leq B$, where
$M$ is the main memory size and $B$ is the block size.
###### Proof.
The correctness follows by closely noticing that we maintain invariants
I.1–I.8, and from those we have that DeleteMin$(Q)$ and FindMin$(Q)$ always
returns the minimum element of $Q$.
The worst case I/O bound of $\mathcal{O}(1)$ is trivial as every operation
only touches $\mathcal{O}(1)$ records. Although Bias is recursive, we notice
that in the case where $|B(Q)|>0$, Bias only calls itself after making
$|B(Q)|=0$, so it will not end up in this case again. Similarly, if $|B(Q)|=0$
and $k_{Q}>1$ there might also be a recursive call to Bias. However, before
the call at least $b$ elements have been taken out of $Q$, and thus the
following recursive call to Bias will ensure at least $b$ more are taken out.
This is enough to stop the recursion, which will have depth at most $3$.
The $\mathcal{O}(1/b)$ amortized I/O bounds, follows from the potential
analysis made throughout the description of each operation. ∎
### 3.2 Concatenating a Sequence of I/O-CPQAs
We describe how to CatenateAndAttrite I/O-CPQAs $Q_{1},Q_{2},\ldots,Q_{\ell}$
into a single I/O-CPQA in $\mathcal{O}(1)$ worst case I/Os, given that
DeleteMin is not called in the sequence of operations. We moreover impose two
more assumptions. In particular, we say that I/O-CPQA $Q$ is in state
$x\in\mathbb{Z}$, if $|C(Q)|=\sum_{i=1}^{k_{Q}}{|D_{i}(Q)|}+k_{Q}-1+x$ holds.
Positive $x$ implies that Bias$(Q)$ will be called after the inequality for
$Q$ is aggravated by $x+1$. Negative $x$ implies that Bias$(Q)$ need to be
called $x$ operations times in order to restore inequality for $Q$. So, we
moreover assume that I/O-CPQAs $Q_{i},i\in[1,\ell]$ are at state at least
$+2$, unless $Q_{i}$ contains only one record in which case it may be in state
$+1$. We call a record $r=(l,p)$ in an I/O-CPQA $Q_{i}$ critical, if $r$ is
accessed at some time during the sequence of operations. In particular, the
critical records for $Q_{i}$ are
$\text{first}(C(Q_{i})),\text{first}(\text{rest}(C(Q_{i}))),\text{last}(C(Q_{i})),\text{first}(B(Q_{i})),\text{first}(D_{1}(Q_{i})),\text{last}(D_{k_{Q_{i}}}(Q_{i}))$,
and $\text{last}(\text{front}(D_{k_{Q_{i}}}(Q_{i})))$ if it exists. Otherwise,
record $\text{last}(D_{k_{Q_{i}}-1}(Q_{i}))$ is critical. So, we moreover
assume that the critical records for I/O-CPQAs $Q_{i},i\in[1,\ell]$ are loaded
into memory.
The algorithm considers I/O-CPQAs $Q_{i}$ in decreasing index $i$ (from right
to left). It sets $Q^{i}=Q_{\ell}$ and constructs the temporary I/O-CPQA
$Q^{i-1}$ by calling CatenateAndAttrite($Q_{i-1}$,$Q^{i}$). This yields the
final I/O-CPQA $Q^{1}$.
###### Lemma 3.1.
I/O-CPQAs $Q_{i},i\in[1,\ell]$ can be CatenateAndAttrited into a single
I/O-CPQA without any access to external memory, provided that:
1. 1.
$Q_{i}$ is in state at least $+2$, unless it contains only one record, in
which case its state is at least $+1$,
2. 2.
all critical records of all $Q_{i}$ reside in main memory.
###### Proof.
To avoid any I/Os during the sequence of CatenateAndAttrites, we ensure that
Bias is not called, and that the critical records are sufficient, and thus no
more records need to be loaded into memory.
To avoid calling Bias we prove by induction the invariant that the temporary
I/O-CPQAs $Q^{i},i\in[1,\ell]$ constructed during the sequence are in state at
least $+1$. Let the invariant hold of $Q^{i+1}$ and let $Q^{i}$ be constructed
by CatenateAndAttrite($Q_{i}$,$Q^{i+1}$). If $Q_{i}$ contains at most two
records, which both reside in dequeue $C(Q_{i})$, we only need to access
record $\text{first}(C(Q^{i+1}))$ and the at most two records of $Q_{i}$. The
invariant holds for $Q^{i}$, since it holds inductively for $Q^{i+1}$ and the
new records were added at $C(Q^{i+1})$. As a result, the inequality of I.6 for
$Q^{i+1}$ can only be improved. If $Q^{i+1}$ consists of only one record, then
either one of the following cases apply or we follow the steps described in
operation CatenateAndAttrite. In the second case, there is no aggravation for
the inequality of 6 and only critical records are used.
In the following, we can safely assume that $Q_{i}$ has at least three records
and its state is at least $+2$. We parse the cases of the CatenateAndAttrite
algorithm assumming that $e=\min(Q^{i+1})$.
* Case 1
The invariant holds trivially since $Q_{i}$ is discarded and no change happens
to $Q^{i}=Q^{i+1}$. Bias is not called.
* Cases 2,3
The algorithm checks whether the first two records of $C(Q_{i})$ are attrited
by $e$. If this is the case, we continue as denoted at the start of this
proof. Otherwise, case 2 of CatenateAndAttrite is applied as is. $Q^{i+1}$ is
in state $0$ after the concatenation and $Q^{i}$ is in state $+1$. Thus the
invariant holds, and Bias is not. Note that all changes take place at the
critical records of $Q_{i}$ and $Q^{i+1}$.
* Case 4
The algorithm works exactly as in case 4 of CatenateAndAttrite, with the
following exception. At the end, $Q^{i}$ will be in state $0$, since we added
the deque $D_{k_{Q^{i+1}}+1}$ with a new record and the inequality of I.6 is
aggrevated by $2$. To restore the invariant we apply case 2(1) of Bias. This
step requires access to records $\text{last}(D_{k_{Q^{i}}-1})$ and
$\text{first}(D_{k_{Q^{i}}})$. These records are both critical, since the
former corresponds to $\text{last}(D_{k_{Q^{i+1}}})$ and the latter to
$\text{first}C(Q^{i+1})$. In addition, Bias$(Q^{i+1})$ need not be called,
since by the invariant, $Q^{i+1}$ was in state $+1$ before the removal of
$\text{first}C(Q^{i+1})$. In this way, we improve the inequality for $Q^{i}$
by $1$ and invariant holds.
∎
## 4 Dynamic Planar Range Skyline Reporting
In this Section we present dynamic I/O-efficient data structures that support
3-sided planar orthogonal range skyline reporting queries.
#### 3-Sided Skyline Reporting
We describe how to utilize I/O-CPQAs in order to obtain dynamic data
structures that support 3-sided range skyline reporting queries and arbitrary
insertions and deletions of points, by modifying the approach of [14] for the
pointer machine model. In particular, let $P$ be a set of $n$ points in the
plane, sorted by $x$-coordinate. To access the points, we store their
$x$-coordinates in an $(a,2a)$-tree $T$ with branching parameter $a\geq 2$ and
leaf parameter $k\geq 1$. In particular, every node has degree within $[a,2a]$
and every leaf contains at most $k$ consecutive by $x$-coordinate input
points. Every internal node $u$ of $T$ is associated with an I/O-CPQA whose
non-attrited elements correspond to the maximal points among the points stored
in the subtree of $u$. Moreover, $u$ contains a representative block with the
critical records of condition 2 in Lemma 3.1 for the I/O-CPQAs associated with
its children nodes.
To construct the structure, we proceed in a bottom up manner. First, we
compute the maximal points among the points contained in every leaf of $T$. In
particular for every leaf, we initialize an I/O-CPQA $Q$. We consider the
points $(p_{x},p_{y})$ stored in the block in increasing $x$-coordinate, and
call InsertAndAttrite($Q,-p_{y}$). In this way, a point $p$ in the block that
is dominated by another point $q$ in the block, is inserted before $q$ in $Q$
and has value $-p_{y}>-q_{y}$. Therefore, the dominated points in the block
correspond to the attrited elements in $Q$.
We construct the I/O-CPQA for an internal node $u$ of $T$ by concatenating the
already constructed I/O-CPQAs $Q_{i}$ at its children nodes $u_{i}$ of $u$,
for $i\in[1,a]$ in Section 3. Then we call Bias to the resulting I/O-CPQA
appropriately many times in order to satisfy condition 1 in Lemma 3.1. The
procedure ends when the I/O-CPQA is constructed for the root of $T$. Notice
that the order of concatenations follows implicitly the structure of the tree
$T$. To insert (resp. delete) a point $p=(p_{x},p_{y})$ to the structure, we
first insert (resp. delete) $p_{x}$ to $T$. This identifies the leaf with the
I/O-CPQA that contains $p$. We discard all I/O-CPQAs from the leaf to the root
of $T$, and recompute them in a bottom up manner, as described above.
To report the skyline among the points that lie within a given 3-sided query
rectangle $[x_{\ell},x_{r}]\times[y_{b},+\infty)$, it is necessary to obtain
the maximal points in a subtree of a node $u$ of $T$ by querying the I/O-CPQA
stored in $u$. Notice, however, that computing the I/O-CPQA of an internal
node of $T$ modifies the I/O-CPQAs of its children nodes. Therefore, we can
only report the skyline of all points stored in $T$, by calling DeleteMin at
the I/O-CPQA stored in the root of $T$. The rest of the I/O-CPQAs in $T$ are
not queriable in this way, since the corresponding nodes do not contain the
version of their I/O-CPQA, before it is modified by the construction of the
I/O-CPQA for their parent nodes. For this reason we render the involved
I/O-CPQAs confluently persistent, by implementing their clean, buffer and
dirty deques as purely functional catenable deques [11]. In fact, $T$ encodes
implicity the directed acyclic version graph of the confluently persistent
I/O-CPQAs, by associating every node of $T$ with the version of the I/O-CPQA
at the time of its construction. Every internal node of $T$ stores a
representative block with the critical records for the versions of the
I/O-CPQAs associated with its children nodes. Finally, the update operation
discards the I/O-CPQA of a node in $T$, by performing in reverse the
operations on the purely functional catenable deques involved in the
construction of the I/O-CPQA (undo operation).
With the above modification it suffices for the query operation to identify
the two paths $p_{\ell},p_{r}$ from the root to the leaves of $T$ that contain
the $x$-successor point of $x_{\ell}$ and the $x$-predecessor point of
$x_{r}$, respectively. Let $R$ be the children nodes of the nodes on the paths
$p_{\ell}$ and $p_{r}$ that do not belong to the paths themselves, and also
lie within the query $x$-range. The subtrees of $R$ divide the query $x$-range
into disjoint $x$-ranges. We consider the nodes of $R$ from left to right. In
particular, for every non-leaf node in $p_{\ell}\cup p_{r}$, we load into
memory the representative blocks of the versions of the I/O-CPQAs in its
children nodes that belong to $R$. We call CatenateAndAttrite on the loaded
I/O-CPQAs and on the resulting I/O-CPQAs for every node in $p_{\ell}\cup
p_{r}$, as decribed in Section 3. The non-attrited elements in the resulting
auxiliary I/O-CPQA correspond to the skyline of the points in the query
$x$-range, that are not stored in the leaves of $p_{\ell}$ and $p_{r}$. To
report the output points of the query in increasing $x$-coordinate, we first
report the maximal points within the query range among the points stored in
the leaf of $p_{\ell}$. Then we call DeleteMin to the auxiliary I/O-CPQA that
returns the maximal points in increasing $x$-coordinate, and thus also in
decreasing $y$-coordinate, and thus we terminate the reporting as soon as a
skyline point with $y$-coordinate smaller than $y_{b}$ is returned. If the
reporting has not terminated, we also report the rest of the maximal points
within the query range that are contained in the leaf of $p_{r}$.
###### Theorem 4.1.
There exist I/O-efficient dynamic data structures that store a set of $n$
planar points and support reporting the $t$ skyline points within a given
3-sided orthogonal range unbounded by the positive $y$-dimension in
$\mathcal{O}(\log_{2B^{\epsilon}}n+t/B^{1-\epsilon})$ worst case I/Os, and
updates in $\mathcal{O}(\log_{2B^{\epsilon}}n)$ worst case I/Os, using
$\mathcal{O}(n/B^{1-\epsilon})$ disk blocks, for a parameter
$0\leq\epsilon\leq 1$.
###### Proof.
We set the buffer size parameter $b$ of the I/O-CPQAs equal to the leaf
parameter $k$ of $T$, and we set the parameters $a=2B^{\epsilon}$ and
$k=B^{1-\epsilon}$ for $0\leq\epsilon\leq 1$. In this way, for a node of $T$,
the representative blocks for all of its children nodes can be loaded into
memory in $\mathcal{O}(1)$ I/Os. Since every operation supported by an
I/O-CPQA involves a $\mathcal{O}(1)$ number of deque operations, I/O-CPQAs can
be made confluently persistent without deteriorating their I/O and space
complexity. Moreover, the undo operation takes $\mathcal{O}(1)$ worst case
I/Os, since the purely functional catenable deques are worst case efficient.
Therefore by Theorem 3.1, an update operation takes
$\mathcal{O}(\log_{2B^{\epsilon}}\frac{n}{B^{1-\epsilon}})=\mathcal{O}(\log_{2B^{\epsilon}}n)$
worst case I/Os. Lemma 3.1 takes $\mathcal{O}(1)$ I/Os to construct the
temporary I/O-CPQAs for every node in the search paths, since they satisfy
both of its conditions. Moreover, by Theorem 3.1, it takes
$\mathcal{O}(\frac{\log_{2B^{\epsilon}}n}{B^{1-\epsilon}})$ I/Os to catenate
them together. Thus, the construction of the auxiliary query I/O-CPQA takes
$\mathcal{O}(\log_{2B^{\epsilon}}n)$ worst case I/Os in total. Moreover, it
takes $\mathcal{O}(1+t/B^{1-\epsilon})$ worst case I/Os to report the output
points. There are $\mathcal{O}(\frac{n}{B^{1-\epsilon}})$ internal nodes in
$T$, and every internal node contains $\mathcal{O}(1)$ blocks. ∎
#### 4-Sided Skyline Reporting
Dynamic I/O-efficient data structures for 4-sided range skyline reporting
queries can be obtained by following the approach of Overmars and Wood for
dynamic rectangular visibility queries [15]. In particular, 4-sided range
skyline reporting queries are supported in $\mathcal{O}(\frac{a\log^{2}n}{\log
a\log{2B^{\epsilon}}}+t/B^{1-\epsilon})$ worst case I/Os, using
$\mathcal{O}(\frac{n}{B^{1-\epsilon}}\log_{a}n)$ blocks, by employing our
structure for 3-sided range skyline reporting as a secondary structure on a
dynamic range tree with branching parameter $a$, built over the $y$-dimension.
Updates are supported in $\mathcal{O}(\frac{\log^{2}n}{\log
a\log{2B^{\epsilon}}})$ worst case I/Os, since the secondary structures can be
split or merged in $\mathcal{O}(\log_{2B^{\epsilon}}n)$ worst case I/Os.
###### Remark 4.1.
In the pointer machine, the above constructions attains the same complexities
as the existing structures for dynamic 3-sided and 4-sided range maxima
reporting [4], by setting the buffer size, branching and leaf parameter to
$\mathcal{O}(1)$.
## 5 Lower Bound for Dominating Minima Reporting
Let $S$ be a set of $n$ points in $\mathbb{R}^{2}$. Let
$\mathcal{Q}=\\{Q_{i}\\}$ be a set of $m$ orthogonal 2-sided query ranges
$Q_{i}\in\mathbb{R}^{2}$. Range $Q_{i}$ is the subspace of $\mathbb{R}^{2}$
that dominates a given point $q_{i}\in\mathbb{R}^{2}$ in the positive $x$\-
and $y$\- direction (the “upper-right” quadrant defined by $q_{i}$). Let
$S_{i}=S\cap Q_{i}$ be the set of all points in $S$ that lie in the range
$Q_{i}$. A dominating minima reporting query $Q_{i}$ contains the points
$\min(S_{i})\in S_{i}$ that do not dominate any other point in $S_{i}$. In
this section we prove that any pointer-based data structure that supports
dominating minima queries in $\mathcal{O}(\log^{\mathcal{O}(1)}{n}+t)$ time,
must use superlinear space. This separates the problem from the easier problem
of supporting dominating maxima queries and the more general 3-sided range
skyline reporting queries. The same trade-off also holds for the symmetric
dominated maxima reporting queries that are the simplest special case of
4-sided range skyline reporting queries that demands superlinear space.
Moreover, the lower bound holds trivially for the I/O model, if no address
arithmetic is being used. In particular, for a query time of
$\mathcal{O}(\frac{\log^{\mathcal{O}(1)}{n}}{B}+\frac{t}{B})$ the data
structure must definitely use
$\Omega(\frac{n}{B}\frac{\log{n}}{\log{\log{n}}})$ blocks of space. In the
following, we prove the lower bound for the dominating minima reporting
queries.
Henceforth, we use the terminology presented in Section 2. Without loss of
generality, we assume that $n=\omega^{\lambda}$, since this restriction
generates a countably infinite number of inputs and thus the lower bound is
general. In our case, $\omega=\log^{\gamma}{n}$ holds for some $\gamma\>0$,
$m=2$ and
$\lambda=\left\lfloor\frac{\log{n}}{1+\gamma\log{\log{n}}}\right\rfloor$. Let
$\rho_{\omega}(i)$ be the integer obtained by writing $0\leq i<n$ using
$\lambda$ digits in base $\omega$, by first reversing the digits and then
taking their complement with respect to $\omega$. In particular, if
$i=i^{(\omega)}_{0}i^{(\omega)}_{1}\ldots i^{(\omega)}_{\lambda-1}$ holds,
then
$\rho_{\omega}(i)=(\omega-i^{(\omega)}_{\lambda-1}-1)(\omega-i^{(\omega)}_{\lambda-2}-1)\ldots(\omega-i^{(\omega)}_{1}-1)(\omega-i^{(\omega)}_{0}-1)$
where $i^{(\omega)}_{j}$ is the $j$-th digit of number $i$ in base $\omega$.
We define the points of $S$ to be the set $\\{(i,\rho_{\omega}(i))|0\leq
i<n\\}$. Figure 3 shows an example with $\omega=4$, $\lambda=2$.
To define the query set $\mathcal{Q}$, we encode the set of points
$\\{\rho_{\omega}(i)|0\leq i<n\\}$ in a full trie structure of depth
$\lambda$. Recall that $n=\omega^{\lambda}$. Notice that the trie structure is
implicit and it is used only for presentation purposes. Input points
correspond to the leaves of the trie and their $y$ value is their label at the
edges of the trie. Let $v$ be an internal node at depth $d$ (namely, $v$ has
$d$ ancestors), whose prefix $v_{0},v_{1},\ldots,v_{d-1}$ corresponds to the
path from $v$ to the root $r$ of the trie. We take all points in its subtree
and sort them by $y$. From this sorted list we construct groups of size
$\omega$ by always picking each $\omega^{\lambda-d-1}$-th element starting
from the smallest non-picked element. Each such group corresponds to the
output of each query. See Figure 3 for an example. In this case, we say that
the query is associated to node $v$.
Figure 3: An example for $\omega=4$ and $\lambda=2$. Two examples of queries
are shown, out of the $8$ possible queries with different output. Connecting
lines represent points whose $L_{1}$ distance is $\omega^{k},1\leq
k\leq\lambda$. All $8$ possible queries can be generated by translating the
blue lines horizontally so that the answers of all $4$ queries are disjoint.
Similarly for the red lines with the exception that we translate them
vertically.
A node of with depth $d$ has $\frac{n}{\omega^{d}}$ points in its subtree and
thus it defines at most $\frac{n}{\omega^{d-1}}$ queries. Thus, the total
number of queries is:
$\left|\mathcal{Q}\right|=\sum_{d=0}^{\lambda-1}{\omega^{d}\frac{n}{\omega^{d+1}}}=\sum_{d=0}^{\lambda-1}{\frac{n}{\omega}}=\frac{\lambda
n}{\omega}$
This means that the total number of queries is
$|\mathcal{Q}|=\frac{\lambda
n}{\omega}=\frac{\log{n}}{1+\gamma\log{\log{n}}}\frac{1}{\log^{\gamma}{n}}n=\frac{n}{\log^{\gamma-1}{n}(1+\gamma\log{\log{n}})}$
The following lemma states that $\mathcal{Q}$ is appropriate for our purposes.
###### Lemma 5.1.
$\mathcal{Q}$ is $(2,\log^{\gamma}{n})$-favorable.
###### Proof.
First we prove that we can construct the queries so that they have output size
$\omega=\log^{\gamma}{n}$. Assume that we take a group of $\omega$ consecutive
points in the sorted order of points with respect to the $y$-coordinate at the
subtree of node $v$ at depth $d$. These have common prefix of length $d$. Let
the $y$-coordinates of these points be
$\rho_{\omega}(i_{1}),\rho_{\omega}(i_{2}),\ldots,\rho_{\omega}(i_{\omega})$
in increasing order, where
$\rho_{\omega}(i_{j})-\rho_{\omega}(i_{j-1})=\omega^{\lambda-d-1},1<j\leq\omega$.
This means that these numbers differ only at the $\lambda-d-1$-th digit. This
is because they have a common prefix of length $d$ since all points lie in the
subtree of $v$. At the same time they have a common suffix of length
$\lambda-d-1$ because of the property that
$\rho_{\omega}(i_{j})-\rho_{\omega}(i_{j-1})=\omega^{\lambda-d-1},1<j\leq\omega$
which comes as a result from the way we chose these points. By inversing the
procedure to construct these $y$-coordinates, the corresponding
$x$-coordinates $i_{j},1\leq j\leq\omega$ are determined. By complementing we
take the increasing sequence
$\bar{\rho}_{\omega}(i_{\omega}),\ldots,\bar{\rho}_{\omega}(i_{2}),\bar{\rho}_{\omega}(i_{1})$,
where $\bar{\rho}_{\omega}(i_{j})=\omega^{\lambda}-\rho_{\omega}(i_{j})-1$ and
$\bar{\rho}_{\omega}(i_{j-1})-\bar{\rho}_{\omega}(i_{j})=\omega^{\lambda-d-1},1<j\leq\omega$.
By reversing the digits we finally get the increasing sequence of
$x$-coordinates $i_{\omega},\ldots,i_{2},i_{1}$, since the numbers differ at
only one digit. Thus, the group of $\omega$ points are decreasing as the
$x$-coordinates increase, and as a result a query $q$ whose horizontal line is
just below $\rho_{\omega}(i_{1})$ and the vertical line just to the left of
$\rho_{\omega}(i_{\omega})$ will certainly contain this set of points in the
query. In addition, there cannot be any other points between this sequence and
the horizontal or vertical lines defining query $q$. This is because all
points in the subtree of $v$ have been sorted with respect to $y$, while the
horizontal line is positioned just below $\rho_{\omega}(i_{1})$, so that no
other element lies in between. In the same manner, no points to the left of
$\rho_{\omega}(i_{\omega})$ exist, when positioning the vertical line of $q$
appropriately. Thus, for each query $q\in\mathcal{Q}$, it holds that $|S\cap
q|=\omega=\log^{\gamma}{n}$.
It is enough to prove that for any two query ranges $p,q\in\mathcal{Q}$,
$|S\cap q\cap p|\leq 1$ holds. Assume that $p$ and $q$ are associated to nodes
$v$ and $u$, respectively, and that their subtrees are disjoint. That is, $u$
is not a proper ancestor or descendant of $v$. In this case, $p$ and $q$ share
no common point, since each point is used only once in the trie. For the other
case, assume without loss of generality that $u$ is a proper ancestor of $v$
($u\neq v$). By the discussion in the previous paragraph, each query contains
$\omega$ numbers that differ at one and only one digit. Since $u$ is a proper
ancestor of $v$, the corresponding digits will be different for the queries
defined in $u$ and for the queries defined in $v$. This implies that there can
be at most one common point between these sequences, since the digit that
changes for one query range is always set to a particular value for the other
query range. The lemma follows. ∎
Lemma 5.1 allows us to apply Lemma 2.1, and thus the query time of
$\mathcal{O}(\log^{\gamma}{n}+t)$, for output size $t$, can only be achieved
at a space cost of $\Omega\left(n\frac{\log{n}}{\log{\log{n}}}\right)$. The
following theorem summarizes the result of this section.
###### Theorem 5.1.
The dominating minima reporting problem can be solved with
$\Omega\left(n\frac{log{n}}{\log{\log{n}}}\right)$ space, if the query is
supported in $\mathcal{O}(\log^{\gamma}{n}+t)$ time, where $t$ is the size of
the answer to the query and parameter $\gamma=\mathcal{O}(1)$.
## 6 Conclusion
We presented the first dynamic I/O-efficient data structures for 3-sided
planar orthogonal range skyline reporting queries with worst case
polylogarithmic update and query complexity. We also showed that the space
usage of the existing structures for 4-sided range skyline reporting in
pointer machine is optimal within doubly logarithmic factors.
It remains open to devise a dynamic I/O-efficient data structure that supports
reporting all $m$ planar skyline points in $\mathcal{O}(m/B)$ worst case I/Os
and updatess in $\mathcal{O}(\log_{B}n)$ worst case I/Os. It seems that the
hardness for reporting the skyline in optimal time is derived from the fact
that the problem is dynamic. The dynamic indexability model of Yi [21] may be
useful to prove a lower bound towards the direction of rendering our structure
for 3-sided range skyline reporting I/O-optimal, as defined by Papadias et
al.[16]. Finally it remains open to obtain a
$\mathcal{O}(\frac{n}{B}\log_{B}n)$ space dynamic I/O-efficient data
structures for 4-sided range skyline reporting with
$\mathcal{O}(\log^{2}_{B}n)$ worst case query and update I/Os, regardless of
the I/O-complexity per reported point.
## References
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* [3] Stephan Börzsönyi, Donald Kossmann, and Konrad Stocker. The skyline operator. In ICDE, pages 421–430, 2001.
* [4] Gerth Brodal and Konstantinos Tsakalidis. Dynamic planar range maxima queries. In Luca Aceto, Monika Henzinger, and Jirí Sgall, editors, Automata, Languages and Programming, volume 6755 of Lecture Notes in Computer Science, pages 256–267. Springer Berlin / Heidelberg, 2011. 10.1007/978-3-642-22006-7_22.
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* [14] Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. Journal of Computer and System Sciences, 23(2):166 – 204, 1981\.
* [15] Mark H. Overmars and Derick Wood. On rectangular visibility. J. Algorithms, 9(3):372–390, September 1988.
* [16] Dimitris Papadias, Yufei Tao, Greg Fu, and Bernhard Seeger. Progressive skyline computation in database systems. ACM Trans. Database Syst., 30(1):41–82, 2005.
* [17] Cheng Sheng and Yufei Tao. On finding skylines in external memory. In Proceedings of the thirtieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, PODS ’11, pages 107–116, New York, NY, USA, 2011. ACM.
* [18] Rajamani Sundar. Worst-case data structures for the priority queue with attrition. Inf. Process. Lett., 31:69–75, April 1989.
* [19] Yufei Tao and Dimitris Papadias. Maintaining sliding window skylines on data streams. IEEE Trans. on Knowl. and Data Eng., 18(3):377–391, 2006.
* [20] Ping Wu, Divyakant Agrawal, Ömer Egecioglu, and Amr El Abbadi. Deltasky: Optimal maintenance of skyline deletions without exclusive dominance region generation. In ICDE, pages 486–495, 2007.
* [21] Ke Yi. Dynamic indexability and lower bounds for dynamic one-dimensional range query indexes. In Proceedings of the twenty-eighth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, PODS ’09, pages 187–196, New York, NY, USA, 2009. ACM.
|
arxiv-papers
| 2012-07-10T13:32:12 |
2024-09-04T02:49:32.866416
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Casper Kejlberg-Rasmussen and Konstantinos Tsakalidis and Kostas\n Tsichlas",
"submitter": "Casper Kejlberg-Rasmussen",
"url": "https://arxiv.org/abs/1207.2341"
}
|
1207.2372
|
# Planar Symmetric Concave Central Configurations in Four-body Problem
Chunhua Deng1 and Shiqing Zhang2
1\. Faculty of Mathematics and Physics, Huaiyin Institute of Technology,
Huai’an 223003, China
chdeng8011@sohu.com
2\. College of Mathematics, Sichuan University, Chengdu 610064, China
Abstract: In this paper, we consider the problem: given a symmetric concave
configuration of four bodies, under what conditions is it possible to choose
positive masses which make it central. We show that there are some regions in
which no central configuration is possible for positive masses. Conversely,
for any configuration in the complement of the union of these regions, it is
always possible to choose positive masses to make the configuration central.
Keywords: four-body problem, central configuration, Celestial mechanics.
## 1\. Introduction and Main Results
The Newtonian $n$-body problem concerns the motion of $n$ mass points with
masses $m_{i}\in\mathbb{R}^{+},i=1,2,\cdots,n$. The motion is governed by
Newton’s law of gravitation:
$m_{i}\ddot{q}_{i}=\sum\limits_{k\neq
i}\frac{m_{k}m_{i}(q_{k}-q_{i})}{|q_{k}-q_{i}|^{3}},i=1,2,\cdots,n,$ (1.1)
where $q_{i}\in\mathbb{R}^{d}(d=1,2,3)$ is the position of $m_{i}$.
Alternatively the system (1.1) can be written
$m_{i}\ddot{q}_{i}=\frac{\partial U(q)}{\partial q_{i}},i=1,2,\cdots,n$ (1.2)
where
$U(q)=U(q_{1},q_{2},\cdots,q_{n})=\sum\limits_{1\leq k<j\leq
n}\frac{m_{k}m_{j}}{|q_{k}-q_{j}|}$ (1.3)
is the Newtonian potential of system (1.1). Let
$C=m_{1}q_{1}+\cdots+m_{n}q_{n},M=m_{1}+\cdots+m_{n},c=C/M$
be the first moment, total mass and center of mass of the bodies,
respectively. The set $\bigtriangleup$ of collision configurations is defined
by
$\bigtriangleup=\\{q\in(\mathbb{R}^{d})^{n}:q_{i}=q_{j}\;\text{for
some}\;i\neq j\\}.$
A configuration
$q=(q_{1},\cdots,q_{n})\in(\mathbb{R}^{d})^{n}\backslash\bigtriangleup$ is
called a central configuration if there exists some positive constant
$\lambda$ such that
$-\lambda(q_{i}-c)=\sum\limits_{j=1,j\neq
i}^{n}\frac{m_{j}(q_{j}-q_{i})}{|q_{j}-q_{i}|^{3}},\quad i=1,2,\cdots,n.$
(1.4)
Furthermore it can be easily verified that $\lambda=U/I$, where $I$ is the
moment of inertial of the system, i.e.
$I=\sum\limits_{i=1}^{N}m_{i}|q_{i}|^{2}$. The set of central configurations
are invariant under three classes of transformations on
$(\mathbb{R}^{d})^{n}$: translations, scalings, and orthogonal
transformations. A configuration $q=(q_{1},\cdots,q_{n})$ is concave if one
mass point is in the interior of the triangle formed by the other three mass
points. For $n=4,q_{i}\in\mathbb{R}^{2}$, Long and Sun [] proved
Lemma 1.1. Let $\alpha,\beta>0$ be any two given real numbers. Let
$q=(q_{1},q_{2},q_{3},q_{4})\in(\mathbb{R}^{2})^{4}$ be a concave non-
collinear central configuration with masses $(\beta,\alpha,\beta,\beta)$
respectively, and with $q_{2}$ located inside the triangle formed by
$q_{1},q_{3}$, and $q_{4}$. Then the configuration $q$ must possess a
symmetry, so either $q_{1},q_{3}$, and $q_{4}$ form an equilateral triangle
and $q_{2}$ is located at the center of the triangle, or $q_{1},q_{3}$, and
$q_{4}$ form an isosceles triangle, and $q_{2}$ is on the symmetrical axis of
the triangle.
In this paper we consider the inverse problem: given a planar symmetric
concave configuration (Figure 1), find the positive mass vectors, if any, for
which it is a central configuration. The equations for the central
configurations can be written as
$\left\\{\begin{array}[]{l}m_{2}\frac{q_{2}-q_{1}}{|q_{2}-q_{1}|^{3}}+m_{3}\frac{q_{3}-q_{1}}{|q_{3}-q_{1}|^{3}}+m_{4}\frac{q_{4}-q_{1}}{|q_{4}-q_{1}|^{3}}=-\lambda(q_{1}-c)\\\
m_{1}\frac{q_{1}-q_{2}}{|q_{1}-q_{2}|^{3}}+m_{3}\frac{q_{3}-q_{2}}{|q_{3}-q_{2}|^{3}}+m_{4}\frac{q_{4}-q_{2}}{|q_{4}-q_{2}|^{3}}=-\lambda(q_{2}-c)\\\
m_{1}\frac{q_{1}-q_{3}}{|q_{1}-q_{3}|^{3}}+m_{2}\frac{q_{2}-q_{3}}{|q_{2}-q_{3}|^{3}}+m_{4}\frac{q_{4}-q_{3}}{|q_{4}-q_{3}|^{3}}=-\lambda(q_{3}-c)\\\
m_{1}\frac{q_{1}-q_{4}}{|q_{1}-q_{4}|^{3}}+m_{2}\frac{q_{2}-q_{4}}{|q_{2}-q_{4}|^{3}}+m_{3}\frac{q_{3}-q_{4}}{|q_{3}-q_{4}|^{3}}=-\lambda(q_{4}-c)\\\
\end{array}\right.$ (1.5)
We can obtain the following results:
Theorem 1.1. Let $q_{1}=(-1,0)$, $q_{2}=(1,0)$, $q_{3}=(0,t)$, $q_{4}=(0,s)$
where $t>s>0$, and assume that the center of mass $c=C/M=q_{4}$. The symmetric
concave configuration $q=(q_{1},q_{2},q_{3},q_{4})$ can be a central
configuration if and only if $t=\sqrt{3},s=\frac{\sqrt{3}}{3}$, and the masses
of $q_{1}$, $q_{2}$ and $q_{3}$ are all equal, i.e. $m_{1}=m_{2}=m_{3}>0$. The
mass of $q_{4}$ can be any positive number $m_{4}>0$.
Theorem 1.2. Let $q_{1}=(-1,0)$, $q_{2}=(1,0)$, $q_{3}=(0,t)$, $q_{4}=(0,s)$,
where $t>s>0$, and assume that the center of mass $c=C/M\neq q_{4}$. There
exists two open bounded regions $C$ and $D$ which can be seen in figure (),
the configuration $q=(q_{1},q_{2},q_{3},q_{4})$ can be a central configuration
with positive masses, where
$\displaystyle
m_{1}=m_{2}=\lambda\frac{2^{3}\sqrt{1+t^{2}}^{3}(t-c_{y})}{2t\sqrt{1+s^{2}}^{3}(t-s)^{3}}\frac{(t-s)^{3}-\sqrt{1+s^{2}}^{3}}{\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3}}$
(1.6)
$\displaystyle m_{3}=\frac{\lambda
s\sqrt{1+t^{2}}^{3}}{\sqrt{1+s^{2}}^{6}(t-s)^{3}}\frac{(\sqrt{1+s^{2}}^{3}-2^{3})(\sqrt{1+s^{2}}^{3}-(t-s)^{3})}{(\frac{t-s}{(t-s)^{3}}+\frac{s}{\sqrt{1+s^{2}}^{3}}-\frac{t}{\sqrt{1+t^{2}}^{3}})((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}$
(1.7)
$m_{4}=\frac{\lambda(t-c_{y})}{(t-s)}\frac{(2^{3}-\sqrt{1+t^{2}}^{3})}{((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}.$
(1.8)
Figure 1: The symmetric concave configuration
## 2\. General Symmetric Concave Central Configurations with Four bodies
Assume the center of mass $c=(c_{x},c_{y})$. Given $q_{1}=(-1,0)$,
$q_{2}=(1,0)$, $q_{3}=(0,t)$, $q_{4}=(0,s)$, where $t>s>0$, the system (1.5)
can be divided into two parts:
$\left\\{\begin{array}[]{l}\frac{2}{2^{3}}m_{2}+\frac{1}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{1}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda(1+c_{x})\\\
\frac{-2}{2^{3}}m_{2}+\frac{-1}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{-1}{\sqrt{1+s^{2}}^{3}}m_{4}=-\lambda(1-c_{x})\\\
\frac{-1}{\sqrt{1+t^{2}}^{3}}m_{1}+\frac{1}{\sqrt{1+t^{2}}^{3}}m_{2}=\lambda
c_{x}\\\
\frac{-1}{\sqrt{1+s^{2}}^{3}}m_{1}+\frac{1}{\sqrt{1+s^{2}}^{3}}m_{2}=\lambda
c_{x}\\\ \end{array}\right.$ (2.1)
and
$\left\\{\begin{array}[]{l}\frac{t}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda
c_{y}\\\
\frac{t}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda
c_{y}\\\
\frac{-t}{\sqrt{1+t^{2}}^{3}}m_{1}+\frac{-t}{\sqrt{1+t^{2}}^{3}}m_{2}+\frac{s-t}{(t-s)^{3}}m_{4}=-\lambda(t-c_{y})\\\
\frac{-s}{\sqrt{1+s^{2}}^{3}}m_{1}+\frac{-s}{\sqrt{1+s^{2}}^{3}}m_{2}+\frac{t-s}{(t-s)^{3}}m_{3}=-\lambda(s-c_{y}).\\\
\end{array}\right.$ (2.2)
In (2.2) the first two equations are identical. The third and the fourth
equations in (2.1) imply that
$(\frac{1}{\sqrt{1+t^{2}}^{3}}-\frac{1}{\sqrt{1+s^{2}}^{3}})(m_{2}-m_{1})=0.$
For $t>s>0$ we have
$m_{1}=m_{2}.$
The first two equations in (2.1) together with $m_{1}=m_{2}$ and positive
number $\lambda>0$ imply that
$c_{x}=0.$
Thus systems (1.5) for central configurations become
$\left\\{\begin{array}[]{l}\frac{2}{2^{3}}m_{2}+\frac{1}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{1}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda\\\
\frac{t}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda
c_{y}\\\
\frac{-2t}{\sqrt{1+t^{2}}^{3}}m_{2}+\frac{s-t}{(t-s)^{3}}m_{4}=-\lambda(t-c_{y})\\\
\frac{-2s}{\sqrt{1+s^{2}}^{3}}m_{2}+\frac{t-s}{(t-s)^{3}}m_{3}=-\lambda(s-c_{y}).\\\
\end{array}\right.$ (2.3)
## 3\. The Proof of Theorem 1.1
In this section, we will find the solution of masses $m_{1},m_{2},m_{3},m_{4}$
with two parameters $s,t$ for the four-body central configuration. We assume
the center of mass $c=C/M=q_{4}$, i.e. $c_{y}=s$. the system (2.3) for central
configurations become
$\left\\{\begin{array}[]{l}\frac{2}{2^{3}}m_{2}+\frac{1}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{1}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda\\\
\frac{t}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda
s\\\
\frac{-2t}{\sqrt{1+t^{2}}^{3}}m_{2}+\frac{s-t}{(t-s)^{3}}m_{4}=-\lambda(t-s)\\\
\frac{-2s}{\sqrt{1+s^{2}}^{3}}m_{2}+\frac{t-s}{(t-s)^{3}}m_{3}=0.\\\
\end{array}\right.$ (3.1)
The fourth equation in (3.1) can be written
$m_{2}=\frac{t-s}{(t-s)^{3}}\frac{\sqrt{1+s^{2}}^{3}}{2s}m_{3}.$ (3.2)
Substituting (3.2) into the third equation in (3.1), we have
$\frac{-2t}{\sqrt{1+t^{2}}^{3}}\frac{t-s}{(t-s)^{3}}\frac{\sqrt{1+s^{2}}^{3}}{2s}m_{3}+\frac{s-t}{(t-s)^{3}}m_{4}=-\lambda(t-s),$
(3.3)
for $t>s>0$, then
$\frac{t}{\sqrt{1+t^{2}}^{3}}\frac{1}{(t-s)^{3}}\frac{\sqrt{1+s^{2}}^{3}}{s}m_{3}+\frac{1}{(t-s)^{3}}m_{4}=\lambda.$
(3.4)
From he second equation in (3.1) and the above equation (3.4), we have
$(t-s)=\sqrt{1+s^{2}}.$ (3.5)
Thus the last three equations in (3.1) is equivalent to
$\left\\{\begin{array}[]{l}(t-s)=\sqrt{1+s^{2}}\\\
m_{4}=\lambda\sqrt{1+s^{2}}^{3}-\frac{2t}{\sqrt{1+t^{2}}^{3}}\frac{\sqrt{1+s^{2}}^{3}}{t-s}m_{2}\\\
m_{3}=\frac{2s}{t-s}m_{2}.\\\ \end{array}\right.$ (3.6)
Substituting (3.6) into the first equation in (3.1) and simplifying, we have
$t=\sqrt{3},s=\frac{\sqrt{3}}{3}.$ (3.7)
Then we have $m_{1}=m_{2}=m_{3}$ and
$m_{4}=\frac{8}{9}\sqrt{3}\lambda-\frac{\sqrt{3}}{3}m_{2}$. Furthermore, for
any positive mass $m_{4}>0$, we can choose suitable $\lambda>0$ such that
$m_{4}=\frac{8}{9}\sqrt{3}\lambda-\frac{\sqrt{3}}{3}m_{2}$. This completes the
proof of Theorem 1.1.
## 4\. The Proof of Theorem 1.2
In this section, we assume the center of mass $c=C/M\neq q_{4}$, i.e.
$c_{y}\neq s$. Combining the second and the third equations in (2.3) and
eliminating $m_{4}$, the following equation is derived
$\frac{2t}{\sqrt{1+t^{2}}^{3}}\frac{s}{\sqrt{1+s^{2}}^{3}}m_{2}+\frac{t}{\sqrt{1+t^{2}}^{3}}\frac{s-t}{(t-s)^{3}}m_{3}=\lambda((\frac{s-t}{(t-s)^{3}}-\frac{s}{\sqrt{1+s^{2}}^{3}})c_{y}+\frac{ts}{\sqrt{1+s^{2}}^{3}}).$
(4.1)
Multiplying both sides of the fourth equation in (2.3) by
$\frac{t}{\sqrt{1+t^{2}}^{3}}$, we have
$\frac{2t}{\sqrt{1+t^{2}}^{3}}\frac{s}{\sqrt{1+s^{2}}^{3}}m_{2}+\frac{t}{\sqrt{1+t^{2}}^{3}}\frac{s-t}{(t-s)^{3}}m_{3}=\lambda(c_{y}-s)\frac{t}{\sqrt{1+t^{2}}^{3}}.$
(4.2)
Then we have the necessary conditions for the solvability of (2.3):
$(\frac{s-t}{(t-s)^{3}}-\frac{s}{\sqrt{1+s^{2}}^{3}})c_{y}+\frac{ts}{\sqrt{1+s^{2}}^{3}}=(c_{y}-s)\frac{t}{\sqrt{1+t^{2}}^{3}},$
(4.3)
then
$c_{y}=(\frac{ts}{\sqrt{1+s^{2}}^{3}}-\frac{ts}{\sqrt{1+t^{2}}^{3}})/(\frac{t-s}{(t-s)^{3}}+\frac{s}{\sqrt{1+s^{2}}^{3}}-\frac{t}{\sqrt{1+t^{2}}^{3}}).$
(4.4)
The system (2.3) for central configurations become
$\left\\{\begin{array}[]{l}c_{y}=(\frac{ts}{\sqrt{1+s^{2}}^{3}}-\frac{ts}{\sqrt{1+t^{2}}^{3}})/(\frac{t-s}{(t-s)^{3}}+\frac{s}{\sqrt{1+s^{2}}^{3}}-\frac{t}{\sqrt{1+t^{2}}^{3}})\\\
\frac{2}{2^{3}}m_{2}+\frac{1}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{1}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda\\\
\frac{t}{\sqrt{1+t^{2}}^{3}}m_{3}+\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda
c_{y}\\\
\frac{-2t}{\sqrt{1+t^{2}}^{3}}m_{2}+\frac{s-t}{(t-s)^{3}}m_{4}=-\lambda(t-c_{y}).\\\
\end{array}\right.$ (4.5)
The third and the fourth equations in (4.5) can be written
$\frac{1}{\sqrt{1+t^{2}}^{3}}m_{3}=\frac{1}{t}(\lambda
c_{y}-\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4}),$ (4.6)
$\frac{2}{2^{3}}m_{2}=(\lambda(t-c_{y})-\frac{t-s}{(t-s)^{3}}m_{4})\frac{\sqrt{1+t^{2}}^{3}}{2t}\frac{2}{2^{3}},$
(4.7)
and substituting the above two equations into the second equation in (4.5), we
obtain
$(\lambda(t-c_{y})-\frac{t-s}{(t-s)^{3}}m_{4})\frac{\sqrt{1+t^{2}}^{3}}{2t}\frac{2}{2^{3}}+\frac{1}{t}(\lambda
c_{y}-\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4})+\frac{1}{\sqrt{1+s^{2}}^{3}}m_{4}=\lambda,$
then
$m_{4}=\frac{\lambda(t-c_{y})}{(t-s)}\frac{(2^{3}-\sqrt{1+t^{2}}^{3})}{((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}.$
(4.8)
Substituting (4.8) into (4.6) and simplifying, we have
$\displaystyle m_{3}=$ $\displaystyle\frac{\sqrt{1+t^{2}}^{3}}{t}(\lambda
c_{y}-\frac{s}{\sqrt{1+s^{2}}^{3}}m_{4})$ (4.9) $\displaystyle=$
$\displaystyle\frac{\lambda
s\sqrt{1+t^{2}}^{3}}{\sqrt{1+s^{2}}^{6}(t-s)^{3}}\frac{(\sqrt{1+s^{2}}^{3}-2^{3})(\sqrt{1+s^{2}}^{3}-(t-s)^{3})}{(\frac{t-s}{(t-s)^{3}}+\frac{s}{\sqrt{1+s^{2}}^{3}}-\frac{t}{\sqrt{1+t^{2}}^{3}})((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}.$
Substituting (4.8) into (4.7) and simplifying, we have
$\displaystyle m_{2}=$
$\displaystyle(\lambda(t-c_{y})-\frac{t-s}{(t-s)^{3}}m_{4})\frac{\sqrt{1+t^{2}}^{3}}{2t}$
(4.10) $\displaystyle=$
$\displaystyle\lambda\frac{2^{3}\sqrt{1+t^{2}}^{3}(t-c_{y})}{2t\sqrt{1+s^{2}}^{3}(t-s)^{3}}\frac{((t-s)^{3}-\sqrt{1+s^{2}}^{3})}{((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}.$
$m_{1}=m_{2}=\lambda\frac{2^{3}\sqrt{1+t^{2}}^{3}(t-c_{y})}{2t\sqrt{1+s^{2}}^{3}(t-s)^{3}}\frac{((t-s)^{3}-\sqrt{1+s^{2}}^{3})}{((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}.$
(4.11)
Thus we give the necessary condition (4.4) for the existence of the solution
of masses, and give the solution of masses explicitly in (4.8-4.11). In the
following we will analyze the mass functions and find the possible region in
$st$-plane such that the mass functions are positive.
Lemma 4.1. The region in which $m_{4}>0$ for $t>s>0$ is the union of $A$ and
$B$ in figure 2 surrounded by curves $t=\sqrt{3}$,
$2(t-s)-\sqrt{1+t^{2}}\sqrt{1+s^{2}}=0$ and $t-s=0$.
Proof. With simple computation, we can find the center of mass
$c=(c_{x},c_{y})=(0,\frac{sm_{4}+tm_{3}}{m_{1}+m_{2}+m_{3}+m_{4}}),$ (4.12)
then $t-c_{y}>0$ for $t>s>0$. For convenience, we denote
$p_{1}=2^{3}-\sqrt{1+t^{2}}^{3}$ and
$p_{2}=(\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3}$. Thus
$m_{4}>0$ is equivalent to $\frac{p_{1}}{p_{2}}>0$. We can show that $p_{2}=0$
give rise a smooth monotone increasing curve above the curve $t=s$, and
bounded from right by $s=\sqrt{3}$.
Figure 2:
$p_{2}=0$ is equivalent to $\sqrt{1+s^{2}}\sqrt{1+t^{2}}=2(t-s)$. We observe
that
$\sqrt{1+s^{2}}\sqrt{1+t^{2}}=2(t-s)<2t,$
then
$\sqrt{1+s^{2}}<\frac{2t}{\sqrt{1+t^{2}}}<2,$
thus
$s<\sqrt{3}.$ (4.13)
Furthermore, from $\sqrt{1+s^{2}}\sqrt{1+t^{2}}=2(t-s)$, we have
$\lim_{t\to+\infty}2(1-\frac{s}{t})=\lim_{t\to+\infty}\sqrt{1+s^{2}}\sqrt{1+\frac{1}{t^{2}}},$
then
$\lim_{t\to+\infty}s=\sqrt{3}.$ (4.14)
Let’s take the derivative of $\sqrt{1+s^{2}}\sqrt{1+t^{2}}=2(t-s)$ with
respect to $s$,
$(2-\frac{t\sqrt{1+s^{2}}}{\sqrt{1+t^{2}}})\frac{dt}{ds}=2+\frac{s\sqrt{1+t^{2}}}{\sqrt{1+s^{2}}}.$
Since
$2-\frac{t\sqrt{1+s^{2}}}{\sqrt{1+t^{2}}}>2-\frac{2t}{\sqrt{1+t^{2}}}>0,$
we have
$\frac{dt}{ds}>0.$
Also the signs of $p_{1},p_{2}$ are shown in the first three pictures of
Figure 2. So the region of $m_{4}>0$ is the union of two nonempty open sets
$A,B$ indicated in the fourth picture of Figure 2.
Figure 3: the sign of $p_{5}$
Lemma 4.2. The region in which $m_{4},m_{3}>0$ for $t>s>0$ is the union of $C$
and $D$ in figure 3.
Proof. For convenience, we denote
$p_{3}=\sqrt{1+s^{2}}^{3}-2^{3},$ $p_{4}=\sqrt{1+s^{2}}^{3}-(t-s)^{3},$
$p_{5}=\frac{t-s}{(t-s)^{3}}+\frac{s}{\sqrt{1+s^{2}}^{3}}-\frac{t}{\sqrt{1+t^{2}}^{3}}.$
Then $m_{3}>0$ is equivalent to $\frac{p_{3}p_{4}}{p_{5}p_{2}}>0$.
By $t>s$ and $t-s<\sqrt{1+t^{2}}$, we have
$\displaystyle
p_{5}=\frac{t-s}{(t-s)^{3}}+\frac{s}{\sqrt{1+s^{2}}^{3}}-\frac{t}{\sqrt{1+t^{2}}^{3}}=$
$\displaystyle
t(\frac{1}{(t-s)^{3}}-\frac{1}{\sqrt{1+t^{2}}^{3}})+s(\frac{1}{\sqrt{1+s^{2}}^{3}}-\frac{1}{(t-s)^{3}})$
(4.15) $\displaystyle>$ $\displaystyle
s(\frac{1}{(t-s)^{3}}-\frac{1}{\sqrt{1+t^{2}}^{3}})+s(\frac{1}{\sqrt{1+s^{2}}^{3}}-\frac{1}{(t-s)^{3}})$
$\displaystyle=$ $\displaystyle
s(\frac{1}{\sqrt{1+s^{2}}^{3}}-\frac{1}{\sqrt{1+t^{2}}^{3}})$ $\displaystyle>$
$\displaystyle 0.$
Figure 4: the sign of $p_{4}$ Figure 5:
The equation $p_{3}=0$ gives rise to a straight line $s=\sqrt{3}$ in the
$st$-plane. Also $p_{3}$ is positive on the right of this line. The equation
$p_{4}=0$ determines a smooth monotone increasing curve $t=s+\sqrt{1+t^{2}}$,
and $p_{4}$ is negative above this curve (Figure 4). With simple computation,
we can find the implicit curves $p_{1}=0$, $p_{2}=0$ and $p_{4}=0$ have only
one intersecting point $(\frac{\sqrt{3}}{3},3)$ with the domain $t>s>0$ which
can be shown in Figure 5. So the region of $m_{4},m_{3}>0$ is the union of two
nonempty open sets $C,D$ indicated in Figure 5.
Lemma 4.3. The region in which $m_{i}>0,i=1,2,3,4$ for $t>s>0$ is just the
union of $C$ and $D$ in figure 5.
Proof. We have obtained
$\displaystyle m_{1}=m_{2}=$
$\displaystyle\lambda\frac{2^{3}\sqrt{1+t^{2}}^{3}(t-c_{y})}{2t\sqrt{1+s^{2}}^{3}(t-s)^{3}}\frac{((t-s)^{3}-\sqrt{1+s^{2}}^{3})}{((\frac{2}{\sqrt{1+s^{2}}})^{3}-(\frac{\sqrt{1+t^{2}}}{t-s})^{3})}$
(4.16) $\displaystyle=$
$\displaystyle-\lambda\frac{2^{3}\sqrt{1+t^{2}}^{3}(t-c_{y})}{2t\sqrt{1+s^{2}}^{3}(t-s)^{3}}\frac{p_{4}}{p_{2}}$
The signs of $p_{4},p_{2}$ decide the sign of $m_{i},i=1,2$. This complete the
proof of Lemma 4.3.
## Acknowledgements
Both authors are supported by NSFC, and the first author is supported by the
the Scientific Research Foundation of Huaiyin Institute of Technology
(HGA1102,HGB1004).
## References
* [1] Albouy, A. and Moeckel R.: 2000, ’The inverse problem for collinear central configuration’, Celestial Mech. Dyn. Astron. 77, 77 C91.
* [2] A. Albouy, The symmetric central configurations of four equal masses. Contemp. Math 198 (1996) 131 C135.
* [3] A. Albouy and A. Chenciner, Le probl eme des n corps et les distances mutuells.Invent. Math. 131 (1998) 151 C184.
* [4] Yingming Long and Shanzhong Sun, Four-body Central Configurations With some Equal Masses, Arch. Rational Mech. Anal. 162(2002) 25-44.
* [5] R. Moeckel, 1990, ’On central con?gurations’, Math. Zeit. 205, 499 C517.
* [6] F. R. Moulton, The straight line solutions of the n-body problem.Ann. of Math. II Ser. 12 (1910) 1 C17.
* [7] D. Saari: 1980, ’On the role and the properties of n-body central configurations’, Celestial Mech. 21, 9 C20.
* [8] D. Schmidt, Central configurations in R2 and R3. Contemp. Math. 81 (1988) 59 C76.
* [9] C. Siegel and J. Moser, Lectures on Celestial Mechanics. Berlin, Springer, 1971.
* [10] S. Smale: 1970, ’Topology and mechanics.II. The planar n-body problem’, Invent. Math. 11, 45 C64.
* [11] A. Wintner, The Analytical Foundations of Celestial Mechanics. Princeton Math. Series 5, 215. Princeton Univ. Press, Princeton, NJ, 1941\.
* [12] Tiancheng Ouyang and Zhifu Xie, Collinear central configuration in four-body problem, Celestial Mechanics and Dynamical Astronomy (2005) 93:147-166.
* [13] Zhang Shiqing and Zhu Changrong, Central configurations consist of two layer twisted regular polygons, Science in China Series A: Mathematics Volume 45, Number 11 (2002), 1428-1438.
|
arxiv-papers
| 2012-07-10T14:34:00 |
2024-09-04T02:49:32.877067
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chunhua Deng and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1207.2372"
}
|
1207.2507
|
# Thermodynamics in the Limit of Irreversible Reactions
A. N. Gorban ag153@le.ac.uk Department of Mathematics, University of
Leicester, Leicester, LE1 7RH, UK E. M. Mirkes Institute of Space and
Information Technologies, Siberian Federal University, Krasnoyarsk, Russia G.
S. Yablonsky Parks College, Department of Chemistry, Saint Louis University,
Saint Louis, MO 63103, USA
###### Abstract
For many real physico-chemical complex systems detailed mechanism includes
both reversible and irreversible reactions. Such systems are typical in
homogeneous combustion and heterogeneous catalytic oxidation. Most complex
enzyme reactions include irreversible steps. The classical thermodynamics has
no limit for irreversible reactions whereas the kinetic equations may have
such a limit. We represent the systems with irreversible reactions as the
limits of the fully reversible systems when some of the equilibrium
concentrations tend to zero. The structure of the limit reaction system
crucially depends on the relative rates of this tendency to zero. We study the
dynamics of the limit system and describe its limit behavior as $t\to\infty$.
If the reversible systems obey the principle of detailed balance then the
limit system with some irreversible reactions must satisfy the extended
principle of detailed balance. It is formulated and proven in the form of two
conditions: (i) the reversible part satisfies the principle of detailed
balance and (ii) the convex hull of the stoichiometric vectors of the
irreversible reactions does not intersect the linear span of the
stoichiometric vectors of the reversible reactions. These conditions imply the
existence of the global Lyapunov functionals and alow an algebraic description
of the limit behavior. The thermodynamic theory of the irreversible limit of
reversible reactions is illustrated by the analysis of hydrogen combustion.
###### keywords:
entropy , free energy , reaction network , detailed balance , irreversibility
###### PACS:
05.45.-a , 82.40.Qt , 82.20.-w , 82.60.Hc
## 1 Introduction
### 1.1 The problem: non-existence of thermodynamic functions in the limit of
irreversible reactions
We consider a homogeneous chemical system with $n$ components $A_{i}$, the
concentration of $A_{i}$ is $c_{i}\geq 0$, the amount of $A_{i}$ in the system
is $N_{i}\geq 0$, $V$ is the volume, $N_{i}=Vc_{i}$, $T$ is the temperature.
The $n$ dimensional vectors $c=(c_{i})$ and $N=(N_{i})$ belong to the closed
positive orthant $\mathbb{R}^{n}_{+}$ in $\mathbb{R}^{n}$.
($\mathbb{R}^{n}_{+}$ is the set of all vectors $x\in\mathbb{R}^{n}$ such that
$x_{i}\geq 0$ for all $i$.)
The classical thermodynamics has no limit for irreversible reactions whereas
the kinetic equations have. For example, let us consider a simple cycle
$A_{1}\underset{k_{-1}}{\overset{k_{1}}{\rightleftharpoons}}A_{2}\underset{k_{-2}}{\overset{k_{2}}{\rightleftharpoons}}A_{3}\underset{k_{-3}}{\overset{k_{3}}{\rightleftharpoons}}A_{1}$
with the equilibrium concentrations $c^{\rm eq}=(c_{1}^{\rm eq},c_{2}^{\rm
eq},c_{3}^{\rm eq})$ and the detailed balance conditions:
$k_{i}c^{\rm eq}_{i}=k_{-i}c^{\rm eq}_{i+1}$
under the standard cyclic convention, here, $A_{3+1}=A_{1}$ and
$c_{3+1}=c_{1}$. The perfect free energy has the form
$F=\sum_{i}RTVc_{i}\left(\ln\left(\frac{c_{i}}{c_{i}^{\rm
eq}}\right)-1\right)+const\,.$
Let the equilibrium concentration $c_{1}^{\rm eq}\to 0$ for the fixed values
of $c_{2,3}^{\rm eq}>0$. This means that
$\frac{k_{-1}}{k_{1}}=\frac{c_{1}^{\rm eq}}{c_{2}^{\rm eq}}\to 0\mbox{ and
}\frac{k_{3}}{k_{-3}}=\frac{c_{1}^{\rm eq}}{c_{3}^{\rm eq}}\to 0\,.$
Let us take the fixed values of the rate constants $k_{1}$, $k_{\pm 2}$ and
$k_{-3}$. Then the limit kinetic system exists and has the form:
$A_{1}{\overset{k_{1}}{\rightarrow}}A_{2}\underset{k_{-2}}{\overset{k_{2}}{\rightleftharpoons}}A_{3}\underset{k_{-3}}{\leftarrow}A_{1}\,.$
It is a routine task to write a first order kinetic equation for this scheme.
At the same time, the free energy function $F$ has no limit: it tends to
$\infty$ for any positive vector of concentrations because the term
$c_{1}\ln({c_{1}}/{c_{1}^{\rm eq}})$ increases to $\infty$. The free energy
cannot be normalized by adding a constant term because the variation of the
term $c_{1}\ln({c_{1}}/{c_{1}^{\rm eq}})$ on an interval $[0,\overline{c}]$
with fixed $\overline{c}$ also increases to $\infty$, it varies from $-c^{\rm
eq}_{1}/e$ (for the minimizer, $c_{1}={c_{1}^{\rm eq}}/e$) to a large number
$\overline{c}(\ln\overline{c}-\ln{c_{1}^{\rm eq}})$ (for
$c_{1}=\overline{c}$).
The logarithmic singularity is rather “soft” and does not cause a real
physical problem because even for ${c_{1}^{\rm eq}}/{c_{1}}=10^{-10}$ the
corresponding large term in the free energy will be just $\sim 23RT$ per mole.
Nevertheless, the absence of the limit causes some mathematical questions. For
example, the density,
$f=F/(RTV)=\sum_{i}c_{i}(\ln(c_{i}/c_{i}^{\rm eq})-1)\,,$ (1)
is a Lyapunov function for a system of chemical kinetics for a perfect mixture
with detailed balance under isochoric isothermal conditions. Here, $c_{i}$ is
the concentration of the $i$th component and $c_{i}^{\rm eq}$ is its
equilibrium concentration for a selected value of the linear conservation
laws, the so-called “reference equilibrium”.
This function is used for analysis of stability, existence and uniqueness of
chemical equilibrium since the work of Zeldovich (1938, reprinted in 1996
[26]). Detailed analysis of the connections between detailed balance and the
free energy function was provided in [19]. Perhaps, the first detailed proof
that $f$ is a Lyapunov function for chemical kinetics of perfect systems with
detailed balance was published in 1975 [22]. Of course, it does not differ
significantly from the Boltzman’s proof of his $H$-theorem (1873 [2]).
For the irreversible systems which are obtained as limits of the systems with
detailed balance, we should expect the preservation of stability of the
equilibrium. Moreover, one can expect existence of the Lyapunov functions
which are as universal as the thermodynamic functions are. The “universality”
means that these functions depend on the list of components and on the
equilibrium concentrations but do not depend on the reaction rate constants
directly.
The thermodynamic potential of a component $A_{i}$ cannot be defined in the
irreversible limits when the equilibrium concentration of $A_{i}$ tends to 0.
Nevertheless, in this paper, we construct the universal Lyapunov functions for
systems with some irreversible reactions. Instead of detailed balance we use
the weaker assumption that these systems can be obtained from the systems with
detailed balance when some constants tend to zero.
### 1.2 The extended form of detailed balance conditions for systems with
irreversible reactions
Let us consider a reaction mechanism in the form of the system of
stoichiometric equations
$\sum_{i}\alpha_{ri}A_{i}\to\sum_{j}\beta_{rj}A_{j}\;\;(r=1,\ldots,m)\,,$ (2)
where $\alpha_{ri}\geq 0$, $\beta_{rj}\geq 0$ are the stoichiometric
coefficients. The reverse reactions with positive rate constants are included
in the list (2) separately (if they exist). The stoichiometric vector
$\gamma_{r}$ of the elementary reaction is $\gamma_{r}=(\gamma_{ri})$,
$\gamma_{ri}=\beta_{ri}-\alpha_{ri}$. We always assume that there exists a
strictly positive conservation law, a vector $b=(b_{i})$, $b_{i}>0$ and
$\sum_{i}b_{i}\gamma_{ri}=0$ for all $r$. This may be the conservation of mass
or of total number of atoms, for example.
According to the generalized mass action law, the reaction rate for an
elementary reaction (2) is (compare to Eqs. (4), (7), and (14) in [14] and Eq.
(4.10) in [7])
$w_{r}=k_{r}\prod_{i=1}^{n}a_{i}^{\alpha_{ri}}\,,$ (3)
where $a_{i}\geq 0$ is the activity of $A_{i}$,
$a_{i}=\exp\left(\frac{\mu_{i}-\mu_{i}^{0}}{RT}\right)\,.$ (4)
Here, $\mu_{i}$ is the chemical potential and $\mu_{i}^{0}$ is the standard
chemical potential of the component $A_{i}$.
This law has a long history (see [6, 24, 13, 7]). It was invented in order to
meet the thermodynamic restrictions on kinetics. For this purposes, according
to the principle of detailed balance, the rate of the reverse reaction is
defined by the same formula and its rate constant should be found from the
detailed balance condition at a given equilibrium.
It is worth mentioning that the free energy has no limit when some of the
reaction equilibrium constants tend to zero. For example, for the ideal gas
the chemical potential is $\mu_{i}(c,T)=RT\ln c_{i}+\mu_{i}^{0}(T)$. In the
irreversible limit some $\mu_{i}^{0}\to\infty$. On the contrary, the
activities remain finite (for the ideal gases $a_{i}=c_{i}$) and the approach
based on the generalized mass action law and the detailed balance equations
$w_{r}^{+}=w_{r}^{-}$ can be applied to find the irreversible limit.
The list (2) includes reactions with the reaction rate constants $k_{r}>0$.
For each $r$ we define $k_{r}^{+}=k_{r}$, $w_{r}^{+}=w_{r}$, $k_{r}^{-}$ is
the reaction rate constant for the reverse reaction if it is on the list (2)
and 0 if it is not, $w_{r}^{-}$ is the reaction rate for the reverse reaction
if it is on the list (2) and 0 if it is not. For a reversible reaction,
$K_{r}=k_{r}^{+}/k_{r}^{-}$
The principle of detailed balance for the generalized mass action law is: For
given values $k_{r}$ there exists a positive equilibrium $a_{i}^{\rm eq}>0$
with detailed balance, $w_{r}^{+}=w_{r}^{-}$.
Recently, we have formulated the extended form of the detailed balance
conditions for the systems with some irreversible reactions [12]. This
extended principle of detailed balance is valid for all systems which obey the
generalized mass action law and are the limits of the systems with detailed
balance when some of the reaction rate constants tend to zero. It consists of
two parts:
* 1.
The algebraic condition: The principle of detailed balance is valid for the
reversible part. (This means that for the set of all reversible reactions
there exists a positive equilibrium where all the elementary reactions are
equilibrated by their reverse reactions.)
* 2.
The structural condition: The convex hull of the stoichiometric vectors of the
irreversible reactions has empty intersection with the linear span of the
stoichiometric vectors of the reversible reactions. (Physically, this means
that the irreversible reactions cannot be included in oriented cyclic
pathways.)
Let us recall the formal convention: the linear span of empty set is
$\\{0\\}$, the convex hull of empty set is empty.
In our previous work [12] we studied the systems with some irreversible
reactions which are the limits of the reversible systems with detailed
balance. The structural and algebraic conditions were found. The present paper
is focused on the dynamical consequences of these conditions. We prove that
the attractors always consist of the fixed points. These limit points
(“partial equilibria”) are situated on the faces of the positive orthant of
concentrations. These faces and the partial equilibria are described in the
paper.
### 1.3 The structure of the paper
In Sec. 2 we study the systems with detailed balance, their multiscale limits
and the limit systems which satisfy the extended principle of detailed
balance. The classical Wegscheider identities for the reaction rate constants
are presented. Their limits when some of the equilibria tend to zero give the
extended principle of detailed balance.
We use the generalized mass action law for the reaction rates. For the
analysis of equilibria for the general systems, the formulas with activities
are the same as for the ideal systems and it is convenient to work with
activities unless we need to study dynamics. The dynamical variables are
amounts and concentrations. In a special subsection 2.3 we discuss the
relations between concentration and activities, formulate the main assumptions
and present formulas for the dissipation rate.
We introduce attractors of the systems with some irreversible reactions and
study them in Sec. 3. It includes the central results of the paper. We fully
characterize the faces of the positive orthant that include $\omega$-limit
sets. On such a face, dynamics is completely degenerated (zero rates) or it is
driven by a smaller reversible system that obeys classical thermodynamics.
Hydrogen combustion is the most studied and very important gas reaction. It
serves as a main benchmark example for many studies of chemical kinetics. This
is already not a toy example but the complexity of this system is not
extremely high: in the usual models there are 6-8 components and $\sim$15-30
elementary reversible reactions. Under various conditions some of these
reactions are practically irreversible. We use this system as a benchmark in
Sec. 4 and give an example of the correct separation of the reactions into
reversible and irreversible part. The limit behavior of this system in time is
described.
In Conclusion we briefly discuss the results with focus on the unsolved
problems.
## 2 Multiscale limit of a system with detailed balance
### 2.1 Two classical approaches to the detailed balance condition
There are two traditional approach to the description of the reversible
systems with detailed balance. First, we can start from the independent rate
constants of the elementary reactions and consider the solvability of the
detailed balance equations as the additional condition on the admissible
values of the rate constants. Here, for $m$ elementary reactions we have $m$
constants ($m$ should be an even number, $m=2\ell$, $\ell=m/2$) and some
equations which describe connections between these constants. This approach
was introduced by Wegscheider in 1901 [23] and developed further by many
authors [20, 4].
Secondly, we can select a “forward” reaction in each pair of mutually reverse
elementary reactions. If a positive equilibrium is known then we can find the
reaction rate constants for the reverse reaction from the constants for
forward reaction and the detailed balance equations. Therefore, the forward
reaction rate constants and a set of the equilibrium activities form the
complete description of the reaction. Here we have $\ell+n$ independent
constants, $\ell=m/2$ rate constants of forward reactions and $n$ (it is the
number of components) equilibrium activities. For these $\ell+n$ constants,
the principle of detailed balance produces no restrictions. This second
approach is popular in applied chemical thermodynamics and kinetics [17, 10,
25] because it is convenient to work with the independent parameters “from
scratch”.
The Wegscheider conditions appear as the necessary and sufficient conditions
of solvability of the detailed balance equations. (See, for example, the
textbook [24]). Let us join the forward and reverse elementary reactions and
write
$\sum_{i}\alpha_{ri}A_{i}\rightleftharpoons\sum_{j}\beta_{rj}A_{j}\;\;(r=1,\ldots,\ell)\,.$
(5)
The stoichiometric matrix is $\boldsymbol{\Gamma}=(\gamma_{ri})$,
$\gamma_{ri}=\beta_{ri}-\alpha_{ri}$ (gain minus loss). The stoichiometric
vector $\gamma_{r}$ is the $r$th row of $\boldsymbol{\Gamma}$ with coordinates
$\gamma_{ri}=\beta_{ri}-\alpha_{ri}$.
Both sides of the detailed balance equations, $w_{r}^{+}=w_{r}^{-}$, are
positive for positive activities. The solvability of this system for positive
activities means the solvability of the following system of linear equations:
$\sum_{i}\gamma_{ri}x_{i}=\ln k_{r}^{+}-\ln k_{r}^{-}=\ln
K_{r}\;\;(r=1,\ldots\ell)$ (6)
($x_{i}=\ln a_{i}^{\rm eq}$). Of course, we assume that if $k_{r}^{+}>0$ then
$k_{r}^{-}>0$ (reversibility) and the equilibrium constant $K_{r}>0$ is
defined for all reactions from (5).
###### Proposition 1.
The necessary and sufficient conditions for existence of the positive
equilibrium $a_{i}^{\rm eq}>0$ with detailed balance is: For any solution
$\boldsymbol{\lambda}=(\lambda_{r})$ of the system
$\boldsymbol{\lambda\Gamma}=0\;\;\left(\mbox{i.e.}\;\;\sum_{r=1}^{\ell}\lambda_{r}\gamma_{ri}=0\;\;\mbox{for
all}\;\;i\right)$ (7)
the Wegscheider identity holds:
$\prod_{r=1}^{\ell}(k_{r}^{+})^{\lambda_{r}}=\prod_{r=1}^{\ell}(k_{r}^{-})^{\lambda_{r}}\,.$
(8)
It is sufficient to use in (8) any basis of solutions of the system (7):
$\boldsymbol{\lambda}\in\\{\boldsymbol{\lambda}^{1},\cdots,\boldsymbol{\lambda}^{q}\\}$.
### 2.2 Multiscale degeneration of equilibria
We consider the systems with some irreversible reactions as the limits of the
fully reversible systems when some reaction rate constants tend to zero. In
the reversible systems, the principle of detailed balance implies the
Wegscheider identities (8). Therefore, the limit system is not arbitrary. Some
consequences of the Wegscheider identities persist though a part of reaction
rate constants in these identities become zero. In [12] we compare these
consequences with the grin of the Cheshire cat: the whole cat (the reversible
system with detailed balance) vanishes but the grin persists.
We can postulate that some reaction rate constants go to zero. However, the
reaction rate constants are not independent. They are connected by the
Wegscheider identities. The rate constants should tend to zero with
preservation of their relations. Therefore, the simple strategy, just to
neglect the rates of some of the reactions, cannot be applied for complex
reactions. Nevertheless, we can change the variables and use the independent
set “reaction rate constants for the forward reactions + equilibrium
activities” (see [17, 10, 25, 12]). Every set of positive values of these
variables corresponds to a reversible system with detailed balance and no
additional restrictions are needed. If the reversible system degenerates to a
system with some irreversible reactions then some of the equilibrium
activities tend to zero. Let us study this process of degeneration of
reversible reactions into irreversible ones starting from the corresponding
degeneration of equilibrium activities to zero.
Let us take a system with detailed balance and send some of the equilibrium
activities to zero: $a_{i}^{\rm eq}\to 0$ when $i\in I$ for some set of
indexes $I$. Immediately we find a surprise: this assumption is not sufficient
to find a limiting irreversible mechanism. It is necessary to take into
account the rates of the convergency to zero of different $a_{i}^{\rm eq}$.
Indeed, let us study a very simple example,
$A_{1}\underset{k_{-1}}{\overset{k_{1}}{\rightleftharpoons}}A_{2}\underset{k_{-2}}{\overset{k_{2}}{\rightleftharpoons}}A_{3}$
when $a_{1}^{\rm eq},a_{2}^{\rm eq}\to 0$.
If $a_{1}^{\rm eq},a_{2}^{\rm eq}\to 0$, $a_{1}^{\rm eq}/a_{2}^{\rm
eq}=const>0$ and $a_{3}^{\rm eq}=const>0$ then the limit system should be
$A_{1}\underset{k_{-1}}{\overset{k_{1}}{\rightleftharpoons}}A_{2}\to A_{3}$
and we can keep $k_{1,-1,2}=const$ whereas $k_{-2}\to 0$.
If $a_{1}^{\rm eq},a_{2}^{\rm eq}\to 0$, $a_{1}^{\rm eq}/a_{2}^{\rm eq}\to 0$
then the limit system should be $A_{1}\to A_{2}\to A_{3}$ and we can keep
$k_{1,2}=const>0$ whereas $k_{-1,-2}\to 0$.
If $a_{1}^{\rm eq},a_{2}^{\rm eq}\to 0$, $a_{2}^{\rm eq}/a_{1}^{\rm eq}\to 0$
then in the limit survives only one reaction $A_{2}\to A_{3}$ (if we assume
that all the reaction rate constants are bounded).
We study asymptotics $a_{i}^{\rm eq}={\rm
const}\times\varepsilon^{\delta_{i}}$, $\varepsilon\to 0$ for various values
of non-negative exponents $\delta_{i}\geq 0$ ($i=1,\ldots,n$). At equilibrium,
each reaction rate in the generalized mass action law is proportional to a
power of $\varepsilon$:
$w_{r}^{{\rm eq}+}=k_{r}^{+}{\rm
const}\times\varepsilon^{\sum_{i}\alpha_{ri}\delta_{i}}\,,\;\;w_{r}^{{\rm
eq}-}=k_{r}^{-}{\rm const}\times\varepsilon^{\sum_{i}\beta_{ri}\delta_{i}}\,.$
According to the principle of detailed balance, $w_{r}^{{\rm eq}+}=w_{r}^{{\rm
eq}-}$ and
$\frac{k_{r}^{+}}{k_{r}^{-}}={\rm
const}\times\varepsilon^{(\gamma_{r},\delta)}\,,$ (9)
where $\delta$ is the vector of exponents, $\delta=(\delta_{i})$.
There are three groups of reactions with respect to the given vector $\delta$:
$1.\,(\gamma_{r},\delta)=0;\;\;2.\,(\gamma_{r},\delta)<0;\;\;3.\,(\gamma_{r},\delta)>0\,.$
In the first group ($(\gamma_{r},\delta)=0$) the ratio
${k_{r}^{+}}/{k_{r}^{-}}$ remains constant and we can take
$k_{r}^{\pm}=const>0$. In the second group ($(\gamma_{r},\delta)<0$) the ratio
$k_{r}^{-}/k_{r}^{+}\to 0$ and we should take $k_{r}^{-}\to 0$ whereas
$k_{r}^{+}$ may remain constant and positive. In the third group
($(\gamma_{r},\delta)>0$), the situation is inverse: $k_{r}^{+}/k_{r}^{-}\to
0$ and we can take $k_{r}^{-}=const>0$, whereas $k_{r}^{+}\to 0$.
These three groups depend on $\delta$ but this dependence is piecewise
constant. For every $\gamma_{r}$, three sets of $\delta$ are defined: (i)
hyperplane $(\gamma_{r},\delta)=0$, (ii) hemispace $(\gamma_{r},\delta)<0$ and
hemispace $(\gamma_{r},\delta)>0$. The space of vectors $\delta$ is split in
the subsets defined by the values of functions ${\rm sign}(\gamma_{r},\delta)$
($\pm 1$ or 0).
We consider bounded systems, hence the negative values of $\delta$ should be
forbidden. At least one equilibrium activity should not vanish. Therefore,
$\delta_{j}=0$ for some $j$. Below we assume that $\delta_{i}\geq 0$ and
$\delta_{j}=0$ for a non-empty set of indices $J_{0}$. Moreover, the atom
balance in equilibrium should be positive. Here, this means that for the set
of equilibrium concentrations $c^{\rm eq}_{i}$ ($i\in J_{0}$) the
corresponding values of all atomic concentrations are strictly positive and
separated from zero.
Let the vector of exponents, $\delta=(\delta_{i})$ be given and the three
groups of reactions be found. For the reactions of the third group (with
$(\gamma_{r},\delta)>0$) the forward reaction vanishes in the limit
$\varepsilon\to 0$. It is convenient to transpose the stoichiometric equations
for these reactions and swap the forward reactions with reverse ones. Let us
perform this transposition. After that, $\alpha_{r}$ changes over $\beta_{r}$,
$\gamma$ transforms into $-\gamma$, and the inequality $(\gamma_{r},\delta)>0$
transforms into $(\gamma_{r},\delta)<0$.
Let us summarize. We use the given vector of exponents $\delta$ and produce a
system with some irreversible reactions from a system of reversible reactions
and detailed balance equilibrium $a_{i}^{\rm eq}$ by the following rules:
1. 1.
if $\delta_{i}>0$ then we assign $a_{i}^{\rm eq}=0$ and if $\delta_{i}=0$ then
$a_{i}^{\rm eq}$ does not change;
2. 2.
if $(\gamma_{r},\delta)=0$ then $k_{r}^{\pm}$ do not change;
3. 3.
if $(\gamma_{r},\delta)<0$ then we assign $k_{r}^{-}=0$ and $k_{r}^{+}$ does
not change;
4. 4.
if $(\gamma_{r},\delta)>0$ then we assign $k_{r}^{+}=0$ and $k_{r}^{-}$ does
not change. (In the last case, we transpose the stoichiometric equation and
swap the forward reaction with reverse one, for convenience, $\gamma_{r}$
changes to -$\gamma_{r}$ and $k_{r}^{-}$ becomes 0. Therefore, this case
transforms into case 3.)
This is a limit system caused by the multiscale degeneration of equilibrium.
The multiscale character of the limit $a_{i}^{\rm
eq}=const\times\varepsilon^{\delta_{i}}\to 0$ (for some $i$) is important
because for different values of $\delta$ reactions may have different dominant
directions and the set of irreversible reactions in the limit may change.
The general form of the kinetic equations for the homogeneous systems is
$\frac{{\mathrm{d}}N}{{\mathrm{d}}t}=V\sum_{r}w_{r}\gamma_{r}\,,$ (10)
where $N_{i}$ is the amount of $A_{i}$, $N$ is the vector with components
$N_{i}$ and $V$ is the volume.
Let us consider a limit system for the degeneration of equilibrium with the
vector of exponents $\delta$. For this system $(\gamma_{r},\delta)\leq 0$ for
all $r$ and, in particular, $(\gamma_{r},\delta)<0$ for all irreversible
reactions and $(\gamma_{r},\delta)=0$ for all reversible reactions.
###### Proposition 2.
A linear functional $G_{\delta}(N)=(\delta,N)$ decreases along the solutions
of kinetic equations (10) for this limit system:
${\mathrm{d}}G_{\delta}(N)/{{\mathrm{d}}t}\leq 0$ and
${\mathrm{d}}G_{\delta}(N){{\mathrm{d}}t}=0$ if and only if all the reaction
rates for the irreversible reactions are zero.
###### Proof.
Indeed,
$\frac{{\mathrm{d}}G_{\delta}(N)}{{\mathrm{d}}t}=V\sum_{r}w_{r}(\gamma_{r},\delta)\leq
0\,,$ (11)
because for reversible reactions $(\gamma_{r},\delta)=0$, and for irreversible
reactions $w_{r}=w_{r}^{+}\geq 0$ and $(\gamma_{r},\delta)<0$. All the terms
in this sum are non-negative, hence it may be zero if and only if each summand
is zero. ∎
This Lyapunov function may be used in a proof that the rates of all
irreversible reactions in the system tend to 0 with time. Indeed, if they do
not tend to zero then on a solution of (10) $G_{\delta}(N(t))\to-\infty$ when
$t\to\infty$ and $N(t)$ is unbounded. Equation (11) and Proposition (2) give
us the possibility to prove the extended principle of detailed balance in the
following form. Let us consider a reaction mechanism that includes reversible
and irreversible reactions. Assume that the reaction rates satisfy the
generalized mass action law (3) and the set of reaction rate constants is
given. Let us ask the question: Is it possible to obtain this reaction
mechanism and reaction rate constants as a limit in the multiscale
degeneration of equilibrium from a fully reversible system with the classical
detailed balance. The answer to this question gives the following theorem
about the extended principle of detailed balance.
###### Theorem 1.
A system can be obtained as a limit in the multiscale degeneration of
equilibrium from a reversible system with detailed balance if and only if (i)
the reaction rate constants of the reversible part of the reaction mechanism
satisfy the classical principle of detailed balance and (ii) the convex hull
of the stoichiometric vectors of the irreversible reactions does not intersect
the linear span of the stoichiometric vectors of reversible reactions.
###### Proof.
Let the given system be a limit of a reversible system with detailed balance
in the multiscale degeneration of equilibrium with the exponent vector
$\delta$. Then for the reversible reactions $(\gamma_{r},\delta)=0$ and for
the irreversible reactions $(\gamma_{r},\delta)<0$. For every vector $x$ from
the convex hull of the stoichiometric vector of the irreversible reactions
$(x,\delta)<0$ and for any vector $y$ from the linear span of the
stoichiometric vectors of the reversible reactions $(y,\delta)=0$. Therefore,
these sets do not intersect. The reaction rate constants for the reversible
reactions satisfy the classical principle of detailed balance because they do
not change in the equilibrium degeneration and keep this property of the
original fully reverse system with detailed balance.
Conversely, let a system satisfy the extended principle of detailed balance:
(i) the reaction rate constants of the reversible part of the reaction
mechanism satisfy the classical principle of detailed balance and (ii) the
convex hull of the stoichiometric vectors of the irreversible reactions does
not intersect the linear span of the stoichiometric vectors of reversible
reactions. Due to the classical theorems of the convex geometry, there exists
a linear functional that separates this convex set from the linear subspace.
(Strong separation of closed and compact convex sets.) This separating
functional can be represented in the form $(x,\theta)$ for some vector
$\theta$. For the reversible reactions $(\gamma_{r},\theta)=0$ and for the
irreversible reactions $(\gamma_{r},\theta)<0$.
It is possible to find vector $\delta$ with this separation property and non-
negative coordinates. Indeed, according to the basic assumptions, there exists
a linear conservation law with strongly positive coordinates. This is a vector
$b$ ($b_{i}>0$) with the property: $(\gamma_{r},b)=0$ for all reactions. For
any $\lambda$, the vector $\theta+\lambda b$ has the same separation property
as the vector $\theta$ has. We can select such $\lambda$ that
$\delta_{i}=\theta_{i}+\lambda b_{i}\geq 0$ and $\delta_{i}=\theta_{i}+\lambda
b_{i}=0$ for some $i$. Let us take this linear combination $\delta$ as a
vector of exponents.
Let us create a fully reversible system from the initial partially
irreversible one. We do not change the reversible reactions and their rate
constants. Because the reversible reactions satisfy the classical principle of
detailed balance, there exists a strongly positive vector of equilibrium
activities $a_{i}^{*}>0$ for the reversible reactions.
For each irreversible reaction with the stoichiometric vector $\gamma_{r}$ and
reaction rate constant $k_{r}=k_{r}^{+}>0$ we add a reverse reaction with the
reaction rate constant
$k_{r}^{-}=k_{r}^{+}\prod_{i}(a_{i}^{*})^{-\gamma_{ri}}\,.$
For this fully reversible system the activities $a_{i}^{*}>0$ provide the
point of detailed balance. In the multiscale degeneration process, the
equilibrium activities depend on $\varepsilon\to 0$ as $a_{i}^{\rm
eq}=a_{i}^{*}\varepsilon^{\delta_{i}}$. For the reactions with
$(\gamma_{r},\delta)=0$ the reaction rate constants do not depend on
$\varepsilon$ and for the reactions with $(\gamma_{r},\delta)<0$ the rate
constant $k_{r}^{-}$ tends to zero as $\varepsilon^{-(\gamma_{r},\delta)}$ and
$k_{r}^{+}$ does not change. We return to the initial system of reactions in
the limit $\varepsilon\to 0$. ∎
This is a particular form of the extended principle of detailed balance. For
more discussion see [12]. Fig. 1 illustrates geometric sense of the extended
detailed balance condition: the convex hull of the stoichiometric vectors of
the irreversible reactions does not intersect the linear span of the
stoichiometric vectors of the reversible reactions. In this illustration,
$\\{\gamma_{r}\,|\,r\in J_{0}\\}$ are the stoichiometric vectors of the
reversible reactions and $\\{\gamma_{r}\,|\,r\in J_{1}\\}$ are the
stoichiometric vectors of the irreversible reactions.
Figure 1: Main operations in the application of the extended detailed balance
conditions. In the concentration space $\mathbb{R}^{n}$ we should find the
subspace spanned by all the stoichiometric vectors
$\\{\gamma_{r}\,|\,r=1,\ldots,{\ell}\\}$ (a). In this subspace we have to
select the internal coordinates. In
span$\\{\gamma_{r}\,|\,r=1,\ldots,{\ell}\\}$ we have to select the subspace
spanned by the stoichiometric vectors of the reversible reactions (b) (the
dashed vectors). The stoichiometric vectors of the irreversible reactions are
in (b,c,d) solid and bold. Due to the extended principle of detailed balance,
span$\\{\gamma_{r}\,|\,r\in J_{0}\\}$ should not intersect
conv$\\{\gamma_{r}\,|\,r\in J_{1}\\}$ (the dotted triangle in Fig.). For
analysis of this intersection, it is convenient to proceed to the quotation
space span$\\{\gamma_{r}\,|\,r=1,\ldots,{\ell}\\}$/span$\\{\gamma_{r}\,|\,r\in
J_{0}\\}$ (c,d). In this quotation space, span$\\{\bar{\gamma}_{r}\,|\,r\in
J_{0}\\}$ is $\\{0\\}$ and two situations are possible: (c)
$0\in\mbox{conv}\\{\bar{\gamma}_{r}\,|\,r\in J_{1}\\}$ (the dotted triangle
includes the origin) or (d) $0\notin\mbox{conv}\\{\gamma_{r}\,|\,r\in
J_{1}\\}$ (the dotted triangle does not include the origin). In the case (c)
the extended detailed balance condition is violated. The case (d) satisfies
this condition.
### 2.3 Activities, concentrations and affinities
To combine the linear Lyapunov functions $G_{\delta}(N)=(\delta,N)$ (11) with
the classical thermodynamic potential and study the kinetic equations in the
closed form we have to specify the relations between activities and
concentrations. We accept the assumption: $a_{i}=c_{i}g_{i}(c,T)$, where
$g_{i}(c,T)>0$ is the activity coefficient. It is a continuously
differentiable function of $c,T$ in the whole diapason of their values. In a
bounded region of concentrations and temperature we can always assume that
$g_{i}>g_{0}>0$ for some constant $g_{0}$. This assumption is valid for the
non-ideal gases and for liquid solutions. It holds also for the “surface gas”
in kinetics of heterogeneous catalysis [24] and does not hold for the solid
reagents (see for example, analysis of carbon activity in the methane
reforming [12]).
The system of units should be commented. Traditionally, $a_{i}$ is assumed to
be dimensionless and for perfect systems $a_{i}=c_{i}/c_{i}^{\circ}$, where
$c_{i}^{\circ}$ is an arbitrary “standard” concentration. To avoid
introduction of unnecessary quantities, we always assume that in the selected
system of units, $c_{i}^{\circ}\equiv 1$.
If the thermodynamic potentials exist then due to the thermodynamic definition
of activity (4), this hypothesis is equivalent to the logarithmic singularity
of the chemical potentials, $\mu_{i}=RT\ln c_{i}+\ldots$ where $\ldots$ stands
for a continuous function of $c,T$ (all the concentrations and the
temperature). In this case, the free energy has the form
$F(N,T,V)=RT\sum_{i}N_{i}(\ln c_{i}-1+f_{0i}(c,T))\,,$ (12)
where the functions $f_{0i}(c,T)$ are continuously differentiable for all
possible values of arguments. Functions $f_{0i}$ in the right hand side of the
representation (12) cannot be restored unambiguously from the free energy
function $F(N,T,V)$ but for a small admixture $A_{i}$ it is possible to
introduce the partial pressure $p_{i}$ which satisfies the law
$p_{i}=RTc_{i}+o(c_{i})$. This is due to the terms $N_{i}\ln c_{i}$ in $F$.
Indeed, $P=-\partial F(N,T,V)/\partial
V=RTc_{i}+o(c_{i})+P\left|{}_{c_{i}=0}\right.$. Connections between the
equation of state, free energy and kinetics are discussed in more detail in
[7, 8].
There are several simple algebraic corollaries of the assumed connection
between activities and concentrations. Let us consider an elementary reaction
$\sum\alpha_{i}A_{i}\to\sum\beta_{i}A_{i}$ with $\alpha_{i},\beta_{i}\geq 0$.
Then, according to the generalized mass action law, for any vector of
concentrations $c$ ($c_{i}\geq 0$)
1. 1.
If, for some $i$, $c_{i}=0$ then $\gamma_{i}w(c)\geq 0$;
2. 2.
If, for some $i$, $c_{i}=0$ and $\gamma_{i}<0$ then $\alpha_{i}>0$ and
$w(c)=0$.
Similarly, for a reversible reaction
$\sum\alpha_{i}A_{i}\rightleftharpoons\sum\beta_{i}A_{i}$
1. 1.
If, for some $i$, $c_{i}=0$ and $\gamma_{i}>0$ then $\beta_{i}>0$ and
$w^{-}(c)=0$;
2. 2.
If, for some $i$, $c_{i}=0$ and $\gamma_{i}<0$ then $\alpha_{i}>0$ and
$w^{+}(c)=0$.
These statements as well as Proposition 3 and Corollary 1 below are the
consequences of the generalized mass action law (3) and the connection between
activities and concentrations without any assumptions about extended principle
of detailed balance.
Each set of indexes $J=\\{i_{1},\ldots,i_{j}\\}$ defines a face of the
positive polyhedron,
$F_{J}=\\{c\,|\,c_{i}\geq 0\mbox{ for all }i\mbox{ and }c_{i}=0\mbox{ for
}i\in J\\}\,.$
By definition, the relative interior of $F_{J}$, $ri(F_{J})$, consists of
points with $c_{i}=0$ for $i\in J$ and $c_{i}>0$ for $i\notin J$.
###### Proposition 3.
Let for a point $c\in ri(F_{J})$ and an index $i\in J$
$\sum_{r}\gamma_{ri}w_{r}(c)=0\,.$
Then this identity holds for all $c\in F_{J}$.
###### Proof.
For convenience, let us write all the forward and reverse reactions separately
and represent the reaction mechanism in the form (2). All the terms in the sum
$\sum_{r}\gamma_{ri}w_{r}(c)$ are non-negative, because $c_{i}=0$. Therefore,
if the sum is zero then all the terms are zero. The reaction rate $w_{r}$ (3)
with non-zero rate constant takes zero value if and only if $\alpha_{rj}>0$
and $a_{j}=0$ for some $j$. The equality $a_{i}=0$ is equivalent to $c_{i}=0$.
Therefore, $w_{r}(c)=0$ for a point $c\in ri(F_{J})$ if and only if there
exists $j\in J$ such that $\alpha_{rj}>0$. If $\alpha_{rj}>0$ for an index
$j\in J$ then $w_{r}(c)=0$ for all $c\in F_{J}$ because $c_{j}=0$ in $F_{J}$.
∎
We call a face $F_{J}$ of the positive orthant $\mathbb{R}^{n}_{+}$ invariant
with respect to a set $S$ of elementary reactions if $\sum_{r\in
S}\gamma_{rj}w_{r}(c)=0\,$ for all $c\in F_{J}$ and every $j\in J$.
Let us consider the reaction mechanism in the form (2) where all the forward
and reverse reactions participate separately.
###### Corollary 1.
The following statements are equivalent:
1. 1.
$\sum_{r\in S}\gamma_{ri}w_{r}(c)=0$ for a point $c\in ri(F_{J})$ and all
indexes $i\in J$;
2. 2.
The face $F_{J}$ is invariant with respect to the set of reactions $S$;
3. 3.
The face $F_{J}$ is invariant with respect to every elementary reaction from
$S$;
4. 4.
For every $r\in S$ either $\gamma_{rj}=0$ for all $j\in J$ or $\alpha_{rj}>0$
for some $j\in J$.
We aim to perform the analysis of the asymptotic behavior of the kinetic
equations in the multiscale degeneration of equilibrium described in Sec. 2.2.
For this purpose, we have to answer the question: how the relations between
activities $a_{i}$ and concentrations $c_{i}$ depend on the degeneration
parameter $\varepsilon\to 0$? We do no try to find the maximally general
appropriate answer to this question. For the known applications, the answer
is: the relations between $a_{i}$ and $c_{i}$ do not depend on $\varepsilon\to
0$. In particular, it is trivially true for the ideal systems. The simple
generalization, $a_{i}=c_{i}g_{i}(c,T,\varepsilon)$, where
$g_{i}(c,T,\varepsilon)>g_{0}>0$ are continuous functions, is not a
generalization at all, because we can use for $\varepsilon\to 0$ the limit
case that does not depend on $\varepsilon$, $g_{i}(c,T)=g_{i}(c,T,0)$.
This independence from $\varepsilon$ implies that the reversible part of the
reaction mechanism has the thermodynamic Lyapunov functions like free energy.
If we just delete the irreversible part then the classical thermodynamics is
applicable and the thermodynamic potentials do not depend on $\varepsilon$.
For the generalized mass action law, the time derivative of the relevant
thermodynamic potentials have very nice general form. Let, under given
condition, the function $\Phi(N,\ldots)$ be given, where by $\ldots$ is used
for the quantities that do not change in time under these conditions. It is
the thermodynamics potential if $\partial\Phi(N,\ldots)/\partial
N_{i}=\mu_{i}$. For example, it is the free Helmholtz energy $F$ for
$V,T=const$ and the free Gibbs energy $G$ for $P,V=const$.
Let us calculate the time derivative of $\Phi(N,\ldots)$ due to kinetic
equation (10). The reaction rates are given by the generalized mass action law
(3) with definition of activities through chemical potential (4). We assume
that the principle of detailed balance holds (it should hold for the
reversible part of the reaction mechanism according to the extended detailed
balance conditions). More precisely, there exists an equilibrium with detailed
balance for any temperature $T$, $a^{\rm eq}(T)$: for all $r$,
$w_{r}^{+}(a^{\rm eq})=w_{r}^{-}(a^{\rm eq})=w_{r}^{\rm eq}(T)$. It is
convenient to represent the reaction rates using these equilibrium fluxes
$w_{r}^{\rm eq}(T)$:
$w^{+}_{r}=w^{\rm eq}_{r}\exp\left(\sum_{i}\frac{\alpha_{ri}(\mu_{i}-\mu^{\rm
eq}_{i})}{RT}\right)\,,\;\;w^{-}_{r}=w^{\rm
eq}_{r}\exp\left(\sum_{i}\frac{\beta_{ri}(\mu_{i}-\mu^{\rm
eq}_{i})}{RT}\right)\,.$
where $\mu^{\rm eq}_{i}=\mu_{i}(a^{\rm eq},T)$.
These formulas give immediately the following representation of the
dissipation rate
$\begin{split}\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}t}&=\sum_{i}\frac{\partial\Phi(N,\ldots)}{\partial
N_{i}}\frac{{\mathrm{d}}N_{i}}{{\mathrm{d}}t}=\sum_{i}\mu_{i}\frac{{\mathrm{d}}N_{i}}{{\mathrm{d}}t}\\\
&=-VRT\sum_{r}(\ln w_{r}^{+}-\ln w_{r}^{-})(w_{r}^{+}-w_{r}^{-})\leq
0\,.\end{split}$ (13)
The inequality holds because $\ln$ is a monotone function and, hence, the
expressions $\ln w_{r}^{+}-\ln w_{r}^{-}$ and $w_{r}^{+}-w_{r}^{-}$ have
always the same sign. Formulas of this kind for dissipation are well known
since the famous Boltzmann $H$-theorem (1873 [2], see also [13]). The entropy
increase in isolated systems has the similar form:
$\frac{{\mathrm{d}}S}{{\mathrm{d}}t}=VR\sum_{r}(\ln w_{r}^{+}-\ln
w_{r}^{-})(w_{r}^{+}-w_{r}^{-})\geq 0\,.$
Let us notice that
$\ln w_{r}^{+}-\ln
w_{r}^{-}=\frac{1}{RT}\sum_{i}\mu_{i}(\alpha_{ri}-\beta_{ri})=-\frac{(\gamma_{r},\mu)}{RT}\,.$
The quantity $-(\gamma_{r},\mu)$ is one of the central notion of physical
chemistry, affinity [5]. It is positive if the forward reaction prevails over
reverse one and negative in the opposite case. It measures the energetic
advantage of the forward reaction over the reverse one (free energy per mole).
The activity divided by $RT$ shows how large is this energetic advantage
comparing to the thermal energy. We call it the normalized affinity and use a
special notation for this quantity:
$\mathbb{A}_{r}=-\frac{(\gamma_{r},\mu)}{RT}$
Let us apply an elementary identity
$\exp a-\exp b=(\exp a+\exp b)\tanh\frac{a-b}{2}$
to the reaction rate, $w_{r}=w_{r}^{+}-w_{r}^{-}$:
$w_{r}=(w_{r}^{+}+w_{r}^{-})\tanh\frac{\mathbb{A}_{r}}{2}\,.$ (14)
This representation of the reaction rates gives immediately for the
dissipation rate:
$\frac{{\mathrm{d}}\Phi}{{\mathrm{d}}t}=-VRT\sum_{r}(w_{r}^{+}+w_{r}^{-})\mathbb{A}_{r}\tanh\frac{\mathbb{A}_{r}}{2}\leq
0\,.$ (15)
In this formula, the kinetic information is collected in the non-negative
factors, the sums of reaction rates $(w_{r}^{+}+w_{r}^{-})$. The purely
thermodynamical multipliers $\mathbb{A}_{r}\tanh({\mathbb{A}_{r}}/{2})$ are
also non-negative. For small $|\mathbb{A}_{r}|$, the expression
$\mathbb{A}_{r}\tanh({\mathbb{A}_{r}}/{2})$ behaves like
$\mathbb{A}_{r}^{2}/2$ and for large $|\mathbb{A}_{r}|$ it behaves like the
absolute value, $|\mathbb{A}_{r}|$.
So, we have two Lyapunov functions for two fragments of the reaction
mechanism. For the reversible part, this is just a classical thermodynamic
potential. For the irreversible part, this is a linear functional
$G_{\delta}(N)=(\delta,N)$. More precisely, the irreversible reactions
decrease this functional, whereas for the reversible reactions it is the
conservation law. Therefore, it decreases monotonically in time for the whole
system.
## 3 Attractors
### 3.1 Dynamical systems and limit points
The kinetic equations (10) do not give a complete representation of dynamics.
The right hand side includes the volume $V$ and the reaction rates $w_{r}$
which are functions (3) of the concentrations $c$ and temperature $T$, whereas
in the left hand side there is $\dot{N}$. To close this system, we need to
express $V$, $c$ and $T$ through $N$ and quantities which do not change in
time. This closure depends on conditions. The simplest expressions appear for
isochoric isothermal conditions: $V,T=const$, $c=N/V$. For other classical
conditions ($U,V=const$, or $P,T=const$, or $H,P=const$) we have to use the
equations of state. There may be more sophisticated closures which include
models or external regulators of the pressure and temperature, for example.
Proposition 2 is valid for all possible closures. It is only important that
the external flux of the chemical components is absent. Further on, we assume
that the conditions are selected, the closure is done, the right hand side of
the resulting system is continuously differentiable and there exists the
positive bounded solution for initial data in $\mathbb{R}^{n}_{+}$ and $V$,
$T$ remain bounded and separated from zero. The nature of this closure is not
crucial. For some important particular closures the proofs of existence of
positive and bounded solutions are well known (see, for example, [22]).
Strictly speaking, such a system is not a dynamical system in
$\mathbb{R}^{n}_{+}$ but a semi-dynamical one: the solutions may lose
positivity and leave $\mathbb{R}^{n}_{+}$ for negative values of time. The
theory of the limit behavior of the semi-dynamical systems was developed for
applications to kinetic systems [9].
We aim to describe the limit behavior of the systems as $t\to\infty$. Under
the extended detailed balance condition the limit behavior is rather simple
and the system will approach steady states but to prove this behavior we need
the more general notion of the $\omega$-limit points.
By the definition, the $\omega$-limit points of a dynamical system are the
limit points of the motions when time $t\to\infty$. We consider a kinetic
system in $\mathbb{R}^{n}_{+}$. In particular, for each solution of the
kinetic equations $N(t)$ the set of the corresponding $\omega$-limit points is
closed, connected and consists of the whole trajectories ([9], Proposition
1.5). This means that the motion which starts from an $\omega$-limit point
remains in $\mathbb{R}^{n}_{+}$ for all time moments, both positive and
negative.
###### Proposition 4.
Let $N(t)$ be a positive solution of the kinetic equation and $x^{*}$ be an
$\omega$-limit point of this solution and $x_{i}^{*}=0$. then at this point
$\dot{x}_{i}|_{x^{*}}=0$.
###### Proof.
Let $x(t)$ be a solution of the kinetic equations with the initial state
$x(0)=x^{*}$. All the points $x(t)$ ($-\infty<t<\infty$) belong to
$\mathbb{R}^{n}_{+}$. Indeed, there exists such a sequence $t_{j}\to\infty$
that $N(t_{j})\to x^{*}$. For any $\tau\in(-\infty,\infty)$, $N(t_{j}+\tau)\to
x(\tau)$. For sufficiently large $j$, $t_{j}+\tau>0$ and the value
$N(t_{j}+\tau)\in\mathbb{R}^{n}_{+}$. Therefore,
$x(\tau)\in\mathbb{R}^{n}_{+}$ ($-\infty<\tau<\infty$) and for any $\tau$ the
point $x(\tau)$ is an $\omega$-limit point of the solution $N(t)$. Let
$x_{i}^{*}=0$ and $\dot{x}_{i}|_{x^{*}}=v\neq 0$. If $v>0$ then for small
$|\tau|$ and $\tau<0$ the value of $x_{i}$ becomes negative, $x_{i}(\tau)<0$.
It is impossible because positivity. Similarly, If $v<0$ then for small
$\tau>0$ the value of $x_{i}$ becomes negative, $x_{i}(\tau)<0$. It is also
impossible because positivity. Therefore, $\dot{x}_{i}|_{x^{*}}=0$. ∎
We use Proposition 4 in the following combination with Proposition 3. Let us
write the reaction mechanism in the form (2).
###### Corollary 2.
If an $\omega$-limit point belongs to the relative interior $riF_{J}$ of the
face $F_{J}\subset\mathbb{R}^{n}_{+}$ then the face $F_{J}$ is invariant with
respect to the reaction mechanism and for every elementary reaction either
$\gamma_{rj}=0$ for all $j\in J$ or $\alpha_{rj}>0$ for some $j\in J$.
###### Proof.
If an $\omega$-limit point belongs to $riF_{J}$ then at this point all
$\dot{c}_{j}=0$ for $j\in J$ due to Proposition 4. Therefore, we can apply
Corollary 1.∎
### 3.2 Steady states of irreversible reactions
Under extended detailed balance conditions, all the reaction rates of the
irreversible reactions are zero at every limit point of the kinetic equations
(10), due to Proposition 2. In this section, we give a simple combinatorial
description of steady states for the set of irreversible reactions. This
description is based on Proposition 2 and, therefore, uses the extended
detailed balance conditions.
We continue to study multiscale degeneration of a detailed balance
equilibrium. The vector of exponents $\delta=(\delta_{i})$ is given,
$\delta_{i}\geq 0$ for all $i$ and $\delta_{i}=0$ for some $i$. There are two
sets of reaction. For one of them, $(\gamma_{r},\delta)=0$ and in the limit
both $k_{r}^{\pm}>0$. In the second set, $(\gamma_{r},\delta)<0$ and in the
limit we assign $k_{r}^{-}=0$ and $k_{r}^{+}$ is the same as in the initial
system (before the equilibrium degeneration). If it is necessary, we transpose
the stoichiometric equations and swap the forward reactions with reverse ones.
For convenience, let us change the notations. Let $\gamma_{i}$ be the
stoichiometric vectors of reversible reactions with $(\gamma_{r},\delta)=0$
($r=1,\ldots,h$), and $\nu_{l}$ be the stoichiometric vectors for the
reactions from the second set, $(\nu_{l},\delta)<0$ ($l=1,\ldots,s$). For the
reaction rates and constants for the first set we keep the same notations:
$w_{r}$, $w_{r}^{\pm}$, $k_{r}^{\pm}$. For the second set, we use for the
reaction rate constants $q_{l}=q_{l}^{+}$ and for the reaction rates
$v_{l}=v_{l}^{+}$. (They are also calculated according to the generalized mass
action law (3).) The input and output stoichiometric coefficients remain
$\alpha_{ri}$ and $\beta_{ri}$ for the first set and for the second set we use
the notations $\alpha_{li}^{\nu}$ and $\beta_{li}^{\nu}$.
Let the rates of all the irreversible reaction be equal to zero. This does not
mean that all the concentrations $a_{i}$ with $\delta_{i}>0$ achieve zero. A
bimolecular reaction $A+B\to C$ gives us a simple example: $w=ka_{A}a_{B}$ and
$w=0$ if either $a_{A}=0$ or $a_{B}=0$. On the plane with coordinates
$a_{A},a_{B}$ and with the positivity condition, $a_{A},a_{B}\geq 0$, the set
of zeros of $w$ is a union of two semi-axes, $\\{a_{A}=0,a_{B}\geq 0\\}$ and
$\\{a_{A}\geq 0,a_{B}=0\\}$. In more general situation, the set in the
activity space, where all the irreversible reactions have zero rates, has a
similar structure: it is the union of some faces of the positive orthant.
Let us describe the set of the steady states of the irreversible reactions.
Due to Proposition 2, if $\sum_{l}v_{l}\nu_{l}=0$ then all $v_{l}=0$. Let us
describe the set of zeros of all $v_{l}$ in the the positive orthant of
activities.
For every $l=1,\ldots,s$ the set of zeros of $v_{l}$ in $\mathbb{R}^{n}_{+}$
is given by the conditions: at least for one $i$ $\alpha_{li}^{\nu}\neq 0$ and
$a_{i}=0$. It is convenient to represent this condition as a disjunction. Let
$J_{l}=\\{i\,|\,\alpha_{li}^{\nu}\neq 0\\}$. Then the set of zeros of $v_{l}$
an a positive orthant of activities is presented by the formula $\bigvee_{i\in
J_{l}}(a_{i}=0)$. The set of zeros of all $v_{l}$ is represented by the
following conjunction form
$\wedge_{l=1}^{s}\left(\vee_{i\in J_{l}}(a_{i}=0)\right)\,.$ (16)
To transform it into the unions of subspaces we have to move to a disjunction
form and make some cancelations. First of all, we represent this formula as a
disjunction of conjunctions:
$\wedge_{l=1}^{s}\left(\vee_{i\in J_{l}}(a_{i}=0)\right)=\vee_{i_{1}\in
J_{1},\ldots,i_{s}\in
J_{s}}\left((a_{i_{1}}=0)\wedge\ldots\wedge(a_{i_{s}}=0)\right)\,.$ (17)
For a cortege of indexes $\\{i_{1},\ldots,i_{s}\\}$ the correspondent set of
their values may be smaller because some values $i_{l}$ may coincide. Let this
set of values be $S_{\\{i_{1},\ldots,i_{s}\\}}$. We can delete from (17) a
conjunction $(a_{i_{1}}=0)\wedge\ldots\wedge(a_{i_{s}}=0)$ if there exists a
cortege $\\{i^{\prime}_{1},\ldots,i^{\prime}_{s}\\}$ ($i^{\prime}_{l}\in
J_{l}$) with smaller set of values, $S_{\\{i_{1},\ldots,i_{s}\\}}\supseteq
S_{\\{i^{\prime}_{1},\ldots,i^{\prime}_{s}\\}}$. Let us check the corteges in
some order and delete a conjunction from (17) if there remain a term with
smaller (or the same) set of index values in the formula. We can also
substitute in (17) the corteges by their sets of values. The resulting
minimized formula may become shorter. Each elementary conjunction represents a
coordinate subspace and after cancelations each this subspace does not belong
to a union of other subspaces. The final form of formula (17) is
$\vee_{j}(\wedge_{i\in S_{j}}(a_{i}=0))\,,$ (18)
where $S_{j}$ are sets of indexes, $S_{j}\subset\\{1,\ldots,n\\}$ and for
every two different $S_{j}$, $S_{p}$ none of them includes another,
$S_{j}\nsubseteq S_{p}$. The elementary conjunction $\wedge_{i\in
S_{j}}(a_{i}=0)$ describes a subspace.
The steady states of the irreversible part of the reaction mechanism are given
by the intersection of the union of the coordinate subspaces (18) with
$\mathbb{R}^{n}_{+}$. For applications of this formula, it is important that
the equalities $a_{i}=0$, $c_{i}=0$ and $N_{i}=0$ are equivalent and the
positive orthants of the activities $a_{i}$, concentrations $c_{i}$ or amounts
$N_{i}$ represent the same sets of physical states. This is also true for the
faces of these orthants: $F_{J}$ for the activities, concentrations or amounts
correspond to the same sets of states. (The same state may corresponds to the
different points of these cones, but the totalities of the states are the
same.)
### 3.3 Sets of steady states of irreversible reactions invariant with
respect to reversible reactions
In this Sec. we study the possible limit behavior of systems which satisfy the
extended detailed balance conditions and include some irreversible reactions.
All the $\omega$-limit points of such systems are steady states of the
irreversible reactions due to Proposition 2 but not all these steady states
may be the $\omega$-limit points of the system. A simple formal example gives
us the couple of reaction: $A\rightleftharpoons B$, $A+B\to C$. Here, we have
one reversible and one irreversible reaction. The conditions of the extended
detailed balance hold (trivially): the linear span of the stoichiometric
vector of the reversible reaction, $(-1,1,0)$, does not include the
stoichiometric vector of the irreversible reaction, $(-1,-1,1)$. For the
description of the multiscale degeneration of equilibrium, we can take the
exponents $\delta_{A}=1,\delta_{B}=1,\delta_{C}=0$.
The steady states of the irreversible reaction are given in
$\mathbb{R}^{n}_{+}$ by the disjunction, $(c_{A}=0)\vee(c_{B}=0)$ but only the
points $(c_{A}=c_{B}=0)$ may be the limit points when $t\to\infty$. Indeed, if
$c_{A}=0$ and $c_{B}>0$ then
${\mathrm{d}}c_{A}/{\mathrm{d}}t=k_{1}^{-}c_{B}>0$. Due to Proposition 4 this
is not an $\omega$-limit point. Similarly, the points with $c_{A}>0$ and
$c_{B}=0$ are not the $\omega$-limit points.
Let us combine Propositions 2, 4 and Corollary 2 in the following statement.
###### Theorem 2.
Let the kinetic system satisfy the extended detailed balance conditions and
include some irreversible reactions. Then an $\omega$-limit point $x^{*}\in
riF_{J}$ exists if and only if $F_{J}$ consists of steady states of the
irreversible reactions and is invariant with respect to all reversible
reactions.
###### Proof.
If an $\omega$-limit point $x^{*}\in riF_{J}$ exists then it is a steady state
for all irreversible reactions (due to Propositions 2). Therefore, the face
$F_{J}$ consists of steady-states of the irreversible reactions (Proposition
4) and is invariant with respect to all reversible reactions (Proposition 4
and Corollary 2). To prove the reverse statement, let us assume that $F_{J}$
consists of steady states of the irreversible reactions and is invariant with
respect to all reversible reactions. The reversible reactions which do not act
on $c_{j}$ for $j\in J$ define a semi-dynamical system on $F_{J}$. The
positive conservation law $b$ defines an positively invariant polyhedron in
$F_{J}$. Dynamics in such a compact set always has $\omega$-limit points.∎
Let us find the faces $F_{J}$ that contain the $\omega$-limit points in their
relative interior $riF_{J}$. According to Theorem 2, these faces should
consist of the steady states of the irreversible reactions and should be
invariant with respect to all reversible reactions. Let us look for the
maximal faces with this property. For this purpose, we always minimize the
disjunctive forms by cancelations. We do not list the faces that contain the
$\omega$-limit points in their relative interior and are the proper subsets of
other faces with this property. All the $\omega$-limit points belong to the
union of these maximal faces.
Let us start from the minimized disjunctive form (18). Equation (18)
represents the set of the steady states of the irreversible part of the
reaction mechanism by a union of the coordinate subspaces $\wedge_{i\in
S_{j}}(c_{i}=0)$ in intersection with $\mathbb{R}^{n}_{+}$. It is the union of
the faces, $\cup_{j}F_{S_{j}}$. If a face $F_{J}$ consists of the steady
states of the irreversible reactions then $J\supseteq S_{j}$ for some $j$.
The following formula (19) is true on a face $F_{J}$ if it contains
$\omega$-limit points in the relative interior $riF_{J}$ (Theorem 2):
$(c_{i}=0)\Rightarrow\left[\left(\wedge_{r,\gamma_{ri}>0}\
\vee_{j,\alpha_{rj}>0}(c_{j}=0)\right)\wedge\left(\wedge_{r,\gamma_{ri}<0}\
\vee_{j,\beta_{rj}>0}(c_{j}=0)\right)\right]\,.$ (19)
Here, $c_{i}=0$ in $F_{J}$ may be read as $i\in J$. Following the previous
section, we use here the notations $\gamma_{ri}$, $\beta_{ri}$ and
$\beta_{ri}$ for the reversible reactions and reserve $\nu_{l}$,
$\alpha_{li}^{\nu}$ and $\beta_{li}^{\nu}$ for the irreversible reactions. The
set of $\gamma_{r}$ in this formula is the set of the stoichiometric vectors
of the reversible reactions.
The required faces $F_{J}$ may be constructed in an iterative procedure. First
of all, let us introduce an operation that transforms a set of indexes
$S\subset\\{1,2,\ldots,n\\}$ in a family of sets,
$\mathfrak{S}(S)=\\{S^{\prime}_{1},\ldots,S^{\prime}_{l}\\}$. Let us take
formula (19) and find the set where it is valid for all $i\in S$. This set is
described by the following formula:
$\wedge_{i\in S}\left[(c_{i}=0)\wedge\left(\wedge_{r,\gamma_{ri}>0}\
\vee_{j,\alpha_{rj}>0}(c_{j}=0)\right)\wedge\left(\wedge_{r,\gamma_{ri}<0}\
\vee_{j,\beta_{rj}>0}(c_{j}=0)\right)\right]\,.$ (20)
Let us produce a disjunctive form of this formula and minimize it by
cancelations as it is described in Sec. 3.2. The result is
$\vee_{j=1,\ldots,k}\left(\wedge_{i\in S_{j}^{\prime}}(c_{i}=0)\right)\,.$
(21)
Because of cancelations, the sets $S^{\prime}_{j}$ do not include one another.
They give the result,
$\mathfrak{S}(S)=\\{S^{\prime}_{1},\ldots,S^{\prime}_{l}\\}$. Each
$S^{\prime}_{j}\in\mathfrak{S}(S)$ is a superset of $S$, $S^{\prime}\supseteq
S$.
Let us extend the operation $\mathfrak{S}$ on the sets of sets
$\mathbf{S}=\\{S_{1},\ldots,S_{p}\\}$ with the property: $S_{i}\not\subset
S_{j}$ for $i\neq j$. Let us apply $\mathfrak{S}$ to all $S_{i}$ and take the
union of the results:
$\mathfrak{S}_{0}(\mathbf{S})=\cup_{i}\mathfrak{S}(S_{i})$. Some sets from
this family may include other sets from it. Let us organize cancelations: if
$S^{\prime},S^{\prime\prime}\in\mathfrak{S}_{0}(\mathbf{S})$ and
$S^{\prime}\subset S^{\prime\prime}$ then retain the smallest set,
$S^{\prime}$, and delete the largest one. We do the cancelations until it is
possible. Let us call the final result $\mathfrak{S}(\mathbf{S})$. It does not
depend on the order of these operations.
Let us start from any family $\mathbf{S}$ and iterate the operation
$\mathfrak{S}$. Then, after finite number of iterations, the sequence
$\mathfrak{S}^{d}(\mathbf{S})$ stabilizes:
$\mathfrak{S}^{d}(\mathbf{S})=\mathfrak{S}^{d+1}(\mathbf{S})=\ldots$ because
for any set $S$ all sets from $\mathfrak{S}(S)$ include $S$.
The problems of propositional logic that arise in this and the previous
section seem very similar to elementary logical puzzles [3]. In the solution
we just use the logical distribution laws (distribution of conjunction over
disjunction and distribution of disjunction over conjunction), commutativity
of disjunction and conjunction, and elementary cancelation rules like
$(A\wedge A)\Leftrightarrow A$, $(A\vee A)\Leftrightarrow A$, $[A\wedge(A\vee
B)]\Leftrightarrow A$, and $[A\vee(A\wedge B)]\Leftrightarrow A$.
Now, we are in position to describe the construction of all $F_{J}$ that have
the $\omega$-limit points on their relative interior and are the maximal faces
with this property.
1. 1.
Let us follow Sec. 3.2 and construct the minimized disjunctive form (18) for
the description of the steady states of the irreversible reactions.
2. 2.
Let us calculate the families of sets $\mathfrak{S}^{d}(\\{S_{j}\\})$ for the
family of sets $\\{S_{j}\\}$ from (18) and $d=1,2,\ldots$, until
stabilization.
3. 3.
Let
$\mathfrak{S}^{d}(\\{S_{j}\\})=\mathfrak{S}^{d+1}(\\{S_{j}\\})=\\{J_{1},J_{2},\ldots
J_{p}\\}$. Then the family of the faces $F_{J_{i}}$ ($i=1,2,\ldots,p$) gives
the answer: the $\omega$-limit points are situated in $riF_{J_{i}}$ and for
each $i$ there are $\omega$-limit points in $riF_{J_{i}}$.
### 3.4 Simple examples
In this Sec., we present two simple and formal examples of the calculations
described in the previous sections.
1\. $A_{1}+A_{2}\rightleftharpoons A_{3}+A_{4}$, $\gamma=(-1,-1,1,1,0)$;
$A_{1}+A_{2}\to A_{5}$, $\nu=(-1,-1,0,0,1)$. The extended principle of
detailed balance holds: the convex hull of the stoichiometric vectors of the
irreversible reactions consists of one vector $\gamma_{2}$ and it is linearly
independent of $\gamma_{1}$. The input vector $\alpha$ for the irreversible
reaction $A_{1}+A_{2}\to A_{5}$ is $(-1,-1,0,0,0)$. The set $J=J_{l}$ from the
conjunction form (16) is defined by the non-zero coordinates of this
$\alpha^{\nu}$: $J=\\{1,2\\}$. The conjunction form in this simple case (one
irreversible reaction) loses its first conjunction operation and is just
$(c_{1}=0)\vee(c_{2}=0)$. It is, at the same time, the minimized disjunction
form (18) and does not require additional transformations. This formula
describes the steady states of the irreversible reaction in the positive
orthant $\mathbb{R}^{n}_{+}$. For this disjunction form, The family of sets
$\mathbf{S}=\\{S_{j}\\}$ consists of two sets, $S_{1}=\\{1\\}$ and
$S_{2}=\\{2\\}$.
Let us calculate $\mathfrak{S}({S_{1,2}})$. For both cases, $i=1,2$ there are
no reversible reactions with $\gamma_{ri}=0$. Therefore, one expression in
round parentheses vanishes in (20). For $S=\\{1\\}$ this formula gives
$(c_{1}=0)\wedge((c_{3}=0)\vee(c_{4}=0))$
and for $S=\\{2\\}$ it gives
$(c_{2}=0)\wedge((c_{3}=0)\vee(c_{4}=0))\,.$
The elementary transformations give the disjunctive forms:
$[(c_{1}=0)\wedge((c_{3}=0)\vee(c_{4}=0))]\Leftrightarrow[((c_{1}=0)\wedge(c_{3}=0))\vee((c_{1}=0)\wedge(c_{4}=0))]\,,$
$[(c_{2}=0)\wedge((c_{3}=0)\vee(c_{4}=0))]\Leftrightarrow[((c_{2}=0)\wedge(c_{3}=0))\vee((c_{2}=0)\wedge(c_{4}=0))]\,.$
Therefore, $\mathfrak{S}(S_{1})=\\{\\{1,3\\},\\{1,4\\}\\}$,
$\mathfrak{S}(S_{2})=\\{\\{2,3\\},\\{2,4\\}\\}$ and
$\mathfrak{S}(\\{S_{1},S_{2}\\})=\\{\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\}\\}\,.$
No cancelations are needed. The iterations of $\mathfrak{S}$ do not produce
new sets from $\\{\\{1,3\\},\\{1,4\\},\\{2,3\\},\\{2,4\\}\\}$. Indeed, if
$c_{1}=c_{3}=0$, or $c_{1}=c_{4}=0$, or $c_{2}=c_{3}=0$, or $c_{2}=c_{4}=0$
then all the reaction rates are zero. More formally, for example for
$\mathfrak{S}(\\{1,3\\})$ formula (20) gives
$[(c_{1}=0)\wedge((c_{3}=0)\vee(c_{4}=0))]\wedge[(c_{3}=0)\wedge((c_{1}=0)\vee(c_{2}=0))]\,.$
This formula is equivalent to $(c_{1}=0)\wedge(c_{3}=0)$. Therefore,
$\mathfrak{S}(\\{1,3\\})=\\{1,3\\}$. The same result is true for $\\{1,4\\}$,
$\\{2,3\\}$, and $\\{2,4\\}$.
All the $\omega$-limit points (steady states) belong to the faces
$F_{\\{1,3\\}}=\\{c\,|,c_{1}=c_{3}=0\\}$,
$F_{\\{1,4\\}}=\\{c\,|,c_{1}=c_{4}=0\\}$,
$F_{\\{2,3\\}}=\\{c\,|,c_{2}=c_{3}=0\\}$, or
$F_{\\{2,4\\}}=\\{c\,|,c_{2}=c_{4}=0\\}$. The position of the $\omega$-limit
point for a solution $N(t)$ depends on the initial state. More specifically,
this system of reactions has three independent linear conservation laws:
$b_{1}=N_{1}+N_{2}+N_{3}+N_{4}+2N_{5}$, $b_{2}=N_{1}-N_{2}$ and
$b_{3}=N_{3}-N_{4}$. For given values of these $b_{1,2,3}$ vector $N$ belongs
to the $2D$ plane in $\mathbb{R}^{5}$. The intersection of this plane with the
selected faces depends on the signs of $b_{2,3}$:
* 1.
If $b_{2}<0$, $b_{3}<0$ then it intersects $F_{\\{1,3\\}}$ only, at one point
$N=(0,-b_{2},0,-b_{3},b_{1}+b_{2}+b_{3})$ ($N_{5}$ should be non-negative,
$b_{1}+b_{2}+b_{3}\geq 0$) .
* 2.
If $b_{2}=0$, $b_{3}<0$ then it intersects both $F_{\\{1,3\\}}$ and
$F_{\\{2,3\\}}$ at one point $N=(0,0,0,-b_{3},b_{1}+b_{3})$ ($N_{5}$ should be
non-negative, $b_{1}+b_{3}\geq 0$).
* 3.
If $b_{2}<0$, $b_{3}=0$ then it intersects both $F_{\\{1,3\\}}$ and
$F_{\\{1,4\\}}$ at one point $N=(0,-b_{2},0,0,b_{1}+b_{2})$ ($N_{5}$ should be
non-negative, $b_{1}+b_{2}\geq 0$).
* 4.
If $b_{2}>0$, $b_{3}<0$ then it intersects $F_{\\{2,3\\}}$ only, at one point
$N=(b_{2},0,0,-b_{3},b_{1}+b_{2}+b_{3})$ ($N_{5}$ should be non-negative,
$b_{1}+b_{2}+b_{3}\geq 0$).
* 5.
If $b_{2}>0$, $b_{3}=0$ then it intersects $F_{\\{2,3\\}}$ and $F_{\\{2,4\\}}$
at the point $N=(b_{2},0,0,0,b_{1}+b_{2})$ ($N_{5}$ is non-negative because
$b_{1}+b_{2}+b_{3}\geq 0$).
* 6.
If $b_{2}<0$, $b_{3}>0$ then it intersects $F_{\\{1,4\\}}$ only, at one point
$N=(0,-b_{2},b_{3},0,b_{1}+b_{2}+b_{3})$ ($N_{5}$ should be non-negative,
$b_{1}+b_{2}+b_{3}\geq 0$).
* 7.
If $b_{2}=0$, $b_{3}>0$ then it intersects $F_{\\{1,4\\}}$ and $F_{\\{2,4\\}}$
at one point $N=(0,0,b_{3},0,b_{1}+b_{3})$ ($N_{5}$ is non-negative because
$b_{1}+b_{3}\geq 0$).
* 8.
If $b_{2}>0$, $b_{3}>0$ then it intersects $F_{\\{2,4\\}}$ only, at one point
$N=(b_{2},0,b_{3},0,b_{1}+b_{2}+b_{3})$ ($N_{5}$ is non-negative because
$b_{1}+b_{2}+b_{3}\geq 0$).
As we can see, the system has exactly one $\omega$-limit point for any
admissible combination of the values of the conservation laws. These points
are the listed points of intersection.
For the second simple example, let us change the direction of the irreversible
reaction.
2\. $A_{1}+A_{2}\rightleftharpoons A_{3}+A_{4}$, $\gamma_{1}=(-1,-1,1,1,0)$,
$A_{5}\to A_{1}+A_{2}$, $\nu=(1,1,0,0,-1)$. The extended principle of detailed
balance holds. The steady-states of the irreversible reactions is given by one
equation, $c_{5}=0$. Formula (20) gives for $\mathfrak{S}(\\{5\\})$ just
$(c_{5}=0)$. The face $F_{\\{5\\}}$ includes $\omega$-limit points in
$riF_{\\{5\\}}$. Dynamics on this face is defined by the fully reversible
reaction system and tends to the equilibrium of the reaction
$A_{1}+A_{2}\rightleftharpoons A_{3}+A_{4}$ under the given conservation laws.
On this face, there exist the border equilibria, where $c_{1}=c_{3}=0$, or
$c_{1}=c_{4}=0$, or $c_{2}=c_{3}=0$, or $c_{2}=c_{4}=0$ but they are not
attracting the positive solutions.
## 4 Example: H2+O2 system
For the case study, we selected the H2+O2 system. This is one of the main
model systems of gas kinetics. The hydrogen burning gives us an example of the
medium complexity with 8 components ($A_{1}=$H2, $A_{2}=$O2, $A_{3}=$OH,
$A_{4}=$H2O, $A_{5}=$H, $A_{6}=$O, $A_{7}=$HO2, and $A_{8}=$H2O2) and 2 atomic
balances (H and O). For the example, we selected the reaction mechanism from
[21]. The literature about hydrogen burning mechanisms is huge. For recent
discussion we refer to [16, 18]. We do not aim to compare the different
schemes of this reaction but use this reaction mechanism as an example and a
benchmark.
Table 1: H2 burning mechanism [21] No | Reaction | Stoichiometric vector
---|---|---
1 | H2 \+ O2 $\rightleftharpoons$ 2OH | (-1,-1,2,0,0,0,0,0)
2 | H2 \+ OH $\rightleftharpoons$ H2O + H | (-1,0,-1,1,1,0,0,0)
3 | OH + O $\rightleftharpoons$ O2 \+ H | (0,1,-1,0,1,-1,0,0)
4 | H2 \+ O $\rightleftharpoons$ OH + H | (-1,0,1,0,1,-1,0,0)
5 | O2 \+ H +M $\rightleftharpoons$ HO2 +M | (0,-1,0,0,-1,0,1,0)
6 | OH + HO2 $\rightleftharpoons$ O2 \+ H2O | (0,1,-1,1,0,0,-1,0)
7 | H + HO2 $\rightleftharpoons$ 2OH | (0,0,2,0,-1,0,-1,0)
8 | O + HO2 $\rightleftharpoons$ O2 \+ OH | (0,1,1,0,0,-1,-1,0)
9 | 2OH $\rightleftharpoons$ H2O + O | (0,0,-2,1,0,1,0,0)
10 | 2H + M $\rightleftharpoons$ H2 \+ M | (1,0,0,0,-2,0,0,0)
11 | 2H + H2 $\rightleftharpoons$ H2 \+ H2 | (1,0,0,0,-2,0,0,0)
12 | 2H + H2O $\rightleftharpoons$ H2 \+ H2O | (1,0,0,0,-2,0,0,0)
13 | OH + H + M $\rightleftharpoons$ H2O + M | (0,0,-1,1,-1,0,0,0)
14 | H + O + M $\rightleftharpoons$ OH + M | (0,0,1,0,-1,-1,0,0)
15 | 2O + M $\rightleftharpoons$ O2 \+ M | (0,1,0,0,0,-2,0,0)
16 | H + HO2 $\rightleftharpoons$ H2 \+ O2 | (1,1,0,0,-1,0,-1,0)
17 | 2HO2 $\rightleftharpoons$ O2 \+ H2O2 | (0,1,0,0,0,0,-2,1)
18 | H2O2 \+ M $\rightleftharpoons$ 2OH + M | (0,0,2,0,0,0,0,-1)
19 | H + H2O2 $\rightleftharpoons$ H2 \+ HO2 | (1,0,0,0,-1,0,1,-1)
20 | OH + H2O2 $\rightleftharpoons$ H2O + HO2 | (0,0,-1,1,0,0,1,-1)
A special symbol “M” is used for the “third body”. It may be any molecule. The
third body provides the energy balance. Efficiency of different molecules in
this process is different, therefore, the “concentration” of the third body is
a weighted sum of the concentrations of the components with positive weights.
The third body does not affect the equilibrium constants and does not change
the zeros of the forward and reverse reaction rates but modifies the non-zero
values of reaction rates. Therefore, for our analysis we can omit these terms.
The elementary reactions 10, 11 and 12 are glued in one,
2H$\rightleftharpoons$H2, after cancelation of the third bodies, and we
analyze the mechanism of 18 reaction.
Under various conditions, some of the reactions are (almost) irreversible and
some of them should be considered as reversible. For example, let us consider
the H2+O2 system at or near the atmospheric pressure and in the temperature
interval 800–1200K. The reactions 1, 2, 4, 18, 19, and 20 are supposed to be
reversible (on the base of the reaction rate constants presented in [21]). The
first question is: if these reactions are reversible then which reactions may
be irreversible?
Due to the general criterion, the convex hull of the stoichiometric vectors of
the irreversible reactions has empty intersection with the linear span of the
stoichiometric vectors of the reversible reactions. Therefore, if the
stoichiometric vector of a reaction belongs to the linear span of the
stoichiometric vectors of the reversible reactions, then this reaction is
reversible. Simple linear algebra gives that
$\gamma_{3,5,9}\in{\rm
span}\\{\gamma_{1},\gamma_{2},\gamma_{4},\gamma_{18},\gamma_{19},\gamma_{20}\\}\,.$
In particular, $\gamma_{3}=-\gamma_{1}+\gamma_{4}$,
$\gamma_{5}=\gamma_{1}-\gamma_{18}+\gamma_{19}$,
$\gamma_{9}=\gamma_{2}-\gamma_{4}$. So, the list of the reversible reactions
should include the reactions 1, 2, 3, 4, 5, 9, 18, 19, and 20. The reactions
6, 7, 8, 10, 11, 12, 13, 14, 15, and 17 may be irreversible. Formally, there
are $2^{8}=256$ possible combinations of the directions of these 8 reactions
(the reactions 10, 11 and 12 have the same stoichiometric vector and, in this
sense, should be considered as one reaction). The general criterion and simple
linear algebra give that there are only two admissible combinations of the
directions of irreversible reactions: either for all of them $k^{-}_{r}=0$ or
for all of them $k^{+}_{r}=0$. Here, the forward and reverse reactions and the
notations $k^{\pm}_{r}$ are selected according to the Table 1. We can
immediately notice that the inverse direction of all reactions is very far
from the reality under the given conditions, for example, it includes the
irreversible dissociation H${}_{2}\to 2$H.
Let us demonstrate in detail, how the general criterion produces this
reduction from the 256 possible combinations of directions of irreversible
reactions to just 2 admissible combinations. We assume that the initial set of
reactions is spit in two: reversible reactions with numbers $r\in J_{0}$ and
irreversible reactions with $r\in J_{1}$,
rank$\\{\gamma_{1},\gamma_{2},\ldots,\gamma_{\ell}\\}=d$,
rank$\\{\gamma_{r}\,|\,r\in J_{0}\\}=d_{0}$. The rank of all vectors
$\gamma_{r}$, $d$, must exceed the rank of the stoichiometric vectors of the
reversible reactions, $d>d_{0}$, because if $d=d_{0}$ then all the reactions
must be reversible and the problem becomes trivial.
According to [12], we have to perform the following operations with the set of
stoichiometric vectors $\gamma_{r}$ (Fig. 1):
1. 1.
Eliminate several coordinates from all $\gamma_{r}$ using linear conservation
laws. This is transfer to the internal coordinates in
span$\\{\gamma_{r}\,|\,r=1,\ldots,{\ell}\\}$;
2. 2.
Eliminate coordinates from all $\gamma_{r}$ ($r\in J_{1}$) using the
stoichiometric vectors of the reversible reactions and the Gauss–Jordan
elimination procedure. This is the map to the quotient space
span$\\{\gamma_{j}\,|\,j=1,\ldots,{\ell}\\}/\mbox{span}\\{\gamma_{j}\,|\,j\in
J_{0}\\}$. Me denote the result as $\overline{\gamma}_{r}$;
3. 3.
Use the linear programming technique and analyze for which combinations of the
signs, the convex hull conv$\\{\pm\overline{\gamma}_{r}\,|\,r\in J_{1}\\}$
does not include 0.
Table 2: Elimination of coordinates of stoichiometric vectors for H2 burning mechanism. The reversible reactions are collected in the upper part of the Table. The reaction in the lower part of the table are irreversible. The group of equivalent reactions 10, 11, 12 is presented by one of them. In the second column, the first two coordinates (which correspond to H2 and O2) are excluded using the atomic balance. In the following columns the results of the coordinates elimination are presented. For each step, the pivot for elimination is underlined and highlighted in bold in the previous column. The eliminated coordinates at each step are named at the top of each column. Their zero values are omitted. No | H2, O2 | OH | H2O2 | H2O | H | O
---|---|---|---|---|---|---
1 | ($\underline{\mathbf{2}}$,0,0,0,0,0) | (0,0,0,0,0) | (0,0,0,0) | (0,0,0) | (0,0) | (0)
2 | (-1,1,1,0,0,0) | (1,1,0,0,0) | ($\underline{\mathbf{1}}$,1,0,0) | (0,0,0) | (0,0) | (0)
3 | (-1,0,1,-1,0,0) | (0,1,-1,0,0) | (0,1,-1,0) | ($\underline{\mathbf{1}}$,-1,0) | (0,0) | (0)
4 | (1,0,1,-1,0,0) | (0,1,-1,0,0) | (0,1,-1,0) | (1,-1,0) | (0,0) | (0)
5 | (0,0,-1,0,1,0) | (0,-1,0,1,0) | (0,-1,0,1) | (-1,0,1) | (-$\underline{\mathbf{1}}$,1) | (0)
9 | (-2,1,0,1,0,0) | (1,0,1,0,0) | (1,0,1,0) | (-1,1,0) | (0,0) | (0)
18 | (2,0,0,0,0,-1) | (0,0,0,0,-$\underline{\mathbf{1}}$) | (0,0,0,0) | (0,0,0) | (0,0) | (0)
19 | (0,0,-1,0,1,-1) | (0,-1,0,1,-1) | (0,-1,0,1) | (-1,0,1) | (-1,1) | (0)
20 | (-1,1,0,0,1,-1) | (1,0,0,1,-1) | (1,0,0,1) | (-1,0,1) | (-1,1) | (0)
6 | (-1,1,0,0,-1,0) | (1,0,0,-1,0) | (1,0,0,-1) | (-1,0,-1) | (-1,-1) | (-2)
7 | (2,0,-1,0,-1,0) | (0,-1,0,-1,0) | (0,-1,0,-1) | (-1,0,-1) | (-1,-1) | (-2)
8 | (1,0,0,-1,-1,0) | (0,0,-1,-1,0) | (0,0,-1,-1) | (0,-1,-1) | (-1,-1) | (-2)
10 | (0,0,-2,0,0,0) | (0,-2,0,0,0) | (0,-2,0,0) | (-2,0,0) | (-2,0) | (-2)
13 | (-1,1,-1,0,0,0) | (1,-1,0,0,0) | (1,-1,0,0) | (-2,0,0) | (-2,0) | (-2)
14 | (1,0,-1,-1,0,0) | (0,-1,-1,0,0) | (0,-1,-1,0) | (-1,-1,0) | (-2,0) | (-2)
15 | (0,0,0,-2,0,0) | (0,0,-2,0,0) | (0,0,-2,0) | (0,-2,0) | (-2,0) | (-2)
16 | (0,0,-1,0,-1,0) | (0,-1,0,-1,0) | (0,-1,0,-1) | (-1,0,-1) | (-1,-1) | (-2)
17 | (0,0,0,0,-2,1) | (0,0,0,-2,1) | (0,0,0,-2) | (0,0,-2) | (0,-2) | (-2)
In the Table 2 we present the results of the step-by-step elimination. First,
the atomic balances give us for every possible stoichiometric vector
$\eta=(\eta_{1},\ldots,\eta_{8})$ two identities:
1. 1.
$2\eta_{1}+\eta_{3}+2\eta_{4}+\eta_{5}+\eta_{7}+2\eta_{8}=0$ or
$\eta_{1}=-\frac{1}{2}(\eta_{3}+2\eta_{4}+\eta_{5}+\eta_{7}+2\eta_{8})$;
2. 2.
$2\eta_{2}+\eta_{3}+\eta_{4}+\eta_{6}+2\eta_{7}+2\eta_{8}=0$ or
$\eta_{2}=-\frac{1}{2}(\eta_{3}+\eta_{4}+\eta_{6}+2\eta_{7}+2\eta_{8})$.
Let us recall that the order of the coordinates $(\eta_{1},\ldots,\eta_{8})$
corresponds to the following order of the components, (H2, O2, OH, H2O, H, O,
HO2, H2O2). Due to these identities, a stoichiometric vector $\eta$ for this
mixture is completely defined by six coordinates $(\eta_{3},\ldots,\eta_{8})$.
In the second column of the Table 2 these 6D vectors are given for all the
reactions from the H2 burning mechanism (the Table 1).
In five columns No. 3-7, the results of the coordinate eliminations are
presented (and the zero-valued eliminated coordinates are omitted). Each
elimination step may be represented as a projection:
$x\mapsto x-x_{i}\frac{1}{\eta_{i}}\eta\,,$
where $\eta_{i}$ is a pivot (highlighted in bold in the column preceding the
result of elimination), and $\eta$ is the vector that includes the pivot (as
the $i$th coordinate). The projection operator is applied to every vector of
the previous column. At the end (the last column), all the stoichiometric
vectors of the reversible reaction are transformed into zero, and the
stoichiometric vectors of the irreversible reactions with the given direction
(from the left to the right) are transformed into the same vector $(-2)$. If
we restore all the zeros, then the corresponding 6D vector is
$(0,0,0,0,-2,0)$. We have to use the atomic balances to return to the 8D
vectors. The coordinate $x_{7}$ corresponds to HO2, $x_{1}$ corresponds to H2,
and $x_{2}$ corresponds to O2, hence, $2x_{1}-2=0$ and $2x_{2}-4=0$. The
restored 8D vector is $(1,2,0,0,0,0,-2,0)$.
A convex combination of several copies of one vector cannot give zero.
Therefore, the structural condition of the extended principle of detailed
balance holds. It holds also for the inverse direction of all the irreversible
reactions. All other distributions of directions can produce zero in the
convex hull and are inadmissible. So, we have the following list of
irreversible reactions that satisfies the extended principle of detailed
balance for given reversible reactions. (We will not discuss the second list
of reverse irreversible reactions because it has not much sense for given
conditions.)
6 | OH + HO2 $\to$ O2 \+ H2O
---|---
7 | H + HO2 $\to$ 2OH
8 | O + HO2 $\to$ O2 \+ OH
10 | 2H $\to$ H2
13 | OH + H $\to$ H2O
14 | H + O $\to$ OH
15 | 2O $\to$ O2
16 | H + HO2 $\to$ H2 \+ O2
17 | 2HO2 $\to$ O2 \+ H2O2.
We assume that all the reaction rate constants for the selected directions are
strictly positive. The rate of all these reaction vanish if and only if
concentration of H, O and HO2 are equal to zero, $c_{5,6,7}=0$. Indeed,
$c_{5}=0$ if and only if $w_{10}=0$, $c_{6}=0$ if and only if $w_{15}=0$,
$a_{7}=0$ if and only if $w_{17}=0$. On the other hand, all other reaction
rates from this list are zeros if $c_{5,6,7}=0$.
Let us reproduce this reasoning using formulas from Sec. 3.2. For the $l$th
irreversible reaction, $J_{l}$ is the set of indexes $i$ for which
$\alpha_{li}\neq 0$. Let us keep for the irreversible reactions their numbers
(6, 7, 8, 10, 13, 14, 15, 16, 17). For them, $J_{6}=\\{3,7\\}$,
$J_{7}=\\{5,7\\}$, $J_{8}=\\{6,7\\}$, $J_{10}=\\{5\\}$, $J_{13}=\\{3,5\\}$,
$J_{14}=\\{5,6\\}$, $J_{15}=\\{6\\}$, $J_{16}=\\{5,7\\}$, $J_{17}=\\{7\\}$.
Formula (18) gives for the steady states of the irreversible reactions:
$\begin{split}&((c_{3}=0)\vee(c_{7}=0))\wedge((c_{5}=0)\vee(c_{7}=0))\wedge((c_{6}=0)\vee(c_{7}=0))\wedge(c_{5}=0)\\\
\wedge&((c_{3}=0)\vee(c_{5}=0))\wedge((c_{5}=0)\vee(c_{6}=0))\wedge(c_{6}=0)\wedge((c_{5}=0)\vee(c_{7}=0))\wedge(c_{7}=0).\end{split}$
It is equivalent to
$(c_{5}=0)\wedge(c_{6}=0)\wedge(c_{7}=0)\,.$
Of course, the result is the same, the face $F_{\\{5,6,7\\}}$ ($c_{5,6,7}=0$,
$c_{i}\geq 0$) is the set of the steady states of all irreversible reaction.
Let us look now on the list of reversible reactions:
1 | H2 \+ O2 $\rightleftharpoons$ 2OH
---|---
2 | H2 \+ OH $\rightleftharpoons$ H2O + H
3 | OH + O $\rightleftharpoons$ O2 \+ H
4 | H2 \+ O $\rightleftharpoons$ OH + H
5 | O2 \+ H $\rightleftharpoons$ HO2
9 | 2OH $\rightleftharpoons$ H2O + O
18 | H2O2 $\rightleftharpoons$ 2OH
19 | H + H2O2 $\rightleftharpoons$ H2 \+ HO2
20 | OH + H2O2 $\rightleftharpoons$ H2O + HO2
If the concentration OH ($c_{3}$) is positive then the component O is produced
in the reaction 9. If the concentrations of H2 ($c_{1}$) and OH ($c_{3}$) both
are positive then the component H is produced in reaction 2. If the
concentrations of H2O2 ($c_{8}$) and OH ($c_{3}$) both are positive then the
component HO2 is produced in reaction 2. Due to the reversible reaction 18 any
of two components H2O2 and OH produces the other component. Moreover, the
first reaction produces OH from H2 \+ O2. This production stops if and only if
either concentration of H2 is zero ($c_{1}=0$) or concentration of O2 is zero
($c_{2}=0$).
This means that the set of zeros of the irreversible reactions, $c_{5,6,7}=0$
($c\geq 0$), is not invariant with respect to the kinetics of the reversible
reactions. This means that from an initial conditions on this set the kinetic
trajectory will leave it unless, in addition, $c_{3}=c_{8}=0$ and either
$c_{1}=0$ or $c_{2}=0$.
The reactions of all irreversible reactions should tend to zero due to
Proposition 2. Therefore, the kinetic trajectory should approach the union of
two planes, $c_{1,3,5,6,7,8}=0$ and $c_{2,3,5,6,7,8}=0$ (under condition
$c\geq 0$). These planes are two-dimensional and the position of the state
there is completely defined by the atomic balances.
If the concentration vector belongs to the first plane, then all the atoms are
collected in O2 and H2O. It is possible if and only if $b_{\rm
O}\geq\frac{1}{2}b_{\rm H}$. In this case, $c_{4}=\frac{1}{2}b_{\rm H}$ and
$c_{2}=\frac{1}{2}(b_{\rm O}-\frac{1}{2}b_{\rm H})$.
If the concentration vector belongs to the second plane, then all the atoms
are collected in H2 and H2O. It is possible if and only if $b_{\rm
O}\leq\frac{1}{2}b_{\rm H}$. In this case, $c_{4}=b_{\rm O}$ and
$c_{1}=\frac{1}{2}(b_{\rm H}-2b_{\rm O})$.
Let us reproduce this reasoning formally using the general formalism of Sec.
3.3. Formula 20 gives for $\mathfrak{S}(\\{5,6,7\\})$
$\begin{split}(c_{5}=0)\wedge&\left(\wedge_{r,\gamma_{r5}>0}\vee_{j,\alpha_{rj}>0}(c_{j}=0)\right)\wedge\left(\wedge_{r,\gamma_{r5}<0}\
\vee_{j,\beta_{rj}>0}(c_{j}=0)\right)\\\
\wedge(c_{6}=0)\wedge&\left(\wedge_{r,\gamma_{r6}>0}\vee_{j,\alpha_{rj}>0}(c_{j}=0)\right)\wedge\left(\wedge_{r,\gamma_{r6}<0}\
\vee_{j,\beta_{rj}>0}(c_{j}=0)\right)\\\
\wedge(c_{7}=0)\wedge&\left(\wedge_{r,\gamma_{r7}>0}\vee_{j,\alpha_{rj}>0}(c_{j}=0)\right)\wedge\left(\wedge_{r,\gamma_{r7}<0}\
\vee_{j,\beta_{rj}>0}(c_{j}=0)\right)\,.\end{split}$ (22)
Vectors $\gamma_{r}$ that participate in this formula are the stoichiometric
vectors of reversible reactions ($r=1,2,3,4,5,9,18,19,20$). From the Table 1
we find that $\gamma_{r5}>0$ for $r=2,3,4$, $\gamma_{r5}<0$ for $r=5,19$,
$\gamma_{r6}>0$ for $r=9$, $\gamma_{r6}<0$ for $r=3,4$, $\gamma_{r7}>0$ for
$r=5,19,20$, and $\gamma_{r7}\not<0$ for all $r$. Formula (22) transforms into
$\begin{split}&(c_{5}=0)\wedge((c_{1}=0)\\!\vee\\!(c_{3}=0))\wedge((c_{3}=0)\\!\vee\\!(c_{6}=0))\wedge((c_{1}=0)\\!\vee\\!(c_{6}=0))\wedge(c_{7}=0)\\\
\wedge&((c_{1}=0)\\!\vee\\!(c_{7}=0))\wedge(c_{6}=0)\wedge(c_{3}=0)\wedge((c_{2}=0)\\!\vee\\!(c_{5}=0))\wedge((c_{3}=0)\\!\vee\\!(c_{5}=0))\\\
\wedge&(c_{7}=0)\wedge((c_{2}=0)\\!\vee\\!(c_{5}=0))\wedge((c_{5}=0)\\!\vee\\!(c_{8}=0))\wedge((c_{3}=0)\\!\vee\\!(c_{8}=0))\,.\end{split}$
After simple transformations it becomes
$(c_{3}=0)\wedge(c_{5}=0)\wedge(c_{6}=0)\wedge(c_{7}=0)\,.$ (23)
Therefore, $\mathfrak{S}(\\{5,6,7\\})=\\{3,5,6,7\\}$. To iterate, we have to
compute $\mathfrak{S}(\\{3,5,6,7\\})$. For this calculation, we have to add
one more line to formula (22), namely,
$\wedge(c_{3}=0)\wedge\left(\wedge_{r,\gamma_{r3}>0}\vee_{j,\alpha_{rj}>0}(c_{j}=0)\right)\wedge\left(\wedge_{r,\gamma_{r3}<0}\vee_{j,\beta_{rj}>0}(c_{j}=0)\right)\,.$
Let us take into account that $\gamma_{r3}>0$ for $r=1,4,18$ and
$\gamma_{r3}<0$ for $r=2,3,9,20$, and rewrite this formula in the more
explicit form
$\begin{split}&(c_{3}=0)\wedge((c_{1}=0)\vee(c_{2}=0))\wedge((c_{1}=0)\vee(c_{6}=0))\wedge(c_{8}=0)\\\
\wedge&((c_{4}=0)\vee(c_{5}=0))\wedge((c_{2}=0)\vee(c_{5}=0))\wedge((c_{4}=0)\vee(c_{6}=0))\wedge(c_{7}=0)\,.\end{split}$
Let us take the conjunction of this formula with (22) taken in the simplified
equivalent form (23) and transform the result to the disjunctive form. We get
$\begin{split}[&(c_{3}=0)\wedge(c_{5}=0)\wedge(c_{6}=0)\wedge(c_{7}=0)\wedge(c_{8}=0)\wedge(c_{1}=0)]\\\
\vee[&(c_{3}=0)\wedge(c_{5}=0)\wedge(c_{6}=0)\wedge(c_{7}=0)\wedge(c_{8}=0)\wedge(c_{2}=0))]\,.\end{split}$
(24)
This means that
$\mathfrak{S}^{2}(\\{5,6,7\\})=\mathfrak{S}(\\{3,5,6,7\\})=\\{\\{1,3,5,6,7,8\\},\\{2,3,5,6,7,8\\}\\}$.
The further calculations show that the next iteration does not change the
result. Therefore, all the $\omega$-limit points belong to two faces,
$F_{\\{1,3,5,6,7,8\\}}$ and $F_{\\{2,3,5,6,7,8\\}}$. The result is the same as
for the previous discussion. The detailed formalization becomes crucial for
more complex systems and for software development.
Let us find the vector of exponents $\delta=(\delta_{i})$ ($i=1,\ldots,8$)
from the Table 2. After all the eliminations, the corresponding linear
functional $\hat{\delta}$ is just a value of the 7th coordinate:
$\hat{\delta}(x)=x_{7}$. Its values are negative ($-2$) for all irreversible
reactions and zero for all reversible reactions (see the last column of the
Table 2). The conditions $(\delta,\gamma)=0$ for the reversible reactions and
$(\delta,\gamma)<0$ for all irreversible reactions do not define the unique
vector: if $\delta$ satisfies these conditions then its linear combination
with the vectors of atomic balances also satisfy them. Such a combination is a
vector
$\lambda\delta+\lambda_{\rm H}(2,0,1,2,1,0,1,2)+\lambda_{\rm
O}(0,2,1,1,0,1,2,2)\,$ (25)
under condition $\lambda>0$. This transformation of $\delta$ does not change
the signs of $\hat{\delta}$ on the stoichiometric vectors because of atomic
balances.
In our case the only coordinate remains not eliminated, $x_{7}$ (the bottom
part of the last column of the Table 2). If, for some reaction mechanism and
selected sets of reversible and irreversible reaction, there remain several
($q$) coordinates, then it is necessary to find $q$ corresponding functionals
$\hat{\delta}$ and the space of possible vectors of exponents is
($q+j$)-dimensional. Here, $j$ is the number of the independent linear
conservation laws for the whole system, $j=n-{\rm rank}\\{\gamma_{r}\\}$, $n$
is the number of the components, $\\{\gamma_{r}\\}$ includes all the
stoichiometric vectors for reversible and irreversible reactions.
To find $\delta$, we apply the elimination procedures from the Table 2 to an
arbitrary vector $y=(y_{i})$ ($i=1,\ldots,8$):
$\begin{split}&(y_{1},y_{2},y_{3},y_{4},y_{5},y_{6},y_{7},y_{8})\mapsto(y_{1},y_{2},0,y_{4},y_{5},y_{6},y_{7},y_{8})\\\
\mapsto&(y_{1},y_{2},0,y_{4},y_{5},y_{6},y_{7},0)\mapsto(y_{1},y_{2},0,0,y_{5}-y_{4},y_{6},y_{7},0)\\\
\mapsto&(y_{1},y_{2},0,0,0,y_{6}+y_{5}-y_{4},y_{7},0)\mapsto(y_{1},y_{2},0,0,0,0,y_{7}+y_{6}+y_{5}-y_{4},0)\,.\end{split}$
(26)
This sequence of transformations gives us the linear functional
$\hat{\delta}(y)=y_{7}+y_{6}+y_{5}-y_{4}\,.$
The corresponding vector of exponents $(0,0,0,-1,1,1,1,0)$ should be corrected
because its coordinates cannot be negative. Let us apply (25) with $\lambda=2$
(for convenience). The coordinates of this combination are non-negative if and
only if $\lambda_{\rm H}\geq 0$, $\lambda_{\rm O}\geq 0$ and $2\lambda_{\rm
H}+\lambda_{\rm O}-2\geq 0$. The solutions of these linear inequality on the
$(\lambda_{\rm H},\lambda_{\rm O})$ plane is a convex combination of the
extreme points (corners) $(1,0)$ and $(0,2)$ plus any non-negative 2D vector:
$(\lambda_{\rm H},\lambda_{\rm
O})=\varsigma(1,0)+(1-\varsigma)(0,2)+(\vartheta_{1},\vartheta_{2})$,
$\vartheta_{1,2}\geq 0$ and $1\geq\varsigma\geq 0$. The corresponding vectors
of exponents are
$(0,0,0,-2,2,2,2,0)+(\varsigma+\vartheta_{1})(2,0,1,2,1,0,1,2)+(1-\varsigma+\vartheta_{2})(0,4,2,2,0,2,4,4)\,.$
At least one of the exponents should be zero. There are only three
possibilities, $\delta_{1}$, $\delta_{2}$ or $\delta_{4}$. For all other $i$,
$\delta_{i}>0$ if $\vartheta_{1,2}\geq 0$ and $1\geq\varsigma\geq 0$.
To provide any necessary atomic balance in the limit $\varepsilon\to 0$ it is
necessary that two of $\delta_{i}$ are zeros. If $b_{\rm
O}\leq\frac{1}{2}b_{\rm H}$, then $\delta_{1}=\delta_{4}=0$. This means that
$\vartheta_{1,2}=0$, $\varsigma=0$ and $\delta=(0,4,2,0,2,4,6,4)$. It is
convenient to divide this $\delta$ by 2 and write
$\delta=(0,2,2,0,2,2,3,2)\,.$
For these exponents, the equilibrium concentrations tend to 0 with the small
parameter $\varepsilon\to 0$ ($\varepsilon>0$) as
$\begin{split}&c^{\rm eq}_{{\rm H}_{2}}=c^{\rm eq}_{1}=const,\,c^{\rm
eq}_{{\rm O}_{2}}=c^{\rm eq}_{2}\sim\varepsilon^{2},\,c^{\rm eq}_{{\rm
OH}}=c^{\rm eq}_{3}\sim\varepsilon^{2},\,c^{\rm eq}_{{\rm H}_{2}{\rm
O}}=c^{\rm eq}_{4}=const,\,\\\ &c^{\rm eq}_{{\rm H}}=c^{\rm
eq}_{5}\sim\varepsilon^{2},\,c^{\rm eq}_{{\rm O}}=c^{\rm
eq}_{6}\sim\varepsilon^{2},\,c^{\rm eq}_{{\rm H}{\rm O}_{2}}=c^{\rm
eq}_{7}\sim\varepsilon^{3},\,c^{\rm eq}_{{\rm H}_{2}{\rm O}_{2}}=c^{\rm
eq}_{6}\sim\varepsilon^{2}\,.\end{split}$ (27)
If $b_{\rm O}\geq\frac{1}{2}b_{\rm H}$, then $\delta_{2}=\delta_{4}=0$. This
means that $\vartheta_{1,2}=0$, $\varsigma=1$ and
$\delta=(2,0,1,0,3,2,3,2)\,.$
For these exponents, the equilibrium concentrations tend to 0 with the small
parameter $\varepsilon\to 0$ ($\varepsilon>0$) as
$\begin{split}&c^{\rm eq}_{{\rm H}_{2}}=c^{\rm
eq}_{1}\sim\varepsilon^{2},\,c^{\rm eq}_{{\rm O}_{2}}=c^{\rm
eq}_{2}=const,\,c^{\rm eq}_{{\rm OH}}=c^{\rm eq}_{3}\sim\varepsilon,\,c^{\rm
eq}_{{\rm H}_{2}{\rm O}}=c^{\rm eq}_{4}=const,\,\\\ &c^{\rm eq}_{{\rm
H}}=c^{\rm eq}_{5}\sim\varepsilon^{3},\,c^{\rm eq}_{{\rm O}}=c^{\rm
eq}_{6}\sim\varepsilon^{2},\,c^{\rm eq}_{{\rm H}{\rm O}_{2}}=c^{\rm
eq}_{7}\sim\varepsilon^{3},\,c^{\rm eq}_{{\rm H}_{2}{\rm O}_{2}}=c^{\rm
eq}_{6}\sim\varepsilon^{2}\,.\end{split}$ (28)
The linear combination $\sum_{i}\delta_{i}N_{i}$ decreases in time due to
kinetic equations. This is true for any vector of exponents presented by a
linear combination (25) ($\lambda\neq 0$) of the initial vector
$(0,0,0,-1,1,1,1,0)$ with the vectors of the atomic balances. At the same
time, any of these combinations give an additional linear conservation law for
the system of reversible reactions.
Below are several versions of this function:
* 1.
The initial version, $\hat{\delta}$, obtained from the Table 2 is
$(\delta,N)=-N_{{\rm H}_{2}{\rm 0}}+N_{\rm H}+N_{\rm O}+N_{\rm H}{{\rm
O}_{2}}$;
* 2.
Vector of exponents, calibrated by adding of the atomic balances (25) to meet
the atomic balance conditions for $b_{\rm O}\leq\frac{1}{2}b_{\rm H}$ in the
limit $\varepsilon\to 0$ is $(\delta,N)=2N_{{\rm O}_{2}}+2N_{\rm OH}+2N_{\rm
H}+2N_{\rm O}+3N_{{\rm H}{\rm O}_{2}}+2N_{{\rm H}_{2}{\rm O}_{2}}$;
* 3.
Vector of exponents, calibrated to meet the atomic balance conditions for
$b_{\rm O}\geq\frac{1}{2}b_{\rm H}$ is, $(\delta,N)=2N_{{\rm H}_{2}}+N_{\rm
OH}+3N_{\rm H}+2N_{\rm O}+3N_{{\rm H}{\rm O}_{2}}+2N_{{\rm H}_{2}{\rm
O}_{2}}$.
All these forms differs by the combinations of the atomic balances (25) and
are, in this sense, equivalent.
## 5 Conclusion
The general principle of detailed balance was formulated in 1925 as follows
[15]: “Corresponding to every individual process there is a reverse process,
and in a state of equilibrium the average rate of every process is equal to
the average rate of its reverse process.” Rigorously speaking, the chemical
reactions have to be considered as reversible ones, and every step of the
complex reaction consists of two reactions, forward and reverse (backward)
one. However, in reality the rates of some forward or reverse reactions may be
negligible. Typically, the complex combustion reactions, in particular,
reactions of hydrocarbon oxidation or hydrogen combustion, include both
reversible and irreversible steps. It is a case in catalytic reactions as
well. In particular, many enzyme reactions are “partially irreversible”.
Although many catalytic reactions are globally irreversible, they always
include some reversible steps, in particular steps of adsorption of gases.
Many enzyme reactions are also “partially irreversible”.
This work aims to solve the problem of the partially irreversible limit in
chemical thermodynamics when some reactions become irreversible whereas some
other reactions remain reversible. The main results in this direction are
1. 1.
Description of the multiscale limit of a system reversible reactions when some
of equilibrium concentrations tend to zero (Sec. 2.2).
2. 2.
Extended principle of detailed balance for the systems with some irreversible
reactions (Theorem 1).
3. 3.
The linear functional $G_{\delta}$ that decreases in time on solutions of the
kinetic equations under the extended detailed balance conditions (Proposition
2 and Eq. (11)).
4. 4.
The entropy production (or free energy dissipation) formulas for the
reversible part of the reaction mechanism under the extended detailed balance
conditions (Eqs. (13), (15)).
5. 5.
Description of the faces of the positive orthant which include the
$\omega$-limit points in their relative interior and, therefore, description
of limiting behavior in time (Theorem 2).
Did we solve the main problem and create the thermodynamic of the systems with
some irreversible reaction? The answer is: we solved this problem partially.
We described the limit behavior but we did not find the global Lyapunov
function that captures relaxation of both reversible and irreversible parts of
the system. The good candidate is a linear combination of the relevant
classical thermodynamic potential and $G_{\delta}$ but we did not find the
coefficients. In that sense, the problem of the limit thermodynamics remains
open.
Nevertheless, one problem is solved ultimately and completely: How to throw
away some reverse reactions without violation of thermodynamics and
microscopic reversibility? The answer is: the convex hull of the
stoichiometric vectors of the irreversible reactions should not intersect with
the linear span of the stoichiometric vectors of the reversible reactions and
the reaction rate constants of the remained reversible reactions should
satisfy the Wegscheider identities (8).
The solution of this theoretical problem is important for the modeling of the
chemical reaction networks. This is because some of reactions are practically
irreversible. Removal of some reverse reaction from the reaction mechanism
cannot be done independently of the whole structure of the reaction network;
the whole reaction mechanism should be used in the decision making.
If the irreversible reactions are introduced correctly then we also know that
the closed system with this reaction mechanism goes to an equilibrium state.
At this equilibrium, all the reaction rates are zero: the irreversible
reaction rates vanish and the rates of the reversible reactions satisfy the
principle of detailed balance. The limit equilibria are situated on the faces
of the positive orthant of concentrations and these faces are described in the
paper.
The oscillatory or chaotic attractors are impossible in closed systems which
satisfy the extended principle of detailed balance. This general statement can
be considered as a simple consequence of thermodynamics. It can be easily
proved if the thermodynamic Lyapunov functions (potentials) are given.
However, the thermodynamic potentials have no limits for the systems with some
irreversible reactions and we do not know a priori any general theorem that
prohibits bifurcations at the zero values of some reaction rate constants. In
this paper we proved, in particular, that the emergency of nontrivial
attractors in systems with some irreversible reactions is impossible if they
are the limits of the reversible systems which satisfy the principle of
detailed balance. In this sense, the thermodynamic behavior is proven for the
systems with some irreversible reactions under the extended detailed balance
conditions. Nevertheless, the general problem of the thermodynamic potentials
in this limit remains open.
## References
* [1] D. Bertsimas, J.N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, Cambridge, MA, USA, 1997.
* [2] L. Boltzmann, Lectures on gas theory, Univ. of California Press, Berkeley, CA, USA, 1964.
* [3] B.R. Clarke, Challenging Logic Puzzles, Sterling Publishing Co., New York, NY, USA, 2003.
* [4] D. Colquhoun, K.A. Dowsland, M. Beato, A.J.R. Plested, How to Impose Microscopic Reversibility in Complex Reaction Mechanisms, Biophysical Journal 86 (6) (2004) 3510–3518.
* [5] T. de Donder, P. van Rysselberghe, Thermodynamic Theory of Affinity. A Book of Principles, Stanford Univ. Press, 1936.
* [6] M. Feinberg. On chemical kinetics of a certain class. Arch. Rat. Mechan. Anal. 46 (1972) 1–41.
* [7] V. Giovangigli, L. Matuszewski, Supercritical fluid thermodynamics from equations of state, Physica D 241 (6) (2012) 649–670.
* [8] A.N. Gorban, Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis, Nauka, Novosibirsk, 1984.
* [9] A.N. Gorban, Singularities of transition processes in dynamical systems: Qualitative theory of critical delays, Electron. J. Diff. Eqns., Monograph 05, 2004\. http://ejde.math.txstate.edu/Monographs/05/abstr.html
* [10] A.N. Gorban, E.M. Mirkes, A.N. Bocharov, V.I. Bykov, Thermodynamic consistency of kinetic data, Combustion, Explosion, and Shock Waves 25 (5) (1989) 593–600.
* [11] A.N. Gorban, M. Shahzad, The Michaelis–Menten–Stueckelberg Theorem, Entropy 13 (5) (2011) 966–1019; arXiv:1008.3296 [physics.chem-ph].
* [12] A.N. Gorban, G.S. Yablonskii, Extended detailed balance for systems with irreversible reactions, Chem. Eng. Sci. 66 (2011) 5388–5399; arXiv:1101.5280 [cond-mat.stat-mech].
* [13] M. Grmela, Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Adv. Chem. Eng. 39 (2010) 75–128.
* [14] M. Grmela, Fluctuations in extended mass-action-law dynamics, Physica D 241 (10) (2012) 976–986.
* [15] G.N. Lewis, A new principle of equilibrium, P.N.A.S. USA 11 (3) (1925), 179–183.
* [16] J. Li, Zh. Zhao, A. Kazakov, and F.L. Dryer, An Updated Comprehensive Kinetic Model of Hydrogen Combustion, Int. J. Chem. Kinet. 36 (2004) 566–575.
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* [18] P. Saxena and F. A. Williams. Testing a small detailed chemical-kinetic mechanism for the combustion of hydrogen and carbon monoxide. Combustion and Flame 145 (2006) 316–23.
* [19] N.Z. Shapiro, L.S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math. 16 (1965) 353–375.
* [20] S. Schuster, R. Schuster, A generalization of Wegscheider’s condition. Implications for properties of steady states and for quasi-steady-state approximation, J. Math. Chem. 3 (1) (1989) 25–42.
* [21] D.G. Vlachos, Reduction of detailed kinetic mechanisms for ignition and extinction of premixed hydrogen/air flames, Chem. Eng. Sci. 51 (16) (1996) 3979–3993.
* [22] A.I. Volpert, S.I. Khudyaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Nijoff. Dordrecht, The Netherlands, 1985. (Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.)
* [23] R. Wegscheider, Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme, Monatshefte für Chemie / Chemical Monthly 32 (8) (1901) 849–906.
* [24] G.S. Yablonskii, V.I. Bykov, A.N. Gorban, V.I. Elokhin, Kinetic Models of Catalytic Reactions, Elsevier, Amsterdam, The Netherlands, 1991.
* [25] J. Yang, W.J. Bruno, W.S. Hlavacek, J. Pearson, On imposing detailed balance in complex reaction mechanisms. Biophys. J. 91 (2006) 1136–1141.
* [26] Y.B. Zeldovich, Proof of the uniqueness of the solution of the equations of the law of mass action. In: Selected Works of Yakov Borisovich Zeldovich; Volume 1, Ed. by J.P. Ostriker; Princeton Univ. Press, Princeton, NJ, USA, 1996; pp. 144–148.
|
arxiv-papers
| 2012-07-10T22:53:05 |
2024-09-04T02:49:32.890474
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. N. Gorban, E. M. Mirkes, G. S. Yablonsky",
"submitter": "Alexander Gorban",
"url": "https://arxiv.org/abs/1207.2507"
}
|
1207.2610
|
11institutetext: Institute of Theoretical Physics, University of Wrocław,
PL–50204 Wrocław, Poland ExtreMe Matter Institute EMMI, GSI, D-64291
Darmstadt, Germany
# Probing QCD chiral cross over transition in heavy ion collisions with
fluctuations
Krzysztof Redlich redlich@ift.uni.wroc.pl
###### Abstract
We argue that by measuring higher moments of the net proton number
fluctuations in heavy ion collisions (HIC) one can probe the QCD chiral cross
over transition experimentally. We discuss the properties of fluctuations of
the net baryon number in the vicinity of the chiral crossover transition
within the Polyakov loop extended quark-meson model at finite temperature and
baryon density. The calculation includes non-perturbative dynamics implemented
within the functional renormalization group approach. We find a clear signal
for the chiral crossover transition in the fluctuations of the net baryon
number. We address our theoretical findings to experimental data of STAR
Collaboration on energy and centrality dependence of the net proton number
fluctuations and their probability distributions in HIC.
QCD phase transition and phase diagram Chiral symmetry and charge
fluctuations Heavy ion collisions
††articletype: Communication
## I QCD phase diagram
Although, the important question addressed in QCD on the existence of a true
2nd order phase transition at finite chemical potential (critical point) has
not been answered yet, nevertheless, there is recently an essential progress
in the quantitative description of the QCD phase diagram. The lattice QCD
(LQCD) provided a final value for the chiral cross over transition temperature
at vanishing chemical potential l1 . The LQCD has also provided arguments that
at small $\mu\simeq 0$ the chiral cross over line is the pseudo critical line
of the 2nd order chiral phase transition belonging to the universality class
of 3-dimensional, O(4) symmetric spin models l2 . Based on the universality
arguments the LQCD has also provided the curvature of the chiral transition
line, $T_{c}(\mu_{c})$ l2 . These results confirm that, at least at small
values of the baryon chemical potential $\mu$, the chiral cross over
transition appears in the near vicinity to the chemical freezeout line f1
obtained from the analysis of particle yields measured in HIC.
The numerical coincidence of thermal parameters for the chiral transition and
the freezeout conditions indicates that the hadron resonance gas partition
function, which describes chemical equilibration of particle yields in HIC,
should describe also the QCD thermodynamics up to a near vicinity to the
transition to a quark gluon plasma phase. Indeed, the equation of state
calculated on the lattice as well as other thermodynamical observables in the
hadronic phase, were shown to be very well quantified by the hadron resonance
gas (HRG) partition function h1 .
Already at vanishing chemical potential, i.e. under conditions realized in the
high energy runs at RHIC or LHC, the question arises to what extent a refined
analysis of freeze-out conditions can establish the existence of a chiral
phase transition. In this lecture we will argue that even at $\mu_{B}/T\simeq
0$ the net baryon number fluctuations and their higher moments can be used to
identify the chiral cross over transition experimentally o1 ; o2 ; o3 ; o4 .
Figure 1: The temperature dependence of kurtosis
$R_{4,2}:=9\chi_{4}^{B}$/$\chi_{2}^{B}$ and the higher order,
$\chi_{6}^{B}$/$\chi_{2}^{B}$ and $\chi_{8}^{B}/\chi_{2}^{B}$, ratios of
cumulants for different $\mu_{q}/T$ calculated in the PQM model within the FRG
approach o2 ; o4 . The $T_{pc}$ is the pseudo-critical temperature obtained in
the model at the physical pion mass.
## II Charge fluctuations and the chiral cross over transition
Due to remnants of O(4) criticality related with the chiral phase transition
observed in LQCD, the free energy ($f$) near the chiral phase transition
temperature $T_{c}$ may be represented in terms of singular ($f_{s}$) and
regular contributions ($f_{r}$) as
$f(T,\mu_{q},m_{q})=f_{s}(T,\mu_{q},m_{q})+f_{r}(T,\mu_{q},m_{q})\;,$ (1)
where in addition to the temperature $T$ we also introduced explicit
dependence on the light quark chemical potential, $\mu_{q}=\mu_{B}/3$, and the
(degenerate) light quark masses $m_{q}\equiv m_{u}=m_{d}$. The singular part
of the free energy may be written as s1
$f_{s}(T,\mu_{q},h)=h^{1+1/\delta}f_{s}(z)\ ,\ z\equiv t/h^{1/\beta\delta}$
(2)
with $\beta,\ \delta$ are critical exponents of the 3-dimensional, $O(4)$
universality class and
$t\equiv\frac{1}{t_{0}}\left(\frac{T-T_{c}}{T_{c}}+\kappa_{q}\left(\frac{\mu_{q}}{T}\right)^{2}\right)$,
$h\equiv\frac{1}{h_{0}}\frac{m_{q}}{T_{c}}$. Here $T_{c}$ is the phase
transition temperature in the chiral limit and $t_{0}$, $h_{0}$ are non-
universal scale parameters. The proportionality constant $\kappa_{q}\simeq
0.06$, has recently been determined from a scaling analysis in (2+1)-flavor
QCD l2 . The scaling function $f_{s}$ and its derivatives have recently been
calculated using high precision Monte Carlo simulations of the 3-dimensional
O(4) spin model e1 .
We want to focus here on properties of moments of net baryon number
fluctuations, which are obtained from Eq. 1 by taking derivatives with respect
to $\hat{\mu}_{B}=\mu_{B}/T$,
$\chi_{n}^{B}=-\frac{1}{3^{n}}\frac{\partial^{n}f/T^{4}}{\partial\hat{\mu}_{q}^{n}}=-\frac{1}{3^{n}}\frac{\partial^{n}f_{r}/T^{4}}{\partial\hat{\mu}_{q}^{n}}-\frac{1}{3^{n}}\frac{\partial^{n}f_{s}/T^{4}}{\partial\hat{\mu}_{q}^{n}}=\chi_{n,r}^{B}+\chi_{n,s}^{B}\;.$
(3)
In the hadronic phase and away from transition temperature the regular part
$\chi_{n,r}^{B}$ should be well described by the hadron resonance gas
partition function which will be a reference for critical fluctuations coming
from the singular part $\chi_{n,s}^{B}$. Thus, any deviations from the regular
i.e. hadronic gas contribution could be an indication of criticality due to
remnants of the chiral O(4) transition.
Higher order moments will become increasingly sensitive to the singular part
of the free energy. From Eq. 2 it is apparent that these moments show a strong
quark mass dependence in the vicinity of the critical temperature,
$\chi_{n,s}^{B}\sim\begin{cases}-(2\kappa_{q})^{n/2}h^{(2-\alpha-n/2)/\beta\delta}f_{s}^{(n/2)}(z)&,\
{\rm for}\ \mu_{q}/T=0,\ {\rm and}\ n\ {\rm even}\\\
-(2\kappa_{q})^{n}\left(\frac{\mu_{q}}{T}\right)^{n}h^{(2-\alpha-n)/\beta\delta}f_{s}^{(n)}(z)&,\
{\rm for}\ \mu_{q}/T>0\end{cases}\ .$ (4)
where we used $2-\alpha=\beta\delta(1+1/\delta)$. As $\alpha=-0.2131(34)$ is
negative in the 3-dimensional, $O(4)$ universality class, the 4th order
moments of the net baryon number fluctuations do not diverge yet in the chiral
limit at the chiral transition temperature, $z=0$. The first divergent moment
is obtained for $n=6$ if $\mu_{q}/T=0$ and for $n=3$ if $\mu_{q}/T>0$.
Figure 2: Left-hand figure: the ratio of quadratic fluctuations and mean net
baryon number ($\sigma^{2}/M$), cubic to quadratic ($S\sigma$) and quartic to
quadratic ($\kappa\sigma^{2}$) baryon number fluctuations calculated in the
HRG model on the freeze-out curve o1 and compared to results obtained by the
STAR Collaboration star . The dashed curves show the approximate
$\tanh(\mu_{B}/T)$ result for $\kappa\sigma^{2}$ and $S\sigma$, respectively.
Middle figure: The probability distributions, uncorrected for event-by-event
counting efficiency, for the net proton number for different centralities
taken by STAR Collaboration in Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV
star . The lines are the Skellam distributions calculated within HRG model o3
. Right-hand figure: Mean (M), variance ($\sigma$), skewness (S) and kurtosis
($\kappa$) of the net proton number calculated from the probability
distributions shown in the middle figure.
In the hadron resonance gas (HRG) the regular part of the fluctuations
$\chi_{n,r}^{B}$ can be directly calculated from the thermodynamic pressure
following Eq. 3. The HRG is a mixture of ideal gases of all particles and
resonances, consequently the thermodynamic pressure exhibits a factorization
of $T$ and $\mu_{B}/T$ dependence. Under the Boltzmann approximation the
pressure in the HRG, $P^{HRG}(T,\mu_{B})\simeq f(T)\cosh(\mu_{B}/T)$, where
$f(T)$ contains contributions from all baryons and baryonic resonances. With
such $P^{HRG}$ there are particular properties of ratios of cumulants,
${{\chi_{2n,r}^{B}}\over{\chi_{2,r}^{B}}}|_{HRG}=1~{}~{},~{}~{}{{\chi_{(2n+1),r}^{B}}\over{\chi_{1,r}^{B}}}|_{HRG}=1~{}~{},~{}~{}{{\chi_{(2n+1),r}^{B}}\over{\chi_{2,r}^{B}}}|_{HRG}\simeq\tanh(\mu_{B}/T)~{}~{},~{}~{}{{\chi_{2n,r}^{B}}\over{\chi_{1,r}^{B}}}|_{HRG}\simeq\coth(\mu_{B}/T),$
(5)
which are independent of the number of baryons, their masses, degeneracy
factors or decay widths.
The above structure of cumulants ratios observed in the HRG will be modified
if the singular part is included in Eq. 3. The $\chi_{n,s}^{B}$ are
increasingly sensitive to the order of cumulants. Thus, ratios of cumulants
with different $n$ should be strongly varying functions of $T$ and $\mu_{B}$
when approaching a chiral cross over transition. Consequently, the observed
deviation from that expected in Eq. 5 could be considered as a signature of
the singular part contribution to the overall fluctuations thus, also of the
chiral cross over transition.
The generic structure of ratios of different cumulants near the chiral cross
over transition is shown in Fig. 1. These ratios were calculated in the
Polyakov loop extended quark-meson (PQM) model at the physical pion mass,
applying the functional renormalization group (FRG) method. In FRG approach
one includes quantum and thermal fluctuations which are needed to preserve the
universal scaling behavior of physical quantities expected in the O(4)
universality class. In the low temperature phase the ratios
${{\chi_{2n,r}^{B}}/{\chi_{2,r}^{B}}}=1$ as expected in the hadron resonance
gas from Eq. 5. For $n=2$, the kurtosis
$R_{4,2}=9{{\chi_{4,r}^{B}}/{\chi_{2,r}^{B}}}$ is not affected by the chiral
critical dynamics since there is no contribution of the singular part to the
first four moments as seen in Eq. 4. The observed in Fig. 1 drop in $R_{4,2}$
is due to ”statistical confinement” property of the PQM model stok . Such
behavior of kurtosis was first observed in LQCD calculations and was
interpreted as being a signature of deconfinement in QCD s1 .
The large deviations of $(R_{4,2}/9)$ from unity in hadronic phase, which are
increasing with $\mu/T$, are due to a singular contribution to the
${\chi_{4}^{B}}$ and ${\chi_{2}^{B}}$ ratio (see Eqs. 3 and 4). With
increasing order of cumulants and the value of the chemical potential their
ratios are dominated by the singular part already deeply below the chiral
cross over transition temperature. Such behavior could be observed in HIC if
freezeout appears near the chiral transition.
Recently, the first data on charge fluctuations and higher order cumulants,
identified through the net-proton fluctuations, were obtained by the STAR
Collaboration in Au- Au collisions at several collision energies star . To
explore possible signs of chiral criticality and a cross over transition, the
STAR data on the first four moments are compared in Fig. 2 to HRG o1 ; o3
following Eq. 5. Different ratios of cumulants of the net proton number can be
directly connected with measured mean (M), variance $(\sigma)$, skewness (S)
and kurtosis $(\kappa)$ o1 .
The basic properties of measured fluctuations and ratios of cumulants are
consistent with that expectated in the HRG model o1 ; o3 . This indicates,
that moments of the net proton number are of thermal origin with respect to
the grand canonical ensemble and that they freezeout along the same chemical
freezout line as particle yields, close to the chiral cross over line.
However, already such first comparison of the HRG model with STAR data reveals
that there are deviations o1 ; o3 ; cp1 . This is seen in Fig. 2 on the level
of different ratios of cumulants as well as by comparing directly the measured
probability distributions with Skellam distribution expected in the HRG for
the net proton number o3 . The HRG model, as seen in Fig. 2, results in a
broader distribution than observed in data, particularly at the most central
collisions. This implies deviations of data on different moments from the HRG
model which are increasing with centrality. Shrinking of measured widths of
the net proton number distribution relative to the HRG results, seen in Fig.
2, is to be already expected due to deconfinement properties of QCD o3 . We
have to stress, however, that the experimental net-proton distributions in
Fig. 2 are not corrected for event-by-event proton/anti-proton counting
efficiency. While the HRG lines in this figure used efficiency corrected mean
multiplicities from the same experiment. If uncorrected data are used in the
Skellam distribution, then the agreement of the HRG model and measure
probability distributions is found to be much better cp2 .
From the model calculations in Fig. 1, as well as from Eqs. 3 and 4, it is
clear that contributions of the O(4) singular part to fluctuations increase
with the order of cummulants. Preliminary data of STAR Collaboration on
$\chi_{6}^{B}/\chi_{2}^{B}$ ratio, taken in Au-Au collisions at the top RHIC
energy, show a strong deviation of this ratio from the HRG model expectations
with a rather moderate deviations of the lower order cumulant ratios.
Recently, the STAR Collaboration has also observed that deviations in central
Au-Au collisions of $\chi_{4}^{B}/\chi_{2}^{B}$ ratio from the HRG is non-
monotonic in energy. Such behaviors of ratios of cumulants are to be expected
due to remnants of criticality related with a chiral cross over transition. It
is a further challenge to understand and quantify the observed energy and
centrality dependence of these deviations from the HRG results.
I am grateful for fruitful collaboration and discussions with P. Braun-
Munzinger, B. Briman, F. Karsch and V. Skokov. Partial support by MEN (Poland)
and EMMI (Germany) and discussion with Nu Xu is also acknowledged.
## References
* (1) Y. Aoki, et al., JHEP 0906, 088 (2009). A. Bazavov et. al. (HotQCD collaboration), arXiv:1111.1710 [hep-lat].
* (2) S. Ejiri, et al., Phys. Rev. D 80, 094505 (2009). O. Kaczmarek, et al., Phys. Rev. D 83, 014504 (2011)
* (3) P. Braun-Munzinger, et al., in Quark-Gluon Plasma 3, Eds. R.C. Hwa and X.N. Wang, (World Scientific Publ., 2004). J. Cleymans, K. Redlich, Phys. Rev. C 60, 054908 (1999); Phys. Rev. Lett. 81, 5284 (1998).
* (4) F. Karsch, et al., Eur. Phys. J. C 29, 549 (2003). C. Ratti, Nucl. Phys. A 855, 253 (2011).
* (5) F. Karsch, K. Redlich, Phys. Lett. B 695, 136 (2011).
* (6) B. Friman, et al., Eur. Phys. J. C 71, 1694 (2011).
* (7) P. Braun-Munzinger, et al., Phys. Rev. C 84, 064911 (2011); arXiv:1111.5063 [hep-ph].
* (8) V. Skokov, et al., Phys. Rev. C 83, 054904 (2011). J. Phys. G 38,124102 (2011). Phys. Lett. B 708, 179 (12).
* (9) See also: Lizhu Chen, these proceedings.
* (10) See also: Xiaofeng Luo and B. Mohanty, these proceedings.
* (11) S. Ejiri, et al., Phys. Lett. B 633, 275 (2006).
* (12) J. Engels, F. Karsch, arXiv:1105.0584 [hep-lat].
* (13) B. Stokic, et al., Phys. Lett. B 673, 192 (2009).
* (14) M. M. Aggarwal et al. [STAR Collaboration], Phys. Rev. Lett. 105, 022302 (2010). X. Luo, et al., [for the STAR Collaboration], arXiv:1106.2926v1.
|
arxiv-papers
| 2012-07-11T12:16:14 |
2024-09-04T02:49:32.915524
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Krzysztof Redlich (University of Wrocaw and ExtreMe Matter Institute\n EMMI/GSI)",
"submitter": "Krzysztof Redlich KR",
"url": "https://arxiv.org/abs/1207.2610"
}
|
1207.2819
|
# A Proof of the Pumping Lemma for Context-Free Languages Through Pushdown
Automata
Antoine Amarilli111École normale supérieure, Paris, France. Marc
Jeanmougin††footnotemark:
###### Abstract
The pumping lemma for context-free languages is a result about pushdown
automata which is strikingly similar to the well-known pumping lemma for
regular languages. However, though the lemma for regular languages is simply
proved by using the pigeonhole principle on deterministic automata, the lemma
for pushdown automata is proven through an equivalence with context-free
languages and through the more powerful Ogden’s lemma. We present here a proof
of the pumping lemma for context-free languages which relies on pushdown
automata instead of context-free grammars.
## 1 Setting
The pumping lemma for regular languages is the following well-known result:
###### Theorem 1.
Let $L$ be a regular language over an alphabet $\Sigma$. There exists some
integer $p\geq 1$ such that, for every $w\in L$ such that $|w|>p$, there
exists a decomposition $w=xyz$ such that:
1. 1.
$|xy|\leq p$
2. 2.
$|y|\geq 1$
3. 3.
$\forall n\geq 0,xy^{n}z\in L$
The pumping lemma for context-free languages [BHPS61], also known as the Bar-
Hillel lemma, is the following similar result:
###### Theorem 2.
Let $L$ be a context-free language over an alphabet $\Sigma$. There exists
some integer $p\geq 1$ such that, for every $w\in L$ such that $|w|>p$, there
exists a decomposition $w=uvxyz$ such that:
1. 1.
$|vxy|\leq p$
2. 2.
$|vy|\geq 1$
3. 3.
$\forall n\geq 0,uv^{n}xy^{n}z\in L$
One would expect the classical proofs of these results to be similar. However,
this is not the case. The pumping lemma for regular languages [HU79] is
usually proved through the equivalence between regular languages and finite
automata by picking a deterministic automaton $A$ which recognizes the
language $L$ ; we can then use the fact that the accepting path of any word
$w$ longer than the number of states of $A$ must pass by the same state twice
(by the pigeonhole principle), yielding the points at which we can decompose
$w$. The pumping lemma for context-free languages, however, is usually derived
from Ogden’s lemma [Ogd68] which is itself proved by examining context-free
grammars (CFGs) and not pushdown automata (using the equivalence of these two
formalisms).
It seems reasonable to hope that the pumping lemma for context-free languages
can be proved directly from the properties of pushdown automata, with no
reference to CFGs. In the next section, we propose such a proof. Though the
underlying ideas that we introduce in this proof are apparently part of the
folklore, we are not aware of any attempt to prove the pumping lemma directly
through pushdown automata. The most relevant existing work that we know of is
a weaker form of the result [Kar12].
Analogous techniques to the one used below can be used to obtain a proof of
Ogden’s lemma. However, it seems that the most natural way to do so is very
similar to a combination of the usual pushdown system encoding to CFGs and the
usual proof of Ogden’s lemma. These further efforts (not included in this
note) suggest that the proof below, though it does not mention CFGs on the
surface, may not differ very much from a CFG-based argument after all.
## 2 Proof
Let $L$ be a context-free language over an alphabet $\Sigma$. Let $A$ be a
pushdown automaton which recognizes $L$, with stack alphabet $\Gamma$. We
denote by $|A|$ the number of states of $A$. To simplify the reasoning, we
will impose the following condition on $A$ (denoted by (*)): all transitions
of $A$ pop the topmost symbol of the stack and either push no symbol on the
stack or push on the stack the previous topmost symbol and some other symbol.
It is easy to see that any pushdown automata which pushes arbitrary sequences
of symbols on the stack can be rewritten in this fashion by replacing its
transitions by an initial pop transition followed by a sequence of
$\epsilon$-transitions pushing the appropriate symbols on the stack. (However,
keep in mind that because of this translation, $|A|$ in what follows does not
refer to the number of states of the original automaton recognizing $A$ but to
that of its translation by this process.)
We define $p^{\prime}=|A|^{2}|\Gamma|$ and define the pumping length to be
$p=|A|(|\Gamma|+1)^{p^{\prime}}$. We will now show that all $w\in L$ such that
$|w|>p$ have a decomposition of the form $w=uvxyz$ such that $|vxy|\leq p$,
$|vy|\geq 1$ and $\forall n\geq 0,uv^{n}xy^{n}z\in L$.
Let $w\in L$ such that $|w|>p$. Let $\pi$ be an accepting path of minimal
length for $w$ (represented as a sequence of transitions of $A$), we denote
its length by $|\pi|$. We can define, for $0\leq i<|\pi|$, $s_{i}$ the size of
the stack at position $i$ of the accepting path. For all $N>0$, we will define
an $N$-level over $\pi$ as a set of three indices $i,j,k$ with $0\leq
i<j<k\leq p$ such that the stack grows by $N$ symbols between $i$ and $j$ and
shrinks by $N$ symbols between $j$ and $k$. Formally, we require that:
1. 1.
$s_{i}=s_{k},s_{j}=s_{i}+N$
2. 2.
for all $n$ such that $i\leq n\leq j$, $s_{i}\leq s_{n}\leq s_{j}$
3. 3.
for all $n$ such that $j\leq n\leq k$, $s_{k}\leq s_{n}\leq s_{k}$.
We define the level $l$ of $\pi$ as the maximal $N$ such that $\pi$ has an
$N$-level. This definition is motivated by the following observation: if the
size of the stack over a path $\pi$ becomes larger than its level $l$, then
the stack symbols more than $l$ levels deep will never be popped. Formally, we
define the configurations of $A$ as the couples of a state of $A$ and a
sequence of $l$ stack symbols (where stacks of size less than $l$ are
represented by padding them to $l$ with a special blank symbol, which is why
we use $|\Gamma|+1$ when defining $p$). By definition, there are
$|A|(|\Gamma|+1)^{l}$ such configurations. Essentially, $A$ acts as a finite
automaton without stack between the configurations.
We can now distinguish two cases: either the level is low and the number of
configurations is small, or the level is high. Formally:
1. 1.
$l<p^{\prime}$ and, by the pigeonhole principle, the same configuration is
encountered twice in the first $p+1$ steps of $\pi$,
2. 2.
$l\geq p^{\prime}$ and, by the pigeonhole principle, we will prove that a
certain notion of _full state_ is repeated for two different stack sizes in
any $l$-level of $w$.
### Case 1.
$l<p^{\prime}$. In this case, the number of configurations is less than $p$.
Hence, in the $p+1$ first steps of $\pi$, the same configuration is
encountered twice at two different positions, say $i<j$. Denote by
$\widehat{i}$ (resp. $\widehat{j}$) the position of the last letter of $w$
read at step $i$ (resp. $j$) of $\pi$. We have $\widehat{i}\leq\widehat{j}$.
Hence, we can factor $w=uvxyz$ with $yz=\epsilon$, $u=w_{0\cdots\widehat{i}}$,
$v=w_{\widehat{i}\cdots\widehat{j}}$, $x=w_{\widehat{j}\cdots|w|}$. (By
$w_{x\cdots y}$ we denote the letters of $w$ from $x$ inclusive to $y$
exclusive.) By construction, $|vxy|\leq p$.
We also have to show that $\forall n\geq 0,uv^{n}xy^{n}z=uv^{n}x\in L$, but
this follows from our observation above: stack symbols deeper than $l$ are
never popped, so there is no way to distinguish configurations which are equal
according to our definition, and an accepting path for $uv^{n}x$ is built from
that of $w$ by repeating the steps between $i$ and $j$, $n$ times.
Finally, we also have $|v|>0$, because if $v=\epsilon$, then, because we have
the same configuration at steps $i$ and $j$ in $\pi$,
$\pi^{\prime}=\pi_{0\cdots i}\pi_{j\cdots|\pi|}$ would be an accepting path
for $w$, contradicting the minimality of $\pi$.
Figure 1: Illustration of the construction for case 2. To simplify the
drawing, the distinction between the path positions and word positions are
omitted.
### Case 2.
$l\geq p^{\prime}$. Let $i,j,k$ be a $p^{\prime}$-level. To any stack size
$h$, $s_{i}\leq h\leq s_{j}$, we associate the last push
$\operatorname*{lp}(h)=\max(\\{y\leq j|s_{y}=h\\})$ and the first pop
$\operatorname*{fp}(h)=\min(\\{y\geq j|s_{y}=h\\})$. By definition,
$i\leq\operatorname*{lp}(h)\leq j$ and $j\leq\operatorname*{fp}(h)\leq k$. We
say that the full state of a stack size $h$ is the triple formed by:
1. 1.
the automaton state at position $\operatorname*{lp}(h)$
2. 2.
the topmost stack symbol at position $\operatorname*{lp}(h)$ (which, by
construction, is also the topmost stack symbol at position
$\operatorname*{fp}(h)$
3. 3.
the automaton state at position $\operatorname*{fp}(h)$
(Observe that there is a link between this definition and what is known as
“Ginsburg triples” when encoding pushdown systems in CFGs.)
There are $p^{\prime}$ possible full states, and $p^{\prime}+1$ stack sizes
between $s_{i}$ and $s_{j}$, so, by the pigeonhole principle, there exist two
stack sizes $g,h$ with $s_{i}\leq g<h\leq s_{j}$ such that the full states at
$g$ and $h$ are the same. Like in Case 1, we define by
$\widehat{\operatorname*{lp}{(}}g)$, $\widehat{\operatorname*{lp}{(}}h)$,
$\widehat{\operatorname*{fp}{(}}h)$ and $\widehat{\operatorname*{fp}{(}}g)$
the positions of the last letters of $w$ read at the corresponding positions
in $\pi$. We factor $w=uvxyz$ where
$u=w_{0\cdots\widehat{\operatorname*{lp}{(}}g)}$,
$v=w_{\widehat{\operatorname*{lp}{(}}g)\cdots\widehat{\operatorname*{lp}{(}}h)}$,
$x=w_{\widehat{\operatorname*{lp}{(}}h)\cdots\widehat{\operatorname*{fp}{(}}h)}$,
$y=w_{\widehat{\operatorname*{fp}{(}}h)\cdots\widehat{\operatorname*{fp}{(}}g)}$,
and $z=w_{\widehat{\operatorname*{fp}{(}}g)\cdots|w|}$.
This factorization ensures that $|vxy|\leq p$ (because $k\leq p$ by our
definition of levels).
We also have to show that $\forall n\geq 0,uv^{n}xy^{n}z\in L$. To do so,
observe that each time that we repeat $v$, we start from the same state and
the same stack top and we do not pop below our current position in the stack
(otherwise we would have to push again at the current position, violating the
maximality of $\operatorname*{lp}(g)$), so we can follow the same path in $A$
and push the same symbol sequence on the stack. By the maximality of
$\operatorname*{lp}(h)$ and the minimality of $\operatorname*{fp}(h)$, while
reading $x$, we do not pop below our current position in the stack, so the
path followed in the automaton is the same regardless of the number of times
we repeated $v$. Now, if we repeat $w$ as many times as we repeat $v$, since
we start from the same state, since we have pushed the same symbol sequence on
the stack with our repeats of $v$, and since we do not pop more than what $v$
has stacked by minimality of $\operatorname*{fp}(g)$, we can follow the same
path in $A$ and pop the same symbol sequence from the stack. Hence, an
accepting path from $uv^{n}xy^{n}z$ can be constructed from the accepting path
for $w$.
Finally, we also have $|vy|>1$, because like in case 1, if $v=\epsilon$ and
$y=\epsilon$, we can build a shorter accepting path for $w$ by removing
$\pi_{\operatorname*{lp}(g)\cdots\operatorname*{lp}(h)}$ and
$\pi_{\operatorname*{fp}(h)\cdots\operatorname*{fp}(g)}$.
Hence, we have an adequate factorization in both cases, and the result is
proved.
## Acknowledgements
We are grateful to Alexander Kartzow for reporting an error in a previous
version of this proof and for pointing us to relevant work.
## References
* [BHPS61] Yehoshua Bar-Hillel, Micha A. Perles, and Eli Shamir. On formal properties of simple phrase structure grammars. Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung, (14):143–172, 1961.
* [HU79] John E. Hopcroft and Jefferey D. Ullman. Introduction to Automata Theory, Languages, and Computation. Adison-Wesley Publishing Company, Reading, Massachusets, USA, 1979.
* [Kar12] Alexander Kartzow. First-order model checking on generalisations of pushdown graphs. CoRR, abs/1202.0137, 2012.
* [Ogd68] William F. Ogden. A helpful result for proving inherent ambiguity. Mathematical Systems Theory, 2(3):191–194, 1968.
|
arxiv-papers
| 2012-07-12T01:01:58 |
2024-09-04T02:49:32.933543
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antoine Amarilli, Marc Jeanmougin",
"submitter": "Antoine Amarilli",
"url": "https://arxiv.org/abs/1207.2819"
}
|
1207.2838
|
# Vascular phyllotaxis transition and an evolutionary mechanism of
phyllotaxis
Takuya Okabe Faculty of Engineering, Shizuoka University, 3-5-1 Johoku,
Hamamatsu 432-8561,Japan ttokabe@ipc.shizuoka.ac.jp
###### Abstract
Leaves of vascular plants are arranged regularly around stems, a phenomenon
known as phyllotaxis. A constant angle between two successive leaves is called
divergence angle. On the one side, the divergence angle $\alpha_{0}$ of an
initial pattern of leaf primordia at a shoot apex is most commonly an
irrational number of about 137.5 degrees, called limit divergence. On the
other side, the divergence $\alpha$ of a final pattern of leaf traces in the
vascular system of a mature stem is expressed in terms of a sequence of
rational numbers, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$, $\frac{3}{8}$,
$\frac{5}{13}$, $\frac{8}{21}$, called phyllotactic fractions. The
mathematical relationship between the initial divergence $\alpha_{0}$, the
final divergence $\alpha$, and the number of internodes traversed by the leaf
traces $n_{c}$ is investigated by means of a theoretical model of vascular
phyllotaxis. It is shown that continuous changes of the trace length $n_{c}$
induce transitions between the fractional orders in the vascular structure.
The vascular phyllotaxis transition suggests an evolutionary mechanism for the
phenomenon of phyllotaxis. To provide supporting evidence for the model and
mechanism, available experimental results for fossil remains of Lepidodendron
and the vascular structure of Linum and Populus are analyzed with the model.
###### keywords:
phyllotaxy; Fibonacci numbers; golden ratio; natural selection; Linum
usitatissimum; Populus deltoides
## 1 Introduction
### 1.1 Review, background, and motivation
Astonishing regularity manifested in plant architecture has fascinated various
fields of scientists for centuries. The regular arrangement of leaves, flowers
and floral organs of higher plants is called phyllotaxis. A constant angle of
rotation between two successive organs is called divergence angle, on which
two apparently irreconcilable concepts have been in general use since the
inception of quantitative investigations on phyllotaxis.
Figure 1: (a) A typical pattern of leaf primordia (points) on a shoot apex
with the initial divergence of $\alpha_{0}=1/(1+\tau)$, or
$360\alpha_{0}\simeq 137.5$ in degrees. The irrational number $\tau\simeq
1.618$ is the golden ratio defined by the proportion equation
$1:\tau=\tau-1:1$. The primordia are numbered in the reverse order of
production. A solid spiral connecting all the primordia in the numerical order
is the genetic spiral. Three dashed spirals (clockwise inward) and five dotted
spirals (counterclockwise inward) are 3 and 5 parastichies, respectively. This
pattern has a parastichy pair $(3,5)$. (b) A typical pattern of leaves on a
mature stem characterized with a divergence fraction of $\alpha=\frac{2}{5}$
($360\alpha=144^{\circ}$). Oblique strands diverging to leaves 1 and 6 are
leaf traces. A solid spiral surrounding the stem is the genetic spiral.
Braun (1831, 1835) and Schimper (1835) noticed that divergence angle is
various but not arbitrary. It is a fraction, or a rational number, a number
that can be expressed as the quotient $\frac{n}{m}$ of two integers $n$ and
$m$. The most widespread is the helical phyllotaxis, also called spiral or
alternate phyllotaxis, in which stems bear a leaf per node. In the helical
phyllotaxis, the numerator $n$ and denominator $m$ of the fraction normally
are two alternate terms of a Fibonacci sequence, 1, 2, 3, 5, 8, 13, 21, 34,
55, 89, $\cdots$. It is generated by the Fibonacci recurrence relation that
each number after the first two terms is the sum of the previous two numbers.
The phyllotactic fractions $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$,
$\frac{3}{8}$, $\frac{5}{13}$, $\frac{8}{21}$, $\frac{13}{34}$, $\cdots$
comprise what is called the main sequence of phyllotaxis. A $\frac{2}{5}$
phyllotaxis is schematically shown in Fig. 1. In multijugate, verticillate or
whorled phyllotaxis, where more than two leaves are borne at each node, the
divergence angle is divided by the number of leaves in a whorl. In general, a
plant stem is partitioned into nodes and internodes. A node is a point at
which a leaf or leaves are attached, and an internode is a section of the stem
between two successive nodes. In the fractional phyllotaxis, there are leaves
aligned vertically above each other along a stem, as represented by leaves 1
and 6 in Fig. 1. A straight line connecting the superposed leaves is called an
orthostichy. In the helical phyllotaxis, the denominator of the phyllotactic
fraction is equal to the number of orthostichies. It is also the number of
internodes between two adjacent leaves on an orthostichy. Thus, the
$\frac{2}{5}$ phyllotaxis in Fig. 1 has five orthostichies, 1-6, 2-7, 3-8, 4-9
and 5-10, and five internodes separate leaves on each orthostichy. An
imaginary spiral connecting all the leaves in the order of production is
called the genetic, fundamental, generative, or ontogenetic spiral. The
numerator of the fraction refers to the number of turns of the genetic spiral
between the two adjacent leaves on an orthostichy. In Fig. 1, a solid spiral
is the genetic spiral. From the leaf 6 to 1, the genetic spiral winds around
the stem twice, the number two being the numerator of $\frac{2}{5}$. As
remarked below, the phyllotactic fraction does not lose its significance even
though vertical alignment is actually not exact but approximate.
In contrast, Bravais and Bravais (1837) suggested that divergence angle is
uniquely and invariably given by an irrational number, that is, a number which
cannot be expressed as a fraction. The most typical angle of $360/(1+\tau)$
degrees is called the golden angle, where the irrational number $\tau$, called
the golden ratio, golden mean, golden section, or extreme and mean ratio, is
defined by the proportional relation $1:\tau=\tau-1:1$. As the positive
solution of the quadratic equation $\tau(\tau-1)=1$, it is given by
$\tau=\frac{\sqrt{5}+1}{2}\simeq 1.61803399\cdots.$ (1)
The defining equation is transformed to $\tau^{-1}=1/(1+\tau^{-1})$. Recursive
substitution of $\tau^{-1}$ in the left-hand side to the right-hand side gives
an infinite continued fraction representation,
$\tau^{-1}=\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\ddots}}},\qquad\tau=1+\tau^{-1}.$
The golden angle $360/(1+\tau)=360/\tau^{2}$ is approximately $137.50776$
degrees. By definition, the golden angle is the smaller angle created by
sectioning the circumference of a circle (360 degrees) according to the golden
ratio $1:\tau$, the golden section. A phyllotactic pattern with divergence
equal to the golden angle is shown in Fig. 1. The ratio of the angle subtended
by 1 and 2 to the angle between 1 and 3 is $\tau$, or $\angle 1O2:\angle
1O3=1:\tau-1=\tau:1$, where $O$ is the origin. Similarly, $\angle 1O4:\angle
2O4=\angle 1O9:\angle 4O9=1:\tau$, and so on. Thus, phyllotactic patterns with
constant divergence equal to the golden angle have harmonious proportions.
Patterns with an irrational divergence angle have no orthostichy in a strict
sense, as no two leaves align vertically or radially. Instead, therefore,
attention is directed to secondary spirals connecting positionally nearby
leaves, called parastichies. Like an orthostichy, a parastichy is
characterized by a difference in number of leaves on it. In Fig. 1, the
genetic spiral, three parastichies and five parastichies are drawn with a
solid curve, dashed curves and dotted curves, respectively. Each of three
parastichies 1-4-7-10, 2-5-8 and 3-6-9, is called a 3-parastichy. Hence there
are three 3-parastichies and five 5-parastichies in Fig. 1, and the pattern in
Fig. 1 is said to have a parastichy pair of $(3,5)$, which is also denoted as
$(3+5)$ or $3:5$. As a remarkable fact, parastichy numbers are almost always
given by Fibonacci numbers. This is a mathematical consequence of the fact
that divergence angle is almost always the special irrational number, the
golden angle. The golden angle is also called the Fibonacci angle, for it is
the limit angle of divergence for the phyllotactic fractions belonging to the
main sequence;
$\begin{array}[]{c}\textstyle 360\times\frac{2}{5}=144,\\\
360\times\frac{3}{8}=135,\\\ 360\times\frac{5}{13}\simeq 138.46,\\\
360\times\frac{8}{21}\simeq 137.14.\end{array}$
The rational angles beyond $\frac{5}{13}$ are practically indistinguishable
from the ‘ideal’ irrational angle of 137.507764$\cdots$ degrees. Therefore, it
is argued that what appear to be different rational angles are nothing but a
single irrational angle disturbed by inevitable random errors.
The seemingly conflicting views on the divergence angle, whether rational
numbers or an irrational number, are a source of inspiration and confusion. In
effect, they are not only compatible but both indispensable. On the one hand,
the irrational number applies to the divergence angle of phyllotactic patterns
of undifferentiated tissues at shoot tips or apical meristems (Church (1904);
Hirmer (1922, 1931)). Let us call it the initial divergence angle. It is
commonly referred to as the ideal or limit divergence angle for the reason
mentioned above. On the other hand, the rational (fractional) divergence
applies to phyllotaxis of leaves, or primary vascular architecture on a mature
stem (Lestiboudois (1848); Nägeli (1858)). In the literature, the majority of
studies discuss the former, i.e., the process of organ initiation, positioning
of the leaf primordia from which leaves will develop, and transitions of
patterns at the shoot apical meristem. In recent years, substantial progress
has been made in understanding plant hormonal factors that influence or
control the formation of leaf primordia and their arrangement on the apical
meristem (Reinhardt (2005); Kuhlemeier (2007)). In striking contrast, the
fractional phyllotaxis of the mature stem have received less scholarly
attention, unfortunately. This is not because the latter is less important
than the former. As a matter of fact, experimental findings on the close
relationship between phyllotactic fraction and vascular organization have been
accumulated without being theorized from a general perspective (Sterling
(1945); Girolami (1953); Jensen (1968); Namboodiri and Beck (1968); Larson
(1977); Beck et al. (1982); Kirchoff (1984)).
Figure 2: Schwendener’s causal model. Right: Contiguous circles with
decreasing radius are stacked on an unrolled surface of a stem cylinder.
Contact parastichy numbers, or differences in the numbers of the circles in
contact, change from (1,1) at the bottom to (5,8) at the top. Left: A
mathematical relation between the divergence angle (the horizontal axis) and
the radius of the contiguous circles (the vertical axis) is indicated with a
solid zigzag curve starting from the top left corner (divergence of 180∘,
corresponding to the bottom part of the right figure) down to the golden angle
137.5∘ (the top of the right figure). The zigzagging is due to shifts in the
contact parastichy numbers from $(1,1)$ through $(1,2)$, $(2,3)$, $(3,5)$,
$(5,8)$, $(8,13)$, $(13,21)$ to $(21,34)$. The top branch for $(1,2)$ extends
from 180∘ to $128^{\circ}34^{\prime}$. Adapted from Schwendener (1883).
Since the influential text by Hofmeister (1868), research into causal or
dynamical mechanisms of primordia initiation has been the central pillar in
the study of phyllotaxis. The empirical observation that new leaf primordia
arise in the largest space between the older primordia is called Hofmeister’s
rule. What was originally a rule of thumb of botanists has been refined and
developed into causal or dynamical models. Airy (1873) speculated on a causal
mechanism in terms of geometrical objects in mechanical action. Schwendener
(1878) put a similar idea on a more solid mathematical basis by regarding
leaves on a stem as solid disks contiguously covering a cylinder surface of
infinite length (Fig. 2). In Schwendener’s model, contiguous circles of a
constant radius are arranged in a periodic pattern characterized with a given
set of contact parastichy numbers. Then it is a purely geometrical problem to
derive various mathematical relations for the divergence angle, the radius of
the circles, the girth of the cylinder, and the set of parastichy numbers. The
radius of the circles is regarded as an independent variable, or a control
parameter of the model. By letting the radius change continuously along the
stem cylinder, the divergence angle varies concomitantly with the contact
parastichy numbers according to the mathematical relations. As a remarkable
result, the divergence angle converges toward the golden angle $137.5^{\circ}$
by decreasing the radius sufficiently slowly from a large initial value to a
small constant value. Before attaining to the golden angle, the model predicts
that the divergence angle oscillates with decreasing amplitude (Fig. 2). The
decrease in the radius corresponds to decrease in relative size of leaf
primordia on the stem or apex. This is a brief summary of Schwendener’s causal
mechanism for the golden angle. The model is referred to as a mechanical or
causal model of phyllotaxis.
Related causal models were discussed in depth by Delpino (1883) and van
Iterson (1907). In recent decades, models of Schwendener and van Iterson have
been elaborated on and developed further mathematically (Adler (1974); Rothen
and Koch (1989); Levitov (1991); Kunz (2001); Atela et al. (2002)) or
numerically (Williams and Brittain (1984); Hellwig et al. (2006)), and even
realized dynamically in a physics laboratory experiment (Douady and Couder
(1996)). The causal models are founded on the basic assumption of causal
determinism that a phyllotactic pattern is a result of causal interaction of
pattern units. In particular, the position of an initiated leaf primordium is
determined by the position of the older primordia according to (supposedly
simple) causal rules. The manner in which the units are arranged depends on
the dynamic history of growth, or particularly on the course of changes in
size of leaf primordia. Thus, a common key factor of the causal models is a
gradual change in size of leaf primordia under mutual repulsion. Accordingly,
there is a variety of causal models in which the repulsive interaction is
ascribed not to the mechanical contact pressure as supposed by Airy and
Schwendener, but to a chemical diffusion process (Schoute (1913); Thornley
(1975); Mitchison (1977); Veen and Lindenmayer (1977); Young (1978); Marzec
and Kappraff (1983); Schwabe and Clewer (1984); Chapman and Perry (1987);
Roberts (1987); Steeves and Sussex (1989); Yotsumoto (1993); Koch and
Meinhardt (1994); Meinhardt et al. (1998)). There are another causal models
based on physical (Hofmeister (1868); Green et al. (1996); Newell et al.
(2008)) and chemical (Cummings and Strickland (1998)) instabilities. Recently,
more intricate models based on molecular-genetic experiments have been
discussed (Smith et al. (2006a, b); Jönsson et al. (2006); Shipman et al.
(2011)), while geometrical models have been used to interpret patterns of real
systems (Malygin (2006); Hotton et al. (2006); Zagórska-Marek and Szpak
(2008)). All these causal models are based on the assumption that divergence
angle is intrinsically variable and determined causally.
In the recent literature, we have had few opportunities of finding
phyllotactic fractions in use. Most theoretical and experimental works attach
importance to parastichy numbers instead, and the phyllotactic fraction is not
mentioned or regarded merely as an approximation even if mentioned (Williams
(1974); Steeves and Sussex (1989); Lyndon (1990); Jean (1994)). The fractional
phyllotaxis is the original problem. There are some reasons for this trend.
First, early researchers did not appreciate the structural significance of the
phyllotactic fraction (Hofmeister (1868); de Candolle (1881); Church (1920);
Hirmer (1922); Richards (1951); Snow (1955)). Second, the studies of vascular
structure organization are comparatively so few in number that they are
overshadowed by intensive research interests directed towards the shoot apical
meristems. Third, Schwendener’s causal model and its descendants are at
variance with the fractional divergence. According to the model, the
divergence angle varies depending on size of leaves, the vertical coordinate
of Fig. 2. Schwendener (1883) guessed that a fractional pattern would be made
secondarily as a result of mechanical straightening of parastichous bundles
connecting initiated leaves. Teitz (1888) confirmed indeed that the fractional
phyllotaxis is accomplished by secondary torsion occurring in the vascular
system during growth of the stem. When there is little or no secondary
distortion for lack of subsequent growth or internodal elongation, the
original pattern at the apex may grow to a similar pattern of mature organs.
This holds true for the most eye-catching patterns of closely packed
reproductive organs, which, therefore, are often compared favorably with
numerically simulated outputs of causal models. Even then, the basic concept
of the phyllotactic fraction may remain significant internally in vascular
connections (Watson and Casper (1984)). In fact, a stem with short internodes
takes a high-order fraction, which can be indistinguishable from the limit
divergence of the undistorted stem. Thus, the phyllotactic fraction may not be
judged by the external appearance alone.
In vascular plants, each leaf is connected to the main stem vascular system
through a strand of fluid-carrying vascular tissue called a leaf trace. A leaf
may have several to many leaf traces. Leaf traces diverge from the stem
vascular system some distance below or very near the nodes at which they enter
the leaves (Lestiboudois (1848); Nägeli (1858); Beck (2010)). At the level of
the shoot apex, leaf traces form parastichous strands winding obliquely round
the stem axis. As the stem elongates, the leaf traces align up along the stem
to make orthostichous bundles by forcing the whole stem to twist slightly from
the original pattern (Fig. 1), thereby a phyllotactic pattern characterized by
a phyllotactic fraction is established. In the final pattern, there is a
definite relationship between the denominator of the phyllotactic fraction and
the number of vascular orthostichies (Kirchoff (1984)). The term orthostichy
might be misleading, because the straightened bundles still may maintain their
tilted course. The fraction neatly represents geometrical arrangement of leaf
traces, and the fractional order need not mean that leaves are positioned
exactly vertically. Developmental sequences in differentiation and
vascularization of leaf primordia are numerically correlated with the
phyllotactic fraction of the shoot (Priestley and Scott (1936); Girolami
(1953); Esau (1965)). Seemingly irregular rhythmical variations in various
lengths of the external structure of a mature plant may be understood as a
consequence of a hidden phyllotactic order in the vascular system (Unruh
(1950); Kumazawa and Kumazawa (1971)). There is evidence for restricted
pathways of translocation of photosynthetic assimilates related to phyllotaxis
(Watson and Casper (1984)). The patterns of translocation are sectorial, or
the phyllotactic fraction has biological significance. The observation most
pertinent to the present work is the significant correlation existing between
the phyllotactic fraction and the number of internodes traversed by leaf
traces: The higher phyllotactic fractions are associated with the longer leaf
traces (Girolami (1953); Esau (1965)). While leaf traces of plants with
helical phyllotaxis typically traverse more than one internode, in distichous
phyllotaxis, a $\frac{1}{2}$ phyllotaxis of two-ranked leaf arrangement, and
in verticillate phyllotaxis, leaf traces are approximately one internode or
less (Beck et al. (1982)). Accordingly, low-order systems of a $\frac{1}{2}$
and $\frac{1}{3}$ phyllotaxis are seen on the stems of plants with long
internodes, while plants with short internodes show high-order fractions, as
remarked above.
The stem vascular bundles, or axial bundles, and associated leaf traces
comprise sympodia, on the nature of which there are two perspectives (Beck
(2010)). In one view, the sympodia are of cauline origin, or derived from stem
vascular tissue (Beck et al. (1982)). In the other view, they are of foliar
origin, or derived from leaf traces (Esau (1965)). There are two different
views on the causal relation between initiation of primordia and development
of leaf traces or procambial strands, the strands differentiating into
vascular bundles of xylem and phloem. In one view, the initiation of leaf
primordia brings about the differentiation of the leaf traces. Hence the
initiation of the leaf traces occurs basipetally, or in the direction from the
leaf primordia toward the vascular system of the stem. In the opposing view,
phyllotaxis of leaf primordia is dictated by the vascular organization that
has been established before the leaf primordia are initiated (Larson (1977,
1983)). The latter is consistent with the observation that an incipient leaf
trace develops acropetally, or in the direction toward the leaf primordium it
serves (Esau (1965); Nelson and Dengler (1997)). Priestley and Scott (1933)
was criticized by Snow and Snow (1934). They both do not cast doubt on
Hofmeister’s rule, that is, they share the causal view that phyllotaxis is a
natural consequence of growth and development of an individual plant. They
differ in what they regard as a basic unit of phyllotaxis. The former adopts
growth units including leaf traces, while the latter places primary emphasis
on leaf primordia at the apex. Accordingly, the former and the latter attach
little importance to the irrational and rational divergence, respectively.
Thus, the causal view has been the paradigm of phyllotaxis.
On the whole, causal models are successful in deriving indefinitely continuing
stable systems, resembling actual phyllotactic patterns. From a computational
point of view, they are particularly appealing in that they provide us with
programmable protocols leading to the golden angle. Irrespective of detailed
mechanisms, however, realistic phyllotactic patterns are derived based on the
following observational facts (Vogel (1979); Rivier et al. (1984);
Prusinkiewicz and Lindenmayer (1991)): (i) Divergence angle is constant. (ii)
The constant is the golden angle. For the sake of argument, the former is
often taken so broadly that the constant may take any value. On this premise,
phyllotaxis is rendered to a geometrical playground of mathematics. There are
mathematical arguments for (ii) based on the generalized hypothesis (i) (de
Candolle (1881); Coxeter (1972); Leigh (1972); Ridley (1982); Marzec and
Kappraff (1983)). It is often stated in this regard that the golden angle is a
special angle at which optimal packing is achieved. As a matter of fact, this
is not true literally, for it is only under the constraint (i) that the golden
angle may be said optimal and there is no a priori reason for the constancy.
For living organisms, the property (i) is far from obvious and no less
astounding than (ii), especially because the angular regularity may persist in
spite of temporal irregularity. A time interval between the formation of
successive leaves is called a plastochron, which is used as a morphological or
developmental time scale. Plants grown in different environmental conditions
may be compared in plastochron units but not in physical time units. The fact
that unit of time is a plastochron and duration of a plastochron is not
constant in physical time poses a problem for realistic causal models based on
physical time.
Phyllotactic patterns at a shoot apex are more regular than those on a mature
stem, because internodes tend to be elongated less regularly on the mature
stem. As a matter of fact, the exact level in the stem at which a bifurcation
or recombination of vascular bundles takes place is not an important
morphological constant (Dormer (1972)). Accordingly, trace lengths may vary
arbitrarily along the stem. For this reason, it is often argued that one
should devote oneself exclusively to the study of the growing apex (Church
(1904); Snow (1955)). Nonetheless, further mathematical relations for the
spiral patterns at the apex can be derived by assuming a stronger mathematical
constraint of exponential growth, according to which the leaf primordia are
arranged on logarithmic spirals in a centric representation (Fig. 1) (Church
(1904); Richards (1951); Thomas (1975); Jean (1994)). In the exponential
growth, the ratio of the distances from the center of the apex to two
successively numbered primordia is a constant, called the plastochron ratio.
For a fixed value of divergence angle, Richards (1951) has advocated the use
of a phyllotaxis index defined in terms of the plastochron ratio (cf. (4)).
The index is used to designate two sets of parastichies intersecting
orthogonally. For instance, for Fig. 1, the plastochron ratio is 1.2, the
phyllotaxis index is 3, and $(3,5)$ parastichies cross at right angles. A
fractional value of the index, such as 2.7, means that no two parastichies are
orthogonal. In this geometrical model, a shift in parastichy numbers, e.g.
from $(3,5)$ to $(5,8)$, a phenomena called rising phyllotaxis, is related to
a variation of the plastochron ratio, or the exponential growth rate. The
model has been generalized to allow for other constant divergence angles than
the golden angle (Richards (1951); Thomas (1975); Jean (1994)). In contrast to
these geometrical models based on constant divergence angle, there exist
geometrical causal models in line with Schwendener’s model, which aim at
deriving the limit divergence angle by assuming variable divergence angles
depending on plastochron, the plastochron ratio, and their own rules (van
Iterson (1907); Williams (1974); Erickson (1983); Williams and Brittain
(1984)). For a vegetative shoot, a plastochron index is defined in terms of
length of leaves, and a leaf on a shoot is labeled with a leaf plastochron
index (Erickson and Michelini (1957)). A developmental index of this kind is
indispensable for the systematic study of plant development (Meicenheimer
(2006)). The exponential growth is a practically useful approximation in
dealing with young organs and early stages of development, although it is
neither essential nor peculiar to phyllotaxis.
Despite the apparent success of causal models, neither their intrinsic
mechanisms nor predictions have yet been subjected to experimental tests
specifically. To name several problems on a descriptive level, existing causal
models that explain all types of observed patterns cannot help predicting also
a multiplicity of unreal or too rare patterns. Even when they are capable of
deriving normal patterns, they are not free from instabilities apparently
irrelevant to living organs. Causal models in general are confronted with a
subtle trade-off. Normal phyllotactic patterns must be stable enough to
account for the current prevalence in nature, while they cannot be quite
stable in order to allow for many other exceptional ideal angles just to such
a degree that they are actually existent. In short, rare patterns should be
neither too common nor too rare. It is not clear how and why this subtle
balance between stability and instability is maintained universally, since
fine control of relative size of phyllotactic units depends not only on
species but individual plants or even on parts of the individual plant. We get
puzzled all the more by the observations of more frequent occurrence of rare
patterns among fossil plants.
Causal models, whether physical or chemical, provide dynamical schemes of
self-adjusting the system under the influence of the older leaf primordia. On
the premise that divergence angles between successive leaves are freely
variable by nature, they aim to derive a special angle, normally the golden
angle, toward which the variable divergence angles tend ultimately. They do
not assume any special constant divergence a priori. For the very reasons,
they are likely to be beset with a fundamental difficulty in protecting the
system against disturbance. In this regard, Hofmeister’s empirical rule is
often overestimated. Observed patterns satisfy Hofmeister’s rule, but
Hofmeister’s rule is not sufficient for observed patterns. Hofmeister’s rule
does not imply the periodic appearance of new primordia (Kirchoff (2003)), nor
does it ensure precise regulation of the divergence of 137.5∘ (cf. Fig. 12).
It is not difficult to draw an unreal pattern according to Hofmeister’s rule.
The remarkable empirical fact is rather that divergence angle during steady
growth seems always regulated stably to one of special angles closely related
to the golden ratio. In fact, if a causal rule is to be strictly applied
throughout, fluctuations in size of the domain of influence of a leaf
primordium should inevitably leave behind everlasting irregularities
propagated in the developing pattern (Snow and Snow (1962)). Mathematically,
the instability is a general consequence of the fact that the number of
possible phyllotactic configurations proliferates as relative size of
phyllotactic units decreases. According to causal interpretations, higher
phyllotaxis becomes more vulnerable. The difficulty may not be obvious if one
were interested only in low-order patterns like a $(2,3)$ and $(3,5)$
phyllotaxis, but it should become conspicuous when dealing with a higher order
pattern which requires higher precision maintenance. Besides this stability
problem for high-order patterns, causal models have another difficulty for
low-order patterns (Sec. 3).
There are apparent geometric correlations between parastichy numbers and
relative size of primordia on the apex (Church (1904); Richards (1951);
Kirchoff (2003)) and between leaf arcs and the plastochron ratio (Rutishauser
(1998)). Causal models implement them as causal relationships with the
intention of proving that a phyllotactic pattern, especially the divergence
angle of 137.5∘, is a necessary consequence of changes in the causal agent,
relative size of leaf primordia. According to this interpretation, divergence
angles and contact parastichies must depend not only on the shape of primordia
but on the geometry of the surface on which they are located. The dependence
has been investigated by van Iterson (1907) on the assumption that all the
primordia keep a common shape while they are allowed to change their sizes. So
far, however, no direct evidence has been provided to support the presumed
causal relationship. As a matter of fact, there are very few studies in which
sufficiently detailed data are obtained to make a close comparison with the
models possible or useful (Erickson (1983)). In particular, the prediction of
causal models that rising phyllotaxis, or change in parastichy numbers, should
accompany wide variations and abrupt turns of divergence angle, as indicated
in Fig. 2, has not been supported unequivocally. On the contrary, the success
of Richards’ model indicates the exponential growth with constant divergence
angle irrespective of whether parastichy numbers rise or fall. Church (1904)
refuted Schwendener’s model by counterexamples showing normal spiral patterns
of circular primordia whose positions are widely separated. In comparing
treated plants, Maksymowych and Erickson (1977) found no significant change in
divergence angle in a correlation diagram for the plastochron ratio and
divergence angle. Statistical analysis of Fujita (1939) has revealed that
divergence angles do not depend so much on parastichy numbers as expected from
causal models (Jean (1986)). There is clear evidence against the basic
assumption that the primordia size is the causal factor of divergence angle. A
plant appears to accomplish geometrical correlations in a phyllotactic pattern
by adapting the size and shape of leafy organs as if it knows the end pattern
at which it aims. Snow and Snow (1962) observe that the secondary extension of
a leaf base adjusts itself so that divergence angle is little affected in
spite of artificial disturbances. This observation, despite the authors’
claim, undermines their space-filling mechanism that the leaf base extension
regulates the phyllotactic pattern. To the contrary, apparent causal changes
in the position, size and shape of leaves or scales in chemical or physical
contact may be just incidental phenomena (Church (1904); Richards (1948); Marc
and Hackett (1991)). No doubt there are cases in which physical or chemical
contact pressure may induce secondary displacement of compactly packed lateral
organs.
Natural selection plays no role in causal interpretations of phyllotaxis. If
one supposes to the contrary that natural selection holds the key to
understanding the golden angle at the shoot apex, then one should investigate
a special effect of the special angle, instead of its cause. In other words,
one should look for the distal or ultimate cause of the special angle, instead
of the proximity cause. This sort of theory intends to explain special traits
not in terms of immediate physiological factors, but in terms of evolutionary
forces acting on them. It aims at a full understanding of the phenomena at a
phenomenological level, independently of whatever physiological mechanisms may
be involved. There is a long history of investigations into selective
advantage of the observed divergences based on the external structure. It goes
as follows: common phyllotactic patterns distribute leaves as evenly as
possible and maximize exposure of leaves to enhance the capacity to intercept
sunlight (Wright (1873)). Such an argument is unpromising because leaves are
aligned in vertical ranks. Indeed, changes in leaf shape and stem length can
compensate for the negative effects of leaf overlap (Niklas (1988, 1998)). For
this obvious reason, it is often argued to the contrary in favor of the ‘most
irrational’ divergence angle; no two leaves lie precisely under one another
when divergence angle is equal to the golden angle (de Candolle (1881);
Wiesner (1875, 1907); Coxeter (1972); Leigh (1972); Takenaka (1994); Pearcy
and Yang (1998); Valladares and Brites (2004); King et al. (2004); Bryntsev
(2004)). There is also a long history of criticism of this view (Thompson
(1917)). In the first place, the golden angle is not a general rule for mature
shoots, and the light-capture mechanism deepens the riddle of the common
occurrence of a $\frac{2}{5}$ phyllotaxis. In general, existing theories tend
to argue for the uses of irrational angles without regard to the uses of
rational angles or vice versa.
### 1.2 Aim and scope of this paper
Let us direct attention to the internal structure, the vascular system.
Mathematical interrelationship between the initial (apical) and the mature
(vascular) phyllotactic pattern seems to have not been discussed
experimentally nor theoretically. This paper develops a theory of vascular
phyllotaxis to fill in the gap between the two distinct but intimately related
phenomena. A physical model has been described mathematically in the previous
paper (Okabe (2011)). However, the model was abstract and its relevance to
real phenomena was not clearly elucidated. The aim of this paper is to develop
the model to show its experimental validity and relevance. This is relevant to
a fundamental problem of phyllotaxis: Is phyllotaxis determined causally or
genetically? In contrast to numerous models holding the causal view, the
present model is based on the genetic perspective that special numbers in
phyllotaxis are primarily of genetic origin, so that it is assumed that
constant primordial divergence angle during steady growth is genetically
determined. According to the model, the effect of constant divergence angle is
investigated, and what value of the constant is advantageous is settled. This
work is not concerned about transient fluctuations of divergence angle during
ontogeny. Therefore, the model is compatible with any physical or chemical
causal models for the positioning of leaf initiation at the shoot apex,
although the limit divergence angle at the apex is interpreted totally
differently. The special angle is not an inevitable consequence of ontogenetic
dynamics, whether physical or chemical. It is regarded as a heritable trait of
a plant. It is supposed that once there was a wide variation in the traits of
individuals, or there have formerly been wide variations of divergence angles.
The special limit divergences found in nature have survived natural selection.
This conforms with the traditional view that biological features that are
under tight genetic control and that have very narrow ranges of variation are
believed to be adaptive (Niklas (1997)). Although the author believes that the
premise of the model, divergence angle as a trait of a plant, is not only
plausible but supported by circumstantial evidence, it has not been
unanimously accepted at present. It may be verified or refuted experimentally
in the future.
For the efficient transport of materials throughout an indefinite number of
leaves attached to a stem of a finite cross section, the leaves should be
aligned along a finite number of ‘orthostichious’ bundles. At this point, a
whole number enters the theory. There are modes of orthostichous order
depending on the initial arrangement and length of leaf traces. The number of
vascular orthostichies may be increased or decreased, but not arbitrarily. By
regarding a leaf primordium and the leaf trace(s) associated with it as the
fundamental unit of phyllotactic patterns, a mathematical correspondence is
derived between the divergence angle of the initial phyllotactic pattern,
$360\alpha_{0}$ degrees, and the phyllotactic fraction $\alpha$ of a mature
pattern, where the number of internodes traversed by the leaf traces, $n_{c}$,
plays a pivotal role. As a general rule, it has been known that phyllotactic
fraction of a vascular plant may vary sequentially during growth (Braun
(1835); Skutch (1927); Allard (1942); Puławska (1965); Larson (1977)). By
means of the mathematical relation between $\alpha_{0}$ (an irrational number)
and $\alpha$ (rational numbers), it is shown that changes in $n_{c}$ cause the
phyllotactic transitions in $\alpha$. As a natural consequence, an
evolutionary mechanism for the phenomenon of phyllotaxis is suggested.
Supporting evidence for the model and the evolutionary mechanism is presented
by analyzing experimental results.
In Sec. 2, a model and results used in the following sections are presented by
means of figures and tables without using mathematics. Tables 1$\sim$18 have
not been presented before.
In Sec. 3, observed precision of the initial divergence $\alpha_{0}$ is
explained by means of a correlation predicted between the range of
$\alpha_{0}$ and the highest-order fraction $\alpha$. In short, divergence
angle $\alpha_{0}$ of a system with a high phyllotactic fraction $\alpha$
should be accurately controlled in order to avoid unnecessary changes in
vascular structure.
In Sec. 4, phyllotaxis of Lepidodendron by Dickson (1871) is analyzed.
Diversity of phyllotaxis is discussed as a result of ineffective selective
pressures.
In Sec. 5, the vascular structure of Linum usitatissimum by Girolami (1953) is
investigated. Various relations between phyllotactic fraction and parastichy
numbers, the phyllotactic fraction $\alpha$ and the length per internode of
leaf traces $n_{c}$, and directions of parastichies and the genetic spiral are
pointed out.
In Sec. 6, the phyllotactic transition of Populus deltoides by Larson (1977)
is analyzed. It is shown that a continuous change in length of leaf traces
causes the discontinuous effect of the phyllotactic transition in the vascular
structure.
In the appendix, a relation between the trace length $n_{c}$ and the
plastochron ratio $a$ is discussed to indicate that the former serves as a
useful developmental index for the mature stem as the latter is used for the
apex.
## 2 Model
Figure 3: Phyllotactic patterns of leaf traces with a length of $n_{c}=4$
before and after secondary torsion are arranged side by side. A dotted and
dashed line of each figure represent a vertical cut of a cylinder surface
unrolled. (a) A pattern with initial divergence of $360\alpha_{0}\simeq
137.5^{\circ}$ ($\alpha_{0}=1/(1+\tau)\simeq 0.382$). (b) The final pattern of
a fractional divergence $\alpha=\frac{2}{5}$ resulting from (a). Leaf traces
in the upper part move rightward while the pattern (a) becomes (b), thereby
five 5-parastichies in (a), such as 1-6-11-16-21, align themselves to make
five orthostichies in (b).
A regular helical pattern of leaf traces is schematically plotted as a lattice
of line segments on an unrolled surface of a cylinder. The divergence angle of
the initial pattern is denoted as $360\alpha_{0}$ in degrees, which is assumed
to be less than 180 degrees, i.e., $0\leq\alpha_{0}\leq\frac{1}{2}$ without
loss of generality. Fig. 3 presents a typical pattern for $360\alpha_{0}\simeq
137.5^{\circ}$ ($\alpha_{0}\simeq 0.382$). The length of leaf traces measured
in internodes is denoted as $n_{c}$ in accordance with the previous notation
(Okabe (2011)). As in Fig. 1, $n_{c}=4$ in Fig. 3. The trace length $n_{c}$
need not be an integer; $n_{c}$ is the average number of leaf traces cut by a
transverse section (Fig. 4). As the number in a section is an integer, this
method gives a good estimate of $n_{c}$ particularly for $n_{c}\gg 1$. The
model comprises two parameters $\alpha_{0}$ and $n_{c}$. For the sake of
argument, patterns with constant values of them are considered below. Effects
of their fluctuations may be discussed based on results to be obtained.
Figure 4: For three transverse sections of a pattern of leaf traces with a
length of $n_{c}=4.3$, the number of the traces in each section is indicated
on the right-hand side below the cut line. The number averaged over sections
should approach $n_{c}$.
The leaf traces repel with each other laterally to arrange themselves in an
orthostichous pattern. The mutual interaction is likely to be regulated by the
plant hormone auxin (Beck (2010)). Fig. 3 is the final pattern resulting from
Fig. 3. Divergence of the final pattern is expressed in terms of the
phyllotactic fraction $\alpha$. The pattern of Fig. 3 is characterized with
$\alpha=\frac{2}{5}$. In Fig. 3, there are five parastichies of 1-6-11-16-21,
2-7-12-17-22, 3-8-13-18-23, 4-9-14-19-24 and 5-10-15-20-25, each of which is
called a 5-parastichy. The five 5-parastichies are lined up vertically to make
five orthostichies of the $\frac{2}{5}$ phyllotaxis in Fig. 3. In the patterns
of Fig. 3, the next visible parastichies are 3-parastichies
(1-4-7-10-13-16-19-22-25, 2-5-8-11-14-17-20-23 and 3-6-9-12-15-18-21-24) and
2-parastichies (1-3-5-7-9-11-13-15-17-19-21-23-25 and
2-4-6-8-10-12-14-16-18-20-22-24). As these parastichies remain conspicuous in
the two patterns, both patterns may be referred to as having a parastichy pair
of $(2,3)$. Thus, for $n_{c}=4$, there is one-to-one correspondence between
$\alpha_{0}\simeq 0.382$ (angle of $360\alpha_{0}\simeq 137.5^{\circ}$) of the
initial pattern and $\alpha=\frac{2}{5}$ of the final pattern. In a similar
manner, $\alpha$ is obtained for arbitrary values of $\alpha_{0}$ and $n_{c}$.
Indeed, we get $\alpha=\frac{2}{5}$ insofar as $3\leq n_{c}<5$ and
$\frac{1}{3}<\alpha_{0}<\frac{1}{2}$ ( see Okabe (2011) for the mathematical
implementation). Below we discuss phyllotactic changes in $\alpha$ that occur
when $n_{c}$ and $\alpha_{0}$ are set out of their respective ranges.
(a)
Figure 5: Change in a phyllotactic pattern of leaf traces with a length of
$n_{c}=7$ (cf. Fig. 3). (a) The initial pattern with
$\alpha_{0}=1/(1+\tau)=1/(2+\tau^{-1})$ ($360\alpha_{0}\simeq 137.5^{\circ}$).
(b) The final pattern with $\alpha=\frac{3}{8}$.
For a fixed value of $\alpha_{0}\simeq 0.382$, Fig. 5(a) is for $n_{c}=7$ in
comparison with Fig. 3 for $n_{c}=4$. As the traces of length longer than five
internodes cannot be aligned in five orthostichies, we obtain
$\alpha=\frac{3}{8}$ for Fig. 5(a), while $\alpha=\frac{2}{5}$ in Fig. 3.
Thus, it is explained that a higher phyllotactic fraction is obtained for a
longer length of leaf traces. Phyllotactic transition from
$\alpha=\frac{2}{5}$ to $\alpha=\frac{3}{8}$ occurs when $n_{c}$ increases
past a threshold value of $n_{c}=5$. Experimental evidence of this transition
is presented below in Fig. 17.
(a)
Figure 6: Change in a phyllotactic pattern of leaf traces with a length of
$n_{c}=4$ (cf. Fig. 3). (a) The initial pattern with
$\alpha_{0}=1/(3+\tau^{-1})\simeq 0.276$ ($360\alpha_{0}\simeq 99.5^{\circ}$).
(b) The final pattern with $\alpha=\frac{2}{7}$.
For a fixed value of $n_{c}$, the phyllotactic fraction $\alpha$ depends on
the initial divergence $\alpha_{0}$. For $n_{c}=4$, Fig. 6 is for
$\alpha_{0}\simeq 0.276$ (angle of $99.5^{\circ}$), which is compared with
Fig. 3 for $\alpha_{0}\simeq 0.382$ $(137.5^{\circ})$. The former leads to a
final pattern of $\alpha=\frac{2}{7}$ in Fig. 6, while the latter gives
$\alpha=\frac{2}{5}$ in Fig. 3. In fact, there are three fractional patterns
conceivable for $n_{c}=4$, namely (a) $\alpha=\frac{1}{5}$ for
$0<\alpha_{0}<\frac{1}{4}$, (b) $\alpha=\frac{2}{7}$ for
$\frac{1}{4}<\alpha_{0}<\frac{1}{3}$ and (c) $\alpha=\frac{2}{5}$ for
$\frac{1}{3}<\alpha_{0}<\frac{1}{2}$.
Figure 7: Tree diagram for the phyllotactic fraction $\alpha$. The horizontal
axis is the initial divergence $\alpha_{0}$, and the vertical axis is the
number of internodes traversed by leaf traces $n_{c}$. Numbers in parentheses
below each fraction are the parastichy pair corresponding to the fraction. By
way of explanation, let us take $\frac{3}{8}$ in the right-bottom quoter as an
example. The fraction $\frac{3}{8}$ with the parastichy pair $(3,5)$ is in
three different positions at $n_{c}=5,6$ and 7. By means of lower order
fractions lying below them, the fraction $\frac{3}{8}$ is bracketed between
$\frac{1}{3}$ and $\frac{2}{5}$. Therefore, we obtain $\alpha=\frac{3}{8}$
with the parastichy pair $(3,5)$ insofar as
$\frac{1}{3}\leq\alpha_{0}<\frac{2}{5}$ and $5\leq n_{c}<8$ (Table 1).
Similarly, we find $\alpha=\frac{5}{12}$ with $(5,7)$ for
$\frac{2}{5}\leq\alpha_{0}<\frac{3}{7}$ and $7\leq n_{c}<12$ (Table 6).
Every phyllotactic fraction for $\alpha$ has its own ranges of values for
$\alpha_{0}$ and $n_{c}$. The mathematical correspondence is presented
succinctly as a tree diagram in Fig. 7. For instance, Fig. 7 gives the
conditions $\frac{1}{4}<\alpha_{0}<\frac{1}{3}$ and $4\leq n_{c}<7$ for
$\alpha=\frac{2}{7}$. For the former inequalities, the boundary fractions
$\frac{1}{4}$ and $\frac{1}{3}$ lie below $\frac{2}{7}$ in Fig. 7. The latter
condition $4\leq n_{c}<7$ is reasoned from the vertical coordinate $n_{c}=4,5$
and $6$ of three $\frac{2}{7}$’s in Fig. 7. The phyllotactic sequence of
fractions derived from an arbitrary value of initial divergence $\alpha_{0}$
may be traced by climbing up the tree of Fig. 7 along the vertical line at
$\alpha_{0}$. For $\alpha_{0}\simeq 0.382$ $(137.5^{\circ})$, the main
sequence
$\frac{1}{2},\frac{1}{3},\frac{2}{5},\frac{3}{8},\frac{5}{13},\frac{8}{21},\cdots$
is obtained in the increasing order of $n_{c}$. The tree diagram extended for
all values of $n_{c}$ includes all conceivable phyllotactic fractions.
For the sake of convenience, let us introduce shorthand notation for the
irrational numbers found in nature,
$\displaystyle\ [n]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{n+\tau^{-1}},$ $\displaystyle\ \qquad[n,m]$
$\displaystyle\equiv$ $\displaystyle\frac{1}{n+\dfrac{1}{m+\tau^{-1}}},$
$\displaystyle\ [n,m,l]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{n+\dfrac{1}{m+\dfrac{1}{l+\tau^{-1}}}},$ (2)
and so on, where $n$, $m$, $l$ are positive integers. With this notation,
$\alpha_{0}=[2]=1/(2+\tau^{-1})=1/(1+\tau)$ gives the main sequence. The last
equality holds by the definition of $\tau$ in (1). Note that the pattern with
$\alpha_{0}=[1]$ ($360\alpha_{0}=222.5^{\circ}$) is nothing but the mirror
image of $\alpha_{0}=[2]$ ($137.5^{\circ}$), because $[1]=1-[2]$ or
$222.5^{\circ}=360^{\circ}-137.5^{\circ}$. For future reference, Tables
1$\sim$18 are provided for the initial divergence $\alpha_{0}$ given by
typical irrational numbers. These are not exhaustive, but they include almost
all phyllotactic fractions observed in nature. The main sequence is presented
in Table 1. In the second column for $\alpha=\frac{2}{5}$, ‘$(2,3)$’ in the
second row represents the parastichy pair corresponding to the fraction
$\frac{2}{5}$, ‘$3\sim$’ in the third row abbreviates $3\leq n_{c}<5$, where 5
for the upper limit is taken from the next column, and
‘$\frac{1}{3}\sim\frac{1}{2}$’ in the fourth row indicates
$\frac{1}{3}<\alpha_{0}<\frac{1}{2}$. As an example, let us take a fraction
$\alpha=\frac{21}{76}$. It is found in the eighth column of Table 2, from
which the conditions $47\leq n_{c}<76$ and
$\frac{8}{29}<\alpha_{0}<\frac{13}{47}$ are read. These results are used in
the next section (Table 20).
The tables show that the denominator of a fraction $\alpha$ and the parastichy
numbers are correlated with the threshold numbers for $n_{c}$. The numbers
comprise a characteristic sequence of integers. The main sequence in Table 1
is characterized with the Fibonacci sequence of 1, 2, 3, 5, 8, 13, 21,
$\cdots$, while Table 6 has a sequence of 1, 2, 3, 2, 5, 7, 12, 19, $\cdots$,
which is sometimes called the lateral sequence. Three consecutive numbers of a
sequence satisfy the Fibonacci recurrence relation ($2+5=7$, $5+7=12$,
$7+12=19$), except for the first several numbers (like 1, 2, 3 in the lateral
sequence). Therefore, each phyllotactic sequence is referred to by a pair of
seed integers for the Fibonacci recurrence formula. The seed pair of each
table, such as $(2,5)$ in Table 6, is highlighted in boldface.
$\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{5}{13}$ | $\frac{8}{21}$ | $\frac{13}{34}$ | $\frac{21}{55}$ | $\frac{34}{89}$ | $\frac{55}{144}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (5,8) | (8,13) | (13,21) | (21,34) | (34,55) | (55,89)
$n_{c}$ | $3\sim$ | $5\sim$ | $8\sim$ | $13\sim$ | $21\sim$ | $34\sim$ | $55\sim$ | $89\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{3}{8}\sim\frac{2}{5}$ | $\frac{3}{8}\sim\frac{5}{13}$ | $\frac{8}{21}\sim\frac{5}{13}$ | $\frac{8}{21}\sim\frac{13}{34}$ | $\frac{21}{55}\sim\frac{13}{34}$ | $\frac{21}{55}\sim\frac{34}{89}$
Table 1: Parastichy numbers (in parentheses) and ranges of $n_{c}$ and $\alpha_{0}$ for the phyllotactic fractions $\alpha$ belonging to the main sequence with the limit divergence of $\alpha_{0}=[2]=1/(2+\tau^{-1})\simeq 0.3820$ ($360\alpha_{0}\simeq 137.5^{\circ}$). The parastichy pairs are generated from the seed pair $(1,2)$ for $\alpha=\frac{1}{3}$ (not shown) by a Fibonacci recurrence relation. The golden angle $\alpha_{0}=[2]$ and the main sequence are called the Fibonacci angle and the Fibonacci sequence. $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{11}$ | $\frac{5}{18}$ | $\frac{8}{29}$ | $\frac{13}{47}$ | $\frac{21}{76}$ | $\frac{34}{123}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (4,7) | (7,11) | (11,18) | (18,29) | (29,47) | (47,76)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 11$\sim$ | 18$\sim$ | 29$\sim$ | 47$\sim$ | 76$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{2}{7}$ | $\frac{3}{11}\sim\frac{2}{7}$ | $\frac{3}{11}\sim\frac{5}{18}$ | $\frac{8}{29}\sim\frac{5}{18}$ | $\frac{8}{29}\sim\frac{13}{47}$ | $\frac{21}{76}\sim\frac{13}{47}$
Table 2: Table for the limit divergence of $\alpha_{0}=[3]=1/(3+\tau^{-1})\simeq 0.2764$ ($360\alpha_{0}\simeq 99.5^{\circ}$). The parastichy pairs are generated from the seed pair $(1,3)$ by a Fibonacci recurrence relation. The sequence 1,3,4,7,11,$\cdots$ is called the Lucas sequence or the first accessory sequence. $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{2}{9}$ | $\frac{3}{14}$ | $\frac{5}{23}$ | $\frac{8}{37}$ | $\frac{13}{60}$ | $\frac{21}{97}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (4,5) | (5,9) | (9,14) | (14,23) | (23,37) | (37,60)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 9$\sim$ | 14$\sim$ | 23$\sim$ | 37$\sim$ | 60$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $\frac{1}{5}\sim\frac{1}{4}$ | $\frac{1}{5}\sim\frac{2}{9}$ | $\frac{3}{14}\sim\frac{2}{9}$ | $\frac{3}{14}\sim\frac{5}{23}$ | $\frac{8}{37}\sim\frac{5}{23}$ | $\frac{8}{37}\sim\frac{13}{60}$
Table 3: Table for the limit divergence of $\alpha_{0}=[4]=1/(4+\tau^{-1})\simeq 0.2165$ ($360\alpha_{0}\simeq 78.0^{\circ}$). The parastichy pairs except (1,3) are generated from the seed pair (1,4) by a Fibonacci recurrence relation. The sequence 1,4,5,9,14,$\cdots$ is called the second accessory sequence. $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{2}{11}$ | $\frac{3}{17}$ | $\frac{5}{28}$ | $\frac{8}{45}$ | $\frac{13}{73}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (1,5) | (5,6) | (6,11) | (11,17) | (17,28) | (28,45)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 6$\sim$ | 11$\sim$ | 17$\sim$ | 28$\sim$ | 45$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $0\sim\frac{1}{5}$ | $\frac{1}{6}\sim\frac{1}{5}$ | $\frac{1}{6}\sim\frac{2}{11}$ | $\frac{3}{17}\sim\frac{2}{11}$ | $\frac{3}{17}\sim\frac{5}{28}$ | $\frac{8}{45}\sim\frac{5}{28}$
Table 4: $\alpha_{0}=[5]=1/(5+\tau^{-1})\simeq 0.1780$ ($360\alpha_{0}\simeq 64.1^{\circ}$). $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{1}{7}$ | $\frac{2}{13}$ | $\frac{3}{20}$ | $\frac{5}{33}$ | $\frac{8}{53}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (1,5) | (1,6) | (6,7) | (7,13) | (13,20) | (20,33)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 6$\sim$ | 7$\sim$ | 13$\sim$ | 20$\sim$ | 33$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $0\sim\frac{1}{5}$ | $0\sim\frac{1}{6}$ | $\frac{1}{7}\sim\frac{1}{6}$ | $\frac{1}{7}\sim\frac{2}{13}$ | $\frac{3}{20}\sim\frac{2}{13}$ | $\frac{3}{20}\sim\frac{5}{33}$
Table 5: $\alpha_{0}=[6]=1/(6+\tau^{-1})\simeq 0.15112$ ($360\alpha_{0}\simeq 54.4^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{5}{12}$ | $\frac{8}{19}$ | $\frac{13}{31}$ | $\frac{21}{50}$ | $\frac{34}{81}$ | $\frac{55}{131}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (5,7) | (7,12) | (12,19) | (19,31) | (31,50) | (50,81)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 12$\sim$ | 19$\sim$ | 31$\sim$ | 50$\sim$ | 81$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{3}{7}$ | $\frac{5}{12}\sim\frac{3}{7}$ | $\frac{5}{12}\sim\frac{8}{19}$ | $\frac{13}{31}\sim\frac{8}{19}$ | $\frac{13}{31}\sim\frac{21}{50}$ | $\frac{34}{81}\sim\frac{21}{50}$
Table 6: $\alpha_{0}=[2,2]=1/(2+1/(2+\tau^{-1}))\simeq 0.4198$ ($360\alpha_{0}\simeq 151.1^{\circ}$). The parastichy pairs except (2,3) are generated from the seed pair (2,5). The sequence 2,5,7,12,19,$\cdots$ is called the first lateral sequence. $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{10}$ | $\frac{5}{17}$ | $\frac{8}{27}$ | $\frac{13}{44}$ | $\frac{21}{71}$ | $\frac{34}{115}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (3,7) | (7,10) | (10,17) | (17,27) | (27,44) | (44,71)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 10$\sim$ | 17$\sim$ | 27$\sim$ | 44$\sim$ | 71$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{3}{10}$ | $\frac{5}{17}\sim\frac{3}{10}$ | $\frac{5}{17}\sim\frac{8}{27}$ | $\frac{13}{44}\sim\frac{8}{27}$ | $\frac{13}{44}\sim\frac{21}{71}$
Table 7: $\alpha_{0}=[3,2]=1/(3+1/(2+\tau^{-1}))\simeq 0.2957$ ($360\alpha_{0}\simeq 106.4^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{4}{9}$ | $\frac{7}{16}$ | $\frac{11}{25}$ | $\frac{18}{41}$ | $\frac{29}{66}$ | $\frac{47}{107}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (2,7) | (7,9) | (9,16) | (16,25) | (25,41) | (41,66)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 9$\sim$ | 16$\sim$ | 25$\sim$ | 41$\sim$ | 66$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{4}{9}$ | $\frac{7}{16}\sim\frac{4}{9}$ | $\frac{7}{16}\sim\frac{11}{25}$ | $\frac{18}{41}\sim\frac{11}{25}$ | $\frac{18}{41}\sim\frac{29}{66}$
Table 8: $\alpha_{0}=[2,3]=1/(2+1/(3+\tau^{-1}))\simeq 0.4393$ ($360\alpha_{0}\simeq 158.1^{\circ}$). The sequence 2,7,9,16,$\cdots$ is called the second lateral sequence. $\alpha$ | $\frac{2}{7}$ | $\frac{3}{10}$ | $\frac{4}{13}$ | $\frac{7}{23}$ | $\frac{11}{36}$ | $\frac{18}{59}$ | $\frac{29}{95}$ | $\frac{47}{154}$
---|---|---|---|---|---|---|---|---
| (3,4) | (3,7) | (3,10) | (10,13) | (13,23) | (23,36) | (36,59) | (59,95)
$n_{c}$ | 4$\sim$ | 7$\sim$ | 10$\sim$ | 13$\sim$ | 23$\sim$ | 36$\sim$ | 59$\sim$ | 95$\sim$
$\alpha_{0}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{1}{3}$ | $\frac{3}{10}\sim\frac{1}{3}$ | $\frac{3}{10}\sim\frac{4}{13}$ | $\frac{7}{23}\sim\frac{4}{13}$ | $\frac{7}{23}\sim\frac{11}{36}$ | $\frac{18}{59}\sim\frac{11}{36}$ | $\frac{18}{59}\sim\frac{29}{95}$
Table 9: $\alpha_{0}=[3,3]=1/(3+1/(3+\tau^{-1}))\simeq 0.3052$ ($360\alpha_{0}\simeq 109.9^{\circ}$). $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{2}{9}$ | $\frac{3}{13}$ | $\frac{5}{22}$ | $\frac{8}{35}$ | $\frac{13}{57}$ | $\frac{21}{92}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (4,5) | (4,9) | (9,13) | (13,22) | (22,35) | (35,57)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 9$\sim$ | 13$\sim$ | 22$\sim$ | 35$\sim$ | 57$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $\frac{1}{5}\sim\frac{1}{4}$ | $\frac{2}{9}\sim\frac{1}{4}$ | $\frac{2}{9}\sim\frac{3}{13}$ | $\frac{5}{22}\sim\frac{3}{13}$ | $\frac{5}{22}\sim\frac{8}{35}$ | $\frac{13}{57}\sim\frac{8}{35}$
Table 10: $\alpha_{0}=[4,2]=1/(4+1/(2+\tau^{-1}))\simeq 0.2282$ ($360\alpha_{0}\simeq 82.2^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{4}{11}$ | $\frac{7}{19}$ | $\frac{11}{30}$ | $\frac{18}{49}$ | $\frac{29}{79}$ | $\frac{47}{128}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (3,8) | (8,11) | (11,19) | (19,30) | (30,49) | (49,79)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 8$\sim$ | 11$\sim$ | 19$\sim$ | 30$\sim$ | 49$\sim$ | 79$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{1}{3}\sim\frac{3}{8}$ | $\frac{4}{11}\sim\frac{3}{8}$ | $\frac{4}{11}\sim\frac{7}{19}$ | $\frac{11}{30}\sim\frac{7}{19}$ | $\frac{11}{30}\sim\frac{18}{49}$ | $\frac{29}{79}\sim\frac{18}{49}$
Table 11: $\alpha_{0}=[2,1,2]=1/(2+1/(1+1/(2+\tau^{-1})))\simeq 0.3672$ ($360\alpha_{0}\simeq 132.2^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{5}{12}$ | $\frac{7}{17}$ | $\frac{12}{29}$ | $\frac{19}{46}$ | $\frac{31}{75}$ | $\frac{50}{121}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (5,7) | (5,12) | (12,17) | (17,29) | (29,46) | (46,75)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 12$\sim$ | 17$\sim$ | 29$\sim$ | 46$\sim$ | 75$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{3}{7}$ | $\frac{2}{5}\sim\frac{5}{12}$ | $\frac{7}{17}\sim\frac{5}{12}$ | $\frac{7}{17}\sim\frac{12}{29}$ | $\frac{19}{46}\sim\frac{12}{29}$ | $\frac{19}{46}\sim\frac{31}{75}$
Table 12: $\alpha_{0}=[2,2,2]=1/(2+1/(2+1/(2+\tau^{-1})))\simeq 0.4133$ ($360\alpha_{0}\simeq 148.8^{\circ}$). $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{11}$ | $\frac{4}{15}$ | $\frac{7}{26}$ | $\frac{11}{41}$ | $\frac{18}{67}$ | $\frac{29}{108}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (4,7) | (4,11) | (11,15) | (15,26) | (26,41) | (41,67)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 11$\sim$ | 15$\sim$ | 26$\sim$ | 41$\sim$ | 67$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{2}{7}$ | $\frac{1}{4}\sim\frac{3}{11}$ | $\frac{4}{15}\sim\frac{3}{11}$ | $\frac{4}{15}\sim\frac{7}{26}$ | $\frac{11}{41}\sim\frac{7}{26}$ | $\frac{11}{41}\sim\frac{18}{67}$
Table 13: $\alpha_{0}=[3,1,2]=1/(3+1/(1+1/(2+\tau^{-1})))\simeq 0.2686$ ($360\alpha_{0}\simeq 96.7^{\circ}$). $\alpha$ | $\frac{1}{4}$ | $\frac{2}{7}$ | $\frac{3}{10}$ | $\frac{5}{17}$ | $\frac{7}{24}$ | $\frac{12}{41}$ | $\frac{19}{65}$ | $\frac{31}{106}$
---|---|---|---|---|---|---|---|---
| (1,3) | (3,4) | (3,7) | (7,10) | (7,17) | (17,24) | (24,41) | (41,65)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 7$\sim$ | 10$\sim$ | 17$\sim$ | 24$\sim$ | 41$\sim$ | 65$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $\frac{1}{4}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{1}{3}$ | $\frac{2}{7}\sim\frac{5}{17}$ | $\frac{2}{7}\sim\frac{5}{17}$ | $\frac{7}{24}\sim\frac{5}{17}$ | $\frac{7}{24}\sim\frac{12}{41}$ | $\frac{19}{65}\sim\frac{12}{41}$
Table 14: $\alpha_{0}=[3,2,2]=1/(3+1/(2+1/(2+\tau^{-1})))\simeq 0.2924$ ($360\alpha_{0}\simeq 105.3^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{4}{11}$ | $\frac{5}{14}$ | $\frac{9}{25}$ | $\frac{14}{39}$ | $\frac{23}{64}$ | $\frac{37}{103}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (3,8) | (3,11) | (11,14) | (14,25) | (25,39) | (39,64)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 8$\sim$ | 11$\sim$ | 14$\sim$ | 25$\sim$ | 39$\sim$ | 64$\sim$
$\alpha_{0}$ | $\frac{1}{2}\sim\frac{1}{3}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{1}{3}\sim\frac{3}{8}$ | $\frac{1}{3}\sim\frac{4}{11}$ | $\frac{5}{14}\sim\frac{4}{11}$ | $\frac{5}{14}\sim\frac{9}{25}$ | $\frac{14}{39}\sim\frac{9}{25}$ | $\frac{14}{39}\sim\frac{23}{64}$
Table 15: $\alpha_{0}=[2,1,3]=1/(2+1/(1+1/(3+\tau^{-1})))\simeq 0.3593$
($360\alpha_{0}\simeq 129.3^{\circ}$).
$\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{4}{9}$ | $\frac{7}{16}$ | $\frac{10}{23}$ | $\frac{17}{39}$ | $\frac{27}{62}$ | $\frac{44}{101}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (2,7) | (7,9) | (7,16) | (16,23) | (23,39) | (39,62)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 9$\sim$ | 16$\sim$ | 23$\sim$ | 39$\sim$ | 62$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{1}{2}$ | $\frac{3}{7}\sim\frac{4}{9}$ | $\frac{3}{7}\sim\frac{7}{16}$ | $\frac{10}{23}\sim\frac{7}{16}$ | $\frac{10}{23}\sim\frac{17}{39}$ | $\frac{27}{62}\sim\frac{17}{39}$
Table 16: $\alpha_{0}=[2,3,2]=1/(2+1/(3+1/(2+\tau^{-1})))\simeq 0.4356$ ($360\alpha_{0}\simeq 156.8^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{7}$ | $\frac{5}{12}$ | $\frac{7}{17}$ | $\frac{9}{22}$ | $\frac{16}{39}$ | $\frac{25}{61}$ | $\frac{41}{100}$
---|---|---|---|---|---|---|---|---
| (2,3) | (2,5) | (5,7) | (5,12) | (5,17) | (17,22) | (22,39) | (39,61)
$n_{c}$ | 3$\sim$ | 5$\sim$ | 7$\sim$ | 12$\sim$ | 17$\sim$ | 22$\sim$ | 39$\sim$ | 61$\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{1}{2}$ | $\frac{2}{5}\sim\frac{3}{7}$ | $\frac{2}{5}\sim\frac{5}{12}$ | $\frac{2}{5}\sim\frac{7}{17}$ | $\frac{9}{22}\sim\frac{7}{17}$ | $\frac{9}{22}\sim\frac{16}{39}$ | $\frac{25}{61}\sim\frac{16}{39}$
Table 17: $\alpha_{0}=[2,2,3]=1/(2+1/(2+1/(3+\tau^{-1})))\simeq 0.4100$ ($360\alpha_{0}\simeq 147.6^{\circ}$). $\alpha$ | $\frac{2}{5}$ | $\frac{3}{8}$ | $\frac{5}{13}$ | $\frac{7}{18}$ | $\frac{12}{31}$ | $\frac{19}{49}$ | $\frac{31}{80}$ | $\frac{50}{129}$
---|---|---|---|---|---|---|---|---
| (2,3) | (3,5) | (5,8) | (5,13) | (13,18) | (18,31) | (31,49) | (49,80)
$n_{c}$ | $3\sim$ | $5\sim$ | $8\sim$ | $13\sim$ | $18\sim$ | $31\sim$ | $49\sim$ | $80\sim$
$\alpha_{0}$ | $\frac{1}{3}\sim\frac{1}{2}$ | $\frac{1}{3}\sim\frac{2}{5}$ | $\frac{3}{8}\sim\frac{2}{5}$ | $\frac{5}{13}\sim\frac{2}{5}$ | $\frac{5}{13}\sim\frac{7}{18}$ | $\frac{12}{31}\sim\frac{7}{18}$ | $\frac{12}{31}\sim\frac{19}{49}$ | $\frac{31}{80}\sim\frac{19}{49}$
Table 18: $\alpha_{0}=[2,1,1,2]=1/(2+1/(1+1/(1+1/(2+\tau^{-1}))))\simeq 0.3876$ ($360\alpha_{0}\simeq 139.5^{\circ}$). $\alpha$ | $\frac{1}{4}$ | $\frac{1}{5}$ | $\frac{1}{6}$ | $\frac{1}{7}$ | $\frac{2}{13}$ | $\frac{3}{19}$ | $\frac{5}{32}$ | $\frac{8}{51}$
---|---|---|---|---|---|---|---|---
| (1,3) | (1,4) | (1,5) | (1,6) | (6,7) | (6,13) | (13,19) | (19,32)
$n_{c}$ | 3$\sim$ | 4$\sim$ | 5$\sim$ | 6$\sim$ | 7$\sim$ | 13$\sim$ | 19$\sim$ | 32$\sim$
$\alpha_{0}$ | $0\sim\frac{1}{3}$ | $0\sim\frac{1}{4}$ | $0\sim\frac{1}{5}$ | $0\sim\frac{1}{6}$ | $\frac{2}{13}\sim\frac{1}{6}$ | $\frac{2}{13}\sim\frac{3}{19}$ | $\frac{5}{32}\sim\frac{3}{19}$ | $\frac{5}{32}\sim\frac{8}{51}$
Table 19: $\alpha_{0}=[6,2]=1/(6+1/(2+\tau^{-1})\simeq 0.1567$
($360\alpha_{0}\simeq 56.4^{\circ}$).
Having prepared the mathematical relationship between the initial divergence
$\alpha_{0}$, the final divergence $\alpha$ and the trace length $n_{c}$, we
are in a position to give an account of what is special about the golden
angle. As shown below in Figs. 16 and 19, discontinuous change in phyllotactic
fraction $\alpha$, or phyllotactic transition, involves reconstruction of the
vascular structure. Therefore, it is advantageous for a plant to suppress the
transitions as few as possible. As internodes vary in length during growth,
the trace length per internode $n_{c}$ may change accordingly. For instance,
$n_{c}$ may depend on the plastochron ratio $a$ (A). Patterns with a fraction
that appears in many places of Fig. 7 are stable against occasional changes in
$n_{c}$. The lowest fraction that appears more than once is $\frac{2}{5}$.
Thus, systems with initial divergence angle giving rise to stable fractions
are most likely to survive. Among all possible values of $\alpha_{0}$, the
initial divergence angle which suffers the least number of phyllotactic
transitions is the golden angle $\alpha_{0}=[2]$ (137.5 degrees). This is a
summary of the evolutionary mechanism for the golden angle (Okabe (2011)).
Fig. 8 shows phyllotactic fractions resulting from various representative
values of $\alpha_{0}$ while $n_{c}$ increases up to eleven. The number of
phyllotactic transitions is indicated by a dashed line. In this example,
initial divergence angles from 135∘ to 154∘ are most likely to be naturally
selected.
Figure 8: Phyllotactic fractions resulting while $n_{c}$ increases to eleven
are arranged vertically for eight representative values of the limit
divergence $\alpha_{0}$. A dashed line is the number of phyllotactic
transitions counted from a $\frac{1}{2}$ phyllotaxis. Initial divergences
within a narrow range around the golden angle $\alpha_{0}=[2]\simeq 0.382$
($137.5^{\circ}$) are most likely to survive.
## 3 Precision of initial divergence angle
As the evolutionary mechanism relies on statistical screening processes, it
does not predict a limit divergence angle with unlimited precision. It is an
empirical fact that divergence angles at the level of the shoot apex are
regulated toward a mean value comparable with an ideal angle given by the
formula (2) after some transient fluctuations (Davies (1939); Snow and Snow
(1962); Barabé et al. (2010)). Excepting initial fluctuations, the precision
with which leaves are organized on the apical meristems is remarkable. It is
undoubtedly controlled by genetics, though it may be slightly affected by
light stimuli depending on the orientation (Kumazawa and Kumazawa (1971)).
Twenty samples of young shoots of Erigeron sumatrensis (Sumatran fleabane)
show mean divergence angles from 137.23∘ to 137.97∘ with the sample average of
$137.499\pm 0.212^{\circ}$ (Kumazawa and Kumazawa (1971)). The mean divergence
angle of the individual plant may deviate statistically significantly from the
ideal limit angle (Maksymowych and Erickson (1977)). Sometimes there occur
other ideal divergence angles than the normal golden angle of 137.5∘.
Phyllotaxis of Musa sapientum (banana) changes with the age of the plant from
$\frac{2}{5}$ through $\frac{3}{7}$ to $\frac{4}{9}$ (Skutch (1927)). This is
consistent with a unique initial divergence of $\alpha_{0}=[2,3]$ (Table 8),
which seems to be true for all species of Musa propagated vegetatively.
Rutishauser (1998) has presented a remarkably exotic pattern of Picea abies
(Norway spruce) showing a $(6,13)$ phyllotaxis (Table 19).
The evolutionary mechanism predicts a correlation between the range of values
of the initial divergence $\alpha_{0}$ and the highest-order fraction $\alpha$
attained in evolutionary or phylogenetic processes. The correlation may seem
strange at first glance, as it appears as an advanced correlation in
developmental or ontogenetic processes of a plant; the precision of divergence
angles on a young shoot is determined by the phyllotactic form at its
maturity. This phylogenetic correlation is contrasted with the instantaneous
correlation that causal models predict between the divergence angle and
parastichies of the standing pattern. In general, divergence angles of a
$(3,5)$ phyllotaxis are widely variable within
$\frac{1}{3}<\alpha_{0}<\frac{2}{5}$, whereas the range is narrowed to
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ when the parastichy pair is raised to
$(5,8)$. Remark that these are general results drawn from regularity of
phyllotactic patterns. The ranges may be restricted further depending on
specific assumptions of models. For instance, consider a regular pattern with
a parastichy pair $(1,2)$, which is realized for any value of divergence
angle. According to Schwendener’s model, however, $(1,2)$ patterns for
$0<\alpha_{0}<0.36$, i.e., from 0 to 128.6 degrees, are not realized, for a
transition to a $(2,3)$ phyllotaxis intervenes at $\alpha_{0}=0.36$ (Adler
(1974); Levitov (1991); Douady and Couder (1996)). See the top branch of the
zigzag path in Fig. 2. The threshold angle $\alpha_{0}=0.36$ specifically
depends on geometrical assumptions, e.g. the circular shape of ‘leaves’ on the
stem cylinder surface. Accordingly, the divergence angles for the parastichy
pair $(1,2)$ is predicted to vary continuously within
$0.36<\alpha_{0}<\frac{1}{2}$, i.e., from 128.6∘ to 180∘. The range is
narrowed substantially but still so wide that it is incompatible with
observations that divergence angles are very close to 137.5∘ even in systems
of low phyllotaxis. Causal models attain a target pattern with 137.5∘ by way
of an almost opposite $(1,2)$ pattern with divergence of about 180∘.
Therefore, they cannot but allow the wide latitude of divergence angles for
the $(1,2)$ pattern, in disagreement with precise control of actual systems
(cf. Fig. 12). This is a very old problem which van Iterson (1907) (p. 247)
was well aware of. Nonetheless, it has been left unnoticed despite a marginal
rise of various causal models in recent years. With reference to experimental
evidence, Church (1904) (p. 340) remarks that already at a $(2,3)$ system the
ideal angle is attained within an error of about one degree. The present model
explains the non-correspondence between divergence angle $\alpha_{0}$ and
parastichy numbers by relating the allowed range of $\alpha_{0}$ not with the
parastichy numbers but with the highest order fraction $\alpha$ that the plant
would attain in its mature state.
By measuring initial divergence angles for thirty species of plants, Fujita
(1939) found that frequency distributions of the divergence angles are almost
independent of the parastichy numbers. The divergence angles cluster in a
narrow range. The width of the range quantifies the remarkable constancy of
the divergence angle (Fig. 12). This results look puzzling from a causal
viewpoint (Jean (1986, 1994)). By contrast, they are consistent with the
evolutionary mechanism in that the initial angle $\alpha_{0}$ is independent
of the parastichy numbers. According to Fujita (1939), initial divergence
angles for the main sequence fall within $138\pm 7^{\circ}$ (Fujita (1939)),
irrespective of the parastichy pair. This corresponds to
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ (135 to 144 degrees), which is as
expected for the highest-order fraction of $\alpha=\frac{5}{13}$ (Table 1).
Similarly, an estimate of $99\pm 4^{\circ}$ for Cunninghamia lanceolata (China
fir) (Fujita (1939)) is consistent with $\frac{1}{4}<\alpha_{0}<\frac{2}{7}$
for $\alpha=\frac{3}{11}$ in Table 2, and a narrow scattering of $151\pm
3^{\circ}$ for $(2,5)$ phyllotaxis at the apex of Cephalotaxus drupacea
(Japanese plum yew) (Fujita (1937)) is consistent with
$\frac{5}{12}<\alpha_{0}<\frac{3}{7}$ for $\alpha=\frac{8}{19}$ (Table 6).
Let us make a general remark that parastichy does not substitute for
divergence angle. The former depends on size and shape of the pattern unit or
on a radial or internodal length scale. Therefore, several different
parastichy pairs may be arbitrarily related to a single divergence angle.
Parastichy numbers given in Tables 1$\sim$18 are the simplest pairs, which
normally represent contact parastichies.
Large fluctuations in the initial divergence $\alpha_{0}$ may cause the
phyllotactic transition in the vascular structure, even if the trace length
$n_{c}$ is fixed constant. To suppress the transition that could happen, the
divergence $\alpha_{0}$ has to be restricted within one of the ranges
determined by $n_{c}$. For a fixed length of $n_{c}=5$, the fraction $\alpha$
is plotted against the initial divergence $\alpha_{0}$ in Fig. 9. To maintain
a $\frac{3}{8}$ phyllotaxis, the initial divergence $\alpha_{0}$ must stay
within $\frac{1}{3}<\alpha_{0}<\frac{2}{5}$ (from $120^{\circ}$ to
$144^{\circ}$); otherwise one would observe occasional excursions to
$\frac{2}{7}$ (for $\alpha_{0}<\frac{1}{3}$) or $\frac{3}{7}$ (for
$\frac{2}{5}<\alpha_{0}$) in the midst of a steady course of the $\frac{3}{8}$
phyllotaxis. Similarly, to maintain a $\frac{5}{13}$ phyllotaxis, the initial
divergence $\alpha_{0}$ has to be kept within
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ (from $135^{\circ}$ to $144^{\circ}$);
otherwise one would find $\frac{4}{11}$ (for $\alpha_{0}<\frac{3}{8}$) or
$\frac{5}{12}$ (for $\frac{2}{5}<\alpha_{0}$) within the mature state of the
$\frac{5}{13}$ phyllotaxis (cf. Fig. 8). Thus, it is explained why the initial
divergence angle has to be ‘quantized’ or fixed around a special constant with
precision determined by the length of leaf traces. For this mechanism to work,
stepwise changes in the fraction $\alpha$ of the vascular order, which are
presumed to occur if the initial divergence angle $\alpha_{0}$ were not
optimum, should incur penalties of extra energy. Thus, efficiency of the
mechanism depends on the energy cost per transition, which should depend on
species. By and large, however, the number of transition may be used as a good
measure of the total cost, at least as a first approximation (Fig. 8).
Figure 9: Phyllotactic fraction $\alpha$ versus initial divergence angle
$\alpha_{0}$ for a fixed length $n_{c}=5$ of leaf traces. The fractional order
changes discontinuously while $\alpha_{0}$ changes continuously. There are
plateaus for five phyllotactic orders with $\frac{1}{6}$, $\frac{2}{9}$,
$\frac{2}{7}$, $\frac{3}{8}$ and $\frac{3}{7}$. The initial divergence
$\alpha_{0}$ is ‘quantized’ within a plateau to avoid the discontinuous
transition. In other words, leaves are initiated regularly with a given
angular precision. The wide plateau for $\alpha=\frac{1}{6}$ is the most
unstable against changes in $n_{c}$, while the plateau at $\alpha=\frac{3}{8}$
is the most stable.
## 4 Fossil record and diversity of phyllotaxis
Dickson (1871) found that nine among thirteen specimens of fossil remains of
Lepidodendron (scale tree) show helical phyllotaxis, of which only three
belong to the main sequence. This is in striking contrast to the current
dominance of the main sequence in existing species (Fujita (1938); Zagórska-
Marek (1985); Jean (1994)). Therefore, Dickson concluded that the phyllotaxis
of Lepidodendron is extremely variable, as much so as that of those most
variable plants like cacti. His results provide us with important information
when they are analyzed in terms of the model.
In the second and third line of Table 20, the phyllotactic fractions and the
parastichy pairs for the nine specimens are presented after Dickson. The
fourth and fifth line are the corresponding ranges of $n_{c}$ and $\alpha_{0}$
according to Tables 1, 2, 3, 6, and 9. The last line is the limit divergence
in terms of the bracket notation defined by (2) in the last section. For
instance, in the second column, the specimen No. 1 has a $\frac{13}{34}$
phyllotaxis $(\alpha=\frac{13}{34})$ with the parastichy pair of $(13,21)$,
for which $21\leq n_{c}<34$ and $\frac{8}{21}<\alpha_{0}<\frac{5}{13}$. The
limit divergence of $\alpha_{0}=[2]$ (137.5∘) satisfies the latter condition.
The specimens Nos. 1-3 belong to the main sequence $\alpha_{0}=[2]$. Fig.
10(a) represents graphically the parameter regions allowed for $n_{c}$ and
$\alpha_{0}$. By comparison, Fig. 10 gives a theoretical result for the most
favored regions in which the number of phyllotactic transitions is minimal
(Okabe (2011)).
No. | 1 | 2 | 3 | 8 | 9,10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---
$\alpha$ | $\frac{13}{34}$ | $\frac{21}{55}$ | $\frac{55}{144}$ | $\frac{21}{76}$ | $\frac{13}{60}$ | $\frac{21}{50}$ | $\frac{18}{59}$ | $\frac{47}{154}$
| (13,21) | (21,34) | (55,89) | (29,47) | (23,37) | (19,31) | (23,36) | (59,95)
$n_{c}$ | [21, 34) | [34,55) | [89,144) | [47,76) | [37,60) | [31,50) | [36,59) | [95,154)
$\alpha_{0}$ | $\frac{8}{21}\sim\frac{5}{13}$ | $\frac{8}{21}\sim\frac{13}{34}$ | $\frac{21}{55}\sim\frac{34}{89}$ | $\frac{8}{29}\sim\frac{13}{47}$ | $\frac{8}{37}\sim\frac{5}{23}$ | $\frac{13}{31}\sim\frac{8}{19}$ | $\frac{7}{23}\sim\frac{11}{36}$ | $\frac{18}{59}\sim\frac{29}{95}$
| [2] | [2] | [2] | [3] | [4] | [2,2] | [3,3] | [3,3]
Table 20: Ranges of $n_{c}$ and $\alpha_{0}$ for the phyllotactic fraction
$\alpha$ and the contact parastichy pair $(n,m)$ of the nine specimens of
Lepidodendron by Dickson (1871). Abbreviations $[21,34)$ and
$\frac{8}{21}\sim\frac{5}{13}$ mean $21\leq n_{c}<34$ and
$\frac{8}{21}<\alpha_{0}<\frac{5}{13}$. The bracket notation in (2) is used
for the limit divergence in the last row. Only the first three specimens
belong to the main sequence $\alpha_{0}=[2]$ (137.5∘).
(a)
Figure 10: (a) Ranges of $\alpha_{0}$ and $n_{c}$ for the specimens of
Dickson (1871) (Table 20) are painted black in the $\alpha_{0}$-$n_{c}$ plane.
(b) A theoretical result for the regions in which the number of phyllotactic
transitions is minimal (adapted from Fig. 13 of Okabe (2011)). The golden
angle $\alpha_{0}\simeq 0.382$ (137.5∘) is singled out for $n_{c}$ below
Fibonacci numbers such as 34, 55, 89 and 144.
According to Fig. 10(a), the trace length $n_{c}$ appears to be independent of
the initial divergence $\alpha_{0}$. Moreover, $n_{c}$ is not as variable as
$\alpha_{0}$. As the order of phyllotaxis is very high, there is considerable
uncertainty in $n_{c}$, while $\alpha_{0}$ is quite accurate. The specimens
may be divided into two groups in terms of $n_{c}$, i.e., one with $n_{c}\sim
50$ and the other with $n_{c}>100$. The fact that the fossil specimens show
various but accurate values of $\alpha_{0}$ strongly suggests the evolutionary
origin of the special divergence angles. It is impossible to tabulate all
phyllotactic fractions for such a large value as $n_{c}=50$ due to lack of
space, but it is mentioned only that the number of possible phyllotactic
fractions at $n_{c}=50$ amounts to 387 $(\simeq 3n_{c}^{2}/\pi^{2}/2)$. Among
them, only the single fraction $\alpha=\frac{21}{55}$ of the main sequence
falls in the optimum regions depicted in Fig. 10, while the specimens Nos.
8-12 do not meet the optimum condition. Nevertheless, all the reported
specimens possess the irrational numbers expressed in the form of (2), as
expected in the evolutionary mechanism (cf. Table 2 of Okabe (2011)).
Phyllotactic patterns for $\alpha_{0}=[3,2]$, $[2,3]$, $[2,1,2]$ and others
are not reported, presumably because of lack of enough samples. Thus,
anomalous patterns are regarded as relics of evolutionary processes.
Figure 11: Three-dimensional fitness landscape. The vertical axis
representing ‘fitness’ is $n_{c}$ minus the number of phyllotactic transition.
The variables on the base plane are $\alpha_{0}$ and $n_{c}$. The number of
transition increases with $n_{c}$. The fitness has a flat bottom minimum in
the worst case of $\alpha_{0}\simeq 0$, whereas there are ‘fitness peaks’ at
$\alpha_{0}=[2]\simeq 0.38$ (the main sequence), $\alpha_{0}=[3]\simeq 0.28$
(an accessory sequence) and others, whose widths decrease as $n_{c}$ increases
(Okabe (2011)).
Figure 12: (a) Frequency distribution of initial divergence angles for a
$(1,2)$ phyllotaxis of Lysimachia clethroides by Fujita (1939). (b) An
enlarged view of a relative ‘fitness’ in Fig. 11 is plotted against the
divergence angle in degrees, $360\alpha_{0}$. The peak plateau extends from
135 to 144 degrees at $n_{c}=12$, and from 135 to 138 degrees at $n_{c}=19$.
It has been an unresolved problem in what quantitative terms normal and
anomalous phyllotaxis are differentiated. The number of phyllotactic
transition during a steady growth provides us with a numerical measure of
relative fitness in evolution. The most fit divergence angles are indicated in
Fig. 10. They are peaks of a ‘fitness landscape’ (Niklas (1997)), shown in
Fig. 11 (Okabe (2011)). A close inspection of the frequency distribution
curves of Fujita (1939) indicates that a primary peak accompanies small
subsidiary peaks at anomalous angles. In Fig. 12, Fujita’s result for
Lysimachia clethroides (gooseneck loosestrife) is arranged along with
transections of the fitness landscape in Fig. 11. Roberts (1984) has discussed
that his chemical contact pressure model explains the anomalous subsidiary
peaks. However, his conclusion is based on circular reasoning that anomalous
systems are less frequent because they are anomalous. Similar fitness curves
are obtained for light absorption efficiency of rosette plants (Niklas (1988,
1998); Pearcy and Yang (1998); King et al. (2004)).
Let us remark incredible precision of the divergence angle. As already
mentioned, it is no less astonishing than the widely noticed fact that
divergence angles converge on one of the special irrational numbers. Let us
take the specimen No. 13 as an example. The divergence angle of the
$\frac{47}{154}$ phyllotaxis is a rational number $360\alpha\simeq
109.870^{\circ}$. This is very close to an irrational, ideal angle of
$\alpha_{0}=[3,3]$, or $360\alpha_{0}\simeq 109.877^{\circ}$. According to
Table 20, the range of $\alpha_{0}$ for the $\frac{47}{154}$ phyllotaxis is
very narrow, that is,
$109.831^{\circ}<360\alpha_{0}<109.895^{\circ},$ (3)
or $360\alpha_{0}\simeq 109.863\pm 0.032$ degrees. The relative precision is
less than about a part per three thousand. For reference, we present results
that would be obtained if $\alpha_{0}$ happens to be off the narrow range of
(3). Instead of $\frac{47}{154}$ and the parastichy pair $(59,95)$ for (3), we
would have obtained $\frac{43}{141}$ and $(59,82)$ if $\alpha_{0}$ were
slightly below the lower limit of (3), or $\alpha=\frac{40}{131}$ and
$(36,95)$ if $\alpha_{0}$ were above the upper limit of (3). Neither of the
last two cases is listed in Tables 1$\sim$18, for they are hardly ever likely
to occur. The plants’ ability to distinguish $\frac{47}{154}$ from
$\frac{43}{141}$ and $\frac{40}{131}$ is due to high precision regulation of
initial divergence angle. The range width of $\alpha_{0}$ depends not so much
on $\alpha_{0}$ as on $n_{c}$. Indeed, we find
$\Delta\alpha_{0}\simeq(\tau/n_{c})^{2}$ according to Eq. (B.39) in Okabe
(2011). The precision as high as the above cannot be attained by a limited
number of cells on the apex (Koch et al. (1998); Meinhardt et al. (1998);
Smith et al. (2006a)). It seems very unlikely that existing causal models can
explain this anomalous phyllotaxis with this precision in this probability of
one out of thirteen specimens.
Diversities of phyllotaxis is considered as a result of selective pressures
being ineffective. In extant plants, the main Fibonacci phyllotaxis is
dominant while some species specifically show very diverse phyllotaxis
(Zagórska-Marek (1994)). In general, the trait diversity will be reduced if
there is selective pressure acting on it. Strength of selective pressure
depends specifically on extra cost required while rearranging phyllotactic
patterns of leaf traces during growth of individual plants. Accordingly, the
diversity may be preserved for some reason or other, e.g., when leaf traces
are so fragile that the energy cost of rearrangement is insignificant. This
view is consistent with recent research on Licopodium revealing a link between
variability of leaf traces and diversity of phyllotaxis (Gola et al. (2007)).
In contrast, the diversity in phyllotaxis of scale trees is considered as a
result of strong selective pressure of insufficient time durations, strong
because divergence angles are highly accurate whereas insufficient because
various angles besides 137.5∘ are still in existence. In discussing diversity
of phyllotaxis, one should make a clear distinction between the variance, or
standard deviation, of divergence angle of an individual and varieties of
divergence angles of individuals. This section was devoted to the latter,
while the former was discussed in the last section.
## 5 Phyllotaxis and vascular organization
Figure 13: Diagrams of the primary vascular system of $\frac{5}{13}$ (left),
$\frac{5}{18}$ (center), $\frac{8}{21}$ (right) phyllotaxis of Linum
usitatissimum. In the inset (top left), dashed lines mark off five parastichy
sectors in a transverse section of the $\frac{5}{13}$ phyllotaxis stem.
Adapted from Girolami (1953).
Girolami (1953) investigated the relation between phyllotaxis and vascular
organization of Linum (flax), whose vascular structures of a $\frac{5}{13}$,
$\frac{5}{18}$ and $\frac{8}{21}$ phyllotaxis are given in the left, center
and right of Fig. 13, respectively. On the one hand, the genetic spirals of
the $\frac{5}{13}$ and $\frac{5}{18}$ phyllotaxis wind up to the right
(counterclockwise), while it goes to the left (clockwise) for the
$\frac{8}{21}$ phyllotaxis. On the other hand, the main parastichies of the
three patterns run in the same direction. That is to say, 5-parastichies for
$\frac{5}{13}$ (1-6-11-16-21, etc.), 7-parastichies for $\frac{5}{18}$
(1-8-15-22-29, etc.) and 8-parastichies for $\frac{8}{21}$ (1-8-15-22-29,
etc.) run steeply from the bottom right to top left (clockwise). The most
direct vascular connection goes along the main parastichies. The vascular
bundles of these parastichies are recognized as sectioned clusters in a
transverse section of the stem, called parastichy sectors. As shown in the
inset of Fig. 13, the $\frac{5}{13}$ phyllotaxis stem is divided into five
parastichy sectors. In what follows, the following points remarked by Girolami
(1953) are analyzed in terms of the model, whereby some useful general rules
are pointed out:
(G1) The number of the parastichy sectors (5, 7 and 8 for $\frac{5}{13}$,
$\frac{5}{18}$ and $\frac{8}{21}$, respectively) agrees with the numerator of
the phyllotactic fraction for $\frac{5}{13}$ and $\frac{8}{21}$ of the main
sequence, but not for $\frac{5}{18}$ of the accessory sequence.
(G2) The length of leaf traces per internode increases with the number of
parastichy sectors, namely 12 for $\frac{5}{13}$, 17 for $\frac{5}{18}$, and
19 for $\frac{8}{21}$ approximately.
(G3) There is no correlation in the relative directions of the genetic spiral
and the parastichies.
As noted in the first point, there is no easy-to-use general formula between
the parastichy numbers and the phyllotactic fraction (see below however), but
the numerical correspondence is immediately read from Fig. 7 and Tables
1$\sim$18. According to Table 2, the parastichy pair of the $\frac{5}{18}$
phyllotaxis is $(7,11)$. The number of parastichy sectors is the small number
of the parastichy pair. Therefore, the number 7 of the parastichy sectors of
the $\frac{5}{18}$ phyllotaxis is obtained. Unlike the numerator, the
denominator satisfies simple rules. Most notably, the denominator of a
fraction is equal to the sum of the contact parastichy pair corresponding to
the fraction (e.g. $18=7+11$). Mathematical relations between various numbers
in phyllotaxis have been investigated since early times on an empirical ground
based on purely mathematical properties of a regular lattice (Bravais and
Bravais (1837); Naumann (1845); Jean (1994)).
On the second point, Tables 1 and 2 give the conditions $8\leq n_{c}<13$,
$11\leq n_{c}<18$ and $13\leq n_{c}<21$ for the phyllotactic fractions
$\frac{5}{13}$, $\frac{5}{18}$ and $\frac{8}{21}$, respectively. The
predictions of the model are supported by the reported values $n_{c}=12,17$
and 19, which satisfy their respective conditions near their upper limits.
Nevertheless, a close look at Fig. 13 indicates that these figures are not
accurate. As a matter of fact, $n_{c}$ appears not constant but somewhat
larger in the upper part of the stem. Changes in length of the leaf traces are
revealed in a more sophisticated analysis of Meicenheimer (1986), where
progressive transitions from $\frac{1}{3}$ through $\frac{2}{5}$ and
$\frac{3}{8}$ up to $\frac{5}{13}$ have been reported. Phyllotactic transition
caused by changes in $n_{c}$ is discussed in the next section.
Figure 14: Three branches from Fig. 7 with which to explain spiral directions
of $\frac{5}{13}$, $\frac{5}{18}$, $\frac{8}{21}$ phyllotaxis. As the fraction
$\frac{5}{18}$ and $\frac{5}{13}$ are numerically bigger than their ‘mother’
fraction $\frac{3}{11}$ and $\frac{3}{8}$, their main parastichies of 7 and 5
are contrary in direction to the genetic spiral. On the contrary, 8
parastichies for $\frac{8}{21}$ are in the same direction as the genetic
spiral. Figure 15: When the final divergence $\alpha$ is numerically bigger
than the initial divergence $\alpha_{0}$, the stem is twisted in the direction
of the genetic spiral. N.B. Divergence angles are less than 180 degrees.
On the third point, a general rule holding between directions of parastichies
and the genetic spiral is presented based on Fig. 7. To this end, it is
convenient to introduce a ‘mother’ fraction of a fraction $\alpha$, which is
defined as the fraction lying immediately below the fraction $\alpha$ in the
tree of Fig. 7. The mother fractions of $\frac{5}{13}$, $\frac{5}{18}$ and
$\frac{8}{21}$ are $\frac{3}{8}$, $\frac{3}{11}$ and $\frac{5}{13}$,
respectively. It is shown that if and only if a phyllotactic fraction $\alpha$
is numerically bigger than its mother fraction, the main parastichies run in
the direction opposite to the genetic spiral. (The main parastichies are
gentle, long spirals characterized by the small number of the contact
parastichy pair.) The fraction $\alpha=\frac{5}{18}$ and $\frac{5}{13}$ are
bigger than the mother fraction $\frac{3}{8}$ and $\frac{3}{11}$,
respectively, while $\alpha=\frac{8}{21}$ is smaller than the mother fraction
$\frac{5}{13}$. The magnitude relations are schematically shown in Fig. 14
extracted from Fig. 7. Thus, the above rule explains Girolami’s observation
consistently. In practice, this rule may be used to identify the direction of
the genetic spiral of a high order phyllotactic pattern for which parastichies
are far easy to follow visually. Some special cases of this general rule have
been remarked (Church (1904)(p. 96), Namboodiri and Beck (1968)) and
occasionally taken up for discussion (Meicenheimer (1986); Fredeen et al.
(2002)). The directional relations between various spirals of a phyllotactic
pattern are also mathematical consequences of the regularity of the
phyllotactic pattern.
The mother fraction enables us to state general rules for the phyllotactic
fraction and the parastichy number: One of the parastichy pair for a fraction
$\alpha$ is equal to the denominator of the mother fraction of $\alpha$; The
other number in the pair is determined such that the sum of the pair is equal
to the denominator of $\alpha$. Consider $\alpha=\frac{5}{18}$, for instance.
One of its parastichy pair is the denominator 11 of the mother fraction
$\frac{3}{11}$, while the other is the difference of the denominators,
$18-11=7$. As a result, the parastichy pair $(7,11)$ is obtained for
$\frac{5}{18}$. Thus, the rules are used to relate the parastichy numbers and
the phyllotactic fraction.
The vascular systems shown in Fig. 13 form closed networks. In each system,
connections between leaf traces are formed along both the paired parastichies,
so that the vascular bundles are divided into parastichy sectors. Among
dicotyledons with helical phyllotaxis, however, an open vascular system is
rather common (Beck et al. (1982)). Primitive angiosperms and many gymnosperms
have open vascular systems (Beck (2010)). According to Beck et al. (1982),
open systems of five sympodia (a $\frac{2}{5}$ phyllotaxis) characterize 67%
of the species with helical phyllotaxy and are clearly a common type among
dicotyledons. In an open system, leaf traces are connected along one
direction. Although the present model determines the basic architecture of
vascular phyllotaxis, it does not specify detailed structure of the reticulate
pattern, whether it remains open or becomes closed. This is not a shortcoming
of the model, because actual linkages between leaf traces are likely to be
secondary events depending on circumstances (Kang et al. (2003)).
To conclude this section, let us remark another obvious correlation between
the direction of the genetic spiral and the secondary torsion of the stem. The
initial divergence $\alpha_{0}$ is related to the fractional divergence
$\alpha$ of a mature pattern by the angle of twist $\alpha-\alpha_{0}$
undergone in the secondary torsion. The direction of the torsion is the same
as the genetic spiral if and only if $\alpha>\alpha_{0}$. This is shown
schematically in Fig. 15. The direction of the secondary torsion would not be
difficult to check experimentally. In most typical cases, the direction is
reversed, or the sign of $\alpha-\alpha_{0}$ changes, as $n_{c}$ crosses a
threshold of phyllotactic transition. Bravais and Bravais (1837) evaluated the
limit divergence $\alpha_{0}$ from mature shoots by correcting the torsion
angle $\alpha-\alpha_{0}$.
## 6 Phyllotactic transition
Figure 16: Transition in the primary vascular system of a cottonwood plant
from a $\frac{2}{5}$ to $\frac{3}{8}$ phyllotaxis. Central, right and left
traces are indicated with crosses, filled and open triangles, respectively.
After Larson (1977).
Larson (1977) has investigated phyllotactic transition in the vascular system
of Populus (cottonwood). His result showing transition from a $\frac{2}{5}$ to
$\frac{3}{8}$ phyllotaxis is reproduced in Fig. 16. Each leaf has three
traces; central, right and left traces are indicated with crosses, filled and
open triangles, respectively. The leaf traces are connected with the stem
vascular bundles to make sympodia. The sympodia are separated from each other,
or the vascular system is open. The three traces leading to each leaf
primordium arise on different sympodia. The number of the sympodia changes
from five in the lower portion to eight in the upper portion of Fig. 16. The
number agrees with the denominator of the phyllotactic fraction in each part.
The region of the $\frac{2}{5}$ phyllotaxis occurs in the basal stem above
some primary leaves, while the $\frac{3}{8}$ phyllotaxis occurs at mid and
upper stem levels, principally in the zone of expanding leaves (Larson
(1977)). In Fig. 16, once the transition is initiated at a point IA on a
sympodium number 2, it progresses through the sympodia at points IB through
IE. Three new central traces to establish the three additional sympodia of the
$\frac{3}{8}$ system are derived from left traces in sequence at points IIA-
IIC. Various interrelations between phyllotaxis and leaf development have been
studied (Larson (1980)). In what follows, a correlation between phyllotactic
transition and lengths of the leaf traces is analyzed by means of the model,
whereby supporting evidence of the model is pointed out.
Figure 17: Length per internode of the leaf traces in Fig. 16 is plotted
against the leaf node number, the vertical axis of Fig. 16. Arrows for a
transition region between the $\frac{2}{5}$ and $\frac{3}{8}$ phyllotaxis and
labels IA-IE and IIA-IIC to indicate initiation of the transition are marked
in accordance with Fig. 16 by Larson. The phyllotactic transition is
consistent with the threshold value of $n_{c}=5$ predicted by the model (Table
1).
The lengths per internode of the leaf traces are optically read from Fig. 16
and plotted in Fig. 17. Arrows indicating the transition region between the
$\frac{2}{5}$ and $\frac{3}{8}$ phyllotaxis in Fig. 17 are marked in
accordance with Fig. 16 after Larson (1977). By comparison, a dashed line at
$n_{c}=5$ is drawn to indicate the theoretical threshold between the
$\frac{2}{5}$ and $\frac{3}{8}$ phyllotaxis (Table 1). In accordance with the
model, the phyllotactic transition is triggered by the increasing length of
the leaf traces crossing a threshold value of five internodes.
According to Table 1, phyllotactic transition is predictable. Transitions of
the main sequence occur whenever the trace length $n_{c}$ crosses Fibonacci
numbers. The trace length, like other parameters of the plant, is predictably
correlated with plant vigor (Larson (1980)). Therefore, in principle, the
model allows us to control phyllotaxis artificially. In Sec. 2, leaf traces
are assumed to have a common length. As noted at the end of the last section,
the direction of the secondary torsion is reversed when $n_{c}$ crosses a
threshold value, so that it may be fixed by a leaf trace of length longer than
the threshold. Fig. 18 schematically shows that long leaf traces 10, 11 and 12
trigger a transition from $\frac{2}{5}$ to $\frac{3}{8}$. In the transition
region of Fig. 17, three left traces (open triangles) of the node number 7, 8
and 9 are the first to cross the threshold at $n_{c}=5$. These are the very
traces labeled with IIA, IIB and IIC by Larson as those from which the three
extra sympodia branch. A close look at Fig. 16 reveals that central traces
below and above the transition region are inclined in the opposite direction.
This is consistent with the prediction of the model, for
$\alpha=\frac{3}{8}<\alpha_{0}<\frac{2}{5}$. Furthermore, five right traces
(filled triangles) striking around $n_{c}\simeq 6$ in Fig. 17 agree with the
special traces labeled with IA through IE. Thus, the observation supports the
special role of the Fibonacci number 5 for the trace length $n_{c}$.
Figure 18: A phyllotactic pattern with $\alpha_{0}=1/(1+\tau)$ (cf. Fig. 3).
Length of leaf traces is $n_{c}=4$ (solid bars) except for 10, 11 and 12 with
$n_{c}=6$ (bold bars). The longer traces can induce a transition from
$\alpha=\frac{2}{5}$ in the lower portion (cf. Fig. 3) to $\alpha=\frac{3}{8}$
in the upper portion (cf. Fig. 5). At the transition, the longer traces
deflect main parastichies (dotted lines), and the parastichy number increases
from 5 to 8. Figure 19: Reconstructed vascular system of a cottonwood plant
showing transition from $\frac{2}{5}$ through $\frac{3}{8}$ to $\frac{5}{13}$
phyllotaxis by Larson (1977). See Fig. 16 for symbols. Figure 20: Length per
internode of the leaf traces read from Fig. 19 is plotted against the leaf
index (the vertical axis of Fig. 19). Two arrows at the top indicate where the
$\frac{3}{8}$ phyllotaxis starts and ends according to Larson (1977) (see Fig.
19). According to the theoretical model, stable regions for the $\frac{2}{5}$,
$\frac{3}{8}$ and $\frac{5}{13}$ phyllotaxis are separated by horizontal
dashed lines at Fibonacci numbers 3, 5 and 8 (cf. Table 1). Thus, Larson’s
estimate of the region of the $\frac{3}{8}$ phyllotaxis agrees with the
theory. Left traces (open triangles) reaching a maximum length of about 10
internodes is consistent with an observation that the highest-order
phyllotactic fraction that this plant attains is $\frac{5}{13}$.
Two-step transition from a $\frac{2}{5}$ to $\frac{5}{13}$ phyllotaxis is
shown in Fig. 19 after Larson (1977), where steady increase in length of leaf
traces is more obvious than Fig. 16. Fig. 20 is obtained from Fig. 19 in the
same manner as Fig. 17 is obtained from Fig. 16. Leaf positions at which the
$\frac{3}{8}$ phyllotaxis starts and ends are marked on the right side of Fig.
19 by Larson (1977), according to which the transient pattern of the
$\frac{3}{8}$ phyllotaxis is maintained for the leaves with plastochron index
from 5 to $-7$. Accordingly, the corresponding positions are marked by arrows
in Fig. 20. On the other hand, horizontal lines at Fibonacci numbers 3, 5, and
8 in Fig. 20 theoretically divide the regions for the $\frac{1}{3}$,
$\frac{2}{5}$, $\frac{3}{8}$ and $\frac{5}{13}$ phyllotaxis (Table 1). Thus,
it is confirmed again that continuous changes in length of leaf traces cause
discontinuous transitions in the vascular structure.
Fig. 20 indicates that $n_{c}$ increases steadily up to an upper bound of
about 10. This observation is consistent with the fact that the $\frac{5}{13}$
phyllotaxis was the stable pattern of the old plant (Larson (1980)). According
to the model, the $\frac{5}{13}$ phyllotaxis is stable insofar as $n_{c}$ lies
between 8 and 13, i.e., there is a 5-internode allowance for the trace length
of the $\frac{5}{13}$ phyllotaxis. The main sequence is special for this wide
clearance between successive threshold values. The interval is denoted as
$\Delta n_{c}$ in Okabe (2011). As shown there, the widest clearances are
achieved for Fibonacci numbers, and a sequence of Fibonacci numbers is
realized when the limit divergence angle is one of the special irrational
numbers related to the golden ratio. As shown in Fig. 8, the number of
transitions encountered while $n_{c}$ grows up to above 10 is kept to a
minimum number insofar as the initial divergence is restricted within
$\frac{3}{8}<\alpha_{0}<\frac{3}{7}$ (from 135∘ to 154∘, as noted at the end
of Sec. 2). When $n_{c}$ becomes larger than 12, the range is narrowed to
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$ (from 135∘ to 144∘). Thus, the normal
phyllotaxis of the main sequence is singled out. Owing to the observation that
the highest-order fraction was $\alpha=\frac{5}{13}$, the model predicts that
the initial divergence $\alpha_{0}$ should be contained within
$\frac{3}{8}<\alpha_{0}<\frac{2}{5}$, just as observed by Fujita (1939) for
other species (Sec. 2). Unfortunately, initial divergences of the cottonwood
plant are not available to us. To support this argument, Puławska (1965) has
reported for Actinidia arguta (hardy kiwi) that initial divergence remains
constant despite changes in the vascular organization between
$\frac{3}{8},\frac{5}{13}$ and $\frac{8}{21}$.
When $n_{c}$ is increased past 8, the model predicts vascular phyllotaxis of
either $\alpha=\frac{5}{13}$ or $\alpha=\frac{5}{12}$ depending on whether the
initial divergence $\alpha_{0}$ is smaller or larger than $\frac{2}{5}$ (angle
of 144∘). Suppose $\alpha_{0}=[3]$ (99.5∘), then one should have five
threshold lines at 2, 3, 4, 7 and 11 (Table 2), instead of four thresholds at
2, 3, 5 and 8 for $\alpha_{0}=[2]$ in Fig. 20. If the initial divergence were
$\alpha_{0}=[5]$ (64.1∘ in Table 4), one should have six threshold lines at
$n_{c}=2$, 3, 4, 5, 6 and 11 separating patterns of
$\alpha=\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{2}{11}$
and $\frac{3}{17}$ (cf. Fig. 8). The vascular phyllotaxis is very unstable.
The instability is energetically unfavorable. Therefore, $\alpha_{0}=[5]$
(64.1∘) is very improbable to survive natural selection because of the
multiplicity of expected transitions. A general remark should be made when
discussing multiple patterns in sequence. In order for a pattern with a
definite value of $\alpha$ to be distinguished as such, the pattern should
consist of more leaves than the denominator of the fraction $\alpha$. This
holds true if $n_{c}$ varies sufficiently gradually; otherwise phyllotaxis
transition may not be distinctly discernible.
Last but not least, whorled phyllotaxis has not been discussed in this paper.
A $J$-jugate pattern with $J$ fundamental spirals is formed when $J$ leaves
are borne at each node. Compared with a helical phyllotaxis with $J=1$,
divergence angles of a $J$-jugate system are divided by $J$ and the parastichy
pairs $(m,n)$ are multiplied by $J$. Therefore, one obtains
$0<J\alpha_{0}<\frac{1}{2}$ and $J(m,n)$ for the divergence angle and
parastichy pair of a $J$-jugate system. It is known that sometimes vascular
structure may change between helical and whorled phyllotaxis during ontogeny.
This type of ‘anomalous’ phyllotactic transition also appears to be caused by
a decrease in length of leaf traces (Jensen (1968); Beck et al. (1982);
Kwiatkowska (1995)). The present model gives $\alpha=\frac{1}{2}$ for $1\leq
n_{c}<2$ and $\alpha=\frac{1}{3}$ for $2\leq n_{c}<3$ irrespective of
$\alpha_{0}$. Correspondingly, it seems natural to consider that a whorled
phyllotaxis is a variation of the most primitive alternate phyllotaxis and
that a whorled phyllotaxis is triggered as $n_{c}$ becomes less than 1.
However, changes in the vascular structure have to be coordinated with changes
in the positioning of initiated leaf primordia while a whorled pattern is
established (Zagórska-Marek (1994); Meicenheimer (1998); Kelly and Cooke
(2003)). The physiological processes involved are unlikely to be amenable to
simple mathematical analysis. Still, a similar transition rule as a helical
pattern should hold for an established whorled pattern in terms of trace
length redefined with a new internode.
## 7 Conclusions
The present work puts forward an important role of Fibonacci numbers as
critical values of the length per internode of leaf traces played in vascular
phyllotaxis transition.
The regular arrangement of leaves and the regularity in divergence angle of
137.5∘ are a result of selective pressure to reduce possible changes in the
vascular structure during growth, i.e., aperiodic arrangements will
necessitate extra nutrients to reconstruct the sectorial or fractional order
of vascular connections.
The phyllotactic fraction $\alpha$ of mature patterns of leaf traces normally
makes transitions through $\frac{1}{2}$, $\frac{1}{3}$, $\frac{2}{5}$,
$\frac{3}{8}$, $\frac{5}{13}$, $\frac{8}{21}$, $\cdots$, whenever the number
of internodes traversed by the leaf traces, $n_{c}$, crosses Fibonacci
numbers, 1, 2, 3, 5, 8, 13, 21, $\cdots$. The Fibonacci numbers make
appearances because initial divergence angle $\alpha_{0}$ of leaves at the
shoot apex is normally the golden angle of about $137.5^{\circ}$ with a good
precision. The golden angle is prevalent because it is the selectively
advantageous angle at which the number of the phyllotactic transition is the
minimum (Fig. 8). The precision of the initial divergence is determined by the
trace length $n_{c}$.
## Acknowledgement
The author would like to thank Prof. Rolf Rutishauser for valuable comments on
Picea abies and others. He would like to thank Prof. Beata Zagórska-Marek for
informing him about a different view on divergence angle.
## Appendix A Relation between the trace length $n_{c}$ and the plastochron
ratio $a$
A point on a cylinder surface is located with the angular coordinate $\varphi$
and the height $z$. Leaves on a stem are represented by a lattice of points
given by $\varphi=2\pi\alpha n$ (in radians) and $z=hn$, where $\alpha$ is a
constant angle of divergence, $h$ an internode length, and $n$ is an integer
index. On the other hand, a point on a plane is located in a polar coordinate
system $(r,\varphi)$, where $r$ and $\varphi$ is the radial distance from the
central axis and the angular coordinate about the axis, respectively. Leaf
primordia at a shoot apex are represented by $r=a^{n}$ and $\varphi=2\pi\alpha
n$, where $a$ is a plastochron ratio. In a conformal growth preserving angles,
the two representations are related by ${2\pi}z={\log r}$. Hence, the
internode length $h$ corresponds to the logarithm of the plastochron ratio,
$\frac{1}{2\pi}\log a$. The number of internodes traversed by the leaf traces
is $n_{c}={Z_{\rm lt}}/{h}=2\pi{Z_{\rm lt}}/{\log a}$, where $Z_{\rm lt}$ is a
length of leaf traces in the stem. Therefore, $n_{c}$ may be regarded as
inversely proportional to $\log a$, or the relative growth rate per
plastochron $\frac{{\rm d}r}{{\rm d}n}/r$. The growth rate should depend on
cell types. Accordingly, $n_{c}$ may change during plant growth.
The plastochron ratio may change as a result of alteration in size of the apex
and primordia. Richards (1951) discussed changing phyllotaxis to the effect
that a continuous shift in the parastichy pair of normal Fibonacci phyllotaxis
is linearly correlated with a double logarithm $\log(\log a)$. He defined the
phyllotaxis index (P.I.) by
${\rm P.I.}=0.38-2.39\log_{10}\log_{10}a,$ (4)
where numerical values are chosen such that the index assumes an integral
value whenever two sets of parastichies in the Fibonacci system intersect
orthogonally. The crossing angle between the contact parastichies changes
continuously as a function of the plastochron ratio. In this descriptive
model, the divergence angle $\alpha_{0}$ is fixed at the golden angle.
Changing phyllotaxis due to change in the plastochron ratio is consistent with
the present model of vascular phyllotaxis. In this model, the divergence
$\alpha$ on the stem changes discontinuously, however. To show a
correspondence between changes in phyllotaxis on the apex and the stem, let us
consider the normal phyllotaxis with an initial divergence of the golden angle
$\alpha_{0}=\tau^{-2}$ (Table 1). Let us introduce the Fibonacci sequence
$F_{n}$ generated from initial integers $F_{1}=1$ and $F_{2}=1$ by the
recurrence relation $F_{n+2}=F_{n+1}+F_{n}$. Accordingly, $F_{n}=1,1,2,3,5,8$
and $13$ for $n=1,2,3,4,5,6$ and $7$, respectively. In terms of $F_{n}$, the
phyllotactic fraction $\alpha=\frac{F_{n}}{F_{n+2}}$ and the parastichy pair
$(F_{n},F_{n+1})$ are obtained for $F_{n+1}\leq n_{c}<F_{n+2}$, or for
$(n+1)\log\tau-\log\sqrt{5}\leq\log n_{c}<(n+2)\log\tau-\log\sqrt{5}$
owing to an approximate formula $F_{n}\simeq\tau^{n}/\sqrt{5}$ valid for large
$n$ (see below (B.31) in Okabe (2011)). Therefore, $\log n_{c}$ is
proportional to the integer index $n$.
To put it concretely, we get $\alpha=\frac{1}{3}$ and the parastichy pair
(1,2) for $2\leq n_{c}<3$ or
$0.7\leq\log n_{c}<1.1,$
$\alpha=\frac{2}{5}$ and $(2,3)$ for $3\leq n_{c}<5$ or
$1.1\leq\log n_{c}<1.6,$
$\alpha=\frac{3}{8}$ and $(3,5)$ for $5\leq n_{c}<8$ or
$1.6\leq\log n_{c}<2.1,$
$\alpha=\frac{5}{13}$ and $(5,8)$ for $8\leq n_{c}<13$ or
$2.1\leq\log n_{c}<2.6,$
and so on. Thus, the shift in the parastichy pair is linearly correlated with
$\log n_{c}\propto\log(\log a)$. This is a general property holding also for
other initial divergences found in nature.
For the systematic study of the mature stem, the index $n_{c}$ is more
usefully regarded as a developmental index than $a$, not only because an
internode is a natural unit of length as the plastochron is the developmental
unit of time, but values of $n_{c}$ allowed for a phyllotactic pattern are
delimited by the special integers traditionally familiar to those who are
enchanted by phyllotaxis; Fibonacci numbers. For a given initial divergence,
the numbers comprise a sequence generated by the Fibonacci recurrence relation
$F_{n+2}=F_{n+1}+F_{n}$ from a pair of different seed integers. The main
sequence, 1, 2, 3, 5, 8, $\cdots$ in Table 1, is generated from the simplest
seed pair $(1,2)$. The next simplest seed integers $(1,3)$ give the accessory
sequence 1, 3, 4, 7, 11, 18 $\cdots$ of Table 2. In this manner, any
phyllotactic sequence is characterized by a pair of seed integers, as well as
the limit divergence $\alpha_{0}$. This is in accordance with accumulated
empirical wisdom of phyllotaxis. Traditionally, these special integers have
been remarked in connection with parastichy numbers (cf. Tables 1$\sim$18).
The present work puts emphasis on these numbers as critical values for the
length per internode of leaf traces. This point has never been remarked
before.
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|
arxiv-papers
| 2012-07-12T03:52:11 |
2024-09-04T02:49:32.943203
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Takuya Okabe",
"submitter": "Takuya Okabe",
"url": "https://arxiv.org/abs/1207.2838"
}
|
1207.3130
|
# Variational Minimizing Parabolic Orbits for the 2-Fixed Center Problems
Ying Lv1 and Shiqing Zhang2
1.School of Mathematics and Statistics, Southwest University, Chongqing
400715, China
2.Mathematical School, Sichuan University, Chengdu 610064, China
ABSTRACT: Using variational minimizing methods,we prove the existence of an
odd symmetric parabolic orbit for the 2-fixed center problems with weak force
type homogeneous potentials.
KEY WORDS: 2-Fixed Center Problems, Odd Symmetric Parabolic Orbits,
Variational Minimizers.
AMS Subject Clasification:34C15,34C25.
## 1 Introduction and Main Results
Sitninkov [1] and Moser [2] and Mathlouthi [3] and Souissi [4]and Zhang [5]
etc. studied the model for the circular restricted 3-body problems: two mass
points of equal mass $m_{1}=m_{2}=\frac{1}{2}$ move in the plane of their
circular orbits such that the center of masses is at rest, and the third small
mass which does not influence the motion of the first two ones moves on the
line perpendicular to the plane containing the first two mass points and going
through the center of mass.
Let $z(t)$ be the coordinate of the third mass point, then $z(t)$ satisfies
$\ddot{z}(t)+\alpha\frac{z(t)}{(|z(t)|^{2}+|r|^{2})^{\alpha/2+1}}=0.$ (1)
Zhang [5] used variational minimizing method to prove:
Theorem 1.1 For the equation (1) with $0<\alpha<2$, there exists one odd
parabolic or hyperbolic orbit .
The 2-fixed center problem is an old problem studied by Euler[6-8] etc. ([9],
[10], [11], [12]): For two masses $1-\mu$ and $\mu$ fixed at $q^{1}=(-\mu,0)$
and $q^{2}=(1-\mu,0)$, the problem is to study the motion $q(t)=(x(t),y(t))$
of the third body with mass $m_{3}>0$. Here, we consider the motion of the
third body attracted by the 2-fixed center masses with general homogeneous
potentials, then it satisfies the following equation:
$\ddot{q}(t)+\frac{\partial V(q)}{\partial q}=0,$ (2)
$V(q)=-\frac{1-\mu}{|q-q^{1}|^{\alpha}}-\frac{\mu}{|q-q^{2}|^{\alpha}}.$ (3)
For $\mu=1/2$, we study the existence for the motion $q(t)=(x(t),y(t))$ of the
third body satisfying $(x(-t),y(-t))=(-x(t),-y(t)),$ here we use variational
minimizing method to prove:
Theorem 1.2 For $(2)-(3)$ with $\mu=\frac{1}{2}$ and $0<\alpha<2$, there
exists an odd symmetrical parabolic-type unbounded orbit.
## 2 Truncation Functional and Its Minimizing Critical Points
In order to find parabolic-type orbit of $(2)-(3)$ , firstly, we restrict
$t\in[-n,n]$ and find solutions of $(2)-(3)$, then let $n\rightarrow+\infty$
to get the parabolic-type orbit. Noticing the symmetry of the equation, we can
find the odd solutions of the following ODE:
$\ddot{q}(t)=\frac{\partial U(q)}{\partial q},$ (4)
$U(q)=\frac{1/2}{|q-q^{1}|^{\alpha}}+\frac{1/2}{|q-q^{2}|^{\alpha}}.$ (5)
We define the functional:
$f(q)=\int_{-n}^{n}(\frac{1}{2}|\dot{q}(t)|^{2}+\frac{1/2}{|q-q^{1}|^{\alpha}}+\frac{1/2}{|q-q^{2}|^{\alpha}})dt,$
(6)
where
$q\in H_{n}=\\{q(t)=(x(t),y(t)):x,y\in
W^{1,2}[-n,n];q(-t)=-q(t),q(t)\not=q^{i},t\in[-n,n]\\}.$ (7)
Since $\forall q\in H_{n},q(0)=0$, then by the famous Hardy-Littlewood-Polya’s
inequality ([9], inequality 256), for $\forall q\in H_{n}$, we have an
equivalent norm :
$\|q\|_{n}=(\int_{-n}^{n}|\dot{q}(t)|^{2}dt)^{1/2}.$
Remark Here we didn’t assume $q(-n)=q(n)=0$ since we want to get the
parabolic-type orbit satisfying
$\max_{t\in R}|q(t)|=+\infty,$ $\min_{t\in R}|\dot{q}(t)|=0.$
We didn’t assume the periodic property for $q(t)$ since we need non-periodic
odd test function in order to get Lemma 2.6.
Lemma 2.1 $f(q)$ is weakly lower semi-continuous(w.l.s.c.) on the closure
$\bar{H}_{n}$ of $H_{n}$.
Proof: (i). It is well-known that the norm and its square are w.l.s.c..
(ii). $\forall\\{q_{m}\\}\subset H_{n}$, if $q_{m}\rightharpoonup q\in H_{n}$
weakly, then by compact embedding theorem, we have the uniformly convergence:
$\max\limits_{-n\leq t\leq n}|q_{m}(t)-q(t)|\rightarrow 0,$
as $m\rightarrow+\infty$, so
$\int_{-n}^{n}\frac{1}{|q_{m}-q^{i}|^{\alpha}}dt\rightarrow\int_{-n}^{n}\frac{1}{|q-q^{i}|^{\alpha}}dt,i=1,2,$
as $m\rightarrow+\infty$. Hence
$\mathop{\underline{\lim}}\limits_{m\rightarrow\infty}f(q_{m})\geq f(q).$
(iii). $\forall\\{q_{m}\\}\subset H_{n}$, if $q_{m}\rightharpoonup
q\in\partial{H}_{n}$ weakly,let
$S=\\{t_{0}\in[-n,n]:q(t_{0})=q_{1}(t_{0}),or,q_{2}(t_{0})\\}$
(1).The Lebesgue measure of $S$ is zero,then $U(q_{m}(t)\rightarrow U(q(t))$
almost everywhere,then by Fatou’s Lemma,$\int_{-n}^{n}U(q)dt$ is $w.l.s.c.$,
it is well-known that the norm and its square are w.l.s.c.,so $f(q)$ is
$w.l.s.c.$.
(2).The Lebesgue measure of $S$:$L(S)>0$ ,then
$\int_{-n}^{n}U(q)dt=+\infty,f(q)=+\infty,$
then by compact embedding theorem, we have the uniformly convergence on $S$:
$\max\limits_{-n\leq t\leq n}|q_{m}(t)-q(t)|\rightarrow 0,$
as $m\rightarrow+\infty$, so on $S$,we have the uniformly convergence:
$\int_{-n}^{n}\frac{1}{|q_{m}-q^{i}|^{\alpha}}dt\rightarrow+\infty,i=1,or,2,$
as $m\rightarrow+\infty$. Hence
$\int_{-n}^{n}U(q_{m}(t))dt\rightarrow+\infty$
$\mathop{\underline{\lim}}\limits_{m\rightarrow\infty}f(q_{m})=+\infty\geq
f(q).$
Lemma 2.2 $f$ is coercive on $\bar{H}_{n}$.
Proof: From the definition of $f(q)$ and Hardy-Littlewood’s inequality,it is
clear that the coercivity
holds($f(q)\rightarrow+\infty,\|q\|\rightarrow+\infty$).
Lemma 2.3(Tonelli, [13], [14]) Let $X$ be a reflexive Banach space, $M\subset
X$ be a weakly closed subset, $f:M\rightarrow R\cup\\{+\infty\\}$, but $f(x)$
is not always $+\infty$ ,suppose $f$ is weakly lower semi-continuous and
coercive($f(x)\rightarrow+\infty,\|x\|\rightarrow+\infty$), then $f$ attains
its infimum on $M$.
Lemma 2.4(Palais’s Symmetry Principle [15]) Let $G$ be a finite or compact
group, $\sigma$ be an orthogonal representation of $G$, let $H$ be a real
Hilbert space, $f:H\rightarrow R$ satisfying
$f(\sigma\cdot x)=f(x),\forall\sigma\in G,\forall x\in H.$
Let
$F\stackrel{{\scriptstyle\triangle}}{{=}}\\{x\in H|\sigma\cdot
x=x,\forall\sigma\in G\\}.$
Then the critical point of $f$ in $F$ is also a critical point of $f$ in $H$.
Lemma 2.5 $f(q)$ attains its infimum on $\bar{H}_{n}$, the minimizer
$\tilde{q}_{\alpha,n}(t)$ is an odd solution.
Proof: Since we had proved Lemmas 2.1-2.2, so in order to apply for Lemma 2.3,
we need to apply for Lemma 2.4 to prove that the critical point of $f(q)$ on
$H_{n}$ is the odd solution of $(4)-(5)$: We define groups
$G_{1}=\\{I_{2\times 2},-I_{2\times 2}\\}$,$G_{2}=\\{1,-1\\}$ and their
actions:
$\sigma_{1}\cdot q(t)=I_{2\times 2}q(t),$ $\sigma_{2}\cdot q(t)=-I_{2\times
2}q(t);$ $\tilde{\sigma}_{1}\cdot q(t)=q(t),$ $\tilde{\sigma}_{2}\cdot
q(t)=q(-t).$
Then it’s easy to prove that $f(q)$ is invariant under
$\sigma_{1},\sigma_{2},\tilde{\sigma}_{1},\tilde{\sigma}_{2},\sigma_{i}\cdot\tilde{\sigma}_{j},\tilde{\sigma}_{j}\cdot\sigma_{i}$
and the fixed point set of the group actions for $G_{1}\times G_{2}$ is just
$H_{n}$, so we can apply for Palais’s Symmetrical Principle.
In order to get the parabolic type solution, we need to prove that
$\tilde{q}_{\alpha,n}(t)\rightarrow\tilde{q}_{\alpha}(t)$
when $n\rightarrow\infty$, and $\tilde{q}_{\alpha}(t)$ has the properties:
$\max_{t\in R}|\tilde{q}_{\alpha}(t)|=+\infty,$ $\min_{t\in
R}|\dot{\tilde{q}}_{\alpha}(t)|=0.$
In order for that, we need some furthermore Lemmas:
Lemma 2.6 There exist constants $c>0$ and $0<\theta<1$ independent of $n$ such
that the variational minimizing value $a_{n}$ for $f(q)$ on $\bar{H_{n}}$
satisfies $a_{n}\leq cn^{\theta}.$
Proof: (i). If $\tilde{q}(t)=(\tilde{x},\tilde{y})\in H_{n}$ is located on
y-axis, then we choose a special odd function defined by
$\tilde{x}=0,\ \tilde{y}=t^{\beta},\ t\in[-n,n],$
where
$\frac{1}{2}<\beta=\frac{l}{m}<\frac{1}{\alpha},$
$l,m$ are odd numbers and $(l,m)=1.$ Then
$\displaystyle f(\tilde{q}(t))$ $\displaystyle=$
$\displaystyle\frac{1}{2}2\int_{0}^{n}\beta^{2}t^{2(\beta-1)}dt+\int_{-n}^{n}[\frac{1/2}{|t^{2\beta}+\frac{1}{4}|^{\alpha/2}}+\frac{1/2}{|t^{2\beta}+\frac{1}{4}|^{\alpha/2}}]dt$
$\displaystyle\leq$
$\displaystyle\frac{\beta^{2}}{2\beta-1}n^{2\beta-1}+\frac{2}{1-\alpha\beta}n^{1-\alpha\beta}.$
Now we define
$\theta=\max(2\beta-1,1-\alpha\beta),$ (8)
$c=\frac{\beta^{2}}{2\beta-1}+\frac{2}{1-\alpha\beta}>0.$ (9)
When
$\frac{1}{2}<\beta=\frac{l}{m}<\frac{1}{\alpha},$
then
$2\beta-1>0,\ 1-\alpha\beta>0$
and $0<\theta<1$. Hence we have
$f(\tilde{q})\leq cn^{\theta}.$
(ii). If $\tilde{q}(t)=(\tilde{x},\ \tilde{y})$ is not on y-axis, we choose a
special odd function on $t$ defined by
$\tilde{x}(t)=t^{\beta},\tilde{y}(t)=0,\ t\in[-n,n],$
where
$\frac{1}{2}<\beta=\frac{l}{m}<\frac{1}{\alpha},$
$l,m$ are odd numbers and $(l,m)=1.$ Then, we have
$\displaystyle f(\tilde{q}(t))$ $\displaystyle\leq$
$\displaystyle\int_{0}^{n}\beta^{2}t^{2(\beta-1)}dt+\int_{0}^{n}[\frac{1}{|t^{\beta}+\frac{1}{2}|^{\alpha}}+\frac{1}{|t^{\beta}-\frac{1}{2}|^{\alpha}}]dt$
$\displaystyle\leq$
$\displaystyle\frac{\beta^{2}}{2\beta-1}n^{2\beta-1}+[\frac{1}{1-\alpha\beta}n^{1-\alpha\beta}+\int_{0}^{n}\frac{1}{|t^{\beta}-\frac{1}{2}|^{\alpha}}dt].$
Now we estimate
$\int_{0}^{n}\frac{1}{|t^{\beta}-\frac{1}{2}|^{\alpha}}dt.$
Let
$t^{\beta}-\frac{1}{2}=\tau^{\beta},$
then $t>\tau$ and
$dt=(\frac{\tau}{t})^{\beta-1}d\tau$
also
$\displaystyle\int_{0}^{n}\frac{1}{|t^{\beta}-\frac{1}{2}|^{\alpha}}dt$
$\displaystyle<$
$\displaystyle\int_{(-\frac{1}{2})^{-\frac{1}{\beta}}}^{(n^{\beta}-\frac{1}{2})^{\frac{1}{\beta}}}{\tau}^{-\alpha\beta}d\tau$
$\displaystyle<$
$\displaystyle\frac{1}{1-\alpha\beta}[n^{1-\alpha\beta}-(-\frac{1}{2})^{-\frac{1}{\beta}(1-\alpha\beta)}].$
Define
$\theta=\max\\{2\beta-1,1-\alpha\beta\\},$
$c=\frac{\beta^{2}}{2\beta-1}+\frac{3}{1-\alpha\beta}>0.$
When
$\frac{1}{2}<\beta=\frac{l}{m}<\frac{1}{\alpha},$
then
$2\beta-1>0,1-\alpha\beta>0$
and $0<\theta<1.$ Hence we also have
$f(\tilde{q})\leq cn^{\theta}.$
Furthermore, for our minimizer, we have
Lemma 2.7 Let $\tilde{q}_{\alpha,n}$ be critical points corresponding to the
minimizing critical values $a_{n}=\mathop{\min}\limits_{H_{n}}f(q),$ then
$\|\tilde{q}_{\alpha,n}\|_{\infty}\rightarrow+\infty,$ when
$n\rightarrow+\infty$.
Proof: By the definition of $f(\tilde{q}_{\alpha,n})$ and Lemma 2.6, we have
$\displaystyle cn^{\theta}$ $\displaystyle\geq$ $\displaystyle
f(\tilde{q}_{\alpha,n})$ $\displaystyle\geq$
$\displaystyle\int_{0}^{n}[\frac{1}{|(x+\frac{1}{2})^{2}+y^{2}|^{\alpha/2}}+\frac{1}{|(x-\frac{1}{2})^{2}+y^{2}|^{\alpha/2}}]dt.$
We notice that
$(x+\frac{1}{2})^{2}+y^{2}\leq 2(x^{2}+y^{2})+\frac{5}{4},$
$(x-\frac{1}{2})^{2}+y^{2}\leq(x^{2}+y^{2})+\frac{1}{4},$
so
$\displaystyle cn^{\theta}$ $\displaystyle\geq$
$\displaystyle\int_{0}^{n}\frac{dt}{(2\|\tilde{q}_{\alpha,n}\|_{\infty}^{2}+\frac{5}{4})^{\alpha/2}}+\frac{dt}{(\|\tilde{q}_{\alpha,n}\|_{\infty}^{2}+\frac{1}{4})^{\alpha/2}}$
$\displaystyle\geq$
$\displaystyle\frac{2n}{(2\|\tilde{q}_{\alpha,n}\|_{\infty}^{2}+\frac{5}{4})^{\alpha/2}}.$
Hence
$\|\tilde{q}_{\alpha,n}\|_{\infty}^{2}\rightarrow+\infty,$ (10)
as $n\rightarrow+\infty$.
Lemma 2.8 $\int_{a}^{b}|\dot{\tilde{q}}_{\alpha,n}|^{2}dt$ is uniformly
bounded on any compact set $[a,b]\subset R$.
Proof: Since the system is autonomous, so for any given $\alpha,n$, along the
solution $\tilde{q}_{\alpha,n}(t)$, the energy $h(t)$ is conservative, i.e., a
constant $h=h(\alpha,n)$:
$\frac{1}{2}|\dot{\tilde{q}}_{\alpha,n}|^{2}-\frac{1/2}{{|\tilde{q}}_{\alpha,n}-q^{1}|^{\alpha}}-\frac{1/2}{{|\tilde{q}}_{\alpha,n}-q^{2}|^{\alpha}}=h.\\\
$ (11)
By the above energy relationship and the definition of the functional $f$, we
have
$\displaystyle f(\tilde{q}_{\alpha,n})$ $\displaystyle=$
$\displaystyle\int_{-n}^{n}(\frac{1}{2}|\dot{\tilde{q}}_{\alpha,n}|^{2}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{2}|^{\alpha}})dt$
$\displaystyle=$
$\displaystyle\int_{-n}^{n}(\frac{1}{2}|\dot{\tilde{q}}_{\alpha,n}|^{2}-\frac{1/2}{{|\tilde{q}}_{\alpha,n}-q^{1}|^{\alpha}}-\frac{1/2}{{|\tilde{q}}_{\alpha,n}-q^{2}|^{\alpha}})dt$
$\displaystyle+$ $\displaystyle
2\int_{-n}^{n}\frac{1/2}{{|\tilde{q}}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{{|\tilde{q}}_{\alpha,n}-q^{2}|^{\alpha}}dt$
$\displaystyle=$ $\displaystyle
2nh+2\int_{-n}^{n}\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{2}|^{\alpha}}dt.$
By Lemma 2.6, we have
$\displaystyle cn^{\theta}$ $\displaystyle\geq$ $\displaystyle
2nh+2\int_{-n}^{n}(\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{2}|^{\alpha}})dt,$
and
$h\leq\frac{c}{2}n^{\theta-1}-\frac{1}{n}\int_{-n}^{n}(\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{2}|^{\alpha}})dt\leq\frac{c}{2}n^{\theta-1}$
(12)
(1).When $n$ is large enough,$|\tilde{q}_{\alpha,n}(t)-q^{i}|$ has uniformly
positive lower bound,that is, $\min_{a\leq t\leq
b}|\tilde{q}_{\alpha,n}(t)-q^{i}|\geq c>0,$ then we have
$\displaystyle\int_{a}^{b}\frac{1}{2}|\dot{\tilde{q}}_{\alpha,n}|^{2}$
$\displaystyle=$ $\displaystyle
h(b-a)+\int_{a}^{b}[\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{2}|^{\alpha}}]dt$
$\displaystyle\leq$ $\displaystyle\frac{c}{2}(b-a)+c^{-\alpha}(b-a).$
(2).There exist $i_{0}=1$ or $2$ and a sequence ${t_{n}}\subset[a,b]$ such
that $\tilde{q}_{\alpha,n}(t_{n})\rightarrow q^{i_{0}}$,then since
$0<\alpha<2$,the potential is weak force potential,so when $n$ is large,we
have
$\int_{a}^{b}[\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha,n}-q^{2}|^{\alpha}}]dt\leq
M,$
$\int_{a}^{b}\frac{1}{2}|\dot{\tilde{q}}_{\alpha,n}|^{2}dt\leq\frac{c}{2}(b-a)+M.$
## 3 Proof of Theorem 1.2
By $\tilde{q}_{\alpha,n}(0)=0$ and Cauchy-Schwarz inequality and Lemma 2.8 we
have
$|\tilde{q}_{\alpha,n}(t)|=|\int_{0}^{t}\dot{\tilde{q}}_{\alpha,n}(s)ds|\leq(b-a)^{1/2}[\int_{a}^{b}|\dot{\tilde{q}}_{\alpha,n}|^{2}ds]^{1/2}\leq
M_{1},$
so we have
(i). $\\{\tilde{q}_{\alpha,n}\\}$ is uniformly bounded on any compact set of
$R$.
By Cauchy-Schwarz inequality and Lemma 2.8 we have
$|\tilde{q}_{\alpha,n}(t_{2})-\tilde{q}_{\alpha,n}(t_{1})|=|\int_{t_{1}}^{t_{2}}\dot{\tilde{q}}_{\alpha,n}(s)ds|\leq[\int_{a}^{b}|\dot{\tilde{q}}_{\alpha,n}|^{2}ds]^{1/2}(t_{2}-t_{1})^{1/2}\leq
M_{2}(t_{2}-t_{1})^{1/2},$
so we have
(ii). $\\{\tilde{q}_{\alpha,n}\\}$ is uniformly equi-continuous on any
$[a,b]\subset R$.
Now we can apply Ascoli-Arzel$\grave{a}$ Theorem, we know
$\\{\tilde{q}_{\alpha,n}\\}$ has a sub-sequence converging uniformly to a
limit $\tilde{q}_{\alpha}(t)$ on any compact set of $R$, and
$\tilde{q}_{\alpha}(t)$ is a solution of (2.2) . By the energy conservation
law and Lemmas 2.7-2.8, we have
$h=\frac{1}{2}|\dot{\tilde{q}}_{\alpha}|^{2}-\frac{1}{2}(\frac{1}{|\tilde{q}_{\alpha}-q^{1}|^{\alpha}}+\frac{1}{|\tilde{q}_{\alpha}-q^{2}|^{\alpha}})=0.$
Then by Corollary 2.3 of [20], we have
$\frac{1}{2}|\dot{\tilde{q}}_{\alpha}|^{2}=\frac{1/2}{|\tilde{q}_{\alpha}-q^{1}|^{\alpha}}+\frac{1/2}{|\tilde{q}_{\alpha}-q^{2}|^{\alpha}}\geq[2^{\frac{\alpha+2}{2}}][2|\tilde{q}_{\alpha}|^{2}+\frac{1}{2}]^{-\alpha/2}.$
(13)
Now we claim
(a).
$max_{t\in R}|\tilde{q}_{\alpha}(t)|=+\infty.$ (14)
In fact, if $\exists\beta>0$ such that
$|\tilde{q}_{\alpha}|<\beta,\forall t\in R.$
By (13), $\exists\gamma>0$ such that
$|\dot{\tilde{q}}_{\alpha}|>\gamma,\forall t\in R.$
Then when $n$ is large,we have
$|\dot{\tilde{q}}_{\alpha,n}|>\gamma,\forall t\in R.$ $\displaystyle
cn^{\theta}\geq\int_{-n}^{n}|\dot{\tilde{q}}_{\alpha,n}|^{2}>2n\gamma^{2},$
which is a contradiction.
Now by (13) we have
(b).
$\min_{t\in R}|{\dot{\tilde{q}}_{\alpha}}(t)|=0.$ (15)
## 4 Acknowledgements
The authors sincerely thank the referee for his/her many valuable comments and
remarks which helped us revising the paper, we aslo thank the supports of NSF
of China and a research fund for the Doctoral program of higher education of
China.
## References
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* [8] M. Euler, Probleme un corps etant attire en raison reciproque quarree des distances vers vers deux points fixes donnes trouver les cas ou la courbe decrite par ce corps sera algebrique. Hist. Acad. Roy. Sci. Bell. Lett. Berlin 1767; 2: 228-49.
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* [18] W. Gordon, A minimizing property of Keplerian orbits Amer. J. Math. 1977; 99: 961-71.
* [19] G. Hardy, J. Littlewood, G. Polya, Inequalities. Second ed. Cambridge Univ. Press: Cambridge 1952.
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* [21] R. McGehee, Parabolic orbits of the restricted three-body problem, Academic Press, New York and London, 1973.
* [22] H. Poincar$\acute{e}$, Les M$\acute{e}$thodes Nouvelles de la M$\acute{e}$canique C$\acute{e}$leste, Gauthier-Villars, Paris, 1899.
* [23] P. H. Rabinnowtz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems, Nonlinear Anal. 11(1987), 595-611.
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|
arxiv-papers
| 2012-07-13T02:14:13 |
2024-09-04T02:49:32.979938
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ying Lv and Shiqing Zhang",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1207.3130"
}
|
1207.3132
|
# On the Automorphism Groups and Equivalence of Cyclic Combinatorial Objects
Kenza Guenda and T. Aaron Gulliver T. Aaron Gulliver is with the Department
of Electrical and Computer Engineering, University of Victoria, PO Box 3055,
STN CSC, Victoria, BC, Canada V8W 3P6. email: agullive@ece.uvic.ca.
###### Abstract
We determine the permutation groups that arise as the automorphism groups of
cyclic combinatorial objects. As special cases we classify the automorphism
groups of cyclic codes. We also give the permutations by which two cyclic
combinatorial objects on $p^{m}$ elements are equivalent.
## 1 Introduction
Let $n$ be a positive integer and $S_{n}$ the group of permutations on $n$
elements. A combinatorial object $\mathcal{C}$ on $n$ elements is called
cyclic if its automorphism group $Aut(\mathcal{C})$ contains the complete
cycle $T=(1,2,\ldots,n)$ of length $n$. Let $C_{p}$ denote the cyclic group of
order $p$.
The class of cyclic objects includes circulant graphs, circulant digraphs,
cyclic designs and cyclic codes. Two cyclic combinatorial objects
$\mathcal{C}$ and $\mathcal{C}^{\prime}$ on $n$ elements in the same category
of objects are said to be equivalent if there exists a permutation $\sigma$ of
the symmetric group $S_{n}$ acting on $\\{0,1,\ldots,n-1\\}$ such that
$\mathcal{C}^{\prime}=\sigma\mathcal{C}$. When $n$ is a prime number, it has
been proven by Bays and Lambossey [4, 28] that circulant graphs are equivalent
by a permutation $\mu$ if and only if $\mu$ satisfies $\mu(i)=ai\pmod{n}$ for
$a\bmod n$ such that $(a,n)=1$. Such permutations are called multipliers.
Alspach and Parson [1] proved that two circulant graphs or digraphs on $pq$
vertices, where $p>q$ are two distinct primes, are equivalent if and only if
they are equivalent by a multiplier. These results hold for all cyclic
combinatorial objects in the case that $Aut(\mathcal{C})$ has a $p$-Sylow
subgroup of order $p$ [1, Case 1]. Muzychuk [34] proved that this result also
holds for circulant graphs in the square-free case. More generally, Palfy
proved that if $n$ is such that $(n,\phi(n))=1$, or $n=4$, then two cyclic
combinatorial objects are equivalent if and only if they are equivalent by a
multiplier. Furthermore, Palfy gave a class of cyclic combinatorial objects on
$n$ elements where $n\neq 4$ and $\gcd(n,\phi(n))\neq 1$ such that the class
contains equivalent objects that are not equivalent by a multiplier.
Brand [10] proved that cyclic combinatorial objects on $p^{m}$ elements can be
equivalent only under a specific set of permutations which depends on the
$p$-Sylow subgroup of $Aut(\mathcal{C})$. Building on Brand’s results [10],
Huffman et al. [23] solved the equivalency problem for cyclic combinatorial
objects and cyclic codes in the case $n=p^{2}$. This was achieved by
explicitly determining the set of permutations under which two cyclic
combinatorial objects or extended cyclic objects can be equivalent. In [23], a
negative answer was given to the generalization of their results to the case
$n=p^{m}$, $m>2$. This is due to the fact that the polynomials of Brand that
are crucial to proving the results do not generate a Sylow subgroup of
$S_{p^{m}}$. More recently, Babai et al. [2] gave an exponential time
algorithm for determining the equivalence of two linear codes. In this paper,
we consider the equivalency problem of cyclic combinatorial objects of length
$p^{m}$. We generalize the results of [23] (which are only for the case
$n=p^{2}$), by explicitly giving the permutations by which two cyclic codes of
length $p^{m}$ are equivalent. This allows us to develop an algorithm which
solves the equivalency problem by checking no more than
$\langle\log_{2}(p-1)\rangle+1$ permutations in the automorphism group. We
also classify the automorphism groups of cyclic combinatorial objects. This
requires knowledge of the $p$-Sylow subgroup of $Aut(\mathcal{C})$. We
consider the special case of cyclic codes. Even though these codes are well
known and have been studied extensively, very little is known about their
automorphism groups, in particular the BCH and Reed-Solomon codes [5, 6, 7].
Beside the theoretical interest in automorphism groups and equivalence, there
are many practical applications. Algorithms capable of determining graph
equivalency can be used in optical character recognition [37] and image
processing. Further, the automorphism groups and equivalency of cyclic codes
can be employed in permutation decoding [8] and determining the weight
distribution of a code [29]. While the equivalency of cyclic design find
application in optical orthogonal codes [15]. The remainder of this paper is
organized as follows. In Section 2, we investigate the automorphisms of cyclic
objects. We also classify the automorphism groups of cyclic combinatorial
objects. Section 3 considers the automorphism groups of cyclic codes. New
results concerning the automorphism groups of cyclic codes of length $p^{m}$
are presented. We also give an algorithm to find cyclic codes of length $p$.
This algorithm requires that only $p-1$ permutations be checked. In Section 4,
we simplify and generalize some results of Huffman et al. [23] from length
$p^{2}$ to length $p^{m}$. This allows us to provide a solution to the
equivalency problem for cyclic combinatorial objects. An algorithm to solve
the equivalency problem is then presented which requires checking at most
$\lceil\log_{2}(p-1)\rceil+1$ permutations.
Throughout this paper, $ord_{n}(q)$ denotes the multiplicative order of $q$
modulo $n$. In other words it is the smallest integer $r$ such that
$q^{r}\equiv 1\pmod{n}$. The group $Aut(\mathcal{C})$ denotes the automorphism
group of the object $\mathcal{C}$ (with elements which are permutations from
$S_{n}$). We denote by $z$ the largest integer such that
$p^{z}|(q^{t^{\prime}}-1)$, where $t^{\prime}$ is the order of $q$ modulo $p$.
## 2 The Automorphism Groups of Cyclic Objects
We begin this section with some well known definitions. Let $n$ be a positive
integer. The set of permutations $AG(n)=\\{\tau_{a,b}:a\neq
0,(a,n)=1,b\in\mathbb{Z}_{n}\\}$ is the subgroup of $S_{n}$ formed by the
permutations defined as follows
$\begin{split}\tau_{a,b}:\mathbb{Z}_{n}&\longrightarrow\mathbb{Z}_{n}\\\
x&\longmapsto(ax+b)\bmod n.\end{split}$ (1)
The group $AG(n)$ is called the group of affine transformations. The affine
transformations $M_{a}=\tau_{a,0}$ are also multipliers. The affine group
$AGL(1,p)$ is the group of affine transformations over $\mathbb{Z}_{p}$. The
projective semi-linear group $P\Gamma L(d,t)$ is the semi-direct product of
the projective linear group $PGL(d,t)$ and the automorphism group
$Z=Gal(\mathbb{F}_{t}/\mathbb{F}_{p})$ of $\mathbb{F}_{t}$, where $t=p^{s}$,
$p$ prime, i.e.
$P\Gamma L(d,t)=PGL(d,t)\rtimes Z.$
The orders of these groups are $|PGL(d,t)|=(d,t-1)|PSL(d,t)|,|Z|=s$ and
$|P\Gamma L(d,t)|=s|PGL(d,t)|$.
###### Remark 2.1
If $(d,t-1)=1$, then $PGL(d,t)=PSL(d,t)$. If $t$ is a prime we have $P\Gamma
L(d,t)=PGL(d,t)$. $\Box$
From the fact that the automorphism group of a cyclic combinatorial object
contains the complete cycle $T$ of length $n$, we can easily prove that this
group is transitive. A transitive group is either primitive or imprimitive. An
interesting class of primitive groups is the class of doubly-transitive
groups. A doubly-transitive group $G$ has a unique minimal normal subgroup $N$
which is either regular and elementary abelian or simple and primitive, and
has centralizer $C_{G}(N)=1$ [13, p. 202]. All simple groups which can occur
as a minimal normal subgroup of a doubly-transitive group are known. This
result is due to the classification of finite simple groups. Using this
classification, McSorley [31] gave the following result.
###### Lemma 2.2
A group $G$ of degree $n$ which is doubly-transitive and contains a complete
cycle has socle $N$ with $N\leq G\leq Aut(N)$, and is equal to one of the
cases in Table 1\. $\Box$
Table 1: The Doubly Transitive Groups that Contain a Complete Cycle $G$ | $n$ | $N$
---|---|---
$AGL(1,p)$ | $p$ | $C_{p}$
$S_{4}$ | $4$ | $C_{2}\times C_{2}$
$S_{n},n\geq 5$ | $n$ | $Alt(n)$
$Alt(n),n\text{ odd and }\geq 5$ | $n$ | $Alt(n)$
$PGL(d,t)\leq G\leq P\Gamma L(d,t)$ | $\frac{t^{d}-1}{t-1}$ | $PSL(d,t)$
$(d,t)\neq(2,2),(2,3),(2,4)$ | |
$PSL(2,11)$ | 11 | $PSL(2,11)$
$M_{11}$(Mathieu) | 11 | $M_{11}$(Mathieu)
$M_{23}$(Mathieu) | 23 | $M_{23}$(Mathieu)
As a direct application of Lemma 2.2, we obtain the following result.
###### Theorem 2.3
Let $\mathcal{C}$ be a cyclic combinatorial objects on $p$ elements. Then
$Aut(\mathcal{C})$ is a primitive group with socle $S$, and one of the
following holds:
* (i)
$Aut(\mathcal{C})=S_{n}$ or $Alt(n)$.
* (ii)
$Aut(\mathcal{C})$ is a solvable group of order $pm$ with $m$ a divisor of
$p-1$ and $S=C_{p}\leq Aut(\mathcal{C})\leq AGL(1,p)$. Furthermore
$Aut(\mathcal{C})$ contains a normal $p$-Sylow group.
* (iii)
$Aut(\mathcal{C})=PSL(2,11)$; of degree $11$.
* (iv)
$Aut(\mathcal{C})=M_{11}$ or $Aut(\mathcal{C})=M_{23}$ of degree $11$ or $23$,
respectively.
* (v)
$S=PSL(d,r^{d^{b}})$ and $PGL(d,r^{d^{b}})\leq Aut(\mathcal{C})\leq P\Gamma
L(d,r^{d^{b}})$ where $d\in\mathbb{N}$, $d\geq 3$ is a prime number such that
$(d,r-1)=1$, and $p=(r^{d^{b+1}}-1)/(r^{d^{b}}-1)$.
Proof. A transitive group of prime degree is a primitive group [35, p. 195].
As a consequence of a result of Burnside [19, Theorem 2], a transitive group
of prime degree is either a subgroup of $AGL(1,p)$ or a doubly-transitive
group. Since the order of $AGL(1,p)$ is $p(p-1)$, the order of any subgroup is
$pm$ where $m|(p-1)$. By Sylow’s Theorem, $Aut(\mathcal{C})$ contains a unique
$p$-Sylow group, so it is a normal subgroup. By [17, Ex. 3.5.1] $G$ is
solvable. The remaining cases follow from Lemma 2.2. A number theory argument
[18, Lemma 3.1] gives that in case (iv) if $p$ is prime, then $d$ must be a
prime such that $(d,r^{a}-1)=1$ and $a=d^{b}$. The result then follows. $\Box$
The following result is obtained by considering the automorphism groups of
cyclic objects of composite length.
###### Theorem 2.4
Let $\mathcal{C}$ be a cyclic combinatorial object on $n$ elements such that
$n$ is a composite number. Then $Aut(\mathcal{C})$ is either
1. (i)
an imprimitive group (in the case that $n=p^{m}$, $p$ prime, the orbit of the
subgroup generated by $T^{p^{m-1}}$ and its conjugate form a complete block
system of $Aut(\mathcal{C})$);
or
2. (ii)
$Aut(\mathcal{C})$ is a doubly-transitive group such that
$PGL(d,r^{a})\leq Aut(\mathcal{C})\leq P\Gamma L(d,r^{a}),\text{ with
}n=\frac{r^{ad}-1}{r^{a}-1}\mbox{ and }a\geq 1.$
Proof. The group $Aut(\mathcal{C})$ contains a complete cycle and has
composite degree. Hence from a theorem of Burnside and Schur [38, p. 65],
$Aut(\mathcal{C})$ is either imprimitive or doubly-transitive. If it is
imprimitive and $n=p^{m}$, by [12, Ch. XVI Theorem VIII] $Aut(C)$ contains an
intransitive normal subgroup generated by $T^{p^{m-1}}$ and its conjugates. By
[38, Proposition 7.1] the orbit of such a subgroup forms a complete block
system of $Aut(\mathcal{C})$.
In the doubly-transitive case, we have from Lemma 2.2 that the only cases when
the socle can be abelian are $N=C_{p}$ and $N=C_{2}\times C_{2}$. In these
cases, $Aut(\mathcal{C})$ must be equal to $AGL(1,p)$ or $S_{4}$, which is
impossible. Since the socle is not abelian and the degree is not prime, this
leads to the only solution given by row six of Table 1. $\Box$
###### Corollary 2.5
([32, Corollary 8.6]) For any circulant graph $\mathcal{C}$ on $n$ elements,
one of the following holds:
1. 1.
$Aut(\mathcal{C})=S_{n}$;
2. 2.
$Aut(\mathcal{C})$ is imprimitive, and the orbit of the subgroup generated by
$T^{p^{m}-1}$ and its conjugate form a complete block system of
$Aut(\mathcal{C})$; or
3. 3.
$n$ is prime and $Aut(\mathcal{C})<AGL(1,p)$.
Proof. If an automorphism group acting on a graph is doubly transitive, then
it takes any ordered pair of vertices to another ordered pair of vertices.
Hence the graph is either complete or empty, and (in either case) the
automorphism group is $S_{n}$. The result then follows from Theorems 2.3 and
2.4. $\Box$
## 3 Automorphism Groups of Cyclic Codes
A linear code $C$ over $\mathbb{F}_{q}$ is cyclic if $T\in Aut(\mathcal{C})$,
where $T=(1,2,\ldots,n)$ is a complete cycle of length $n$. In the case of
cyclic codes we have the following results concerning the group
$Aut(\mathcal{C})$. We begin with the following useful remark.
###### Remark 3.1
The zero code, the entire space, and the repetition code and its dual are
called elementary codes. The permutation group of these codes is $S_{n}$ [24,
p. 1410]. Furthermore, it was proven in [24, p. 1410] that there is no cyclic
code with permutation group equal to $Alt(n)$. $\Box$
The following lemma can be proved using arguments similar to those for the
binary case [9, Theorem E Part 3].
###### Lemma 3.2
Let $\mathcal{C}$ be a non-elementary cyclic code of length
$n=\frac{t^{d}-1}{t-1}$ over a finite field $\mathbb{F}_{q}$, where
$q=r^{\alpha}$ and $t$ is a prime power. If $Aut(\mathcal{C})$ satisfies
$PGL(d,t)\leq Aut(\mathcal{C})\leq P\Gamma L(d,t),$
then $t=r^{a}$ for some $a\geq 1$, $d\geq 3$, and $Aut(\mathcal{C})=P\Gamma
L(d,t)$.
Proof. Assume $d=2$. As the group $PGL(2,t)$ acts 3-transitively on the
$1$-dimensional projective space $\mathbb{P}^{1}(\mathbb{F}_{t})$, we deduce
from [33, Table 1 and Lemma 2] that the underlying code is elementary, which
is a contradiction. Hence $d\geq 3,$ and from [33, Table 1 and Lemma 2], it
must be that since $C$ is non-elementary, $t$ must be equal to $r^{a}$. Now
let $V$ denote the permutation module over $\mathbb{F}_{p}$ associated with
the natural action of $PGL(d,t)$ on the $(d-1)$ dimensional projective space
$\mathbb{P}^{d-1}(\mathbb{F}_{t})$. Let $U_{1}$ be a $PGL(d,t)$-submodule of
$V$. Hence $U_{1}$ is $P\Gamma L(d,t)$-invariant. This is because, if $\sigma$
is a generator of the cyclic group $P\Gamma L(d,t)/PGL(d,t)\simeq
Gal(\mathbb{F}_{t}/\mathbb{F}_{p})$, then $U_{2}=U_{1}^{\sigma},$ regarded as
a $PGL(d,t)$-module, is simply a twist of $U_{1}$. Let
$\overline{\mathbb{F}}_{r}$ be the algebraic closure of $\mathbb{F}_{r}$. Then
the composition factors of the $\overline{\mathbb{F}}_{p}PGL(d,t)$-modules
$\overline{U}_{1}=\overline{\mathbb{F}}_{r}\otimes U_{1}$ and
$\overline{U}_{2}=\overline{\mathbb{F}}_{r}\otimes U_{2}$ are the same. The
submodules of the $\overline{\mathbb{F}}_{r}PGL(d,t)$-module
$\overline{V}=\overline{\mathbb{F}}_{r}\otimes V$ are uniquely determined by
their composition factors [3]. Then we have
$\overline{U}_{1}=\overline{U}_{2}$, which implies that $U_{1}=U_{2}$, and
therefore $Aut(\mathcal{C})=P\Gamma L(d,t)$. $\Box$
###### Theorem 3.3
Let $\mathcal{C}$ be a non-elementary cyclic code of length $n$ over
$\mathbb{F}_{q}$, where $q=r^{\alpha}$, $\alpha\geq 1$. Then if $n=p$ is a
prime number we have that $Aut(\mathcal{C})$ is a primitive group, and one of
the following holds:
* (i)
$Aut(\mathcal{C})$ is a solvable group of order $pm$ with $m$ a divisor of
$p-1$ and $C_{p}\leq Aut(\mathcal{C})\leq AGL(1,p)$, with $p\geq 5$.
Furthermore $Aut(\mathcal{C})$ contains a normal $p$-Sylow group.
* (ii)
If $p=q$, then $Aut(\mathcal{C})=AGL(1,p)$.
* (iii)
$Aut(\mathcal{C})=PSL(2,11)$ and $q$ is a power of 3. $C$ is either an
$[11,6]$ or $[11,5]$ code that is equivalent to the $[11,6,5]$ ternary Golay
code or its dual, respectively.
* (iv)
$Aut(\mathcal{C})=M_{23}$ and $q$ is a power of 2. $C$ is either a $[23,12]$
or $[23,11]$ code that is equivalent to the $[23,12,7]$ binary Golay code or
its dual, respectively.
* (v)
$Aut(C)=P\Gamma L(d,r^{d^{b}})$ where $b\in\mathbb{N}$, $d\geq 3$ is a prime
number such that $(d,r^{d^{b}}-1)=1$, and $p=(r^{d^{b+1}}-1)/(r^{d^{b}}-1)$.
If $n$ is a composite number then $Aut(\mathcal{C})$ is either
* (vi)
an imprimitive group (in the case that $n=p^{m}$, $p$ a prime, the orbit of
the subgroup generated by $T^{p^{m-1}}$ and its conjugate form a complete
block system of $Aut(\mathcal{C})$);
or
* (vii)
$Aut(\mathcal{C})$ is a doubly-transitive group equal to $P\Gamma
L(d,r^{a}),\text{ with }n=\frac{r^{ad}-1}{r^{a}-1}\mbox{ and }a\geq 1$.
Proof. From Theorem 2.3 we have that $Aut(\mathcal{C})$ is either a subgroup
of $AGL(1,p)$ which is solvable of order $pm$, $m$ a divisor of $p-1$, or a
doubly transitive group. For the first case if $p=2$ or $3$, then
$AGL(1,p)=S_{p}$, and $\mathcal{C}$ is elementary by Remark 3.1. For part
(ii), if $q=p$, Roth and Seroussi [36] proved that any cyclic code of prime
length $p$ over $\mathbb{F}_{p}$ must be an MDS code equivalent to an extended
Reed–Solomon code. Berger [5] proved that the permutation group of such codes
is the affine group $AGL(1,p)$. For parts (iii) and (iv), as $\mathcal{C}$ is
non-elementary of prime length $p$, by Lemma 2.2, Remark 3.1 and Lemma 3.2, we
have that $Aut(\mathcal{C})$ is one of $M_{11}$ with $p=11$, $PSL(2,11)$ with
$p=11$, $M_{23}$ with $p=23$, or $P\Gamma L(d,t)$ of degree
$p=(t^{d}-1)/(t-1)$ and $t$ a prime power. If $Aut(\mathcal{C})=M_{11}$, from
[33, Table 1, Lemma 2] $C$ must be elementary, which is a contradiction. If
$Aut(C)=PSL(2,11)$, from [33, Table 1, Lemma 2 and (J)] $q$ must be a power of
3, and there is a unique non-elementary code over $\mathbb{F}_{q}$ contained
in the dual of the repetition code. The $[11,5,6]$ dual of the ternary Golay
code is contained in the repetition code and has permutation group
$PSL(2,11)$; its dual, an $[11,6,5]$ code, intersects the dual of the
repetition code in this $[11,5,6]$ code and also has permutation group
$PSL(2,11)$. Part (ii) then follows. Part (iii) is obtained in an analogous
way from [33, Table 1, Lemma 2 and (I)]. $\Box$
Now we give an algorithm to find the automorphism group of specific cyclic
codes. Let $\mathcal{C}$ be a non elementary cyclic code over $\mbox{\msbm
F}_{r}$ of length $p$, different from the binary and ternary Golay codes.
Further, assume that there are no integers $b$ and $d$ such that
$p=(r^{d^{b+1}}-1)/(r^{d^{b}}-1)$. Then from Theorem 2.3 and Remark 2.1, we
have that $\mathcal{C}\leq AG(p)$. Let $a\in\mbox{\msbm Z}_{p}^{*}$ and
$\mu_{a}$ be the associated multiplier. Hence if
$\mu_{a}(\mathcal{C})=\mathcal{C}$ then $\mu_{a}\in Aut(\mathcal{C})$. From
Remark 4.8, for all $b\in\mbox{\msbm Z}_{p}$ we also have that $\tau_{a,b}\in
Aut(\mathcal{C})$. This suggest the following algorithm to find
$Aut(\mathcal{C})$. To summarize, it is assumed that $\mathcal{C}$ is not
elementary, $p\neq 11$ and $p\neq 23$, and there are no integers $b$ and $d$
such that $p=(r^{d^{b+1}}-1)/(r^{d^{b}}-1)$.
Algorithm A:
1. 1.
Find $A=\\{a\in\mbox{\msbm Z}_{p}^{*}\text{ such that
}\mu_{a}(\mathcal{C})=\mathcal{C}\\}$.
2. 2.
If $A=\mbox{\msbm Z}_{p}^{*}$, then $Aut(\mathcal{C})=AG(p)$.
3. 3.
Otherwise, $Aut(\mathcal{C})=\\{\tau_{a,b},a\in A,b\in\mbox{\msbm Z}_{p}\\}$.
In Table 2, we give examples of permutation groups of BCH codes of length $p$
over $\mathbb{F}_{q}$. $Aut(C)$ (respectively $Aut(C_{2})$ and $Aut(C_{3})$),
denote the permutation groups of narrow sense ($b=1$) BCH codes with designed
distance $\delta$ (respectively BCH codes with designed distance $\delta$ and
$b=2$ and $b=3$).
$q$ | $p$ | $\delta$ | $Aut(C)$ | $Aut(C_{2})$ | $Aut(C_{3})$
---|---|---|---|---|---
2 | 17 | 2 | $C_{8}\ltimes C_{17}$ | $S_{17}$ | $S_{17}$
2 | 23 | 3 | $M_{23}$ | $M_{23}$ | $M_{23}$
2 | 41 | 2 | $C_{20}\ltimes C_{41}$ | $C_{20}\ltimes C_{41}$ | $C_{20}\ltimes C_{41}$
2 | 41 | 3 | $C_{20}\ltimes C_{41}$ | $S_{41}$ | $S_{41}$
2 | 43 | 5 | $C_{14}\ltimes C_{43}$ | $C_{14}\ltimes C_{43}$ | $C_{14}\ltimes C_{43}$
2 | 43 | 7 | $C_{14}\ltimes C_{43}$ | $S_{43}$ | $S_{43}$
3 | 13 | 2 | $C_{3}\ltimes C_{13}$ | $C_{3}\ltimes C_{13}$ | $C_{3}\ltimes C_{13}$
3 | 13 | 4 | $P\Gamma L(3,3)$ | $C_{3}\ltimes C_{13}$ | $C_{3}\ltimes C_{13}$
3 | 13 | 5 | $C_{3}\ltimes C_{13}$ | $C_{3}\ltimes C_{13}$ | $C_{3}\ltimes C_{13}$
3 | 23 | 3 | $C_{11}\ltimes C_{23}$ | $C_{11}\ltimes C_{23}$ | $C_{11}\ltimes C_{23}$
3 | 41 | 5 | $C_{8}\ltimes C_{41}$ | $C_{8}\ltimes C_{41}$ | $C_{8}\ltimes C_{41}$
4 | 43 | 9 | $C_{7}\ltimes C_{43}$ | $S_{43}$ | $S_{43}$
5 | 11 | 5 | $C_{5}\ltimes C_{11}$ | $C_{5}\ltimes C_{11}$ | $C_{5}\ltimes C_{11}$
11 | 5 | 3 | $C_{5}$ | $C_{2}\ltimes C_{5}$ | $C_{5}$
Table 2: Permutation Groups of some BCH Codes of Length $p$
### 3.1 The Automorphism Groups of Cyclic Codes of Length $p^{m}$
In the previous section, the automorphism groups of cyclic codes were
determined. In this section, we provide additional results on these cyclic
combinatorial objects in the case $n=p^{m}$, where $p$ is an odd prime and
$m\geq 1$.
###### Lemma 3.4
Let $q$ be a prime power, $p$ an odd prime, and $z$ the largest integer such
that $p^{z}|(q^{t}-1)$, with $t$ the order of $q$ modulo $p$. If $z=1$ we have
$ord_{p^{m}}(q)=p^{m-1}t.$
Proof. Let $t$ be the order of $q$ modulo $p$, and $u=q^{t}\equiv 1\bmod p$.
Assume that $z=1$, or equivalently $u\neq 1\bmod p^{2}$. It is well known from
elementary number theory [16, p. 87] that $u\bmod p^{m}$ is an element of
order $p^{m-1}$ in the group $\left(\mathbb{Z}_{p^{m}}\right)^{\ast}$ if and
only if $u\neq 1\bmod p^{2}$. Hence $ord_{p^{m}}(q)=p^{m-1}t$. $\Box$
Note that according to Brillhart et al. [11], it is unusual to have $z>1$.
###### Proposition 3.5
Let $\mathcal{C}$ be a cyclic object on $p^{m}$ elements with $m>1$. Hence a
$p$-Sylow subgroup of $Aut(\mathcal{C})$ has order $p^{s}$ such that
$m\leq s\leq p^{m-1}+p^{m-2}+\cdots+1.$ (2)
We consider $\mathcal{C}$ to be a cyclic code of length $p^{m}$ over
$\mbox{\msbm F}_{q}$ with $q=r^{\alpha}$ a prime power and $(q,p)=1$. Let
$\mu_{q}$ be the multiplier defined by $\mu_{q}(i)=iq\bmod p^{m}$. Then the
group $Aut(\mathcal{C})$ contains the subgroup $K=<T,\mu_{q}>$ of order
$p^{m}ord_{p^{m}}(q)$. Let $p^{l}$, $l\geq m$, be the $p-$part of the order of
$K$. Then a $p$-Sylow subgroup $P$ of $Aut(\mathcal{C})$ has order $p^{s}$
such that
$l\leq s\leq p^{m-1}+p^{m-2}+\cdots 1.$
If $z=1$, then $s\geq 2m-1$. If $s=2m-1$, then $P$ is a transitive group of
$K$.
Proof. From the definition of a cyclic code, we have that $T\in
Aut(\mathcal{C})$. It is obvious that each cyclotomic class modulo $n$ over
$\mathbb{F}_{q}$ is invariant under the permutation $\mu_{q}$. This can be
deduced from the fact that the polynomial $f(x)\in\mathbb{F}_{q}[x]$ satisfies
$f(x^{q})=f(x)^{q}$. Thus $\mu_{q}\in Aut(\mathcal{C})$. The order of
$\mu_{q}$ is equal to $|Cl(1)|=ord_{p^{m}}(q)$, and hence $K=<T,M_{q}>$ is a
subgroup of $Aut(\mathcal{C})$ of order ${p^{m}}ord_{p^{m}}(q)$. Then the
order of $K$ has $p-$part $p^{l}$ with $l\leq m$. Let $P$ be a $p$-Sylow
subgroup of $Aut(\mathcal{C})$ which contains $T$ (this can always be assumed
since any $p$ group is contained in a $p$-Sylow subgroup). Then $P$ is a $p$
group of $S_{p^{m}}$. From Sylow’s Theorem, $P$ is contained in a $p$-Sylow
subgroup of $S_{p^{m}}$. It is well known that a $p$-Sylow subgroup of
$S_{p^{m}}$ has order $p^{p^{m-1}+p^{m-2}+\cdots+1}$ [35, Kalužnin’s Theorem].
Since $P$ also contains the subgroup of $K$ of order $p^{l}$, then $l\leq
s\leq p^{m-1}+p^{m-2}+\cdots+1$. If $z=1$, then by Lemma 3.4 the order of the
group $K$ is $ord_{p}(q)p^{2m-1}$. This gives that $p^{2m-1}$ divides
$|Aut(\mathcal{C})|$, so $Aut(\mathcal{C})$ contains a $p$ subgroup of order
at least $p^{2m-1}$. If $s=2m-1$, we can assume that $P\leq K$ because we have
$T\in K$ and $K\leq Aut(\mathcal{C})$. Thus $P$ is a transitive subgroup of
$K$. $\Box$
###### Theorem 3.6
Let $\mathcal{C}$ be a non elementary cyclic code of length $p^{m}$ over
$\mathbb{F}_{r^{\alpha}}$ with $m\geq 1$. Then the following holds:
* (i)
If $p\nmid\alpha$ and $p\nmid(d,r^{a}-1)$, then $Aut(\mathcal{C})=P\Gamma
L(d,r^{a})$, $a\geq 1$, $d\geq 3$, if and only if the $p$-Sylow subgroup of
$Aut(C)$ is of order $p^{m}$.
* (ii)
If $p\geq 5$, $\alpha=1$ and $r=p$, $m>1$, then $Aut(\mathcal{C})$ is an
imprimitive group which admits a complete system formed by the orbit of the
subgroup generated by $T^{p^{m-1}}$ and its conjugate. It also contains a
transitive normal $p$-Sylow subgroup of order $p^{s}$ with $m<s\leq
p^{m-1}+p^{m-2}+\ldots+1$.
* (iii)
If $z=1$, $p\nmid\alpha$ and $p\nmid(d,r^{a}-1)$, then $Aut(\mathcal{C})$ is
an imprimitive group which contains a transitive normal $p$-Sylow subgroup of
order $p^{s}$, with $2m-1\leq s\leq p^{m-1}+p^{m-2}+\ldots+1$. Furthermore,
$Aut(\mathcal{C})$ admits a complete block system formed by the orbit of the
subgroup generated by $T^{p^{m-1}}$ and its conjugate.
Proof. For part (i), we know that the socle of $P\Gamma L(d,r^{a})$ is the
group $PSL(d,r^{a})$ of order
$\frac{r^{ad(d-1)/2}}{(d,r^{a}-1)}\prod_{i=2}^{d}(r^{ai}-1)$. From a lemma of
Zsigmondy [25, Ch. IX, Theorem 8.3], except for the cases $d=2$,
$r^{a}=2^{b}-1$ and $d=6,r^{a}=2$, there exists a prime $q_{0}$ such that
$q_{0}$ divides $r^{ad}-1$, but does not divide $r^{ai}-1$, for $1\leq i<d$.
From Lemma 3.2, we cannot have $d=2$. The case $d=6$ and $r^{a}=2$ does not
give a prime power. Hence if $n=p^{m}=\frac{r^{ad}-1}{r^{a}-1}$, there is a
$q_{0}$ which divides $(r^{ad}-1)=(r^{a}-1)p^{m}$. Since $q_{0}$ does not
divide $r^{a}-1$, $q_{0}$ must divide $p^{m}$, and hence $q_{0}=p$ and $p^{m}$
is the $p-$part of the order of $PSL(d,r^{a})$. Also, since $p\nmid r^{a}-1$,
we have that $p\nmid(d,r^{a}-1)$. Hence if $(\alpha,p)=1$, $p^{m}$ is also in
the $p-$part of the order of $P\Gamma L(d,r^{a})$, and the result follows.
Conversely, if $Aut(\mathcal{C})$ contains a $p$-Sylow group $P$ of order
$p^{m}$, we can assume that $T\in P$, which gives the equality $P=\langle
T\rangle$. Assume that in this case $Aut(\mathcal{C})$ is imprimitive. Then by
[19, Theorem 33], $P$ is normal. $P$ is then the minimal normal subgroup which
is transitive and abelian. From [38, p. 17] $Aut(\mathcal{C})$ is primitive,
which is impossible. Thus if $P=\langle T\rangle$, the group
$Aut(\mathcal{C})$ is equal to $P\Gamma L(d,r^{a})$, which is possible only if
$[P\Gamma L(d,r^{a}):PSL(d,r^{a})]$ is prime to $p$, i.e., $(p,\alpha)=1$ and
$p\nmid(d,r^{a}-1)$.
For part (ii), from Theorem 2.4 if $Aut(\mathcal{C})$ is primitive, then it is
doubly-transitive and equal to $P\Gamma L(d,r^{a})$ with
$n=\frac{r^{ad}-1}{r^{a}-1},d\geq 3$ and $a\geq 1$. From [19, Lemma 22], if
$Aut(\mathcal{C})$ is doubly-transitive with a non abelian socle, then
$Soc(Aut(\mathcal{C}))=Alt(p^{m})$. Hence from Remark 3.1 the code is
elementary. Since $Aut(\mathcal{C})$ is imprimitive, from part (i) the order
of the $p$-Sylow group is $p^{s}$ with $s>m$. The second inequality then
follows from Proposition 3.5.
For part (iii), if $z=1$ then from Proposition 3.5, the order of a $p$-Sylow
subgroup of $Aut(\mathcal{C})$ is at least $p^{2m-1}$. If $Aut(\mathcal{C})$
is doubly transitive, by Theorem 2.4 it is equal to $P\Gamma L(d,r^{a})$ with
$d\geq 3$. By assuming $p\nmid\alpha$ and $p\nmid(d,r^{a}-1)$, we obtain from
part (i) that a $p$-Sylow group of $Aut(C)$ has order $p^{m}$, which is
impossible. Hence $Aut(\mathcal{C})$ is an imprimitive group. From [19,
Theorem 33], $Aut(\mathcal{C})$ contains a transitive normal $p$-Sylow
subgroup. The result then follows. $\Box$
###### Example 3.7
The narrow sense BCH code of length 25 over $\mathbb{F}_{3}$ with designed
distance 3 has an automorphism group which is the imprimitive group $S_{5}\wr
S_{5}$. The narrow sense BCH code of length 9 over $\mathbb{F}_{5}$ with
designed distance 2 has an automorphism group which is the imprimitive group
$S_{3}\wr S_{3}$. The binary $[7,4,3]$ Hamming code has automorphism group
$P\Gamma L(3,2)$, which contains a $7$-Sylow subgroup of order 7.
## 4 Equivalence of Cyclic Combinatorial Objects on $p^{m}$ Elements
Let $\mathcal{C}$ be a cyclic object of length $p^{m}$ where $p$ is an odd
prime, $m>1$ and $P$ is a $p$-Sylow subgroup of $Aut(\mathcal{C})$. The
following subset of $S_{p^{r}}$ was introduced by Brand [10]
$H(P)=\\{\sigma\in S_{p^{m}}|\sigma^{-1}T\sigma\in P\\}.$
The set $H(P)$ is well defined since $<T>$ is a subgroup of $Aut(\mathcal{C})$
of order $p^{m}$, hence it is a $p$-group of $Aut(\mathcal{C})$. From Sylow’s
Theorem, there exists a $p$-Sylow subgroup $P$ of $Aut(\mathcal{C})$ such that
$<T>\;\leq P$. Furthermore, in some cases the set $H(P)$ is a group.
###### Lemma 4.1
([10, Lemma 3.1]) Let $\mathcal{C}$ and $\mathcal{C}^{\prime}$ be cyclic
objects on $p^{m}$ elements. Let $P$ be a $p$-Sylow subgroup of
$Aut(\mathcal{C})$ which contains $T$. Then $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ are equivalent if and only if $\mathcal{C}$ and
$\mathcal{C}^{\prime}$ are equivalent by an element of $H(P)$.
Let $p$ be an odd prime. For $n<p$, we define the following subsets of
$S_{p^{m}}$:
$\begin{array}[]{l}Q^{n}=\\{f:\mathbb{Z}_{p^{m}}\rightarrow\mathbb{Z}_{p^{m}}|{\displaystyle
f(x)=\sum_{i=0}^{n}a_{i}x^{i}},a_{i}\in\mathbb{Z}_{p^{m}}\mbox{ for each
}i,(p,a_{1})=1,\\\ \hskip 36.135pt\mbox{ and }p^{m-1}\mbox{ divides
}a_{i}\mbox{ for }i=2,3,\ldots,n\\}.\end{array}$ $Q_{1}^{n}=\\{f\in
Q^{n}|f(x)=\sum_{i=0}^{n}a_{i}x^{i},\text{ with }a_{1}\equiv 1\bmod
p^{m-1}\\}.$
The sets $Q^{n}$ and $Q_{1}^{n}$ are subgroups of $S_{p^{m}}$ [10, Lemma 2.1].
Note that $Q^{1}=AG(p^{m})$.
###### Lemma 4.2
Let $\mathcal{C}$ be a cyclic object on $p^{m}$ elements, where $p$ is odd and
$m>1$. Let $P$ be a Sylow subgroup of $Aut(\mathcal{C})$ which contains $T$.
If $1\leq n<p$, then
* (i)
$|Q^{n}|=(p-1)p^{2m+n-2}$ and $|Q_{1}^{n}|=p^{m+n}$.
* (ii)
$AG(p^{m})=N_{S_{p^{m}}}(<T>)\subset H(P)$ .
* (iii)
$Q^{n+1}=H(Q_{1}^{n})$.
* (iv)
$N_{S_{p^{m}}}(Q_{1}^{n})=Q^{n+1}$.
Proof. For part (i), from [10, Lemma 3.2] we have the map
$(a_{0},\ldots,a_{n})\longrightarrow f$ where $f(x)=\sum_{i=0}^{n}a_{i}x^{i}$
is injective if $n<p-1$. Thus in $Q^{n}$, the coefficients of $a_{0}$ can take
$p^{m}$ different values, and $a_{1}$ can take $p^{m-1}(p-1)$ values. For
$2\leq i\leq n$, $a_{i}$ can take $p$ values. From these results we have
$|Q^{n}|=p^{2m+n-2}(p-1)$. For $Q_{1}^{n}$, the coefficients of $a_{0}$ can
take $p^{m}$ different values, and $a_{i}$ for $1\leq i\leq n$ can take $p$
values, hence $|Q_{1}^{n}|=p^{m+n}$.
Now we prove that $AG(p^{m})=N_{S_{p^{m}}}(<T>)$. Let $\sigma$ be an element
of $N_{S_{p^{m}}}(<T>)$. Then there is a $j\in\mathbb{Z}_{n}\setminus\\{0\\}$
such that $\sigma T\sigma^{-1}=T^{j}$, or equivalently $\sigma T=T^{j}\sigma$.
Hence $\sigma T(0)=\sigma(1)=T^{j}\sigma(0)=\sigma(0)+j$ and $\sigma
T(1)=\sigma(1)+j=\sigma(0)+2j$, so that $\sigma(k)=\sigma(0)+kj$ for any
$k\in\mathbb{Z}_{n}$. Then $(j,n)=1$ follows from the fact that the order of
$T$ equals the order of $T^{j}$. The last inclusion is obvious.
Part (iii) follows from [10, Lemma 3.7].
For the proof of part (iv), we begin with the $\leq$ condition. Let $h\in
N_{p^{m}}(Q_{1}^{n})$ and $g=h^{-1}Th$. As $T\in Q_{1}^{n}$, it must be that
$g\in Q_{1}^{n}$. Since the order of $g$ is equal to the order of $T$ which is
$p^{m}$, from [10, Lemma 3.6] there exists $f\in Q^{n+1}$ such that
$f^{-1}gf=T$. Thus $f^{-1}h^{-1}Thf=T$. The only elements of $S_{p^{m}}$ which
commute with $T$ (a complete cycle of length $p^{m}$), are the powers of $T$.
Thus $hf=T^{j}$ for some $j$. Since $Q^{n+1}$ is a subgroup of $S_{p^{m}}$ and
$\langle T\rangle\leq Q^{n+1}$, then $h\in Q^{n+1}$, and hence
$N_{p^{m}}(Q_{1}^{n})\leq Q^{n+1}.$
Now consider the $\geq$ condition. Let $h\in Q_{1}^{n}$, where
$h(x)=\sum_{i=0}^{n}h_{i}x^{i}$ with $h_{1}\equiv 1\bmod p^{m-1}$ and
$p^{m-1}|h_{i}$ for $2\leq i\leq n$. Let $g\in Q^{n+1}$ where
$g(x)=\sum_{i=0}^{n+1}g_{i}x^{i}$ with $p\nmid g_{1}$ and $p^{m-1}|g_{i}$ for
$2\leq i\leq n$. We have
$hg(x)=\sum_{i=0}^{n}h_{i}\left(\sum_{j=0}^{n+1}g_{j}x^{j}\right)^{i}=h_{0}+h_{1}\sum_{i=0}^{n+1}g_{j}x^{j}+\sum_{i=2}^{n}h_{i}\left(\sum_{j=0}^{n+1}g_{j}x^{j}\right)^{i}.$
Since $p^{m-1}|h_{i},$ for $i\geq 2$ and $p^{m-1}|g_{j}$ for $j\geq 2$, any
terms in $\sum_{i=2}^{n}h_{i}\left(\sum_{j=0}^{n+1}g_{j}x^{j}\right)^{i}$
involving $g_{j}$ for $j\geq 2$ vanish modulo $p^{m}$, so that
$hg(x)=h_{0}+h_{1}\sum_{j=0}^{n+1}g_{j}x^{j}+\sum_{i=2}^{n}h_{i}\left(g_{0}+g_{1}x\right)^{i}.$
By [10, Lemma 2.1]
$g^{-1}(x)=\sum_{i=1}^{n+1}b_{i}x^{i},\text{ with }b_{1}=g_{1}^{-1}\text{ and
}b_{i}=-g_{i}g_{1}^{-(i+1)}\text{ for }2\leq j\leq n+1.$ (3)
We now determine $g^{-1}hg$ in order to prove that it is in $Q_{1}^{n}$. This
is given by
$g^{-1}hg(x)=\sum_{k=1}^{n+1}b_{k}\left(h_{0}+h_{1}\sum_{j=0}^{n+1}g_{j}x^{j}+\sum_{i=2}^{n}h_{i}(g_{0}+g_{1}x)^{i}-g_{0}\right)^{k}$
$=b_{1}\left(h_{0}+h_{1}\sum_{j=0}^{n+1}g_{j}x^{j}+\sum_{i=2}^{n}h_{i}\left(g_{0}+g_{1}x\right)^{i}-g_{0}\right)$
$\hskip
36.135pt+\sum_{k=2}^{n+1}b_{k}{\left(h_{0}+h_{1}\sum_{j=0}^{n+1}g_{j}x^{j}+\sum_{i=2}^{n}h_{i}{(g_{0}+g_{1}x)}^{i}-g_{0}\right)}^{k}.$
As $p^{m-1}|g_{j}$ for $j\geq 2$, hence $p^{m-1}|b_{k}$ for $k\geq 2$.
Furthermore, we have $p^{m-1}|h_{i}$ for $i\geq 2$, and thus
$g^{-1}hg(x)=b_{1}\left(h_{0}+h_{1}\sum_{j=0}^{n+1}g_{j}x^{j}+\sum_{j=0}^{n+1}h_{i}(g_{0}+g_{1}x)^{i}-g_{0}\right)+\sum_{k=2}^{n+1}b_{k}\left(h_{0}+h_{1}\left(g_{0}+g_{1}x\right)-g_{0}\right)^{k}.$
Let $g^{-1}hg(x)=\sum_{m=0}^{n+1}c_{m}x^{m}$, and note that
$c_{n+1}=b_{1}h_{1}g_{n+1}+b_{n+1}(h_{1}g_{1}){n+1}$. Then replacing the
$b_{i}$ with their values from (3), we obtain
$c_{n+1}=g_{1}^{-1}h_{1}g_{n+1}-g_{n+1}g_{1}^{-(n+2)}h_{1}^{n+1}g_{1}^{n+1}=g_{1}^{-1}h_{1}(g_{n+1}-g_{n+1}h_{1}^{n}).$
As $h_{1}\equiv 1\bmod p^{m-1}$, we have that $h_{1}^{n}\equiv 1\bmod
p^{m-1}$. In addition, as $p^{m-1}|g_{n+1}$, it must be that
$g_{n+1}h_{1}^{n}\equiv g_{n+1}\bmod p^{m}$. Therefore, $c_{n+1}=0$, and
$p^{m-1}|c_{i}$ for $2\leq i\leq n$. Then we only need to show that
$c_{1}\equiv 1\bmod p^{m-1}$. As $g_{j}\equiv 0\bmod p^{m-1}$ for $j\geq 2$,
$h_{i}\equiv 0\bmod p^{m-1}$ for $i\geq 2$, and $b_{k}\equiv 0\bmod p^{m-1}$
for $k\geq 2$, then $c_{1}\equiv b_{1}h_{1}g_{1}\bmod p^{m-1}$. Finally, since
$b_{1}=g_{1}^{-1}$, we have that $c_{1}\equiv h_{1}\equiv 1\bmod p^{m-1}$.
$\Box$
###### Lemma 4.3
Let $1\leq n<p-1$. If $P$ is a $p$ group of $S_{p^{m}}$ with $Q_{1}^{n}\lneq
P\leq Q^{n+1}$, then $P=Q_{1}^{n+1}$.
Proof. By part (ii) of Lemma 4.2, we have $Q_{1}^{n}\lhd Q^{n+1}$. Hence we
can consider $\overline{Q}=Q^{n+1}/Q_{1}^{n}$, which has order $p^{m-1}(p-1)$
by Lemma 4.2. Let $N$ be the number of $p$-Sylow subgroups of $\overline{Q}$.
Then by Sylow’s Theorem, $N\equiv 1\bmod p$ and $N$ divides $p^{m-1}(p-1)$.
Hence $N=1$, so there exists a unique $p$-Sylow subgroup
$\overline{P^{\prime}}$ of $\overline{Q}$ which is normal. From the condition
on $P$ above, the image $\overline{P}$ of $P$ in $\overline{Q}$ is also a
$p$-Sylow subgroup of $\overline{Q}$. Since there is a unique $p$-Sylow
subgroup $\overline{P^{\prime}}=\overline{P}$, by Lemma 4.2 the image
$\overline{Q}_{1}^{n+1}$ of $Q_{1}^{n+1}$ in $\overline{Q}$ is a $p$-Sylow
subgroup of $\overline{Q}$. Hence
$\overline{Q}_{1}^{n+1}=\overline{P}=\overline{P^{\prime}}$. As
$Q_{1}^{n}\lneq P$ and $Q_{1}^{n}\leq Q_{1}^{n+1}$, the result follows. $\Box$
Now we prove that the group $Q_{1}^{1}$ is a special subgroup of $S_{p^{m}}$.
###### Theorem 4.4
The group $Q_{1}^{1}$ is a normal subgroup of $Q^{1}$ and is the unique
subgroup of $S_{p^{m}}$ of order $p^{m+1}$ which contains $T$.
Proof. It is obvious that $T\in Q_{1}^{1}$, and from Lemma 4.2,
$|Q_{1}^{1}|=p^{m+1}$. Consider now an element $g$ of $Q^{1}$,
$g(x)=b_{0}+b_{1}x$, with $b_{0},b_{1}\in\mathbb{Z}_{p^{m}}$ and
$(b_{1},p)=1$. It is not difficult to determine that the inverse of $g$ in
$Q^{1}$ is given by $g^{-1}(x)=-b_{1}^{-1}b_{0}+b_{1}^{-1}x$. Consider $f\in
Q_{1}^{1}$, so that $f(x)=a_{0}+a_{1}x$ with
$a_{0},a_{1}\in\mathbb{Z}_{p^{m}}$, $(a_{1},p)=1$ and $a_{1}\equiv 1\bmod
p^{m}$. We then have
$g^{-1}fg(x)=g^{-1}(a_{0}+a_{1}(b_{0}+b_{1}x))=(-b_{0}+a_{0}+a_{1}b_{0})b_{1}^{-1}+a_{1}x$.
This proves that $g^{-1}fg(x)\in Q_{1}^{1}$. Hence $Q_{1}^{1}$ is normal in
$Q^{1}$. Now let $S$ be a subgroup of $Q^{1}$ of order $p^{m+1}$ which
contains $T$. Thus $<T>$ has index $p$ in $S$, and so $<T>$ is maximal in $S$.
Furthermore, $<T>\lhd S$, because any subgroup of a $p$-group of index $p$
must be normal. Therefore, we have $S=N_{S}(T)\leq N_{S_{p^{m}}}(T)$, and by
Lemma 4.2, $S\leq N_{S_{p^{m}}}(T)=AG(p^{m})=Q^{1}$. Thus, such an $S$ must be
a subgroup of $Q^{1}$. It is clear that $Q_{1}^{1}$ is not abelian, and $S$
cannot be abelian since it is a transitive group. If this were the case it
would have to be a regular group [35, Theorem 1.6.3], and thus $|S|=p^{m}$,
which is impossible. Furthermore, the $p$ groups which contain a cyclic
maximal subgroup are known [35, Theorem 5.3.4]. If these groups are not
abelian or $p\neq 2$, they have the following special forms
$Q_{1}^{1}=<x,T|x^{p}=1;\,x^{-1}Tx=T^{1+p^{m-1}}>,$
and
$S=<y,T|y^{p}=1;\,y^{-1}Ty=T^{1+p^{m-1}}>.$
However, the conditions on $x$ and $y$ give that
$x^{-1}Tx=y^{-1}Ty\iff Tyx^{-1}=yx^{-1}T,$
so the only elements of $S_{p^{m}}$ which commute with $T$ (a complete cycle
of length $p^{m}$), are the powers of $T$. Thus $yx^{-1}=T^{j}$ for some $j$.
Since the order of $yx^{-1}$ is $p$, the only choices for $j$ are $j=p^{m}$ or
$j=p^{m-1}$. For both choices we get $S=Q_{1}^{1}$, namely $j=p^{m}$ gives
that $x=y^{-1}$ (so $S=Q_{1}^{1}$), and $j=p^{m-1}$ gives that
$x=T^{-p^{m-1}}y$. Thus we have $x\in<y,T>$, so that $<x,T>=<y,T>$, and hence
$S=Q_{1}^{1}$. $\Box$
###### Theorem 4.5
Let $G$ be a subgroup of $S_{p^{m}}$ and $P$ a $p$-Sylow subgroup of $G$ of
order $p^{s}$ such that $T\in P$. Then the following holds:
1. (a)
If $s=m,P=<T>$.
2. (b)
If $m<s\leq p+m-1$, then we have $P=Q_{1}^{s-m}$.
Proof. From Lemma 3.5 we have that $m\leq s\leq p^{m-1}+p^{m-2}+\cdots 1$. For
the case $m=s$, it is obvious that $P=<T>$. Now, let $s$ be such that $m<s\leq
p+m-1$. Hence $P$ contains a $p$-subgroup $P^{\prime}$ of order $p^{m+1}$. By
Theorem 4.4, $P^{\prime}=Q_{1}^{1}$. Let $j\geq 1$ be the largest integer such
that $Q_{1}^{j}\leq P$. If $j=p-1$, by Lemma 4.2 we have that
$|Q_{1}^{p-1}|=p^{p+m-1}$. Hence the assumption $s\leq p+m-1$ leads to the
unique solution $P=Q_{1}^{p-1}$. Thus, assume that $1\leq j<p-1$. If
$Q_{1}^{j}\neq P$, $N_{P}(Q_{1}^{j})$ properly contains $Q_{1}^{j}$ and by
Lemma 4.2, $N_{P}(Q_{1}^{j})\leq Q_{1}^{j+1}$. As $Q_{1}^{j}\leq
N_{P}(Q_{1}^{j})\leq Q_{1}^{j+1}$ and $Q_{1}^{j}\neq N_{P}(Q_{1}^{j})$, by
Lemma 4.3 $N_{p}(Q_{1}^{j})=Q_{1}^{j+1}$, a contradiction of the choice of
$j$. $\Box$
###### Theorem 4.6
Let $p$ be an odd prime, $q=r^{\alpha}$ a prime power, $C$ a cyclic code over
$\mathbb{F}_{q}$ of length $p^{m}$, $m>1$ and $P$ a $p$-Sylow subgroup of
$Aut(\mathcal{C})$ of order $p^{s}$ such that $T\in P$. Then the following
holds:
1. (a)
If $p\nmid\alpha$ and $p\nmid(d,r^{a}-1)$, then $s=m$, and $P=<T>$ if and only
if $Aut(\mathcal{C})=P\Gamma L(d,r^{a})$, $d\geq 3$.
2. (b)
If $p\geq 5$, $\alpha=1$ and $r=p$, $m>1$, then $Aut(\mathcal{C})$ is an
imprimitive group and $P$ is normal of order $p^{s}$, $s>m$. If $m<s\leq
p+m-1$, then we have $P=Q_{1}^{s-m}$.
3. (c)
If $z=1$, $p\nmid\alpha$ and $p\nmid(d,r^{a}-1)$, then $Aut(\mathcal{C})$ is
an imprimitive group and $P$ is normal of order $p^{s}\geq p^{2m-1}$.
Furthermore, if $2m-1<s\leq p+m-1$, then we have $P=Q_{1}^{s-m}$.
Proof. Statement (a) and the first parts of (b) and (c) follow from Theorem
3.6. We need only prove that if $s\leq p+m-1$, then $P=Q_{1}^{s-m}$. Assume
$s\leq p+m-1$, so that $P$ contains a $p$ subgroup $P^{\prime}$ of order
$p^{m+1}$. By Theorem 4.4, we obtain $P^{\prime}=Q_{1}^{1}$. Let $j\geq 1$ be
the largest integer such that $Q_{1}^{j}\leq P$. If $j=p-1$, by Lemma 4.2 we
have that $|Q_{1}^{p-1}|=p^{p+m-1}$. Thus $Q_{1}^{p-1}$ is a subgroup of $P$
of the same order as $P$, and hence $P=Q_{1}^{p-1}$, so we can assume that
$1\leq j<p-1$. If $Q_{1}^{j}\lneq P$, then $Q_{1}^{j}\lneq N_{P}(Q_{1}^{j})$
and by Lemma 4.2, $N_{P}(Q_{1}^{j})\leq Q_{1}^{j+1}$. Since $Q_{1}^{j}\lneq
N_{P}(Q_{1}^{j})\leq Q_{1}^{j+1}$, by Lemma 4.2
$N_{p}(Q_{1}^{j})=Q_{1}^{j+1}$, which contradicts the choice of $j$. $\Box$
###### Corollary 4.7
Let $\mathcal{C}$ and $\mathcal{C^{\prime}}$ be two cyclic combinatorial
objects on $p^{m}$ elements, and let $P$ be a $p$-Sylow subgroup of
$\mathcal{C}$ such that $T\in P$. If $|P|=p^{s}$ and $s\leq p+m-1$, then
$\mathcal{C}$ and $\mathcal{C^{\prime}}$ can be equivalent only under the
permutation of the following subgroups of $S_{p^{m}}$:
1. (i)
$AG(p^{m})$ if $s=m$;
2. (ii)
$Q^{s-m+1}$ if $s>m$.
Proof. The result follows from Lemma 4.1, Theorem 4.6 and Lemma 4.2. $\Box$
###### Remark 4.8
Since each affine transformation can be written as the product of a power of
$T$ and a multiplier, and since $T\in Aut(\mathcal{C})$ the power of $T$ is
absorbed in $Aut(\mathcal{C})$. Hence the permutation given in part (i) of
Corollary 4.7 is reduced to a multiplier.
In order to solve the isomorphism problem for cyclic combinatorial objects, we
must know the $p$-Sylow subgroup of $Aut(\mathcal{C})$. To determine this,
consider the following polynomial permutations $f_{1}=T$ and
$f_{i}(x)=1+x+p^{m-1}(x^{2}+\ldots+x^{i})$ for $2\leq i\leq p-2$.
###### Corollary 4.9
Let $G$ be a subgroup of $S_{p^{m}}$ with a $p$-Sylow subgroup $P$, and let
$I$ be the largest value of $i$ such that $f_{i}\in G$. If $I<p-2$, then we
have $P=Q_{1}^{I}$.
Proof. Assume that $I$ is the largest $i$ such that $f_{i}\in G$ and $I<p-2$.
Let $P$ be a $p$-Sylow subgroup of $G$ of order $s$. Let $s$ be such that
$I+m\leq s<p+m-1$. From Theorem 4.6, we have that a $p$-Sylow subgroup of any
subgroup of $G\leq S_{p^{m}}$ which contains $T=f_{1}$ and has order $p^{s}$
with $m\leq s\leq p+m-1$. Then we have $P=T$ when $s=m$ or $P=Q_{1}^{s-m}$, so
in this case $s-m=I$. Now, if $s\leq I+m\leq p+m-1$, we have from Theorem 4.6
that $P=Q_{1}^{s-m}$, so $Q_{1}^{I}\cap G\leq Q_{1}^{s-m}$. The assumption on
$I$ gives $I=s-m$.
Assume now that $s>p+m-1$. Since $I<p-2$, we have that $s>p+m-1>m+I$. We will
prove that this case cannot occur. We have $T=f_{1}\in Q_{1}^{1}$. From
Theorem 4.4, $Q_{1}^{1}$ is the unique subgroup of $S_{p^{m}}$ of order
$p^{m+1}$ which contains $T$. Hence $Q_{1}^{1}\lneq P$. Since $Q_{1}^{1}\lneq
Q_{1}^{2}$, it must be that $Q_{1}^{1}\lneq Q_{1}^{2}\cap P\leq Q^{2}$. Hence
from Lemma 4.3 we obtain $Q_{1}^{2}\cap P=Q_{1}^{2}$ which gives
$Q_{1}^{2}\leq P$. Using the same approach for $2\leq i\leq I$, we obtain
$Q_{I}\leq P$. The assumption on $s$ gives that $Q_{I}\lneq P$. Hence
$Q_{1}^{I}\lneq Q_{1}^{I+1}\cap P\leq Q^{I+1}$ ($Q^{I+1}$ can be considered
since it was assumed that $I<p-2$). Hence from Lemma 4.3 we obtain
$Q_{1}^{I+1}\cap P=Q_{1}^{I+1}$. This contradicts the assumption on $I$.
$\Box$
This corollary suggest the following algorithm for $I<p-2$.
Algorithm B: Let $p$ be an odd prime, and $\mathcal{C}$ and
$\mathcal{C^{\prime}}$ be two cyclic combinatorial objects from the same
category. Then the equivalence of $\mathcal{C}$ and $\mathcal{C^{\prime}}$ can
be determined as follows.
Step 1: Find the order of the Sylow subgroup of $Aut(\mathcal{C})$ as follows.
Find the largest $I$ such that $f_{I}\in Aut(\mathcal{C})$. Then $s=i+m$, and
do Step 2.
Step 2, find $f\in Q^{I+1}$ such that $\mathcal{C^{\prime}}=f\mathcal{C}$.
###### Remark 4.10
To find the required $I$ in Algorithm B we can use (for example) a binary
search which requires checking at most $\lceil\log_{2}(p-1)\rceil+1$ of the
$f_{i}$. Furthermore, the cardinality of $Q^{I+1}$ is $(p-1)p^{2m+I-2}$.
###### Definition 4.11
The cycle graph on $n$ vertices is the graph $\mathcal{C}_{n}$ with vertex set
$\\{0,1\ldots,n-1\\}$ and $i$ adjacent to $j$ if and only if $j-i\equiv\pm 1$.
###### Corollary 4.12
Two cycle graphs $\mathcal{C}_{n}$ and $\mathcal{C^{\prime}}_{n}$ on $p^{m}$
vertices can be isomorphic only by a multiplier.
Proof. The automorphism group of a cycle graph on $n$ vertices has order $2n$
[21, Ex. 2 p. 30]. Hence if $n=p^{m}$, there a is a unique Sylow subgroup of
$Aut(\mathcal{C}_{n})$ of order $p^{m}$. Then the result follows from
Corollary 4.7 and Remark 4.8. $\Box$
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|
arxiv-papers
| 2012-07-13T02:21:40 |
2024-09-04T02:49:32.987083
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kenza Guenda and T. Aaron Gulliver",
"submitter": "Guenda Kenza",
"url": "https://arxiv.org/abs/1207.3132"
}
|
1207.3208
|
ICFP'12, September 9–15, 2012, Copenhagen, Denmark.
-1.5ex [t]6.45cmTo the extent possible under law, Brian Huffman has waived all copyright and related or neighboring rights to “Formal Verification of Monad Transformers”. This work is published from: Germany.
Formal Verification of Monad Transformers
Brian HuffmanInstitut für Informatik, Technische Universität Münchenhuffman@in.tum.de
We present techniques for reasoning about constructor classes that (like the monad class) fix polymorphic operations and assert polymorphic axioms. We do not require a logic with first-class type constructors, first-class polymorphism, or type quantification; instead, we rely on a domain-theoretic model of the type system in a universal domain to provide these features.
These ideas are implemented in the Tycon library for the Isabelle theorem prover, which builds on the HOLCF library of domain theory. The Tycon library provides various axiomatic type constructor classes, including functors and monads. It also provides automation for instantiating those classes, and for defining further subclasses.
We use the Tycon library to formalize three Haskell monad transformers: the error transformer, the writer transformer, and the resumption transformer. The error and writer transformers do not universally preserve the monad laws; however, we establish datatype invariants for each, showing that they are valid monads when viewed as abstract datatypes.
F.3.1Logics and Meanings of ProgramsSpecifying and Verifying and Reasoning about Programs – mechanical verification.
§ INTRODUCTION
As a pure functional language, Haskell promises to work well for equational reasoning and proofs. Having programs and libraries that satisfy equational laws is important, because it lets programmers think about the correctness of their code in a modular and composable way.
Type classes are a valuable abstraction mechanism for writing reusable code in Haskell. Many Haskell type classes also have laws associated with them. Haskell programs that use these type classes often rely on the assumption that the laws hold. For example, a library might implement a datatype of balanced search trees, with elements of type $\tA$. To permit comparisons between elements, the search tree operations use the class constraint $\hsc{Ord}\:\tA$, which provides the comparison operator $(\le)::\tA\to\tA\to\hsc{Bool}$. But just having an operation of the right type is not enough: For the operations to work correctly, the library requires $(\le)$ to satisfy some additional properties, e.g. that $(\le)$ is a total order.
Much Haskell code is written with equational properties in mind: Programs, libraries, and class instances may be expected to satisfy some laws, but unfortunately, there is no formal connection between programs and properties in Haskell. Haskell compilers are not able to check that properties hold. One way to get around this limitation is to verify our Haskell programs in an interactive proof assistant, or theorem prover.
Isabelle/HOL (or simply “Isabelle”) is a generic interactive theorem prover, with tools and automation for reasoning about inductive datatypes and terminating functions in higher-order logic [13]. Isabelle has an ML-like type system extended with axiomatic type classes [17]. In Isabelle, a type class fixes one or more overloaded constants, just like in Haskell. But Isabelle also allows us to specify additional class axioms about those constants.
As an example, here we have an axiomatic class Ord that fixes an order relation $(\le)$ and asserts that it is a total order:
\begin{align*}
& \kwd{class}\:\hsc{Ord}\:\tA\:\kwd{where} \\[-\jot]
& \hspace{8pt} (\le)::\tA\to\tA\to\hsc{Bool} \\
& \hspace{8pt}
\begin{aligned}
& %\forall\,{x},
{x}\le{x} \\[-\jot]
& %\forall\,{x}\,{y}\,{z},
{x}\le{y}\;\wedge\; {y}\le{z}\implies{x}\le{z} \\[-\jot]
& %\forall\,{x}\,{y},
{x}\le{y} \;\wedge\; {y}\le{x}\implies{x}={y} \\[-\jot]
& %\forall\,{x}\,{y},
{x}\le{y} \;\vee\; {y}\le{x}
\end{aligned}
\end{align*}
(Note that free variables appearing in class axioms are treated as universally quantified.) To establish an instance of class Ord in Isabelle, a user must not only provide definitions of the class operations, but also proofs that the operations satisfy the class axioms.
Isabelle/HOLCF is a library of domain theory, formalized within the logic of Isabelle/HOL [12, 5]. It is designed to support denotational reasoning about programs written in pure functional languages like Haskell. HOLCF can deal with programs that are beyond the scope of Isabelle/HOL's automation: HOLCF provides tools for defining and working with (possibly lazy) recursive datatypes, general recursive functions, partial and infinite values, and least fixed-points. These features make Isabelle/HOLCF a useful system for reasoning about a significant subset of Haskell programs. With the combination of HOLCF and axiomatic classes, users can directly formalize many Haskell programs that use ad hoc overloading, and verify generic programs that may rely on laws for class instances.
Type constructor classes.
In addition to ordinary type classes, Haskell also supports type constructor classes. An ordinary type constraint like $\hsc{Ord}\:\tA$ involves a type variable $\tA :: *$. The operations in such a type class have relatively simple types like $(\le) :: (\hsc{Ord}\:\tA) \To \tA \to \tA \to \hsc{Bool}$, where no other type variables besides the one in the class constraint are mentioned. That is, for a specific class instance, the operations are monomorphic: e.g. to define an instance $\hsc{Ord}\;\hsc{Int}$, we have an operation $(\le^\hsc{Int}) :: \hsc{Int} \to \hsc{Int} \to \hsc{Bool}$. (In other words, a dictionary for class Ord contains only monomorphic functions.) On the other hand, a constructor class like $\hsc{Functor}\:\tT$ fixes a type variable of a higher kind, in this case $\tT :: * \to *$. Furthermore, the operations in a constructor class are usually polymorphic. For example, $\fmap :: (\hsc{Functor}\:\tT) \To (\tA \to \tB) \to \tT\:\tA \to \tT\:\tB$ also is polymorphic over the type variables $\tA$ and $\tB$. The laws for the functor class are likewise polymorphic: For functors we usually assume that $\fmap\;\hsid = \hsid$ and $\fmap\;(f \circ g) = \fmap\;f \circ \fmap\;g$. For a specific functor class instance, these laws can be instantiated at various types. For a proper, law-abiding functor, we expect these laws to hold at all possible type instantiations.
These additional requirements pose some real challenges for formal verification. While Isabelle has built-in support for ordinary axiomatic type classes, its type system does not natively support axiomatic constructor classes or type quantification—in fact, it does not even support higher-kinded type variables at all. Other interactive theorem provers exist with stronger type systems (e.g. Coq), but switching would mean giving up all the special support for reasoning about strictness, partial values, and general recursion in HOLCF. Coq and similar provers use a logic of total, terminating functions; thus proofs conducted in them are only applicable to the total, terminating fragment of the Haskell language.
Using a universal domain and a domain-theoretic model of types, we construct a library for Isabelle/HOLCF that gives users first-class type constructors and axiomatic constructor classes. Users can instantiate constructor classes by defining the constants and proving the class axioms at a single type. Using a combination of type coercions and naturality laws, theorems can then be transferred automatically to other type instances.
This work builds upon and improves an earlier formalization of constructor classes in Isabelle, which was joint work with Matthews and White [7]. While some concepts (e.g. representable types, the type application operator, and coercions) remain unchanged, this paper also introduces several new contributions:
* New simplified definition of class Functor
* Fully automatic tools for constructing Functor class instances
* A general, practical method for defining subclasses of Functor
* Automation for transferring theorems between types, using coercions and naturality laws
* Verification of error and writer monad transformers as abstract datatypes
The remainder of the paper is organized as follows. We begin by reviewing relevant information about HOLCF: After a summary of basic domain-theoretic concepts (<ref>), we discuss the deflation model used to represent types in HOLCF (<ref>). The next sections cover the implementation of the Tycon library: We show how to define the various constructor classes (<ref>), and then how to instantiate them (<ref>). Next we discuss the verification of monad transformers with Tycon (<ref>). Finally, we conclude with a discussion of related and future work (<ref>).
§ DOMAIN THEORY IN HOLCF
We now review the basic domain theory definitions used in HOLCF. A partial order is a set or type with a binary relation $(\below)$ that is reflexive, transitive, and antisymmetric. A chain is a countable increasing sequence: $x_0 \below x_1 \below x_2 \below \dots$ A complete partial order (cpo) is a partial order where every chain has a least upper bound (lub). An admissible predicate $P$ holds for the lub of a chain whenever it holds over the entire chain: $\forall{n}.\,P(x_n) \Longrightarrow P\left(\bigsqcup_n x_n\right)$. A continuous function $f$ preserves lubs of chains: $f\left(\bigsqcup_n x_n\right) = \bigsqcup_n f(x_n)$. Note also that every continuous function is monotone. A pointed cpo (pcpo) or “domain” is a cpo with a least element $\bot$. Every continuous function $f$ on a pcpo has a least fixed-point $\mathit{fix}(f) = f(\mathit{fix}(f)) = \bigsqcup_n f^n(\bot)$. In this paper we also use the binder notation $\mu\hair{x}.\,f(x)$ to denote the least fixed-point of $f$.
HOLCF provides a few primitive type constructors, which correspond to basic domain constructions. First, we have the continuous function space $\tA\to\tB$, which consists of the continuous functions from $\tA$ to $\tB$ ordered pointwise; this type is used to model Haskell's function space. Other constructions include strict sums, strict products, and lifting. They correspond to the Haskell datatype definitions here:
\begin{align*}
& \kwd{data}\:\tA\oplus\tB = \hsc{SLeft}\:!\tA \mid \hsc{SRight}\:!\tB \\
& \kwd{data}\:\tA\otimes\tB = \hsc{SPair}\:!\tA\:!\tB \\
& \kwd{data}\:\tA_\bot = \hsc{Lift}\:\tA
\end{align*}
Note that the constructors for $\tA\oplus\tB$ and $\tA\otimes\tB$ are strict, but the constructor for type $\tA_\bot$ is non-strict: $\hsc{Lift}\:\bot \neq \bot$. Finally, HOLCF provides the type $\hsone$, with two elements $\bot \below ()$; this models Haskell's unit type $()$.
§.§ The Domain package
Constructing recursive datatypes is one important application of domain theory. In HOLCF, user-defined recursive datatypes can be specified using the Domain package [5]. It can model many of the same datatypes that we can define in Haskell, e.g., lazy lists:
\begin{equation*}
\kwd{data}\:List\:\tA = \hsc{Nil} \mid \hsc{Cons}\:\tA\:(\hsc{List}\:\tA)
\end{equation*}
Given this datatype specification, the job of the Domain package is to construct a solution to the corresponding domain equation.
\begin{equation}
\hsc{List}\:\tA \cong \hsone \oplus (\tA_\bot \otimes (\hsc{List}\:\tA)_\bot)
\end{equation}
Since Isabelle 2011, the Domain package is completely definitional: It explicitly constructs a solution to this equation and defines $\hsc{List}\:\tA$ without introducing any new axioms [5]. In addition to the type itself, the Domain package also defines the constructor functions and several other related constants. It also generates a large collection of useful lemmas and rewrite rules, including injectivity and exhaustiveness of constructors, and rules for order comparisons like this one:
\begin{equation}
\hsc{Cons}\;x\;xs \below \hsc{Cons}\;y\;ys \Longleftrightarrow x \below y \wedge xs \below ys
\end{equation}
We also get some induction rules generated for us: Every type gets a low-level induction principle in the form of an approximation lemma [8]. For polynomial types (i.e., those expressible as a sum of products) we also get a high-level induction rule, with cases for each constructor plus a case for $\bot$. Induction rules for lazy datatypes have an admissibility side-condition.
\begin{equation}
\inferrule
{\mathrm{admissible}(P) \\ P(\bot) \\ P(\hsc{Nil}) \\
\forall{x}\;\hsc{xs}.\;P(\hsc{xs}) \Longrightarrow P(\hsc{Cons}\;x\;\hsc{xs})}
\end{equation}
The Domain package can handle any datatype expressible in Haskell—subject to the limitations of Isabelle's type system, of course. It supports both strict and lazy constructors, mutual recursion, indirect recursion, and even negative recursion.
\begin{align*}
& \kwd{data}\;\hsc{StrictList}\;\tA =
\hsc{SNil} \mid \hsc{SCons}\;\tA\;!(\hsc{StrictList}\;\tA) \\
& \kwd{data}\;\hsc{Indirect}\;\tA =
\hsc{Leaf}\;\tA \mid \hsc{Node}\;(\hsc{List}\;(\hsc{Indirect}\;\tA)) \\
& \kwd{data}\;\hsc{Neg} =
\hsc{App}\;\hsc{Neg}\;\hsc{Neg} \mid \hsc{Lam}\;(\hsc{Neg} \to \hsc{Neg})
\end{align*}
For formalizing Haskell record definitions, it also conveniently supports selector functions for constructor arguments.
§.§ Notation
We avoid using Isabelle notation as much as possible, favoring a Haskell-style syntax for datatype and function definitions. We also use Haskell-style notation $(\dots) \To \dots$ for class constraints on type variables. Isabelle's syntax follows Standard ML in writing type constructors postfix; however, for consistency we use Haskell-style prefix type application throughout. Isabelle type constructors with multiple arguments are shown as tupled.
We consistently use Greek letters $\tA$, $\tB$, $\tC$ for type variables of kind $*$, and $\tT$ for kind $* \to *$. Latin letters $a, b, c$ are program variables, with $f, g, h, k$ referring specifically to functions. We often use sub- and superscripts to annotate polymorphic functions with their types; e.g., $\fmap^\tT_{\tA,\,\tB}$ means $\fmap :: (\tA \to \tB) \to \tT\;\tA \to \tT\;\tB$.
§ DEFLATION MODEL OF TYPES
HOLCF provides a special domain $\D$ whose values are deflations, a certain kind of idempotent functions. Deflations are used to model types: To each “representable” domain type in HOLCF, we associate a representation of type $\D$. The primary reason for having this model in HOLCF is to implement the Domain package: The deflation model of types lets us reason about the existence of solutions to domain equations, because we can construct recursively defined deflations to represent them.
The Tycon library takes advantage of this existing model of types to derive further benefits. The deflation model gives us a way to express the relationship between different type instances of polymorphic functions, letting us reason about polymorphism. It also lets us reason about type quantification, by quantifying over deflations.
In the remainder of this section, we describe the underlying concepts behind the deflation model, as well as its implementation in HOLCF.
§.§ Embedding-projection pairs and deflations
Some cpos can be embedded within other cpos. The concept of an embedding-projection pair (often shortened to ep-pair) formalizes this notion.
Continuous functions $e :: \tA\to\tB$ and $p :: \tB\to\tA$ form an embedding-projection pair (or ep-pair) if $p \circ e = \hsc{id}_\tA$ and $e \circ p \below \hsid_\tB$. In this case, we write $(e, p) : \tA \eppair \tB$.
Ep-pairs have many useful properties: $e$ is injective, $p$ is surjective, both are strict, each function uniquely determines the other, and the image of $e$ is a sub-cpo of $\tB$. The composition of two ep-pairs yields another ep-pair: If $(e_1, p_1) : \tA \eppair \tB$ and $(e_2, p_2) : \tB \eppair \tC$, then $(e_2 \circ e_1, p_1 \circ p_2) : \tA \eppair \tC$. Ep-pairs can also be lifted over many type constructors, including strict sums, products, and continuous function space.
A continuous function $d :: \tA \to \tA$ is a deflation if it is idempotent and below the identity function: $d \circ d = d \below \hsid_\tA$.
Deflations and ep-pairs are closely related. Given an ep-pair $(e, p) : \tA \eppair \tB$, the composition $e \circ p$ is a deflation on $\tB$ whose image is isomorphic to $\tA$. Conversely, every deflation $d :: \tB \rightarrow \tB$ also gives rise to an ep-pair. Let the cpo $\tA$ be the image of $d$; also let $e$ be the inclusion map from $\tA$ to $\tB$, and let $p = d$. Then $(e, p)$ is an embedding-projection pair. So saying that there exists an ep-pair from $\tA$ to $\tB$ is equivalent to saying that there exists a deflation on $\tB$ whose image is isomorphic to $\tA$. Figure <ref> shows the relationship between ep-pairs and deflations.
[>=stealth, point/.style=coordinate, circle, fill, inner sep=0, outer sep=0.4mm, minimum size=1.2mm, xscale=1.0]
[xshift=-3cm] (0,0) – (-2.5, 2.5) – (-0.5, 2.5) – cycle;
(0, 0) – (-4, 4) – (1.5, 4) – cycle;
[dashed] (0, 0) – (-0.5, 2.5) – (-2.5, 2.5);
(1a) [point] at (-0.6, 1.1) ;
(1b) [point, xshift=-3cm] at (-0.6, 1.1) ;
(2a) [point] at (-1.5, 2.5) ;
(2b) [point, xshift=-3cm] at (-1.5, 2.5) ;
(3a) [point] at (-1.6, 3.7) ;
(1a) edge [->, bend right=20, above] node $p$ (1b);
(1b) edge [->, bend right=20, below] node $e$ (1a);
(2a) edge [->, bend right=20, above, pos=0.65] node $p$ (2b);
(2b) edge [->, bend right=20, below] node $e$ (2a);
(3a) edge [->, bend right=35, above, near end] node $p$ (2b);
(3a) edge [dotted, right] node $\sqsubseteq$ (2a);
[>=stealth, point/.style=coordinate, circle, fill, inner sep=0, outer sep=0.4mm, minimum size=1.2mm, xscale=1.1]
(0, 0) – (-4, 4) – (1.5, 4) – cycle;
[dashed] (0, 0) – (-0.5, 2.5) – (-2.5, 2.5);
(1a) [point] at (-0.6, 1.2) ;
(2a) [point] at (-1.5, 2.5) ;
(3a) [point] at (-1.6, 3.7) ;
(1a) edge [->, loop below, right] node $d$ (1a);
(2a) edge [->, loop below, right] node $d$ (2a);
(3a) edge [->, right] node $d$ (2a);
Embedding-projection pairs and deflations
A deflation is a function, but it can also be viewed as a set: Just take the image of the function, or equivalently, its set of fixed points—for idempotent functions they are the same. The dashed outline in Fig. <ref> shows the set defined by the deflation $d$. Every deflation on a cpo $\tA$ gives a set that is a sub-cpo, and contains $\bot$ if $\tA$ has a least element. Not all sub-cpos have a corresponding deflation, but if one exists then it is unique. The set-oriented and function-oriented views of deflations also give the same ordering: For any deflations $f$ and $g$, $f \sqsubseteq g$ if and only if $\mathrm{Im}(f) \subseteq \mathrm{Im}(g)$.
§.§ Representable types
We say that a type $\tA$ is representable in domain $\U$ if there exists an ep-pair from $\tA$ to $\U$, or equivalently if there exists a deflation $d$ on $\U$ whose image $\mathrm{Im}(d)$ is isomorphic to $\tA$. We say that $\U$ is a universal domain for some class of cpos if every cpo in the class is representable in $\U$. Isabelle/HOLCF provides such a universal domain type $\U$, which can represent any bifinite domain—this is a large class of cpos that includes (but is not limited to) all Haskell datatypes [4].
HOLCF defines an axiomatic class of representable domains. The class fixes operations $\hsemb$ and $\hsproj$, and assumes that they form an ep-pair into the universal domain.
\begin{align*}
& \kwd{class}\:\hsc{Rep}\:\tA\:\kwd{where} \\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsemb &&:: \tA \to \U \\[-\jot]
& \hsproj &&:: \U \to \tA
\end{alignedat} \\
& \hspace{8pt}
\begin{alignedat}{2}
& \hsproj \circ \hsemb = \hsid_\tA \\[-\jot]
& \hsemb \circ \hsproj \below \hsid_\U
\end{alignedat}
\end{align*}
The universal domain type itself is trivially representable, using identity functions. For other base types like $\hsone$, the HOLCF universal domain library provides appropriate ep-pairs (Fig. <ref>).
\begin{align*}
\begin{aligned}
& \kwd{instance}\:\hsRep\:\U\:\kwd{where} \\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsemb &&= \hsid_\U \\[-\jot]
& \hsproj &&= \hsid_\U
\end{alignedat}
%\displaybreak[0] \\[\jot]
\end{aligned}
\begin{aligned}
& \kwd{instance}\:\hsRep\:\hsone\:\kwd{where} \\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsemb &&= \embedding_\hsone \\[-\jot]
& \hsproj &&= \projection_\hsone
\end{alignedat}
\end{aligned}
\end{align*}
HOLCF defines the domain $\D$ of deflations over the universal domain as a subtype of $\U\to\hair\U$. (In the Isabelle formalization, explicit conversions between types $\D$ and $\hair\U\to\hair\U$ are always required, but we will keep those implicit here.)
\begin{equation}
\kwd{typedef}\;\D = \{\,d::\U\to\hair\U \mid {d}\circ{d}={d}\below\hsid_\U\,\}
\end{equation}
Given any representable type $\tA$, we can construct its representation (a deflation of type $\D$) by composing emb and proj. We denote the mapping from types to deflations as follows:
\begin{equation*}
\REP{\tA} \defeq \hsemb_\tA \circ \hsproj_\tA
\end{equation*}
Note that the representation of the universal domain $\REP{\hair\U}$ is therefore the identity deflation, which is maximal among all deflations. Thus we have $\REP{\tA} \below \REP{\hair\U}$ for all representable types $\tA$.
\begin{align*}
& \embedding_\to :: (\U\to\hair\U)\to\hair\U &
& \projection_\to :: \U\to(\U\to\hair\U) \\
& \embedding_\otimes :: (\U\otimes\hair\U)\to\hair\U &
& \projection_\otimes :: \U\to(\U\otimes\hair\U) \\
& \embedding_\oplus :: (\U\oplus\hair\U)\to\hair\U &
& \projection_\oplus :: \U\to(\U\oplus\hair\U) \\
& \embedding_\bot :: \U_\bot\to\hair\U &
& \projection_\bot :: \U\to\hair\U_\bot \\
& \embedding_\hsone :: \hsone\to\hair\U &
& \projection_\hsone :: \U\to\hsone
\end{align*}
Embedding-projection pairs provided by the universal domain library in HOLCF
§.§ Representable type constructors
While types can be represented by deflations, type constructors (which are like functions from types to types) can be represented as functions from deflations to deflations. We say that a type constructor $F :: *\to*$ is representable in $\U$ if there exists a continuous function $F_\D :: \D \to \D$ such that $\REP{F(\tA)} = F_\D\,\REP{\tA}$. Such deflation combinators can be used to build deflations for recursive datatypes <cit.>. This is precisely the technique used by recent versions of the Domain package <cit.>; we will also use the same technique for creating new type constructors in Sec. <ref>.
We have a recipe for setting up a primitive HOLCF type as a representable type constructor: We just need an ep-pair to the universal domain and a map function. (The HOLCF universal domain library provides a selection of suitable ep-pairs; see Fig. <ref>.) We now demonstrate the recipe using the strict product type.
\begin{align*}
& \mapProd :: (\tA \to \tA'\!,\:\tB \to \tB') \to \tA \otimes \tB \to \tA'\! \otimes \tB' \\[-\jot]
& \mapProd\,(f, g)\,(\hsc{SPair}\:x\:y) = \hsc{SPair}\,(f\:x)\,(g\:y)
\\[\jot]
& (\otimes_\D) :: \D \to \D \to \D \\[-\jot]
& a \otimes_\D b = \embedding_\otimes \circ \mapProd\,(a, b) \circ \projection_\otimes
\\[\jot]
& \kwd{instance}\:(\hsRep\:\tA, \hsRep\:\tB) \To \hsRep\,(\tA \otimes \tB)\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsemb &&= \embedding_\otimes \circ \mapProd\,(\hsemb_\tA, \hsemb_\tB)
\\[-\jot]
& \hsproj &&= \mapProd\,(\hsproj_\tA, \hsproj_\tB) \circ \projection_\otimes
\end{alignedat}
\end{align*}
The reader can verify that $(\otimes_\D)$ does in fact preserve deflations, that $\hsemb$ and $\hsproj$ do form an ep-pair for type $\tA\otimes\tB$, and that $(\otimes_\D)$ actually does represent the strict product type constructor: $\REP{\tA\otimes\tB} = \REP{\tA} \otimes_\D \REP{\hair\tB}$.
Most other HOLCF type constructors work exactly like the strict product. However, the continuous function space is special because it is contravariant in its first argument.
\begin{align*}
& \mapFun :: (\tA'\!\to \tA,\:\tB \to \tB') \to (\tA \to \tB) \to (\tA'\! \to \tB') \\[-\jot]
& \mapFun\,(f, g)\:h = g \circ h \circ f
\\[\jot]
& (\to_\D) :: \D \to \D \to \D \\[-\jot]
& a \to_\D b = \embedding_\to \circ \mapFun\,(a, b) \circ \projection_\to
\\[\jot]
& \kwd{instance}\:(\hsRep\:\tA, \hsRep\:\tB) \To \hsRep\,(\tA \to \tB)\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsemb &&= \embedding_\to \circ \mapFun\,(\hsproj_\tA, \hsemb_\tB)
\\[-\jot]
& \hsproj &&= \mapFun\,(\hsemb_\tA, \hsproj_\tB) \circ \projection_\to
\end{alignedat}
\end{align*}
Due to contravariance, the first argument to $\mapFun$ has type $\tA'\!\to\tA$ instead of $\tA\to\tA'$. Also note that in the $\hsRep$ instance, $\hsemb$ calls $\hsprj$ and vice versa. Otherwise everything works similarly to the other types.
§.§ Coercion
We can write a function to coerce between any two representable types: First embed into the universal domain $\U$, and then project out to a different type.
\begin{align*}
& \hscoerce :: (\hsRep\:\tA, \hsRep\:\tB) \To \tA \to \tB
\\[-\jot]
& \hscoerce_{\tA,\,\tB} = \hsproj_\tB \circ \hsemb_\tA
\end{align*}
Our primary use for coercion will be to relate different type instances of polymorphic functions. In the remainder of the paper, we will often need to prove properties about coerced values; to facilitate this, we assemble a collection of simplification rules. First of all, $\hscoerce$ may reduce to $\hsemb$, $\hsprj$, or $\hsid$, depending on the type:
\begin{align}
\hscoerce_{\tA,\tA} &= \hsid_\tA \\
\hscoerce_{\tA,\U} &= \hsemb_\tA \\
\hscoerce_{\U,\tA} &= \hsproj_\tA
\end{align}
Other properties about $\hscoerce$ depend on the relative “sizes” of the source and target types. A coercion from a smaller to a larger type is injective (an embedding, in fact). Coercing twice in a row is the same as coercing once, as long as the intermediate type is larger than one of the source or target types.
\begin{gather}
\inferrule
{\REP{\tA}\below\REP{\hair\tB} \vee \REP{\tC} \below \REP{\hair\tB}}
{\hscoerce_{\tB,\tC} \circ \hscoerce_{\tA,\,\tB} = \hscoerce_{\tA,\tC}}
\end{gather}
Coercing between similar datatypes is the same as mapping $\hscoerce$ over the elements. (As an exercise, the reader may wish to verify Eq. (<ref>) by expanding the definitions given earlier in Sec. <ref>.) Using these rules, it is easy to verify that $\hscoerce$ commutes with each data constructor.
\begin{align}
\label{eq:coerce-sprod}
\hscoerce_{\tA\otimes\tB,\tC\otimes\tD} &= \mapProd(\hscoerce_{\tA,\tC}, \hscoerce_{\tB,\tD}) \\
\hscoerce_{\tA\oplus\tB,\tC\oplus\tD} &= \mapSum(\hscoerce_{\tA,\tC}, \hscoerce_{\tB,\tD}) \\
\hscoerce_{\tA_\bot,\,\tB_\bot} &= \mapLift(\hscoerce_{\tA,\,\tB})
\end{align}
A similar rule holds for coercions between two function types. The expanded form in Eq. (<ref>) will be particularly useful for simplifying coercions in later proofs.
\begin{align}
\hscoerce_{(\tA\to\tB),(\tC\to\tD)} &= \mapFun(\hscoerce_{\tC,\tA}, \hscoerce_{\tB,\tD}) \\
\label{eq:coerce-cfun}
\hscoerce_{(\tA\to\tB),(\tC\to\tD)}\,f &= \hscoerce_{\tB,\tD} \circ f \circ \hscoerce_{\tC,\tA}
\end{align}
A note about the ubiquity of the Rep class: For the remainder of this paper, we will assume that all types $\tA, \tB, \tC, \dots$ are in class Rep, without writing Rep class constraints explicitly. The reader may treat emb, proj, and coerce as if they were completely polymorphic. (HOLCF achieves a similar effect using the “default sort” mechanism, assigning all type variables to class Rep unless annotated otherwise.)
§ TYPE CONSTRUCTOR CLASSES IN THE TYCON LIBRARY
§.§ Class Tycon and type application
In the Haskell type expression $\tT\:\tA$, the two type variables have different kinds: Say $\tA$ is an ordinary type of kind $*$; then $\tT$ may be a type constructor of kind $*\to*$. Isabelle's type system was not designed to be this expressive: All type variables in Isabelle represent ordinary types (corresponding to Haskell kind $*$).
Our solution to this limitation (originally introduced in [7]) is to define a binary Isabelle type constructor $(-\cdot-)$ that models Haskell type application. The right argument must be in the Isabelle class $\hsRep$, which models Haskell kind $*$. The left argument must be in a new class Tycon, which models Haskell kind $*\to*$.
Class Tycon is defined as follows. It has no axioms, but fixes a single constant which is a deflation constructor.[In the Isabelle formalization, we express the dependence of $\TC{\tT}$ on type $\tT$ by adding a dummy function argument whose type is a phantom type mentioning $\tT$.]
\begin{equation*}
\kwd{class}\:\hsc{Tycon}\:\tT\:\kwd{where}\:\TC{\tT} :: \D \to \D
\end{equation*}
Now we want to define type $\tT\cdot\tA$ so that $\REP{\tT\cdot\tA} = \TC{\tT}\REP{\tA}$. We therefore define $\tT\cdot\tA$ as a subtype of $\U$, consisting of the image (or equivalently, the set of fixed-points) of the deflation $\TC{\tT}\REP{\tA}$.
\begin{align*}
& \kwd{typedef}\:(\hsc{Tycon}\:\tT, \hsc{Rep}\:\tA) \To \tT\cdot\tA
\\[-\jot]
& \hspace{8pt}
= \{\,u::\U \mid \TC{\tT}\REP{\tA}\,{u}={u}\,\}
\\[\jot]
& \kwd{instance}\:(\hsc{Tycon}\:\tT, \hsc{Rep}\:\tA) \To \hsc{Rep}\,(\tT\cdot\tA)\:
\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsemb\:x = x \\[-\jot]
& \hsprj\:u = \TC{\tT}\REP{\tA}\,u
\end{alignedat}
\end{align*}
Note that the definitions of $\hsemb$ and $\hsproj$ contain implicit coercions between $\U$ and $\tT\cdot\tA$. The desired representation property then follows directly from the definitions of $\hsemb$ and $\hsproj$:
\begin{equation}
\REP{\tT\cdot\tA} = \TC{\tT}\REP{\tA}
\end{equation}
It is worth pointing out that while the construction refers to the deflation combinator $\TC{\tT}$, actual values of type $\tT$ are never used anywhere. This is consistent with Haskell, where there are no values inhabiting higher-kinded types.
§.§ Class Functor
The Haskell Functor class is for types that can be mapped over.
\begin{align*}
& \kwd{class}\:\hsc{Functor}\:\tT\:\kwd{where} \\[-\jot]
& \hspace{8pt} \fmap :: (\tA \to \tB) \to (\tT\:\tA \to \tT\:\tB)
\end{align*}
Each Haskell Functor instance should satisfy the identity and composition laws:
\begin{align}
\fmap\:\hsid &= \hsid \\
\fmap\:(f \circ g) &= \fmap\:f \circ \fmap\:g
\end{align}
How close are we to being able to formalize this in Isabelle? Using the type application machinery from Sec. <ref>, we can at least express the class constraint $\hsc{Functor}\:\tT$ and the result type of $\fmap$. However, there are still some problems. First, let us examine the type of $\fmap$ more closely. The type constructor variable $\tT$ is fixed, but types $\tA$ and $\tB$ are actually universally quantified:
\begin{equation*}
\fmap^\tT :: \forall\,\tA\:\tB.\,(\tA \to \tB) \to (\tT\cdot\tA \to \tT\cdot\tB)
\end{equation*}
The problem is that Isabelle's class system does not allow polymorphic class constants. Isabelle's type system does not support first-class polymorphism, and the type of class functions are only allowed to contain one free type variable, i.e., the one mentioned in the class constraint [17].
The solution is to move the polymorphism out of the class declaration. We replace the polymorphic $\fmap^\tT$ with a single, monomorphic constant representing $\fmap^\tT_{\U,\U}$, the “largest” type instance of $\fmap^\tT$. (We use the underlined name $\fmapU^\tT$ to refer to this monomorphic version.) We then define the polymorphic $\fmap^\tT$ by coercion from $\fmapU^\tT$.
\begin{align*}
& \kwd{class}\:(\hsc{Tycon}\:\tT) \To \hsc{Functor}\:\tT\:\kwd{where}
\\[-\jot]
& \hspace{8pt} \fmapU^\tT :: (\U \to \U) \to (\tT\cdot\U \to \tT\cdot\U)
\\[\jot]
& \fmap :: (\hsc{Functor}\:\tT) \To (\tA \to \tB) \to \tT\cdot\tA \to \tT\cdot\tB
\\[-\jot]
& \fmap^\tT_{\tA,\,\tB} = \hscoerce\:\fmapU^\tT
\end{align*}
In Haskell, polymorphically typed functions like $\fmap^\tT$ are always parametrically polymorphic. That is, parametricity (a meta-property of the type system) ensures that all of the different type instances of $\fmap^\tT_{\tA,\,\tB}$ behave uniformly [16]. Isabelle's type system does not provide any automatic parametricity guarantees, but by defining all type instances of $\fmap^\tT$ by coercion from a single constant, we ensure a similar kind of uniformity across type instances in the our library.
The formalization of class Functor is yet incomplete: We have the constant, but not the functor laws. We need to find a set of class axioms about $\fmapU^\tT$ that will let us derive the polymorphic functor laws about $\fmap^\tT$.
As a first try, we might just write down the functor laws with all the types specialized to type $\U$:
\begin{align}
\label{eq:fmapu-id}
\fmapU^\tT\:\hsid_\U &= \hsid_{\tT\cdot\U} \\
\label{eq:fmapu-compose}
\fmapU^\tT (f \circ g) &= \fmapU^\tT f \circ \fmapU^\tT g
\end{align}
However, we shall treat these as tentative until we see whether they are sufficient to derive the polymorphic functor laws.
For any $\tT$ in class Functor and representable type $\tA$, the functor identity law holds:
\begin{equation*}
\fmap^\tT_{\tA,\tA}\:\hsid_\tA = \hsid_{\tT\cdot\tA}
\end{equation*}
We start by unfolding the definition of $\fmap$ and rewriting with properties of $\hscoerce$.
\begin{align*}
& \fmap^\tT_{\tA,\tA}\:\hsid_\tA \\
&= (\hscoerce\:\fmapU^\tT)\:\hsid_\tA \\
&= \hscoerce_{(\tT\cdot\hair\U,\tT\cdot\tA)} \circ \fmapU^\tT (coerce\:\hsid_\tA) \circ \hscoerce_{(\tT\cdot\tA,\tT\cdot\hair\U)} \\
&= \hscoerce_{(\tT\cdot\hair\U,\tT\cdot\tA)} \circ \fmapU^\tT \REP{\tA} \circ \hscoerce_{(\tT\cdot\tA,\tT\cdot\hair\U)}
\end{align*}
At this point we are stuck. A law about $\fmapU^\tT\:\hsid_\U$ does not help here, because coercing $\hsid_\tA$ to type $\U\to\U$ does not yield $\hsid_\U$; it gives $\REP{\tA}$ instead. What we really need is a rewrite rule for $\fmapU^\tT$ applied to an arbitrary deflation. The class axiom must assert that the map function $\fmapU^\tT$ “agrees” with the deflation combinator $\TC{\tT}$ in a certain sense.
We say that a function $f$ on a representable type $\tA$ agrees with a deflation $d$ on the universal domain, if $f$ coerced to type $\U\to\U$ is equal to $d$ (regarded as a function).
\begin{equation*}
(f::\tA\to\tA) \isodefl (d::\D) \overset{\mathrm{def}}{\Longleftrightarrow} \hsc{emb}_\tA \circ f \circ \hsc{proj}_\tA = d
\end{equation*}
This agreement relation is already present in HOLCF: It is used internally by the Domain package for relating deflation combinators to map functions, for proving identity laws and deriving induction rules [5]. So it is fitting that it should appear in this situation, where we are again proving functor identity laws.
We replace Eq. (<ref>) with this generalized class axiom, shown here also in its unfolded form:
\begin{gather}
\fmapU^\tT\,(d::\D) \isodefl \TC{\tT}\,d \\
\hsemb_{\tT\cdot\U} \circ \fmapU^\tT\,(d::\D) \circ \hsprj_{\tT\cdot\U} = \TC{\tT}\,d
\end{gather}
Now we can continue where we left off:
\begin{align*}
& \hscoerce_{(\tT\cdot\hair\U,\tT\cdot\tA)} \circ \fmapU^\tT \REP{\tA} \circ \hscoerce_{(\tT\cdot\tA,\tT\cdot\hair\U)} \\
& = \hsc{proj}_{\tT\cdot\tA} \circ \hsc{emb}_{\tT\cdot\hair\U} \circ \fmapU^\tT \REP{\tA} \circ \hsc{proj}_{\tT\cdot\hair\U} \circ \hsc{emb}_{\tT\cdot\tA} \\
&= \hsc{proj}_{\tT\cdot\tA} \circ \TC{\tT}\REP{\tA} \circ \hsc{emb}_{\tT\cdot\tA} \\
&= \hsc{proj}_{\tT\cdot\tA} \circ \REP{\tT\cdot\tA} \circ \hsc{emb}_{\tT\cdot\tA} \\
&= \hsc{proj}_{\tT\cdot\tA} \circ \hsc{emb}_{\tT\cdot\tA} \circ \hsc{proj}_{\tT\cdot\tA} \circ \hsc{emb}_{\tT\cdot\tA} \\
&= \hsid_{\tT\cdot\tA} \circ \hsid_{\tT\cdot\tA} \\
&= \hsid_{\tT\cdot\tA}\qedhere
\end{align*}
For any $\tT$ in class Functor and functions $f::\tB\to\tC$ and $g::\tA\to\tB$, the functor composition law holds:
\begin{equation*}
\fmap^\tT_{\tA,\tC}\,(f \circ g) = \fmap^\tT_{\tB,\tC}\:f \circ \fmap^\tT_{\tA,\,\tB}\:g
\end{equation*}
We rewrite both sides of the equation, trying to reduce it to a trivial equality. We start by unfolding the definition of $\fmap$.
\begin{align*}
\fmap^\tT_{\tA,\tC}\,(f \circ g) &= \fmap^\tT_{\tB,\tC}\:f \circ \fmap^\tT_{\tA,\,\tB}\:g \\
(\hscoerce\:\fmapU^\tT)(f \circ g) &= (\hscoerce\:\fmapU^\tT)\,f \circ (\hscoerce\:\fmapU^\tT)\,g
\end{align*}
After rewriting using Eq. (<ref>), we have
\begin{multline*}
\hscoerce \circ \fmapU^\tT(\hscoerce\,(f \circ g)) \circ \hscoerce \\[-\jot]
= \hscoerce \circ \fmapU^\tT(\hscoerce\:f) \circ \hscoerce \\[-\jot]
\circ \hscoerce \circ \fmapU^\tT(\hscoerce\:g) \circ \hscoerce
\end{multline*}
In the middle of the right-hand side we have two adjacent coercions, where we go from type $\tT\cdot\U$ to $\tT\cdot\tB$ and back. Since the intermediate type $\tT\cdot\tB$ is smaller, they do not cancel completely. It turns out that they reduce to an application of $\fmapU^\tT$:
\begin{align}
\notag & \hscoerce_{\tT\cdot\tB,\tT\cdot\U} \circ \hscoerce_{\tT\cdot\U,\tT\cdot\tB} \\
\notag & = \hsprj_{\tT\cdot\U} \circ \hsemb_{\tT\cdot\tB} \circ \hsprj_{\tT\cdot\tB} \circ \hsemb_{\tT\cdot\U} \\
\notag & = \hsprj_{\tT\cdot\U} \circ \REP{\tT\cdot\tB} \circ \hsemb_{\tT\cdot\U} \\
\notag & = \hsprj_{\tT\cdot\U} \circ \TC{\tT}\REP{\hair\tB} \circ \hsemb_{\tT\cdot\U} \\
\notag & = \hsprj_{\tT\cdot\U} \circ \hsemb_{\tT\cdot_\U} \circ \fmapU^\tT\REP{\hair\tB} \circ \hsprj_{\tT\cdot\U} \circ \hsemb_{\tT\cdot\U} \\
\notag & = \hsid_{\tT\cdot\U} \circ \fmapU^\tT\REP{\hair\tB} \circ \hsid_{\tT\cdot\U} \\
\label{eq:coerce-fmap} & = \fmapU^\tT\REP{\hair\tB}
\end{align}
Rewriting the right-hand side with Eq. (<ref>) yields three occurrences of $\fmapU^\tT$ composed together. Using the composition rule (<ref>) to collapse these, we get
\begin{multline*}
\hscoerce \circ \fmapU^\tT(\hscoerce\,(f \circ g)) \circ \hscoerce \\[-\jot]
= \hscoerce \circ \fmapU^\tT(\hscoerce\:f \circ \REP{\hair\tB} \circ \hscoerce\:g) \circ \hscoerce
\end{multline*}
Finally, it only remains to show that the arguments to $\fmapU^\tT$ on each side are equal. We work from right to left.
\begin{align*}
& \hscoerce_{(\tB\to\tC),(\U\to\U)}\,f \circ \REP{\hair\tB} \circ \hscoerce_{(\tA\to\tB),(\U\to\U)}\,g \\
& = \hsemb_\tC \circ f \circ \hsprj_\tB \circ \REP{\hair\tB} \circ \hsemb_\tB \circ g \circ \hsprj_\tA \\
& = \hsemb_\tC \circ f \circ \hsprj_\tB \circ \hsemb_\tB \circ \hsprj_\tB \circ \hsemb_\tB \circ g \circ \hsprj_\tA \\
& = \hsemb_\tC \circ f \circ \hsid_\tB \circ \hsid_\tB \circ g \circ \hsprj_\tA \\
& = \hsemb_\tC \circ f \circ g \circ \hsprj_\tA \\
& = \hscoerce_{(\tA\to\tC),(\U\to\U)}\,(f \circ g) \qedhere
\end{align*}
The final formulation of class Functor, complete with the generalized identity law, is shown in Fig. <ref>.
We should note that while transfer proofs like Theorem <ref> may look lengthy on paper, they are actually highly automated in Isabelle: Most such proofs require only a single call to Isabelle's simplifier, as long as the appropriate extra rewrite rules like Eq. (<ref>) are in place. This is important for usability of the library, because users will need to perform similar transfer proofs often—not just when defining new constructor classes, but also when instantiating them.
We present proofs here in a point-free style, with liberal use of the function composition operator $(\circ)$, because it makes the proofs easier to read. However, Isabelle is not so good at reasoning modulo the associativity of function composition. Automatic proofs by rewriting work better with nested function applications, e.g. $f(g(h(x)))$ rather than $f \circ g \circ h$. Therefore, the rewrite rules and other theorems in our library are actually formalized using fully applied functions instead of function composition.
\begin{align*}
& \kwd{class}\:\hsc({Tycon}\:\tT) \To \hsc{Functor}\:\tT\:\kwd{where} \\[-\jot]
& \hspace{8pt} \fmapU^\tT :: (\U \to \U) \to (\tT\cdot\U \to \tT\cdot\U) \\[\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \fmapU^\tT\,(d::\D) &&\isodefl \TC{\tT}\,d \\[-\jot]
& \fmapU^\tT (f \circ g) &&= \fmapU^\tT f \circ \fmapU^\tT g
\end{alignedat}
\\[\jot]
& \fmap :: (\hsc{Functor}\:\tT) \To (\tA \to \tB) \to \tT\cdot\tA \to \tT\cdot\tB
\\[-\jot]
& \fmap^\tT_{\tA,\tB} = \hscoerce\:\fmapU^\tT
\end{align*}
Isabelle Functor class
§.§ Generic theorems about functors
Now that we have a functor class, we can prove further theorems about $\fmap$. Here is an example theorem, about its strictness. The proof uses only the functor laws and basic properties of domain theory; the result is applicable to any valid functor instance.
If $f::\tA\to\tB$ is a strict function, then $\fmap\:f$ is also strict: $f\:\bot = \bot \implies \fmap\:f\:\bot = \bot$.
Fix $f :: \tA\to\tB$, and assume $f\:\bot_\tA = \bot_\tB$. Let $g :: \tB\to\tA$ be the constant bottom function, $g\:x = \bot_\tA$. From the strictness of $f$, it follows that $f \circ g = \hsc{const}\:\bot \below id_\tB$. We can now show the goal by antisymmetry and transitivity reasoning:
\begin{align*}
\fmap\:f\:\bot
&\below \fmap\:f\:(\fmap\:g\:\bot) & \justification{monotonicity with $\bot$} \\
&= \fmap\:(f \circ g)\:\bot & \justification{composition law} \\
&\below \fmap\:\hsid\:\bot & \justification{monotonicity, $f \circ g \below \hsid$} \\
&= \bot & \justification{identity law}
\end{align*}
Thus we have $\fmap\:f\:\bot \below \bot$, which implies $\fmap\:f\:\bot = \bot$.
§.§ Subclasses of Functor
Users of the Tycon library can easily formalize additional constructor classes that are subclasses of Functor. The library already contains several examples, and they all follow the same general process.
A constructor class may fix some number of polymorphic constants, and assume a set of polymorphic class axioms. The formalized constructor class fixes a monomorphic version of each polymorphic function, with type variables instantiated to $\U$. Similarly, the formalized class assumes a monomorphic version of each class axiom. The polymorphic version of each functions is defined separately, using coercion. In general, we will also add a naturality law for each polymorphic function, which is related to the parametricity property, or free theorem, derived from its type [16]. The naturality laws are necessary for transferring properties about the monomorphic constants to the polymorphic ones.
The Functor class is a special case: No extra naturality law was needed for $\fmap$, because the functor composition law is the naturality law for $\fmap$. The Monad class is perhaps the primary motivation for this work, but the interactions and redundancies between its laws also make it a bit of a special case. The general principles are best illustrated with a more regular example. So here we present a class FunctorPlus, which fixes a binary append operation for combining functor values:
\begin{align*}
& \kwd{class}\:(\hsc{Functor}\:\tT) \To \hsc{FunctorPlus}\:\tT\:\kwd{where} \\[-\jot]
& \hspace{8pt} (\hsapp)::\tT\:\tA\to\tT\:\tA\to\tT\:\tA
\end{align*}
Each instance of FunctorPlus should also ensure that $(\hsapp)$ is associative:
\begin{equation}
\label{eq:append-assoc}
\end{equation}
Any implementation of $(\hsapp)$ should also satisfy a naturality condition, which essentially states that it commutes with $\fmap$. The form of this law is derived from the polymorphic type of $(\hsapp)$; it holds in Haskell as a consequence of parametricity.
\begin{equation}
\label{eq:fmap-append}
\fmap\:f\:(x \hsapp y) = (\fmap\:f\:x) \hsapp (\fmap\:f\:y)
\end{equation}
We formalize class FunctorPlus in Isabelle according to the general pattern outlined above; the code is shown in Fig. <ref>.
\begin{align*}
& \kwd{class}\:(\hsc{Functor}\:\tT) \To \hsc{FunctorPlus}\:\tT\:\kwd{where}
\\[-\jot]
& \hspace{8pt} (\hsappU^\tT) :: \tT\cdot\U \to \tT\cdot\U \to \tT\cdot\U \\
& \hspace{8pt}
\begin{alignedat}{2}
& \fmapU^\tT\,f\:(x \hsappU^\tT y) &&= (\fmapU^\tT\,f\:x) \hsappU^\tT (\fmapU^\tT\,f\:y)
\\[-\jot]
& (x \hsappU^\tT y) \hsappU^\tT z &&= x \hsappU^\tT (y \hsappU^\tT z)
\end{alignedat}
\\[\medskipamount]
& (\hsapp) :: (\hsc{FunctorPlus}\:\tT) \To \tT\cdot\tA \to \tT\cdot\tA \to \tT\cdot\tA
\\[-\jot]
& (\hsapp^\tT_\tA) = \hscoerce\:(\hsappU^\tT)
\end{align*}
Isabelle FunctorPlus class
The need for the naturality law becomes apparent when transferring laws to the polymorphic version of $(\hsapp)$. When we transfer the associativity law, we get a situation similar to what we had with the proof of Theorem <ref>: Between the two occurrences of $(\hsapp)$, we get a pair of coercions from $\tT\cdot\U$ to $\tT\cdot\tA$ and back; these reduce to $\fmapU^\tT\,\REP{\tA}$. The naturality law lets us push the $\fmapU^\tT$ into the arguments of the inner append, bringing the two appends together so that the monomorphic associativity rule can be applied. In the end, we are able to prove the polymorphic version of the associativity law with one call to the simplifier. Similarly, we can also derive the polymorphic version of the naturality law in one step.
§.§ Class Monad
The definition of the Monad class should be familiar to every Haskell programmer.
\begin{align*}
&\kwd{class}\:\hsc{Monad}\:\tT\:\kwd{where} \\[-\jot]
\begin{alignedat}{2}
& \hsc{return}&&::\tA\to\tT\:\tA \\[-\jot]
& (\hsbind)&&::\tT\:\tA\to(\tA\to\tT\:\tB)\to\tT\:\tB
\end{alignedat}
\end{align*}
The standard monad laws are left unit, right unit, and associativity.
\begin{align}
\label{eq:left-unit}
\hsc{return}\:{a}\hsbind{k} &= {k}\:{a} \\
\label{eq:right-unit}
{m}\hsbind\hsc{return} &= {m} \\
\label{eq:bind-bind}
\end{align}
To translate this Haskell class definition into Isabelle, we can follow the standard process established in Sec. <ref>: Replace the polymorphic operations with monomorphic ones, where each type variable is instantiated to $\U$; specialize the types in the class axioms to $\U$; and add naturality laws for each of the constants.
Below are the naturality laws for the monad operations, derived from their type signatures. Note that $(\hsbind)$ has two naturality laws, because its type has two polymorphic variables.
\begin{align}
\label{eq:fmap-return}
\fmap\:f\,(\hsc{return}\:a) &= \hsc{return}\,(f\:a) \\
\label{eq:bind-fmap}
(\fmap\:f\:m) \hsbind k &= m \hsbind (k \circ f) \\
\label{eq:fmap-bind}
\fmap\:f\,(m \hsbind k) &= m \hsbind (\fmap\:f \circ k)
\end{align}
Three monad laws plus three naturality laws would make six class axioms in total. However, it is possible to reduce this number. Using Eqs. (<ref>) and (<ref>), we can derive a simple definition of $\fmap$ in terms of $(\hsbind)$ and return:
\begin{equation}
\label{eq:monad-fmap}
\fmap\:f\:m = m \hsbind (\hsc{return} \circ f)
\end{equation}
This definition of $\fmap$ is often referred to as a fourth monad law; it is expected to hold for any Haskell type that is an instance of both the Functor and Monad classes.
It is simple to verify that from Eqs. (<ref>), (<ref>), and (<ref>), we can derive all of the other monad and naturality laws. Thus we can use these three to formalize our Monad class (see Fig. <ref>).
\begin{align*}
& \kwd{class}\:(\hsc{Functor}\:\tT)\To\hsc{Monad}\:\tT\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \univ{\hsc{return}}^\tT&&::\U\to\tT\cdot\U \\[-\jot]
& (\hsbindU^\tT)&&::\tT\cdot\U\to(\U\to\tT\cdot\U)\to\tT\cdot\U
\end{alignedat} \\
& \hspace{8pt}
\begin{alignedat}{3}
& \univ{\hsc{return}}^\tT\:{u}\hsbindU^\tT{k} &&= {k}\:{u} \\[-\jot]
& ({m} \hsbindU^\tT {h})\hsbind{k}
&&= {m} \hsbindU^\tT (\lambda{x}.\:{h}\:{x} \hsbindU^\tT {k}) \\[-\jot]
& \fmapU^\tT\,f\:m &&= m \hsbindU^\tT (\univ{\hsc{return}}^\tT \circ f)
\end{alignedat}
\\[\medskipamount]
& \hsc{return} :: (\hsc{Monad}\:\tT) \To \tA\to\tT\cdot\tA \\[-\jot]
& \hsc{return}^\tT_\tA = \hscoerce\:\univ{\hsc{return}}^\tT \\
& (\hsbind) :: (\hsc{Monad}\:\tT) \To \tT\cdot\tA \to (\tA \to \tT\cdot\tB) \to \tT\cdot\tB
\\[-\jot]
& (\hsbind^\tT_{\tA,\tB}) = \hscoerce\:(\hsbindU^\tT)
\end{align*}
Isabelle Monad class
From the class axioms, we re-derive the rest of the original six laws for the monomorphic constants. Then we transfer all of the laws to the polymorphic constants, using the automated method described previously in Sec. <ref>.
§.§ Generic theorems about monads
Using the polymorphic monad laws, we can proceed to prove further theorems about arbitrary monads—for example, a property about the strictness of the bind operator.
Bind is strict in its first argument, if its second argument is also strict: $k\:\bot = \bot \Longrightarrow \bot \hsbind k = \bot$.
By antisymmetry, it suffices to show $\bot \hsbind k \below \bot$.
\begin{align*}
\bot \hsbind k
&\below \hsc{return}\:\bot \hsbind k & \justification{monotonicity} \\
&= k\:\bot & \justification{left unit law} \\
&= \bot & \justification{strictness of $k$} & \qedhere
\end{align*}
Within the context of class Monad, we can also define derived monadic constants, such as join.
\begin{align*}
& \hsc{join} :: (\hsc{Monad}\;\tT) \To \tT\cdot(\tT\cdot\tA) \to \tT\cdot\tA
\\[-\jot]
& \hsc{join}\;m = m \hsbind \hsid
\end{align*}
We can derive a collection of standard lemmas about join by unfolding its definition and rewriting with the monad laws. These lemmas will then be valid for any type in the Monad class.
§ INSTANTIATING TYPE CONSTRUCTOR CLASSES
Type constructor classes like Functor and Monad are already useful on their own: For example, we can use them to formalize generic Haskell monadic operations like sequence and foldM, and prove properties about them. Using the ordinary HOLCF Domain package with the right class constraints, we can also define higher-order type constructors:
\begin{equation*}
\kwd{data}\:\hsc{Tree}(\tT,\tA) = \hsc{Tip} \mid \hsc{Node}\:\tA\:(\tT\cdot(\hsc{Tree}(\tT,\tA)))
\end{equation*}
This is good, but sooner or later, we will want to populate our constructor classes with some concrete instances. To show how a Tycon library user can define new functors and monads, we will now demonstrate the process with a recursive lazy list datatype.
\begin{equation*}
\kwd{data}\:\hsc{List}\:\tA = \hsc{Nil} \mid \hsc{Cons}\:\tA\:(\hsc{List}\:\tA)
\end{equation*}
The Domain package can handle this definition with no trouble. However, we do not want List to be an ordinary Isabelle type constructor, which can only appear in fully applied form. We want List as a first-class type constructor, i.e., an instance of class Tycon. We really want to write this definition instead, which uses the type application operator:
\begin{equation*}
\kwd{data}\:\hsc{List}\cdot\tA = \hsc{Nil} \mid \hsc{Cons}\:\tA\:(\hsc{List}\cdot\tA)
\end{equation*}
The Tycon library now provides full automation for such type definitions, in the form of a new user-level type definition command. It works much like the HOLCF Domain package, and is implemented using much of the same code.
The process by which the Domain package defines new datatypes can be broken down roughly into four steps:
* Define a deflation combinator, and use it to define a representable domain satisfying the domain equation.
* Define constructors and related functions; generate theorems.
* Define take function; derive induction rules.
* Define map function; relate it to the deflation combinator.
Defining a usable Tycon involves essentially the same four steps. However, some of the steps are adapted slightly to deal with the Tycon instance and the type application operator. We now describe how our new command completes each of the four steps to make List into a Tycon and Functor.
Just like the Domain package, it constructs a deflation as a least fixed-point, based on the recursive domain equation. However, instead of defining a type $\hsc{List}\:\tA$ directly from this deflation, it defines List as a singleton type, and makes it an instance of class Tycon. The constructed deflation is used to define $\TC{\hsc{List}}$.
\begin{align}
\TC{\hsc{List}}(a) & = (\mu\hair{t}.\:\hsone_\D \oplus_\D (a_{\bot_\D} \otimes_\D t_{\bot_\D})) \\
\label{eq:repiso}
\REP{\hsc{List}\cdot\tA} & = \REP{\hsone \oplus (\tA_\bot \otimes (\hsc{List}\cdot\tA)_\bot)}
\end{align}
By unfolding the fixed-point, the desired domain equation (<ref>) is derived. It then follows that the coercions absList and repList, defined as shown here, form an isomorphism.
\begin{align*}
\hsc{absList}_\tA & = \hscoerce_{(\hsone \oplus (\tA_\bot \otimes (\hsc{List}\cdot\tA)_\bot), \hsc{List}\cdot\tA)} \\
\hsc{repList}_\tA & = \hscoerce_{(\hsc{List}\cdot\tA, \hsone \oplus (\tA_\bot \otimes (\hsc{List}\cdot\tA)_\bot))}
\end{align*}
Using these isomorphism theorems, a component of the Domain package is called to generate the multitude of definitions and theorems related to the constructors Nil and Cons. This step works exactly the same as with ordinary domain definitions.
A call to another Domain package component generates a chain of listTake functions:
\begin{align*}
\begin{alignedat}{3}
& \hsc{listTake} :: \hsc{Nat} \to \hsc{List}\cdot\tA \to \hsc{List}\cdot\tA
\end{alignedat}
\\[-\jot] &
\begin{alignedat}{3}
& \hsc{listTake}\:0 && \hsc{xs} &&= \bot \\[-\jot]
& \hsc{listTake}\,(n+1)\:&&\hsc{Nil} &&= \hsc{Nil}\\[-\jot]
& \hsc{listTake}\,(n+1)\:&&(\hsc{Cons}\:x\:\hsc{xs})
&&= \hsc{Cons}\:x\:(\hsc{listTake}\:n\:\hsc{xs})
\end{alignedat}
\end{align*}
By reasoning about the deflation agreement relation $(\isodefl)$, we can show
$\bigsqcup_n \hsc{listTake}\:n = \hsid$ from the definitions of listTake and the deflation combinator. From this, the approximation lemma [8] and induction rules are then derived, just as they are in the Domain package.
The final step is to instantiate the Functor class. The $\fmapU$ function is defined in a stylized way, which exactly matches the structure of the definition of $\TC{\hsc{List}}$.
\begin{multline}
\fmapU^\hsc{List}f = (\mu\hair{t}.\:\hsc{absList} \circ{} \\[-\jot]
\mapSum(\hsid_\hsone, \mapProd(\mapLift\:f, \mapLift\:t)) \circ \hsc{repList})
\end{multline}
The Domain package would normally generate the same definition, but would define it as a separate constant mapList.
f d
_, ∘f ∘_, d
f d(f) (d__)
f_1 d_1
f_2 d_2(f_1, f_2) (d_1 ⊕_d_2)
f_1 d_1
f_2 d_2(f_1, f_2) (d_1 ⊗_d_2)
f d^ f d
Agreement rules between map functions and deflations
The Functor class requires a proof of the agreement law $\fmapU^\hsc{List}d \isodefl \TC{\hsc{List}}\,d$. Because the definitions of $\fmapU^\hsc{List}$ and $\TC{\hsc{List}}$ have the same structure, the proof can be discharged using a collection of structural rules, some of which are listed in Fig. <ref>. The Domain package maintains this list of rules for use in its own internal proofs <cit.>.
It is not always possible to automatically prove the functor composition law: For some strict datatypes, the composition law can fail when used with non-strict functions. To avoid this difficulty, we split off a separate Prefunctor superclass that asserts only the identity law. Our new command can then always succeed in generating a Prefunctor instance for each new datatype; we leave it to the user instantiate the Functor class by proving the composition law separately.
For the List type constructor, composition can be proved using the ordinary HOLCF technique of induction over the datatype.
Further class instantiations.
Compared to Tycon and Functor, instantiations of subclasses like FunctorPlus and Monad are relatively straightforward. We write definitions of $(\hsappU)$, $\univ{\hsc{return}}$, and $(\hsbindU)$ using ordinary user-level methods: the standard Isabelle definition command for non-recursive functions, and the HOLCF Fixrec package [5] for the recursive ones. The class axioms for these subclasses are all ordinary equations, so they can be proved using ordinary techniques like induction.
Transferring theorems.
We now have a type constructor List with instances of the Functor, FunctorPlus, and Monad classes. This means that we can use the polymorphic functions $\fmap$, $(\hsapp)$, return, and $(\hsbind)$ at type $\hsc{List}\cdot\tA$. We can also apply any generic theorems from those classes to the List type.
However, we do not have any List-specific theorems about the polymorphic functions yet. For example, if $\hsc{Cons}\;x\;\hsc{xs} \hsappU \hsc{ys} = \hsc{Cons}\;x\;(\hsc{xs} \hsappU \hsc{ys})$ is one of the defining equations for $(\hsappU)$, we should like to have a version of this theorem for $(\hsapp)$ as well.
To obtain the polymorphic versions of such lemmas, we need to do a transfer process, much like we did with Theorem <ref> and for the class axioms in Sec. <ref>. The proofs can generally be completed with one call to the simplifier, using a collection of simplification rules for coercions. To transfer theorems that mention Nil or Cons, we must first prove some additional simplification rules stating that coerce commutes with those data constructors. These proofs are also simple, and potentially could be generated automatically.
§ VERIFYING MONAD TRANSFORMERS
In addition to simple type constructors like List, the Tycon library can also be used to define Tycon instances with additional type parameters, some of which may be type constructors themselves. In particular, this means that we can define a monad transformer—i.e., a monad that is parameterized by another inner monad.
The resumption monad transformer was covered in our previous work [7], but we have some improvements here. With the improved class definitions and better proof automation, we can now prove more with less effort: In addition to instantiating the monad class, we also proceed to define an interleaving operator and prove laws about it.
The new automation provided by the Tycon library has made it easier to test out definitions of new type constructors. Experimentation with the error and writer monad transformers has revealed that neither one truly preserves the monad laws. However, we have also found that the monad laws for both of those types actually are preserved for values constructible from standard operations. That is, it is possible to view each as an abstract datatype whose operations maintain an invariant; in this abstract view, each one actually does form a lawful monad.
§.§ Resumption monad transformer
The resumption monad transformer [14] augments an inner monad with the ability to suspend, resume, and interleave threads of computations. The Haskell definitions for the resumption monad transformer are shown in Fig. <ref>. (Note that although we call it a monad transformer, the Monad instance only requires $\tT$ to be a functor.)
\begin{align*}
& \kwd{data}\;\hsc{ResT}\;\tT\;\tA = \hsc{Done}\;\tA \mid \hsc{More}\;(\tT\;(\hsc{ResT}\;\tT\;\tA))
\\[\jot]
& \kwd{instance}\;(\hsc{Functor}\;\tT) \To \hsc{Monad}\;(\hsc{ResT}\;\tT)\;\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsc{return}\;x &&= \hsc{Done}\;x \\[-\jot]
& \hsc{Done}\;x \hsbind k &&= k\;x \\[-\jot]
& \hsc{More}\;m \hsbind k &&= \hsc{More}\;(\fmap\;(\lambda{r}.\;r \hsbind k)\;m)
\end{alignedat}
\end{align*}
Haskell definition of ResT monad transformer
The constructor $\hsc{Done}\:x$ represents a computation that has run to completion, yielding the result $x$. $\hsc{More}\:c$ represents a suspended computation that still has more work to do: When $c$ is evaluated, it may produce some side effects (according to the monad $\tT$) and eventually returns a new resumption of type $\hsc{ResT}\:\tT\cdot\tA$. Resumptions are a bit like threads in a cooperative multitasking system: A running thread may either terminate ($\hsc{Done}\;x$) or voluntarily yield to the operating system, waiting to be resumed later ($\hsc{More}\;c$).
We formalize the Haskell type $\hsc{ResT}\;\tT\;\tA$ as $\hsc{ResT}\;\tT\cdot\tA$ in our library. The type constructor definition generates an $\fmap$ function satisfying these rules:
\begin{align*}
\fmap\:f\:(\hsc{Done}\;x) &= \hsc{Done}\:(f\:x) \\[-\jot]
\fmap\:f\:(\hsc{More}\;m) &= \hsc{More}\:(\fmap\:(\fmap\:f)\:m)
\end{align*}
From the low-level principle of take induction, we derive a high-level induction rule for type $\hsc{ResT}\:\tT\cdot\tA$:
\begin{equation}
\inferrule
\\ \forall{x}.\,P(\hsc{Done}\:x)
\\ P(\bot)
\\ \forall{m}\,{f}.\,(\forall{r}.\,P(f\:r)) \Longrightarrow P(\hsc{More}\:(\fmap\:f\:m))}
\end{equation}
We then proceed to instantiate the Monad class for $\hsc{ResT}\;\tT$; the proofs of the monad laws are all proved using the high-level induction rule. With this class instance, we have shown that ResT is a valid monad transformer.
\begin{align*}
& \hair(\apRT) :: (FunctorPlus\;\tT) \To {} \\[-\jot]
& \hspace{16pt}
\hsc{ResT}\;\tT\;(\tA\to\tB) \to \hsc{ResT}\;\tT\;\tA \to \hsc{ResT}\;\tT\;\tB
\\[-\jot]
\begin{alignedat}{3}
& \hsc{Done}\;f &&\apRT \hsc{Done}\;x &&= \hsc{Done}\;(f\;x)
\\[-\jot]
& \hsc{Done}\;f &&\apRT \hsc{More}\;v &&=
More\;(\fmap\;(\lambda{r}.\;\hsc{Done}\;f \apRT r)\;v)
\\[-\jot]
& \hsc{More}\;u &&\apRT \hsc{Done}\;x &&=
More\;(\fmap\;(\lambda{r}.\;r \apRT \hsc{Done}\;x)\;u)
\\[-\jot]
& \hsc{More}\;u &&\apRT \hsc{More}\;v &&=
\hsc{More}\;(\fmap\;(\lambda{r}.\; \hsc{More}\;u \apRT r)\;v
\\[-\jot]
&&&&& \hspace{22pt} {} \hsapp \fmap\;(\lambda{r}.\;r \apRT More\;v)\;u)
\end{alignedat}
\end{align*}
Haskell definition of interleaving operator for ResT
Some new features of our library are nicely demonstrated by the definition and verification of an interleaving operator for resumptions [14]. The Haskell definition can be seen in Fig. <ref>. If both arguments are Done, then we combine the results and terminate.[We combine the results with function application so that we get an applicative functor; in other contexts a pair constructor might make more sense.] While either argument is More, we nondeterministically choose one such argument, run it for one step, and then recurse. Note that the definition uses a FunctorPlus class constraint—a type class whose formalization was made possible by the new Tycon library.
It turns out that $(\apRT)$ satisfies all the laws of an applicative functor [11]. The trickiest to prove is the associativity law:
\begin{equation}
\hsc{Done}\;(\circ) \apRT u \apRT v \apRT w = u \apRT (v \apRT w)
\end{equation}
The proof proceeds by nested inductions on $u$, $v$, and $w$; subproofs for the non-trivial cases rely on the naturality and associativity laws from the FunctorPlus class. A formalization of the same theorem was presented in the author's Ph.D thesis [5], although there it was defined with a fixed inner monad. This version is more general and more abstract. We assume exactly what we need to about the type constructor $\tT$, nothing more.
§.§ Error monad transformer
The error monad transformer appears in Andy Gill's mtl library, inspired by Jones [9]. It is simply a composition of the inner monad with an ordinary error monad. The Haskell definition of the Error monad that we use is shown in Fig. <ref>. It is parameterized by an extra type $\tE$, the type of error values.
\begin{align*}
& \kwd{data}\;\hsc{Error}\;\tE\;\tA =
\hsc{Err}\;\tE \mid \hsc{Ok}\;\tA
\\[\jot]
& \kwd{instance}\:\hsc{Functor}\:(\hsc{Error}\:\tE)\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \fmap\:f\:(\hsc{Err}\;e) &&= \hsc{Err}\;e \\[-\jot]
& \fmap\:f\:(\hsc{Ok}\;a) &&= \hsc{Ok}\;(f\:a)
\end{alignedat}
\\[\jot]
& \kwd{instance}\:\hsc{Monad}\:(\hsc{Error}\:\tE)\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsc{return}\;a &&= \hsc{Ok}\;a \\[-\jot]
& \hsc{Err}\;e \hsbind k &&= \hsc{Err}\;e \\[-\jot]
& \hsc{Ok}\;a \hsbind k &&= k\;a
\end{alignedat}
\end{align*}
Haskell definition of Error monad
\begin{align*}
& \kwd{newtype}\:\hsc{ErrorT}\;\tE\;\tT\;\tA = \hsc{ErrorT}\:\{ \runET::\tT\:(\hsc{Error}\;\tE\;\tA) \}
\\[\jot]
& \kwd{instance}\:(\hsc{Monad}\:\tT) \To \hsc{Functor}\:(\hsc{ErrorT}\:\tE\:\tT)\:\kwd{where}
\\[-\jot]
& \hspace{8pt} \fmap\:f\:(\hsc{ErrorT}\:t) = \hsc{ErrorT}\,(\fmap\:(\fmap\:f)\:t)
\\[\jot]
& \kwd{instance}\:(\hsc{Monad}\:\tT) \To \hsc{Monad}\:(\hsc{ErrorT}\:\tE\:\tT)\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsc{return}\:a &&= \hsc{ErrorT}\,(\hsc{return}\,(\hsc{Ok}\;a)) \\[-\jot]
& m \hsbind k &&= \hsc{ErrorT}\,(\runET\:m \hsbind \lambda\,n. \\[-\jot]
& && \hspace{24pt} \mathrm{case}\;n\;\mathrm{of}\;
\begin{aligned}[t]
\hsc{Err}\;e &\to \hsc{return}\,(\hsc{Err}\;e) \\[-\jot]
\hsc{Ok}\;a &\to \runET\,(k\;a))
\end{aligned}
\end{alignedat}
\end{align*}
Haskell definition of ErrorT monad transformer
We define an instance $\hsc{Monad}\:(\hsc{Error}\:\tE)$ using the standard procedure outlined in Sec. <ref>. The formal proofs of the monad laws proceed as expected. The resulting error monad type satisfies the following domain equation:
\begin{equation}
\REP{\hsc{Error}\:\tE\cdot\tA} = \REP{\tE_\bot\oplus\hair\tA_\bot}
\end{equation}
Using the error monad type, we can now proceed to define the error monad transformer.
We follow the Haskell definitions from Fig. <ref>, defining ErrorT as a newtype (i.e., a datatype with a single strict constructor).
\begin{align*}
& \kwd{newtype}\:(\hsc{Monad}\:\tT) \To \hsc{ErrorT}(\tE,\tT)\cdot\tA \\[-\jot]
& \hspace{8pt} = \hsc{ErrorT}\:\{\,\runET :: \tT\cdot(\hsc{Error}\:\tE\cdot\tA)\,\}
\end{align*}
The HOLCF error transformer type satisfies the following domain equation. Note that as a newtype, the right-hand side of its domain equation is not lifted.
\begin{equation}
\REP{\hsc{ErrorT}(\tE,\tT)\cdot\tA} = \REP{\tT\cdot(\hsc{Error}\:\tE\cdot\tA)}
\end{equation}
Building an instance of $\hsc{Functor}\,(\hsc{ErrorT}(\tE,\tT))$ in the standard way, we get a definition of $\fmap$ that satisfies the following rule, as we would expect:
\begin{multline}
\fmap^{(\hsc{ErrorT}(\tE,\tT))} f\,(\hsc{ErrorT}\:t) =\\[-\jot]
%\fmap\:f\:(\hsc{ErrorT}\:t) =\\
\hsc{ErrorT}\,(\fmap^\tT (\fmap^{\hsc{Error}(\tE)} f)\:t)
\end{multline}
Problems with monad instance.
Unfortunately, we run into difficulty when trying to prove an instance of $\hsc{Monad}\,(\hsc{ErrorT}(\tE,\tT))$. Not all of the class axioms are provable. The Monad class will not let us define constants return and $(\hsbind)$ that do not satisfy the laws, so instead we define the return and $(\hsbind)$ from Fig. <ref> as separate constants unitET and bindET. These and other HOLCF definitions for the error monad transformer type are shown in Fig. <ref>.
\begin{align*}
& \hsc{unitET}::(\hsc{Monad}\:\tT)\To\tA\to\hsc{ErrorT}(\tE,\tT)\cdot\tA
\\[-\jot]
& \hsc{unitET}\:a = \hsc{ErrorT}\,(\hsc{return}^\tT\,(\hsc{Ok}\;a))
\\[4pt]
& \hsc{bindET}::(\hsc{Monad}\:\tT)\To\hsc{ErrorT}(\tE,\tT)\cdot\tA\to(\tA\to\hsc{ErrorT}(\tE,\tT)\cdot\tB)\to\hsc{ErrorT}(\tE,\tT)\cdot\tB
\\[-\jot]
& \hsc{bindET}\:m\:k = \hsc{ErrorT}\,(\runET\:m \hsbind^\tT \lambda\,{x}.\:
\mathrm{case}\;x\;\mathrm{of}\;\hsc{Err}\;e \to \hsc{return}^\tT\,(\hsc{Err}\;e); \hsc{Ok}\;a \to \runET\:(k\;a))
\\[4pt]
& \hsc{liftET}::(\hsc{Monad}\:\tT) \To \tT\cdot\tA\to\hsc{ErrorT}(\tE,\tT)\cdot\tA
\\[-\jot]
& \hsc{liftET}\:t = \hsc{ErrorT}\,(\fmap^\tT\,\hsc{Ok}\:t)
\\[4pt]
& \hsc{throwET}::(\hsc{Monad}\:\tT)\To\tE\to\hsc{ErrorT}(\tE,\tT)\cdot\tA
\\[-\jot]
& \hsc{throwET}\:e = \hsc{ErrorT}\,(\hsc{return}^\tT\,(\hsc{Err}\;e))
\\[4pt]
& \hsc{catchET}::(\hsc{Monad}\:\tT)\To\hsc{ErrorT}(\tE,\tT)\cdot\tA\to(\tE\to\hsc{ErrorT}(\tE,\tT)\cdot\tA)\to\hsc{ErrorT}(\tE,\tT)\cdot\tA
\\[-\jot]
& \hsc{catchET}\:m\:h = \hsc{ErrorT}\:(\runET\:m \hsbind^\tT \lambda\,{x}.\:
\mathrm{case}\;x\;\mathrm{of}\;\hsc{Err}\;e \to \runET\,(h\;e); \hsc{Ok}\;a \to \hsc{return}^\tT\,(\hsc{Ok}\;a))
\end{align*}
Isabelle definitions of error monad transformer operations
Using this collection of non-overloaded constants, we can examine in detail the situations where the laws fail. In fact, most of the expected laws, e.g. the left unit law, do hold in general. All of the lemmas shown below can be proven by showing that $\runET$ applied to each side yields the same value.
\begin{gather}
\hsc{bindET}\:(\hsc{unitET}\:a)\:k = k\;a \\
\hsc{catchET}\:(\hsc{throwET}\:e)\:h = h\;e \\
\hsc{bindET}\:(\hsc{throwET}\:e)\:k = \hsc{throwET}\:e \\
\hsc{catchET}\:(\hsc{unitET}\:a)\:h = \hsc{unitET}\:a \\
\hsc{liftET}\:(\hsc{return}^\tT\:a) = \hsc{unitET}\:a \\
\hsc{liftET}\:(t \hsbind^\tT k) = \hsc{bindET}\:(\hsc{liftET}\:t)\:(\hsc{liftET} \circ k)
\end{gather}
A more involved proof shows that associativity also holds for bindET.
The error monad transformer satisfies the monad associativity law.
\begin{equation*}
\hsc{bindET}\:(\hsc{bindET}\:m\;h)\:k
= \hsc{bindET}\:m\:(\lambda{a}.\:\hsc{bindET}\:(h\;a)\:k)
\end{equation*}
Let $R(k)$ abbreviate the lambda expression in the definition of bindET, so that $\hsc{runET}\:(\hsc{bindET}\:m\:k) = \hsc{runET}\:m \hsbind R(k)$. Also note that $R(k)$ is strict. The proof then proceeds by applying runET to both sides of the equation. After simplification, we have:
\begin{multline*}
(\hsc{runET}\:m \hsbind R(h)) \hsbind R(k) \\[-\jot]
= \hsc{runET}\:m \hsbind R(\lambda{a}.\:\hsc{bindET}\:(h\:a)\:k)
\end{multline*}
After rewriting the left-hand side with the associativity law, both sides have the form $\hsc{runET}\:m \hsbind f$. It then suffices to show that the functions on both sides are equal for all arguments:
\begin{equation*}
\forall{x}.\: R(h)\:x \hsbind R(k) = R(\lambda{a}.\:\hsc{bindET}\:(h\:a)\:k)\:x
\end{equation*}
We proceed by cases on $x$. If $x = \bot$, then using Theorem <ref> we see that both sides reduce to $\bot$. If $x = \hsc{Err}\;e$, then both sides reduce to $\hsc{return}^\tau\,(\hsc{Err}\;e)$. Finally, if $x = \hsc{Ok}\;a$, then both sides evaluate to $\hsc{runET}\,(h\:a) \hsbind R(k)$.
On the other hand, the right unit monad law is not satisfied in general. Unless the inner monad $\tT$ has a strict return function, $m = \hsc{ErrorT}\,(\hsc{return}\:\bot)$ is a counterexample to the right unit law.
The error monad transformer satisfies the right unit law if and only if the inner monad has a strict return.
\begin{equation*}
(\forall{m}.\:\hsc{bindET}^\tT\,m\;\hsc{unitET}^\tT = m) \Longleftrightarrow
(\hsc{return}^\tT\,\bot = \bot)
\end{equation*}
Case $(\Longrightarrow)$: If we instantiate $m = \hsc{ErrorT}\:(\hsc{return}\:\bot)$, then the equation reduces to $\bot = \hsc{return}\:\bot$. Case $(\Longleftarrow)$: As above, let $R(k)$ abbreviate the lambda expression in the definition of bindET. We proceed to show $\hsc{bindET}\:m\:\hsc{unitET} = m$ by applying runET to both sides. After simplification, we get:
\begin{equation*}
\hsc{runET}\:m \hsbind R(\hsc{unitET}) = \hsc{runET}\:m
\end{equation*}
After expanding the right-hand side with the right unit law, both sides have the form $\hsc{runET}\:m \hsbind f$. It then suffices to show that the functions on both sides are equal for all arguments:
\begin{equation*}
\forall{x}.\:R(\hsc{unitET})\:x = \hsc{return}\;x
\end{equation*}
If $x = \bot$, then the equation reduces to $\bot = \hsc{return}\:\bot$, which we solve by assumption. In case $x = \hsc{Err}\;e$ or $x = \hsc{Ok}\;a$, then the equation reduces to a trivial equality.
We could prove a monad class instance for the error transformer by creating a subclass for monads-with-strict-return, and putting a stronger constraint on type $\tT$:
\begin{equation*}
\kwd{instance}\:(\hsc{StrictMonad}\:\tT) \To \hsc{Monad}\:(\hsc{ErrorT}(\tE,\tT))
\end{equation*}
However, this is not very useful in practice, because most monads do not have a strict return function (although there are a few that do, e.g. the Identity monad and some varieties of powerdomains).
Data abstraction to the rescue.
It turns out that it is impossible to construct the offending value $\hsc{ErrorT}\:(\hsc{return}\:\bot)$ using only the standard operations listed in Fig. <ref>. Furthermore, we can show that for all values constructible using those operations, the monad laws do always hold. This means that when viewed as an abstract datatype, we could still consider ErrorT to be a valid monad.
We define an inductive set $\INV$ that includes all values that can be constructed with functions in the abstract interface (see Fig. <ref>). We must also include rules for $\bot$ and lubs, to ensure that the set $\INV$ is a pcpo: In Haskell it is possible to define recursive values (i.e., least fixed-points) at any type, abstract or not.
unitET a ∈
m ∈
∀a. k a ∈bindET m k ∈
throwET e ∈
m ∈
∀e. h e ∈catchET m h ∈
liftET t ∈
∀i. m_i ∈_i m_i ∈
Inductive invariant based on ErrorT abstract interface
Finally, we can prove a restricted form of the right unit law by induction on $\INV$. The proof is straightforward, and uses techniques similar to those used for Theorems <ref> and <ref>.
\begin{equation}
m \in \INV \implies \hsc{bindET}\;m\;\hsc{unitET} = m
\end{equation}
Besides using an inductive set, there is another, more direct way of defining the invariant. We can define $\INV$ simply as the set of all values satisfying the right unit law:
\begin{equation}
\INV = \{\,m \mid \hsc{bindET}\;m\;\hsc{unitET} = m\,\}
\end{equation}
It turns out that $(\lambda{m}.\:\hsc{bindET}\;m\;\hsc{unitET})$ is actually a deflation, of which this version of $\INV$ is the corresponding set. (The reader may wish to verify that it is idempotent and below $\hsid$.)
Conveniently, we are already using deflations as our model of types. Therefore, we can use this deflation to define a new representable subtype of $\hsc{ErrorT}(\tE,\tT)\cdot\tA$ that is isomorphic to the set $\INV$. The representation of the new type $\hsc{ErrorT'}(\tE,\tT)\cdot\alpha$ as a deflation is therefore as follows:
\begin{equation}
\REP{\hsc{ErrorT'}(\tE,\tT)\cdot\tA}
= \hsemb \circ (\lambda\,m.\:\hsc{bindET}\:m\;\hsc{unitET}) \circ \hsprj
\end{equation}
We have implemented such a type definition using the Tycon library, and proven a Monad class instance for it. However, we do not yet have a principled technique for transferring definitions or theorems between the ErrorT and ErrorT' types, so working with such subtypes is impractical for casual users. Exploring ways to automate this process will be an area for future research.
§.§ Writer monad transformer
The writer monad allows a program to output a string (or more generally, any Monoid type) along with its ordinary result [9]. The bind operation of the monad concatenates the strings output by each sub-computation. The writer monad transformer composes the writer monad with an inner monad, extending the inner monad with a string output capability. The Haskell definitions are shown in Fig. <ref>.
\begin{align*}
& \kwd{class}\:\hsc{Monoid}\:\tW\:\kwd{where} \\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \mempty &&:: \tW \\[-\jot]
& (\mappend) &&:: \tW \to \tW \to \tW
\end{alignedat}
\\[\jot]
& \kwd{data}\;\hsc{Writer}\;\tW\;\tA = \hsc{Result}\;\tW\;\tA
\\[\jot]
& \kwd{newtype}\:\hsc{WriterT}\:\tW\:\tT\:\tA
= \hsc{WriterT}\:\{ \hsc{runWT} :: \tT\:(\hsc{Writer}\:\tW\:\tA) \}
\\[\jot]
& \kwd{instance}\:(\hsc{Monoid}\:\tW, \hsc{Monad}\:\tT) \To \hsc{Monad}\:(\hsc{WriterT}\:\tW\:\tT)\:\kwd{where}
\\[-\jot]
& \hspace{8pt}
\begin{alignedat}{2}
& \hsc{return}\;a &&= \hsc{WriterT}\,(\hsc{return}\,(\hsc{Result}\;\mempty\;a)) \\[-\jot]
& m \hsbind k &&= \hsc{WriterT}\,(
\begin{aligned}[t]
& \hsc{runWT}\:m \hsbind \lambda(\hsc{Result}\;w_1\;a). \\[-\jot]
& \hsc{runWT}\:(k\;a) \hsbind \lambda(\hsc{Result}\;w_2\;b). \\[-\jot]
& \hsc{return}\:(\hsc{Result}\;(w_1 \mappend w_2)\;b))
\end{aligned}
\end{alignedat}
\\[\jot]
& \hsc{tell} :: \tW \to \hsc{WriterT}\;\tW\;\tT\;() \\[-\jot]
& \hsc{tell}\;w = \hsc{WriterT}\;(\hsc{return}\;(\hsc{Result}\;w\;()))
\\[\jot]
& \hsc{listen} :: \hsc{WriterT}\;\tW\;\tT\;\tA \to \hsc{WriterT}\;\tW\;\tT\;(\hsc{Writer}\;\tW\;\tA) \\[-\jot]
& \hsc{listen}\;m = \hsc{WriterT}\:(
\begin{aligned}[t]
& runWT\;m \hsbind \lambda(\hsc{Result}\;w\;a). \\[-\jot]
& \hsc{Result}\;w\;(\hsc{Result}\;w\;a))
\end{aligned}
\end{align*}
Haskell definition of WriterT monad transformer
The Haskell Monoid class has a set of customary axioms: Instances should ensure that $(\mappend)$ is associative, with $\mempty$ as the identity element, so $\mempty \mappend x = x \mappend \mempty = x$. Note that Monoid is not a constructor class, so we can formalize it as an ordinary Isabelle type class.
The formalization of the writer monad transformer works out in almost exactly the same way as the error monad transformer: The type definitions and Functor instances work fine, but the monad instance fails because neither the left nor the right unit law holds in general. To reason about return and bind without a Monad class instance, we define functions unitWT and bindWT according to the definitions in Fig. <ref>.
The writer monad transformer satisfies the right unit law if and only if the inner monad has a strict return.
\begin{equation*}
(\forall{m}.\;\hsc{bindWT}^\tT\:m\;\hsc{unitWT}^\tT = m) \Longleftrightarrow
(\hsc{return}^\tT\,\bot = \bot)
\end{equation*}
Similar to Theorem <ref>. In the case that return is not strict, instantiating $m = \hsc{WriterT}\:(\hsc{return}\:\bot)$ gives the counterexample.
The writer monad transformer satisfies the left unit law if and only if the inner monad has a strict return.
\begin{equation*}
(\forall{x}\:{k}.\;\hsc{bindWT}^\tT\:(\hsc{unitWT}^\tT\:x)\;k = k\;x) \Longleftrightarrow
(\hsc{return}^\tT\,\bot = \bot)
\end{equation*}
Similar to Theorem <ref>. In case return is not strict, instantiating $k = \lambda{x}.\:\hsc{WriterT}\:(\hsc{return}\:\bot)$ gives the counterexample.
As with the error monad transformer, we can define a subset of type consisting of those values that satisfy the right unit law:
\begin{equation}
\INV = \{\,m \mid \hsc{bindWT}\;m\;\hsc{unitWT} = m\,\}
\end{equation}
It is straightforward to check that all writer transformer operations preserve this invariant, including unitWT, bindWT, and the formalized versions of tell and listen.
The reader may verify that the function $\lambda{m}.\;\hsc{bindWT}\:m\:\hsc{unitWT}$ is indeed a deflation. But we are not quite done showing that the subtype defined by $\INV$ is a monad: Because the left unit law does not hold universally for the writer transformer, we must also verify that all values in $\INV$ satisfy the left unit law as well.
\begin{equation}
k\;x \in \INV \implies \hsc{bindWT}\;(\hsc{unitWT}\;x)\;k = k\;x
\end{equation}
Unfolding the definition of $\INV$, we see that it is sufficient to show $\hsc{bindWT}\;(\hsc{unitWT}\;x)\;k = \hsc{bindWT}\;(k\;x)\;\hsc{unitWT}$. This can easily be proven by applying runWT to both sides and simplifying.
In summary, we have seen that the writer monad transformer is not quite a true monad, because the type contains values that do not respect the monad laws. But when we view it as an abstract datatype, with an interface that exports only operations that preserve the datatype invariant, it is valid to treat it as a real monad.
§ CONCLUSIONS AND RELATED WORK
The Tycon library for Isabelle/HOLCF is now available at the Archive of Formal Proofs [6]. It allows users to define, reason about, and instantiate constructor classes with little effort. It models polymorphism using coercion from a universal domain, which allows it to work in ordinary higher-order logic.
A different domain-theoretic model of polymorphism is presented by Amadio and Curien [1]. Here, polymorphic functions are modeled as functions from types (i.e. deflations) to values. However, this model allows non-parametric polymorphic functions that depend non-trivially on the type argument. Also, building a Tycon library around this model would also require users to write explicit type abstractions and applications when instantiating constructor classes; it is not clear whether this would be practical for users.
Sozeau and Oury [15] have recently developed a type class mechanism for the Coq theorem prover. Coq has a powerful dependent type system that allows reasoning about type constructors, first-class polymorphic values and type quantification. They define a monad class, including monad laws. Their system has the capability to formalize the whole monad class hierarchy, and it appears that it could be used to verify monad transformers; however, we are unaware of any published work in that direction.
Formalizing monad transformers in Coq does have some limitations compared to the Tycon library. For example, Coq does not accept the type definition of the resumption monad transformer: To ensure the strict positivity requirement, indirect recursion can only be used with known type constructors, not a monad parameter. Another difference (not necessarily a limitation) is that Coq is a logic of terminating functions, and does not include notions of bottoms, strictness, or partial values. Results proved in such a logic must be interpreted differently.
Our earlier formalization of axiomatic constructor classes [7] could express many of the same definitions as the current work. However, it did not provide as many features or as much automation for users. Instead of naturality laws, it used a deflation membership relation, written $x ::: d$, to express the fact that polymorphic functions had the right type. For example, the Monad class there had a rule for $\returnU$ that stated $\forall{x}:::d.\; \returnU\;x ::: \TC{\tau}(d)$. Transfer proofs to establish polymorphic laws were lengthy and unprincipled, making subclass definitions impractical for users. Automation for instantiating the Functor and Monad classes was present, but it required users to define $\fmapU$ on a separate copy of the datatype first. Users were then left without a good way to transfer properties to the new Tycon version of the type.
Automation for theorem transfer in the Tycon library is much smoother than it was in our earlier work, but there is still room for further improvement. Currently we rely on a set of rewrite rules, which works well in practice so far. However, the behavior of such rewriting strategies is often hard to predict, the rules were assembled in an ad hoc fashion, and we have no convincing reason to trust that the method will work in all situations.
A better approach would be to use a more principled theorem transfer method, like the quotient packages developed recently by Homeier [3] and Kaliszyk & Urban [10]. For functions respecting an equivalence relation, theorems can be transferred from the underlying “raw” type to a quotient type. In HOLCF, type $\U$ could be considered as a raw type, with a representable type $\tA$ as a quotient type; the $\hsprj$ function induces an equivalence relation on $\U$. The naturality laws for operations on $\U$ could then serve as the respectfulness theorems required by the quotient package.
Thanks to John Matthews for many discussions about HOLCF which helped to develop the ideas in this paper. Thanks also to Jasmin Blanchette for reading an early draft and providing helpful comments.
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Proving in Higher Order Logics (TPHOLs '05), volume 3603 of LNCS,
pages 130–146. Springer-Verlag, 2005.
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Wenzel, editors, Proceedings of the 22nd International Conference on
Theorem Proving in Higher Order Logics (TPHOLs '09), volume 5674 of LNCS, pages 260–275. Springer, 2009.
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Ph.D. thesis, Portland State University, 2012.
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Programming Techniques, volume 925 of LNCS, Båstad, Sweden, May
1995. Springer-Verlag.
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Cezary Kaliszyk and Christian Urban.
Quotients revisited for Isabelle/HOL.
In William C. Chu, W. Eric Wong, Mathew J. Palakal, and Chih-Cheng
Hung, editors, Proc. of the 26th ACM Symposium on Applied Computing
(SAC'11), pages 1639–1644. ACM, 2011.
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volume 2283 of LNCS.
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Matthieu Sozeau and Nicolas Oury.
First-class type classes.
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editors, Theorem Proving in Higher Order Logics, 21st International
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|
arxiv-papers
| 2012-07-13T11:53:44 |
2024-09-04T02:49:32.998192
|
{
"license": "Public Domain",
"authors": "Brian Huffman",
"submitter": "Brian Huffman",
"url": "https://arxiv.org/abs/1207.3208"
}
|
1207.3214
|
# Symmetric cones, the Hilbert and Thompson metrics
Bosché Aurélien Institut Fourier, 100 rue des maths, BP 74, 38402 St Martin
d’Hères cedex, France
###### Abstract.
Symmetric cones can be endowed with at least two interesting non Riemannian
metrics: the Hilbert and the Thompson metrics. It is trivial that the linear
maps preserving the cone are isometries for those two metrics. Oddly enough
those are not the only isometries in general. We give here a full description
of the isometry groups for both the Hilbert and the Thompson metrics using
essentially the theory of euclidean Jordan algebras. Those results were
already proved for the symmetric cone of complexe positive hermitian matrices
by L. Molnár in [7]. In this paper however we do not make any assumption on
the symmetric cone under scrutiny (it could be reducible and contain
exceptional factors).
## 1\. Preliminaries
A cone is a subset $\mathcal{C}$ of some euclidean space $\mathbf{R}^{n}$ that
is invariant by positive scalar exterior multiplication. A convex cone is a
cone that is also a convex subset of $\mathbf{R}^{n}$. A cone $\mathcal{C}$ is
proper (resp. open) if its closure contains no complete line (resp. if it’s
interior is not empty). In this paper we deal exclusively with open proper
cones and $\mathcal{C}$ will always be such a set. The product of two cones is
just the usual product of sets. A cone $\mathcal{C}\in\mathbf{R}^{n}$ is
reducible if $\mathbf{R}^{n}$ splits orthogonally as the sum of two subspaces
$A$ and $B$ each containing a cone $\mathcal{C}_{A}$ and $\mathcal{C}_{B}$
such that $\mathcal{C}$ is the product of $\mathcal{C}_{A}$ and
$\mathcal{C}_{B}$, i.e. the set of all sums of elements of $\mathcal{C}_{A}$
and $\mathcal{C}_{B}$. Otherwise we say that $\mathcal{A}$ is irreducible. To
a cone $\mathcal{C}\in\mathbf{R}^{n}$ we attach the set
$\mathop{\text{Aut}}(\mathcal{C})$ of all linear isomorphisms of
$\mathbf{R}^{n}$ that preserve $\mathcal{C}$. This is a group called the
automorphism group of $\mathcal{C}$. We associate to a cone $\mathcal{C}$
another cone $\mathcal{C}^{*}$ called its dual and defined by
$\mathcal{C}^{*}=\left\\{x\in\mathbf{R}^{n}\mid\forall y\in\mathcal{C},\
\langle x,y\rangle>0\right\\}.$
We say that a cone is self-dual if it is equal to its dual. A cone is
symmetric if and only if its automorphism group acts transitively on it (or
equivalently if it acts transitively on the set of rays of $\mathcal{C}$) and
if it is self-dual. A pointed cone is a couple $(\mathcal{C},e)$ where
$\mathcal{C}$ is a cone and $e\in\mathcal{C}$. It is known that for every
pointed symmetric cone $(\mathcal{C},e)$, $\mathcal{C}\subset\mathbf{R}^{n}$,
one can canonically construct a euclidean Jordan structure $J$ on the ambient
space $\mathbf{R}^{n}$, and that reciprocally, when $\mathbf{R}^{n}$ is
endowed with a euclidean Jordan structure one can canonically construct a
pointed symmetric cone $(\mathcal{C},e)$. Those two operations are inverse to
each other, and a cone is irreducible if and only if the Jordan algebra
associated to it (after choosing a base point) is simple (and this does not
depend on the choice of the base point). We recommend [3] for the general
theory of Jordan algebras and symmetric cones. In this paper, unless otherwise
stated, $J$ will always denote a euclidean Jordan algebra and
$(\mathcal{C},e)$ the symmetric cone associated to it. Remark that $e$ is then
the identity of $J$.
Let now $\mathcal{C}$ be any proper open convex cone of $\mathbf{R}^{n}$. For
$P$, $Q\in\mathcal{C}$ we define $\mathop{\text{M}}(P,Q)=\inf\left\\{\,t>0\mid
tQ-P\in\mathcal{C}\,\right\\}$, and then
$\displaystyle d_{T}(P,Q)$
$\displaystyle=\log\max\left(\mathop{\text{M}}(P,Q),\mathop{\text{M}}(Q,P)\right),$
$\displaystyle d_{H}(P,Q)$
$\displaystyle=\log\mathop{\text{M}}(P,Q)\mathop{\text{M}}(Q,P).$
Then $d_{T}(\cdot,\cdot)$ is a metric on $\mathcal{C}$ whereas
$d_{H}(\cdot,\cdot)$ is only a pseudo-metric (i.e. it is not definite) on
$\mathcal{C}$. Since the condition $d_{H}(P,Q)=0$ and $Q=\lambda P$ for some
$\lambda>0$ are equivalent the pseudo-metric $d_{H}(\cdot,\cdot)$ induces a
metric on the set of rays through $\mathcal{C}$ i.e. on the projectification
of $\mathcal{C}$.
A euclidean Jordan algebra $J$ is a finite dimensional linear space endowed
with a (not necessarily alternative) bilinear commutative product such that
for all $(a,b)\in V$ we have $a\cdot(b\cdot a^{2})=(a\cdot b)\cdot a^{2}$ and
such that $a^{2}+b^{2}=0$ implies $a=b=0$. Such an an algebra is always unital
and we shall denote $e$ its unit. Hence by assumption a euclidean Jordan
algebra is commutative but associativity fails in general. This failure of
associativity in turn creates some “non-cummutativity” effects (this is
certainly the reason why Jordan investigated those algebras for their possible
use in quantum theories). For example we define a center which might very well
not be trivial
###### Definition 1.1.
The center of a Jordan algebra $J$ is the subalgebra consisting of all
elements $x\in J$ satisfying
$\forall a,b\in J,\ x(ab)=(xa)b.$
The set of squares of a euclidean Jordan algebra defines a closed proper cone
and its interior, the connected component of the unit in the set of invertible
elements of the algebra, is a symmetric cone. An element $p\in J$ is an
idempotent if $p^{2}=p$. The map $p\mapsto e-p$ is a bijection of the set of
idempotents of $J$. The image of $p$ under this map will be written
$p^{\prime}$. Two idempotents $p$ and $q$ are orthogonal if $pq=0$. An
idempotent is primitive if it cannot be written as the sum of two orthogonal
elements.
###### Definition 1.2.
A Jordan frame is a family $(p_{i})_{i=1}^{r}$ of mutually orthogonal
primitive idempotents such that $\sum_{1\leq i\leq r}p_{i}=e$. The cardinality
$r$ of a Jordan frame is independent of the Jordan frame and is called the
rank of the algebra.
###### Proposition 1.3.
To each $x\in J$ is associated a Jordan frame $(p_{i})_{i=1}^{r}$ and a family
of real numbers $(\lambda_{i})_{i=1}^{r}$ such that $x=\sum_{1\leq i\leq
r}\lambda_{i}p_{i}$. The $\lambda_{i}$ only depend (up to reordering) on $x$.
We say that $x$ is regular if the $\lambda_{i}$ are distinct. Under those
circumstances the idempotents $p_{i}$ are also well defined (up to reordering,
the reordering being the same as the one alluded to for the $\lambda_{i}$).
Let us now give some notations
###### Definition 1.4.
If $x=\sum_{1\leq i\leq r}\lambda_{i}p_{i}$ is the spectral decomposition of
$x\in J$ then we define $\exp(x)=\sum_{1\leq i\leq r}\exp(\lambda_{i})p_{i}$,
$\mathop{\text{Tr}}(x)=\sum_{1\leq i\leq r}\lambda_{i}$, $\|x\|=\sup_{1\leq
i\leq r}|\lambda_{i}|$, and
$|x|_{\sigma}=\sup_{i,j}|\lambda_{i}-\lambda_{j}|$. The $\lambda_{i}$ are the
eigenvalues of $x$ and the set of the eigenvalues is called the spectrum of
$x$, noted $\operatorname{spec}{x}$. The spectral norm of $x$ is $\|x\|$. We
will also call it the JB-norm.
###### Definition 1.5.
The set of elements of a euclidean Jordan algebra $J$ with positive
eigenvalues is equal to the image of $J$ by the exponential. This is by
definition the (symmetric) cone associated to the Jordan algebra $J$.
We are now ready to define a scalar product on $J$
###### Definition 1.6.
We define $(x,y)=\mathop{\text{Tr}}(xy)$ for $x$, $y\in J$.
###### Proposition 1.7.
The scalar product $(\cdot,\cdot)$ is associative, i.e. satisfies
$\forall x,y\in J,\ (xz,y)=(x,zy).$
###### Definition 1.8.
To each $x\in J$, the linear endomorphism $y\mapsto xy$ of $J$ is noted
$L(x)$. To each such $x$ we associate another linear endomorphism
$\mathop{\text{P}}(x)$ of $J$ called the quadratic representation of $x$ and
defined by $\mathop{\text{P}}(x)=2L^{2}(x)-L(x^{2})$. If $x\in\mathcal{C}$
then $\mathop{\text{P}}(x)$ is a positive definite operator for the natural
scalar product.
###### Remark 1.9.
Hence $x\in J$ lies in the center of $J$ if and only if $L(x)$ commutes with
every $L(y)$, $y\in J$.
###### Definition 1.10.
A Jordan algebra is simple if its only strict ideal is the trivial ideal. A
Jordan algebra is semi-simple if it is a direct sum of simple Jordan algebras.
The following Proposition can be found for example in [3] for example
###### Proposition 1.11.
Every semi-simple Jordan algebra decomposes uniquely as the direct sum of
simple Jordan algebras. Euclidean Jordan algebras are semi-simple.
Let us remind the reader that the simple euclidean Jordan algebras have been
classified.
Let us turn our attention to the isometry group of the Thompson metric and of
the Hilbert semi-metric. Let us begin with the easy
###### Proposition 1.12.
The automorphism group $\mathop{\text{Aut}}(\mathcal{C})$ of a convex proper
open cone $\mathcal{C}$ acts isometrically on $\mathcal{C}$ for both the
Thompson metric and the Hilbert semi-metric. Indeed if
$g\in\mathop{\text{Aut}}(\mathcal{C})$ and $(P,Q)\in\mathcal{C}$ then
$\mathop{\text{M}}(P,Q)=\mathop{\text{M}}(g(P),g(Q))$. This action is faithful
in the case of the Thompson metric, and induces an isometric action on the set
of rays through $\mathcal{C}$ with kernel the subgroup of positive dilatation
$\\{\,\lambda I_{n},\ \lambda>0\,\\}$ in the case of the Hilbert pseudo-
metric.
We will see that this group is not always the full isometry group
$\mathop{\text{Iso}}(\mathcal{C})$ for the chosen metric (Hilbert or
Thompson), but that it is always a subgroup of finite index of it.
Every cone associated to a euclidean Jordan algebra also carries a Riemannian
symmetric structure of negative Ricci curvature. For convenience we remind the
definition of its first fondamental form
###### Definition 1.13.
The scalar product $\langle\cdot,\cdot\rangle_{x}$ at $x\in\mathcal{C}$ is
given by (as usual we identify the tangent space at $x$ with the vector space
obtained by forgetting the algebra structure of $J$)
$\displaystyle\forall u,v\in J,\ \langle
u,v\rangle_{x}=(\mathop{\text{P}}(x)^{-1}u,v)=(\mathop{\text{P}}(x)^{-1/2}u,\mathop{\text{P}}(x)^{-1/2}v).$
This Riemannian structure is complete (locally symmetric manifolds are always
complete) and we note $d_{R}(\cdot,\cdot)$ the associated metric. We write
$i_{x}$ for the geodesic inversion at $x\in\mathcal{C}$ for this Riemannian
structure.
This Riemannian structure is non-positively curved and simply-connected. In
other words it is a Hadamard manifold. Consequently there is exactly one
geodesic joining any two points and one can hence define the midpoint of such
a pair. Remark that this is in sharp contrast to what happens for both the
Hilbert and the Thompson metric. Indeed, putting aside the trivial case where
$\mathcal{C}$ is reduced to a half line, the Thompson metric is never locally
uniquely geodesic, and the Hilbert metric is locally uniquely geodesic if and
only if it is isometric to the model space of constant curvature $-1$.
###### Definition 1.14.
The midpoint of two points $a$ and $b$ for the Riemannian metric associated to
$J$ is written $a\\#b$. In fact we have
$\displaystyle
a\\#b=\mathop{\text{P}}(a^{1/2})\left((\mathop{\text{P}}(a^{-1/2})b)^{1/2}\right)$
and $a\\#b$ is the unique solution of $\mathop{\text{P}}(x)(a^{-1})=b$.
For every $u\in J$, $\det(\exp(u))=\exp(\mathop{\text{Tr}}(u))$. But then the
set of points $a\in\mathcal{C}$ such that $\det{a}=1$ is the image by the
exponential of the kernel of the linear form $u\mapsto\mathop{\text{Tr}}(u)$
().
###### Definition 1.15.
Let $J_{0}$ be the set of $u\in J$ satisfying $\mathop{\text{Tr}}(u)=0$. Then
$J_{0}$ is a linear subspace of $J$ but not a subalgebra in general.
###### Definition 1.16.
Let $\mathcal{C}_{0}$ be the image of $J_{0}$ by the exponential map. Then
$\mathcal{C}_{0}$ is the set of all $a\in\mathcal{C}$ such that $\det{a}=1$.
It is also also a global section of the projectification of $\mathcal{C}$
(because the determinant is positive on $\mathcal{C}$)
Let us remind the expression of the Riemannian geodesics starting at $e$
###### Proposition 1.17.
The constant speed geodesics starting at $e$ are exactly the curves of the
form $t\mapsto\exp(tu)$ with $u\in J^{*}$. The speed of this geodesic it
precisely $\mathop{\text{Tr}}(u^{2})^{1/2}$, i.e. the square root of the sum
of the squares of the eigenvalues of $u$ with multiplicities.
We already introduced the geometric mean in general euclidean Jordan algebras.
We now introduce a new mean called the spectral mean (see [5] )
###### Definition 1.18.
The spectral mean $a\mu b$ of $(a,b)\in\mathcal{C}$ is
$\mathop{\text{P}}(a^{-1}\\#b)^{1/2}a$. It is the unique solution in $J$ of
the equation
$(a^{-1}\\#b)^{1/2}=a^{-1}\\#x.$
Let us introduce a new concept before stating the next Proposition
###### Definition 1.19.
Two elements $a$ and $b$ of $J$ are simultaneously diagonalisable if they are
diagonal in the same Jordan frame i.e. if for some Jordan frame
$(e_{i})_{1\leq i\leq r}$ there exists $(\lambda_{i})_{1\leq i\leq
r}\in\mathbf{R}^{r}$ and $(\mu_{i})_{1\leq i\leq r}\in\mathbf{R}^{r}$ such
that $a=\sum_{1\leq i\leq r}\lambda_{i}e_{i}$ and $b=\sum_{1\leq i\leq
r}\mu_{i}e_{i}$.
###### Remark 1.20.
Two simultaneously diagonalisable primitive idempotents are obviously either
equal or orthogonal.
The following proposition is proved in [5]
###### Proposition 1.21.
For $a$, $b\in J$ the following three conditions are equivalent
* •
$a$ and $b$ are simultaneously diagonalisable,
* •
$\exp(a)$ and $\exp(b)$ are simultaneously diagonalisable,
* •
the geometric mean and the spectral mean of $\exp(a)$ and $\exp(b)$ are equal.
## 2\. More on the Hilbert and Thompson metrics
In [1] and [7] the expression of the Hilbert and Thompson metrics for the
simple euclidean Jordan algebra of complex hermitian matrices is derived. The
computations work in full generality and we include a proof for the ease of
the reader
###### Proposition 2.1.
Let us consider the cone associated to a euclidean Jordan algebra $J$. Its
associated Hilbert metric $d_{H}$ and Thompson metric $d_{T}$ are given by
$\displaystyle d_{H}(a,b)$
$\displaystyle=\operatorname{diam}\log\operatorname{spec}\left(\mathop{\text{P}}(a^{-1/2})b\right),$
$\displaystyle d_{T}(a,b)$
$\displaystyle=\|\log\left(\mathop{\text{P}}(a^{-1/2})b\right)\|,$
where $\|\cdot\|$ is the spectral norm (that is the JB-norm).
Proof: Remember that if $a\in\mathcal{C}$ then $P(a)$ preserves $\mathcal{C}$,
so that
$\displaystyle\log\mathop{\text{M}}(a,b)$ $\displaystyle=\log\inf\\{\,t>0\mid
tb-a\in\mathcal{P}\,\\}$ $\displaystyle=\log\inf\\{\,t>0\mid
t-\mathop{\text{P}}(b^{-1/2})a\in\mathcal{P}\,\\}$
$\displaystyle=\log\sup\operatorname{spec}{\mathop{\text{P}}(b^{-1/2})a},$
and similarly
$\displaystyle\log\mathop{\text{M}}(b,a)$ $\displaystyle=\log\inf\\{\,t>0\mid
ta-b\in\mathcal{P}\,\\}$ $\displaystyle=\log 1/\sup\\{\,t>0\mid
a-tb\in\mathcal{P}\,\\}$ $\displaystyle=\log
1/\sup\\{\,t>0\mid\mathop{\text{P}}(b^{-1/2})(a)-t\in\mathcal{P}\,\\}$
$\displaystyle=-\log\inf\operatorname{spec}\mathop{\text{P}}(b^{-1/2})(a).$
The proposition is a direct consequence of those computations. ∎
###### Proposition 2.2.
The constant speed geodesics for the Riemannian metric on $\mathcal{C}$ (resp.
on $\mathcal{C}_{0}$) are constant speed geodesics for the Thompson metric
(resp. for the Hilbert metric). Consequently the Riemannian midpoints are also
midpoints for those two other metrics.
Proof: Since the isometry group acts transitively for the three metrics we can
concentrate on the Riemannian geodesics emanating from the identity. If
$c:t\mapsto\exp(tu)$, $u\in J$, is a such a geodesic then for every $s$,
$t\in\mathbf{R}$.
$\displaystyle d_{H}(\exp(su),\exp(tu))$
$\displaystyle=\operatorname{diam}{\log{\operatorname{spec}{\exp((t-s)u)}}}$
$\displaystyle=\operatorname{diam}{\operatorname{spec}{(t-s)u)}}$
$\displaystyle=|t-s|\operatorname{diam}{\operatorname{spec}{u}}.$
Similarly
$\displaystyle d_{T}(\exp(su),\exp(tu))$
$\displaystyle=\|\log{\exp((t-s)u)}\|$ $\displaystyle=\|(t-s)u)\|$
$\displaystyle=|t-s|\|u\|.$
∎
The Riemannian geodesics are not only geodesics for those two other metrics
but even play a special role among all the geodesics as we shall see in the
next section. The reason why it is so is basically the following lemma
###### Lemma 2.3.
The Riemannian geodesic inversions are isometries for both the Thompson and
the Hilbert metrics.
Proof: Since we already found a transitive isometry common to the three
metrics it is enough to prove that the geodesic inversion at $e$ is an
isometry. But the Riemannian inversion at the identity is just the algebra
inversion $a\mapsto a^{-1}$. But one easily proves that
$\mathop{\text{M}}(a,b)=\mathop{\text{M}}(b^{-1},a^{-1})$ for $a$ and $b$ in
$\mathcal{C}$. ∎
We now carry out a construction that we shall need later. For $\lambda>0$ let
us define
$\begin{array}[]{rcl}\Phi_{\lambda}:J&\rightarrow&\mathcal{C}\\\
u&\mapsto&\exp(\lambda u).\end{array}$
Then $\Phi_{\lambda}$ is a homeomorphism. Using $\Phi_{\lambda}$ it is
possible to push back the metric $d_{T}/\lambda$ from $\mathcal{C}$ to $J$. We
call $d_{T,\lambda}(\cdot,\cdot)$ this metric. Using the double restriction
$(\Phi_{\lambda})_{|J_{0}}^{|\mathcal{C}_{0}}$ we can do the same with the
Hilbert metric and construct a metric $d_{H,\lambda}(\cdot,\cdot)$ on $J_{0}$.
It so happens that those metrics converge to norms when $\lambda$ converges to
$0$. To prove this we will need the following Lemmas
###### Lemma 2.4.
For $(u,v)\in J^{2}$ we have
$\displaystyle\mathop{\text{P}}(\exp(-tu/2))\exp(tv)=e+t(v-u)+o(t).$
Proof: The map $(a,b)\mapsto P(a)b$ is differentiable and its differential at
$(e,e)$ is $(x,y)\mapsto 2P(e,x)e+P(e)y=2x+y$. But then the differential of
$t\mapsto P(\exp(-tu/2))\exp(tv)$ at $0$ is $2(-u/2)+v=v-u$. Since $P(e)e=e$
the Lemma is proved. ∎
###### Lemma 2.5.
The spectrum is continuous on any Jordan algebra.
Proof: The characteristic polynomial of $a\in J$ is continuous on $a$ and the
roots of a polynomial depend continuously on the polynomial. ∎
###### Proposition 2.6.
The metrics $d_{T,\lambda}(\cdot,\cdot)$ and $d_{H,\lambda}(\cdot,\cdot)$
converge when $\lambda$ converges to $0$, and the limit metrics are both given
by a norm. For $(u,v)\in J^{2}$ and $(u_{0},v_{0})\in J_{0}^{2}$ we have
$\displaystyle\lim_{\lambda\rightarrow 0}d_{T,\lambda}(u,v)$
$\displaystyle=\|v-u\|,$ $\displaystyle\lim_{\lambda\rightarrow
0}d_{H,\lambda}(u_{0},v_{0})\ =\|u_{0}-v_{0}\|_{\sigma}.$
Proof: For $\lambda>0$ we have
$\displaystyle d_{T,\lambda}(u,v)$ $\displaystyle=d_{T}(\exp(\lambda
u),\exp(\lambda v))/\lambda$
$\displaystyle=\|\log\left(\mathop{\text{P}}(\exp(-\lambda u/2))\exp(\lambda
v)\right)\|/\lambda$
$\displaystyle=\|\log\left(e+\lambda(v-u)+o(\lambda)\right)\|/\lambda$
$\displaystyle=\|v-u\|+o(1),$
and
$\displaystyle d_{H,\lambda}(u_{0},v_{0})$ $\displaystyle=d_{H}(\exp(\lambda
u_{0}),\exp(\lambda v_{0}))/\lambda$
$\displaystyle=\operatorname{diam}\log\operatorname{spec}\left(\mathop{\text{P}}(\exp(\lambda-
u_{0}/2))\exp(\lambda v_{0})\right)/\lambda$
$\displaystyle=\operatorname{diam}\operatorname{spec}\log\left(e+\lambda(v_{0}-u_{0})+o(\lambda)\right)/\lambda$
$\displaystyle=\operatorname{diam}\operatorname{spec}\left(\lambda(v_{0}-u_{0})+o(\lambda)\right)/\lambda$
$\displaystyle=\operatorname{diam}\operatorname{spec}(v_{0}-u_{0})+o(1).$
∎
## 3\. Isometries fixing the identity
We begin by the following fundamental result which was already used in the
space case of hermitian definite positive complexe matrices in [7]
###### Proposition 3.1.
Every isometry $g$ of the Thompson or the Hilbert metric preserves the
Riemannian midpoints, that is satisfies $g(a\\#b)=g(a)\\#g(b)$.
Proof: See the Lemma in [7] and how it is applied to show that isometries for
the Hilbert and Thompson preserve the Riemannian midpoints (since the proof is
exactly the same as in [7] we do not duplicate it here). ∎
From this we infer
###### Proposition 3.2.
Every isometry $g$ of the Thompson or the Hilbert metric preserves the
Riemannian geodesics and in particular Riemannnian geodesic lines.
Proof: An isometry for any of those metrics must be a homeomorphism of the
underlying symmetric space because the Thompson and the Hilbert metrics
generate the topology of the underlying manifold. If $[a,b]$ is a compact
geodesic then if we put $M_{0}(a,b)=\\{a,b\\}$ and define inductively
$M_{n+1}(a,b)$ for $n>0$ to be the union of $M_{n}(a,b)$ and the midpoints of
pairs of points of $M_{n}(a,b)$ then $M(a,b)=\cup_{n\geq 0}M_{n}$ is dense in
$[a,b]$. But since $g$ preserves the midpoints we must have
$g(M(a,b))=M(g(a),g(b))$ and by density $g([a,b])=[g(a),g(b)]$. ∎
Assume now that $g$ is an isometry for either the Thompson or the Hilbert
metric and that $g$ fixes $e$. Let $d$ be the metric for which $g$ is an
isometry. If $\lambda>0$ then $g$ must be an isometry for $d/\lambda$ too. If
$d$ is the Thompson metric, let $g_{\lambda}$ be the push-back by
$\Phi_{\lambda}$ of $g$ and $d_{\lambda}$ the push-back of the metric
$d/\lambda$ by the same homeomorphism. If $d$ is the Hilbert metric, replace
$\Phi_{\lambda}$ by its double restriction
$(\Phi_{\lambda})_{|J_{0}}^{|\mathcal{C}_{0}}$ and proceed similarly. Then
obviously $g_{\lambda}$ must be an isometry of $d_{\lambda}$. Let us compute
$g_{\lambda}$
###### Definition 3.3.
If $g$ is an isometry of $d_{T}$ (resp. of $d_{H}$) fixing the identity then
$g$ sends a Riemannian constant speed geodesic $t\mapsto\exp(tu)$, $u\in J$
(resp. $u\in J_{0}$), to another Riemannian constant speed geodesic and hence
there exists a well-defined $v\in J$ (resp. $v\in J_{0}$) such that
$g(\exp(tu))=\exp(tv)$ for every $t\in\mathbf{R}$. We write $g_{*}(u)$ for
this $v$. Obviously $g_{*}$ is homogeneous of degree one and for every $u\in
J$ (resp. $u\in J_{0}$) and $t\in\mathbf{R}$ we have
$g(\exp(tu))=\exp(tg_{*}(u)).$
We have
###### Proposition 3.4.
$g_{\lambda}$ is constant equal to $g_{*}$. In particular the $g_{\lambda}$
converge to $g_{*}$ and $g_{*}$ is a surjective isometry of the limit norm
$\lim_{\lambda\rightarrow 0}d_{\lambda}$.
Proof:
$\displaystyle g_{\lambda}(u)=\frac{1}{\lambda}\log g(\exp{\lambda
u})=\frac{1}{\lambda}\log\exp{\lambda g_{*}(u)}=g_{*}(u).$
$g_{*}$ must be surjective because so is $g$. It is also a consequence of the
fact that it is an isometry of a finite dimensional normed space. ∎
###### Proposition 3.5.
$g_{*}$ is a linear isomorphism of $J$ if $d$ is the Thompson metric and of
$J_{0}$ if $d$ is the Hilbert metric.
Proof: Direct consequence of the Mazur-Ulam Theorem. ∎
We proved that isometries for the Thompson or the Hilbert metric are well-
behaved with respect to the geometric mean. In fact they also behave nicely
with respect to the spectral mean as the following Proposition shows
###### Proposition 3.6.
Every isometry $g$ of the Thompson or the Hilbert metric preserves the
spectral mean i.e. satisfy $g(a\mu b)=g(a)\mu g(b)$ for every
$(a,b)\in\mathcal{C}^{2}$.
Proof: We can assume that $g$ fixes the identity. Since $g$ preserves
midpoints it must preserve the inversion and the square root. But since the
spectral mean of $a$ and $b\in\mathcal{C}$ is the only solution $x$ in
$\mathcal{C}$ of $(a^{-1}\\#b)^{1/2}=a^{-1}\\#x$, $g$ must also preserve it. ∎
## 4\. Case of the Thompson metric
###### Definition 4.1.
A symmetry of an algebra $\mathcal{A}$ is an element $s\in\mathcal{A}$ such
that $s^{2}=1$. It is called central if it lies in the center of the algebra
$\mathcal{A}$.
The following is proved in [4]. Let us recall that a euclidean algebra endowed
with the already defined $JB$-norm is a (finite dimensional) $JB$-algebra.
###### Proposition 4.2.
The isometries of a (not necessarily simple) JB-algebra are exactly the maps
$x\mapsto b\cdot\Phi(x)$ where $b$ is a central symmetry and $\Phi$ is a
Jordan isomorphism. For unital isometries (i.e. preserving the unit $e$) we
have $b=e$.
Now let $g$ be any isometry for the Thompson metric. After composing $g$ on
the right by some element $h$ of the transitive isometry group
$\mathop{\text{Aut}}(J)$ we get an isometry fixing the identity $e$. From now
on we hence assume that $g$ fixes $e$. We proved in the preceding section that
$g_{*}$ is then an isometry of the JB-algebra $J$. Assume first that it fixes
the identity. Then according to the proposition above it must be an algebra
isomorphism of $J$, and so
$\displaystyle\forall u\in\mathcal{C},\
g(\exp(u))=\exp(g_{*}(u))=g_{*}(\exp(u)).$
But then $g$ is the restriction to $\mathcal{C}$ of an algebra isomorphism of
$J$. Let us come back to the general case. Then we can fix a central symmetry
$b$ and an algebra isomorphism $\Phi$ such that $g_{*}(x)=b\Phi(x)$ for all
$x\in J$. The following Lemma is certainly well known but since we did not
find any proof in the existing litterature we include one that does not use
the classification of simple euclidean Jordan algebras
###### Lemma 4.3.
The center of a simple euclidean algebra is $\mathbf{R}e$ where $e$ is the
unit element.
Proof: Let $z$ be in the center and let us write $(x,y)$ for the associative
bilinear form $\mathop{\text{Tr}}(xy)$. Then the bilinear form
$(x,y)\mapsto\langle zx,y\rangle$ is clearly also an associative bilinear
form. But then according to the Proposition III.$4$.$1$. of [3] this form must
be a multiple of the original one, i.e. for some $\lambda\in\mathbf{R}$ we
have
$\displaystyle\forall x,y\in J,\ (zx,y)=\lambda(x,y).$
Choosing $x=e$ and $y$ arbitrary we obtain that $z-\lambda e$ is in the
radical of $J$. Since the radical is reduced to $\\{0\\}$ by assumption we
must have $z=\lambda e$. ∎
###### Lemma 4.4.
Consider the decomposition $J=J_{1}\times\cdots J_{n}$ into simple euclidean
algebras, and let $e_{i}$ be the multiplicative unit of $J_{i}$. Then the
central symmetries are exactly the $\sum_{i=1}^{n}\epsilon_{i}e_{i}$ where
$\epsilon_{i}\in\\{\pm 1\\}$.
Proof: The elements of the form $\sum_{i=1}^{n}\epsilon_{i}e_{i}$ with
$\epsilon_{i}\in\\{\pm 1\\}$ are obviously central symmetries. Reciprocally if
$u=\sum_{i=1}^{n}u_{i}$ is a central symmetry of $J$ where the $u_{i}$ lie in
$J_{i}$ then each $u_{i}$ must be a central symmetry of $J_{i}$. According to
the Lemma 4.3 the center of $J_{i}$ is $\mathbf{R}e_{i}$, and so the $u_{i}$
must be equal to either $e_{i}$ or $-e_{i}$. ∎
$\Phi$ being an algebra isomorphism must permute isometric simple factors.
Hence after composing on the right by the corresponding permutation algebra
isomorphism $\sigma$ we can assume that $\Phi$, and hence $g_{*}$, preserves
each irreducible factor (remark that
$\sigma\in\mathop{\text{Aut}}(\mathcal{C})$). We just proved
###### Proposition 4.5.
Let $g$ be an isometry for the Thompson metric. Then after composing $g_{*}$
on the right by some algebra isomorphism $\sigma^{*}$ we get $x\mapsto bx$ for
some central symmetry $b$. For $g$ this means that after composing by some
$\sigma\in\mathop{\text{Aut}}(\mathcal{C})$ we get a map $a=\sum_{1\leq i\leq
n}a_{i}\mapsto\sum_{1\leq i\leq n}a_{i}^{\epsilon_{i}}$ for some
$\epsilon_{i}\in\\{\pm 1\\}$.
###### Remark 4.6.
The map $a_{i}\mapsto a_{i}^{-1}$ is just the geodesic inversion at $e_{i}$ of
the symmetric space associated to the simple factor $J_{i}$ with unit $e_{i}$.
###### Remark 4.7.
Let $n$ be the number of distinct isomorphism classes of the simple factors of
$J$. Let us order those isomorphism classes arbitrarily from $1$ to $n$.
Suppose that there are $k_{i}\geq 1$ distinct simple factors of $J$ that
represent the class numbered $i$. Then the automorphism group is easily seen
to have index at most
$\prod_{1\leq i\leq n}\sum_{0\leq j\leq k_{i}}(k_{i}+1)=k+n,$
where $k$ is the number of simple factors of $J$. Indeed this is a upper bound
on the number of central symmetries with disjoint orbits under permutation of
isometric simple factors of $J$.
We will see in the next section that there are less isometries for the Hilbert
metric as soon as the algebra is not simple. This comes from the fact that
products of Thompson isometries are again Thompson isometries, whereas the
analogous statement does not hold for Hilbert metrics. Indeed, for the
Thompson metric, we have the
###### Proposition 4.8.
Let $\mathcal{C}$ be a product of cones $\mathcal{C}_{i}$, $1\leq i\leq n$.
Let $d$ (resp. $d_{i}$) be the Thompson metric associated with $\mathcal{C}$
(resp. associated with $\mathcal{C}_{i}$). Then
$d=\sup_{1\leq i\leq n}d_{i}.$
Proof: Direct consequence of the following computations
$\displaystyle\mathop{\text{M}_{C}}(P,Q)$ $\displaystyle=\inf\\{\,t>0\mid
tQ-P\in\mathcal{C}\,\\}$ $\displaystyle=\inf\\{\,t>0\mid\forall 1\leq i\leq
n,\ tQ_{i}-P_{i}\in\mathcal{C}_{i}\,\\}$ $\displaystyle=\sup_{1\leq i\leq
n}\inf\\{\,t>0\mid tQ_{i}-P_{i}\in\mathcal{C}_{i}\,\\}$
$\displaystyle=\sup_{1\leq i\leq n}\mathop{\text{M}_{C_{i}}}(P_{i},Q_{i}).$
∎
###### Remark 4.9.
It follows from this Proposition that the map $a=\sum_{1\leq i\leq
n}a_{i}\mapsto\sum_{1\leq i\leq n}a_{i}^{\epsilon_{i}}$ (for some
$\epsilon_{i}\in\\{\pm 1\\}$) is an isometry for the Thompson metric (because
geodesic inversions are). From this it follows easily that the index of the
automorphism group in the full isometry group is exactly equal to $k+n$ (the
notations are those of the remark 4.7 ).
## 5\. Case of the Hilbert metric
This case requires substantially more work than the case of the Thompson
metric. The reason is that though we associated some linear map $h^{*}$ to
every isometry fixing the origin $e$, this map is not defined on $J$ but on
the hyperplane $J_{0}$ of $J$. Moreover, even though $h^{*}$ is an isometry
for some norm, this norm is not the restriction to $J_{0}$ of the JB-norm of
$J$. This problem was already encountered in [6] and we will begin our proof
likewise. However the Jordan algebra considered in [6] is both simple and
exceptional so we have to proceed differently. Remark that our proof is not
considerably longer than the one in the aforementioned paper.
###### Definition 5.1.
We note $\bar{J}$ the quotient vector space $J/(\mathbf{R}e)$. The class of
$u\in J$ is noted $[u]$.
$\bar{J}$ is naturally linearly isomorphic to $J_{0}$ but it is sometimes
better to work with $\bar{J}$. We provide $\bar{J}$ with a norm through this
identification.
###### Lemma 5.2.
The lower (resp. upper) eigenvalue of $L(x)$ is equal to that of $x$.
Proof: It is well known (see [3] for example) that the Lemma holds when $x$ is
an idempotent since then (putting the two trivial cases aside) the eigenvalues
of $x$ are $0$ and $1$ and those of $L(x)$ are among $0$, $1/2$ and $1$. The
general case follows from this remark and the spectral decomposition Theorem.
∎
###### Remark 5.3.
In fact the eigenvalues of $L(x)$ can be deduced from that of $x$. Indeed in
the Proposition $2$.$1$ of [5] it is proved that when $J$ is simple the
eigenvalues of $L(x)$ are precisely the $(\lambda_{i}+\lambda_{j})/2$ for
$i\neq j$ (where the $\lambda_{i}$ are the eigenvalues of $x$). The general
case follows from this one by splitting $J$ into simple algebras. Indeed if
$J=J_{1}\times\cdots J_{n}$ is such a splitting and $x=(x_{1},\ldots,x_{n})\in
J$ then the eigenvalues of $x$ are those of the $x_{i}$ and the eigenvalues of
$L(x)$ are those of the $L(x_{i})$.
###### Corollary 5.4.
If $x\in J$ has eigenvalues contained in $[0,1]$ then, with respect to the
usual partial ordering of symmetric operators, $0\leq L(x)\leq Id$ (where $Id$
is teh identity mapping of $J$) i.e.
$\displaystyle\forall y\in J,\ 0\leq(L(x)y,y)\leq\|y\|^{2}.$
###### Proposition 5.5.
The extremal points of the unit ball of $\bar{J}$ are exactly the classes
$[p]$ of the non trivial idempotents of $J$.
Proof: Adapted from the Lemma $2$ in [6]. Let $p$ be an idempotent and let us
show that $[p]$ is an extreme point of the unit ball. Let us write
$[p]=t[a]+(1-t)[b]$ for some $t\in]0,1[$ and $(a,b)\in J^{2}$ with
$|a|_{\sigma}=|b|_{\sigma}=1$. Hence for some additional
$\lambda\in\mathbf{R}$ we have $p=ta+(1-t)b+\lambda e$ and we can always
assume that the spectrum of both $a$ and $b$ is contained in $[0,1]$ and that
$0$ and $1$ are eigenvalues of both $a$ and $b$. Hence according to the
Corollary 5.4 we have $0\leq L(a)\leq Id$ and $0\leq L(b)\leq Id$, from which
we deduce that
$\displaystyle(L(p)p,p)=1$
$\displaystyle=t(L(a)p,p)+(1-t)(L(b)p,p)+\lambda\|p\|^{2}$ $\displaystyle\leq
t\|p\|^{2}+(1-t)\|p\|^{2}+\lambda\|p\|^{2}$ $\displaystyle\leq
1+\lambda\|p\|^{2},$
and so $\lambda\geq 0$. Since $p$ is not trivial, there exists some non
trivial idempotent $q\in J$ such that $pq=0$ (take $q=p^{\prime}=e-p$ for
example). But then using the Corollary 5.4 again we obtain this time
$(L(p)q,q)=0=t(L(a)q,q)+(1-t)(L(b)q,q)+\lambda\|q\|^{2}\geq\lambda\|q\|^{2},$
so that finally $\lambda=0$ and $p=ta+(1-t)b$. But it is proved in [8] (see
also the remark after the Theorem $1$.$1$ in [4] ) that the projections are
the extreme points of the interval $[0,1]$ (i.e. of elements with spectrum
contained in $[0,1]$) so that we must have $p=a=b$ and $p$ is indeed an
extreme point of the unit ball.
Any class with unit norm of $\bar{J}$ is represented by an element $v\in J$
with sprectum contained in $[0,1]$ and containing both $0$ and $1$. Moreover
the class $[v]$ of such an element contains an idempotent if and only if $v$
is itself idempotent. Assume that $v$ is not. Then some of its eigenvalues lie
in $]0,1[$ and we can write $v=\lambda_{1}p_{1}+\sum_{2\leq i\leq
r}\lambda_{i}p_{i}$ with $0<\lambda_{1}<1$ and $0\leq\lambda_{i}\leq 1$. But
then $v=\alpha d+(1-\alpha)f$ where $\alpha=1-\lambda_{1}$,
$f=\lambda_{1}/2p_{1}+\sum_{2\leq i\leq r}\lambda_{i}p_{i}$ and
$d=(\lambda_{1}/2+1/2)p_{1}+\sum_{2\leq i\leq r}\lambda_{i}p_{i}$. Obviously
$[d]$ and $[f]$ are still in the unit ball of $\bar{J}$ and are distinct so
that $[v]$ is not an extreme point of the unit ball. ∎
###### Lemma 5.6.
Assume that some element $u\in J$ can be written $u=p+\lambda e$ for some non-
trivial idempotent $p$. Then $p$ and $\lambda$ are well defined and depend
continuously on $u$.
Proof: Remark that $p^{\prime}=e-p$ is a non-trivial idempotent orthogonal to
$p$ and that
$\displaystyle u=p+\lambda e=p+\lambda(p+(e-p))=(1+\lambda)p+\lambda(e-p).$
Hence the eigenvalues of $u$ are exactly $\lambda$ and $1+\lambda$. So
$\lambda$ is the smallest eigenvalue of $u$ and depends continuously on $u$ by
continuous dependence of the roots of polynomials. But then $p=u-\lambda e$
also depends continuously on $u$. ∎
Let $h_{*}$ be any isometry of $J_{0}$. Let us extend $h_{*}$ linearly to $J$
by sending $e$ to itself and write $\hat{h}_{*}$ for the extended map. Then
$\hat{h}_{*}$ is still an isometry (on the whole of $J$) for
$|\cdot|_{\sigma}$, though $|\cdot|_{\sigma}$ is not a norm anymore (it is
degenerate since for example $|e|_{\sigma}=0$). If $p$ is a non trivial
idempotent of $J$ then we can write $\hat{h}_{*}(p)=q+\lambda e$ for some non
trivial idempotent q and $\lambda\in\mathbf{R}$, and $q$ depends continuously
on $p$ according to the previous Lemma. Let us write $q=f(p)$ for convenience.
Then $f$ is a continuous function from the set $\mathcal{\text{P}}(J)$ of non
trivial idempotents of $J$ to itself. The same reasoning with the inverse map
$h_{*}^{-1}$ immediately yields that $f$ is a homeomorphism of
$\mathcal{\text{P}}(J)$ and hence preserves the connected components of
$\mathcal{\text{P}}(J)$.
###### Definition 5.7.
For any Jordan algebra $J$ let $\mathcal{P}_{k}(J)$ be the set of idempotent
of fixed rank $k$, where $k$ is an integer and $J$ is any Jordan algebra. For
convenience we put $\mathcal{P}_{k}(J)=\\{0\\}$ if $k\leq 0$ and
$\mathcal{P}_{k}(J)=\\{e\\}$ if $k\geq\operatorname{rank}{J}$.
###### Lemma 5.8.
Let $J=J_{1}\times\cdots J_{n}$ be the decomposition into simple Jordan
algebras of $J$. Then the connected components of $\mathcal{\text{P}}(J)$ are
exactly the products $\prod_{i=1}^{n}\mathcal{P}_{k_{i}}(J_{i})$ where $k_{i}$
is an arbitrary integer. In particular the connected components of
$\mathcal{\text{P}}(J_{i})$, $1\leq i\leq n$, is the set of idempotents with
fixed rank.
Proof: One only needs consider the case of a simple Jordan algebra. But then
the connected component $K$ of the identity in the isomorphism group of the
Jordan algebra $J$ acts transitively on the set of idempotents with fixed rank
(See the Proposition IV.$3$.$1$, (iii) in [3] ). The Lemma follows since $K$
is connected. ∎
###### Proposition 5.9.
Let $k\in[0,r]$. Then the image by $f$ of a connected component of
$\mathcal{P}_{k}(J)$ lies in $\mathcal{P}_{k}(J)$ or in
$\mathcal{P}_{r-k}(J)$.
###### Lemma 5.10.
Let $m=(m_{i})_{1\leq i\leq r}$ be a Jordan frame and $p=\sum_{1\leq i\leq
r}\lambda_{i}m_{i}$, $\lambda_{i}\in\\{0,1\\}$, be a non trivial idempotent
which is diagonal in this Jordan frame. Then the set $\mathfrak{S}(p,e)$ of
idempotents $q$ which are diagonal in the same Jordan frame $(m_{i})_{1\leq
i\leq r}$ and that satisfy $|p-q|_{\sigma}=1$ has cardinality
$2^{\operatorname{rank}{p}}+2^{r-\operatorname{rank}{p}}-2$.
Proof of the Lemma: The set $\mathfrak{S}(p,m)$ is the disjoint union of the
two sets $\mathfrak{S}_{1}(p,m)$ and $\mathfrak{S}_{2}(p,m)$ where
$\mathfrak{S}_{1}(p,m)$ consists of the non-zero idempotents
$q\in\mathfrak{S}(p,e)$ satisfying $q<p$ (i.e. if $q=\sum_{1\leq i\leq
r}\mu_{i}m_{i}\in\mathfrak{S}(p,e)$ then $q\neq p$ and $\lambda_{i}=0$ implies
$\mu_{i}=0$) and $\mathfrak{S}_{2}(p,m)=p+\mathfrak{S}_{1}(e-p,m)$ (i.e. if
$q=\sum_{1\leq i\leq r}\mu_{i}m_{i}\in\mathfrak{S}_{2}(p,m)$ then $q\neq p$
and $\mu_{i}=0$ implies $\lambda_{i}=0$). Then one has
$\displaystyle\operatorname{card}\mathfrak{S}_{1}(p,m)=2^{k}-1=2^{\operatorname{rank}{p}}-1,$
and so
$\operatorname{card}\mathfrak{S}_{2}(p,m)=2^{\operatorname{rank}(e-p)}-1=2^{r-\operatorname{rank}{p}}-1$.
The Lemma follows since $\mathfrak{S}_{1}(p,m)$ and $\mathfrak{S}_{2}(p,m)$
are disjoint. ∎
Proof of the Proposition: The set of regular elements is an open dense subset
of $J$ and by continuity so must be its pre-image by $\hat{h}_{*}$. The
intersection of those two open dense sets is certainly not empty and so we can
fix a regular element $x\in J$ such that $\hat{h}_{*}(x)$ is also regular. Let
$m=(m_{i})_{1\leq i\leq r}$ (resp. $m^{\prime}=(m^{\prime}_{i})_{1\leq i\leq
r}$) be a Jordan frame in which $x$ is diagonal (resp. $\hat{h}_{*}(x)$), and
remark that by regularity the elements that can be diagonalised in the same
frame as $x$ (resp. in the same frame as $\hat{h}_{*}(x)$) are exactly those
that are diagonal in the given frame $m$ (resp. in $m^{\prime}$). We have
already proved that $\hat{h}_{*}$ preserves simultaneous diagonalisation so
elements which are diagonal in the frame $m$ are mapped to elements which are
diagonal in $m^{\prime}$. Moreover the set of idempotents diagonal in the
frame $m$ intersects all connected components of $\mathcal{P}(J)$. Since
$\hat{h}_{*}$ preserves the semi-norm $|\cdot|_{\sigma}$, so does $f$, and we
can now infer that if $0<k<r$
$\displaystyle f(\mathfrak{S}(\sum_{1\leq i\leq
k}m_{i},m))=\mathfrak{S}(f(\sum_{1\leq i\leq k}m_{i}),m^{\prime}),$
and in particular those two sets have the same cardinality $2^{k}+2^{r-k}-2$.
But one easily checks that for $(x,y)\in[0,r]^{2}$,
$\displaystyle 2^{x}+2^{r-x}-2=2^{y}+2^{r-y}-2\Leftrightarrow
x=y\quad\textrm{or}\quad x=r-y.$
Indeed, the function $x\mapsto 2^{x}+2^{r-x}-2$ is symmetric around $r/2$,
strictly decreasing on $]-\infty,r/2[$ and strictly increasing on
$]r/2,+\infty[$. Hence $f(\sum_{1\leq i\leq k}m_{i})$ has rank $k$ or $r-k$
and we are done. ∎
Let now $J=J_{1}\times...\times J_{s}$ be the decomposition of $J$ into simple
factors where $J_{i}$ as unit $e_{i}$. Let also $r$ (resp. $r_{k}$) be the
rank of $J$ (resp. of $J_{k}$) and $\mathcal{P}_{i}(J)$ (resp. $P_{i}(J_{k})$)
be the set of idempotents of rank $i$ in $J$ (resp. in $J_{k}$). Define
$\displaystyle
Q_{1}(J_{k})=\\{e_{1}\\}\times\cdots\\{e_{k-1}\\}\times\mathcal{P}_{{r_{k}}-1}(J_{k})\times\\{e_{k+1}\\}\times\cdots\times\\{e_{s}\\}.$
Then the connected components of $\mathcal{P}_{1}(J)$ (resp.
$\mathcal{P}_{r-1}(J)$) are exactly the $P_{1}(J_{k})$ (resp. the
$Q_{1}(J_{k})$). Moreover $x\mapsto e-x$ is a diffeomorphism between
$P_{1}(J_{k})$ and $Q_{1}(J_{k})$ . Now take $a\in[1,s]$. There are two
possibilities for $\mathcal{P}(J_{a})$ according to the Proposition 5.9
Case one: $f$ sends $\mathcal{P}_{1}(J_{a})$ onto $\mathcal{P}_{1}(J_{b})$ for
some $b$. Since $h_{*}$ is trace preserving we must have
$h_{*}(\mathcal{P}_{1}(J_{a}))=\mathcal{P}_{1}(J_{b})$ and by linearity
$h_{*}(J_{a})=J_{b}$.
Case two: $f$ sends $\mathcal{P}_{1}(J_{a})$ onto $Q_{1}(J_{b})$ for some $b$.
But then if $p\in\mathcal{P}(J_{a})$ we have
$\mathop{\text{Tr}}(f(p))=\mathop{\text{Tr}}(e)-\mathop{\text{Tr}}(p)=\mathop{\text{Tr}}(e)-1$
and from $h_{*}(p)=f(p)+\lambda e$ we then deduce that
$\lambda=2/\mathop{\text{Tr}}(e)-1$. The decomposition of $h_{*}(p)$
corresponding to the splitting $J=J_{0}\otimes(\mathbf{R}e)$ is then
(1) $\displaystyle
h_{*}(p)=(1/\mathop{\text{Tr}}(e)e+f(p)-e)+1/\mathop{\text{Tr}}(e)e.$
Consider now the first factor $J_{1}$. Assume that we are in the second case
above, i.e. that $\mathcal{P}_{1}(J_{1})=Q_{1}(J_{b})$ for some
$b\in[1,s]$.Then if we compose $h$ by the inversion $x\mapsto x^{-1}$, $h_{*}$
is composed by $x\mapsto-x$ and we easily deduce that the extension
$\hat{f}_{*}$ of $h_{*}$ is replaced by (see (1) )
$h_{*}(p)=-(1/\mathop{\text{Tr}}(e)e+f(p)-e)+1/\mathop{\text{Tr}}(e)e=e-f(p),$
so that after composing $h$ with the inversion the factor
$\mathcal{P}_{1}(J_{1})$ is in the first case above. Since the inversion is an
isometry and preserves $J_{0}$ it follows that we can always assume that
$f(\mathcal{P}_{1}(J_{1}))=\mathcal{P}_{1}(J_{b})$ for some $b\in[1,s]$.
###### Lemma 5.11.
Either all the factors are in the first case above or they are all in the
second.
Proof: We can always assume that $s\geq 2$. From the previous discussion we
can always assume that $J_{1}$ is such that we have
$f(\mathcal{P}_{1}(J_{1}))=\mathcal{P}_{1}(J_{b})$ for some $b\in[1,s]$.
Let us prove first that we must have
$f(\mathcal{Q}_{1}(J_{1}))=\mathcal{Q}_{1}(J_{b})$. If $c\in[1,s]$ is
different from $b$ then it is easy to see that the spectrum of $q-p$ is
independent of the choice of $q\in\mathcal{Q}_{1}(J_{c})$ and
$p\in\mathcal{P}_{1}(J_{b})$ and contains exactly $r-2$ ones and $2$ zeros.
But, for $p\in\mathcal{P}_{1}(J_{b})$, $q=e-p\in\mathcal{Q}_{1}(J_{c})$ and
the spectrum of $q-p=e-2p$ contains at least one $1$ and one $-1$ (because
$s\geq 2$). Hence $b$ is the only element of $c\in[1,s]$
$\mathcal{Q}_{1}(J_{b})$ such that we do not have $\|q-p\|_{\sigma}=1$ for
every $q\in\mathcal{Q}_{1}(J_{c})$ and $p\in\mathcal{P}_{1}(J_{b})$. Since $f$
preserves $\|\cdot\|_{\sigma}$ and $\mathcal{P}_{1}(J_{b})$ we must have
$f(\mathcal{Q}_{1}(J_{1}))=\mathcal{Q}_{1}(J_{b})$.
Let us assume that $J_{2}$ is such that
$f(\mathcal{P}_{1}(J_{2}))=\mathcal{Q}_{1}(J_{c})$ for some $c\in[1,s]$. We
must have $c\neq b$ because $f$ is injective. If
$p_{1}\in\mathcal{P}_{1}(J_{1})$ and $p_{2}\in\mathcal{P}_{1}(J_{2})$ then
$\|p_{1}-p_{2}\|_{\sigma}=2.$
However it is easy to see that we must have $\|f(p_{1})-f(p_{2})\|_{\sigma}=1$
because $c\neq b$: contradiction. ∎
Hence after possibly composing $h$ with the inversion we can assume that all
the factors are in the first case, i.e. that $f$ preserves the set of
irreducible idempotents of each simple factor. Composing again $h$ by the
Jordan automorphism that permutes isometric factors we can moreover assume
that $h$ preserves each simple factor. We now prove that $h$ must be linear.
###### Proposition 5.12.
$h$ is linear (i.e. is the restriction to $\mathcal{C}$ of a linear
isomorphism of $\mathbf{R}^{n}$).
Proof: We already know that $h_{*}$ preserves the set of primitive
idempotents. Let us show that $h_{*}$ sends orthogonal primitive idempotents
to orthogonal primitive idempotents. But two primitive idempotents are
orthogonal if and only if they are simultaneously diagonalisable and distinct.
Since $h_{*}$ preserves simultaneously diagonalisable pairs and is injective
it must also preserve pairs of orthogonal primitive idempotents. It follows
easily that $h_{*}$ is a Jordan isomorphism. But then $h_{*}$ commutes with
the exponential and
$\displaystyle\forall x\in J,\ h(\exp x)$ $\displaystyle=\exp(h_{*}x)$
$\displaystyle=h_{*}\exp(x).$
Hence $h$ is the restriction of $h_{*}$ to the symmetric cone and is linear. ∎
###### Corollary 5.13.
The automorphism group is a subgroup of index two or zero in the isometry
group of a symmetric cone for the Hilbert metric. The automorphism group is
equal to the isometry group only for the Lorentz cones, i.e. only when the
underlying Jordan algebra has rank at most two. To be more precise and closer
to the spirit of the classification of euclidean simple Jordan algebras,
equality between the two groups appear only in the following three cases, the
third being an infinite family
1. (1)
$\mathcal{C}$ is a half-line (i.e. has rank one),
2. (2)
$\mathcal{C}$ is the positive quadrant of $\mathbf{R}^{2}$, i.e. the set of
points with positive coordinates (this is the direct product of two half-
lines),
3. (3)
$n=\dim(J)\geq 3$ and $\mathcal{C}$ is the irreducible Lorentz cone i.e. the
set of points $(x_{1},\ldots,x_{n})$ satisfying
$x_{1}^{2}>x_{2}^{2}+\cdots+x_{n}^{2}$.
Proof: First let us show that when the rank is two the geodesic inversion at
$e$ is in $\mathop{\text{Aut}}(\mathcal{C})$. But in that case if
$(e_{1},e_{2})$ is a Jordan frame and $u=\lambda_{1}e_{1}+\lambda_{2}e_{2}$
(for $\lambda_{1}$, $\lambda_{2}>0$) we have
$\displaystyle u^{-1}=\frac{1}{\lambda_{1}}e_{1}+\frac{1}{\lambda_{2}}e_{2}$
$\displaystyle=\frac{1}{\lambda_{1}\lambda_{2}}\left(\lambda_{2}e_{1}+\lambda_{1}e_{2}\right)$
$\displaystyle=\frac{1}{\lambda_{1}\lambda_{2}}\left((\lambda_{1}+\lambda_{2}-\lambda_{1})e_{1}+(\lambda_{1}+\lambda_{2}-\lambda_{2})\right)e_{2}$
$\displaystyle=\frac{1}{\lambda_{1}\lambda_{2}}\left(\mathop{\text{Tr}}(u)e-u\right),$
which is clearly a projective transformation (if we only consider the
restriction of the inversion to the set of $x$ satisfying $\det(x)=1$ then it
is even “linear”). The rank one case is trivial so their only remains to prove
that when the rank is at least three the inversion is not projective. This can
easily be shown directly but the following argument (due to M. Crampon) gives
more insight into what happens near the boundary of the cone.
First remark that the inversion is homogeneous (of degree $-1$) and so we can
work with rays instead of restricting the inversion to $\mathcal{C}_{0}$. For
$x\in\mathcal{C}$ let $[x]$ be the ray through $x$. Let $e_{1}$, $e_{2}$ and
$e_{3}$ be three orthogonal idempotents. Then if $n>0$ the inverse of
$u^{1}_{n}=ne_{1}+e_{2}+e_{3}$ is $u^{-1}_{n}=(1/n)e_{1}+e_{2}+e_{3}$.
Similarly the inverse of $u^{2}_{n}=ne_{1}+2e_{2}+e_{3}$ is
$u_{n}^{-2}=(1/n)e_{1}+(1/2)e_{2}+e_{3}$. But the rays $[u^{1}_{n}]$ and
$[u^{2}_{n}]$ defined by $u^{1}_{n}$ and $u^{2}_{n}$ both converge to the ray
$[e_{1}]$. If the inversion were a projective map, the rays $[u^{-1}_{n}]$ and
$[u^{-2}_{n}]$ would also converge to the same ray. But $[u^{-1}_{n}]$
converges to $[e_{2}+e_{3}]$ and $[u^{-2}_{n}]$ converges to
$[(1/2)e_{2}+e_{3}]$. Since those two rays differ the inversion cannot be
projective. ∎
###### Remark 5.14.
For the rank two case we could equally have used the fact that the cone is
then strictly convex and so the isometry group for the Hilbert metric is
reduced to $\mathop{\text{Aut}}(\mathcal{C})/\mathbf{R}$ (see [2] ).
###### Remark 5.15.
In fact we showed that the inversion does not have a continuous prolongation
to the boundary. See the paper of De la Harpe [2] where he investigates the
“blow off” near the boundary for the simplicial cone of $\mathbf{R}^{3}$ (this
shows that isometries need not admit a prolongation to the boundary of the
convex).
###### Question 5.16.
If a map $f:J\rightarrow J$ preserves the semi-norm $\|\cdot\|_{\sigma}$ and
the trace then it acts as the identity on $\mathbf{R}e$ (because $\mathbf{R}e$
is the set of elements with zero $\|\cdot\|_{\sigma}$ semi-norm and on this
set the trace is injective) and its double-restriction to $J_{0}$ must be
linear by the Mazur-Ulam Theorem since on this set the semi-norm
$\|\cdot\|_{\sigma}$ is definite. In fact since the projection onto the second
factor of the decomposition $J=J_{0}\oplus\mathbf{R}e$ is
$x\mapsto\mathop{\text{Tr}}(x)/\mathop{\text{Tr}}(e)e$ and $f$ commutes with
this operator, $f$ always preserves the second factor. From this it follows
that $f$ must itself be linear and preserve the decomposition (indeed if
$x_{0}=a+\lambda_{0}e$ is the decomposition of $x_{0}$ in the direct sum
$J=J_{0}\oplus\mathbf{R}e$ then $f(x_{0})=b+\lambda_{0}e$ for some $b\in
J_{0}$ that must itself satisfy $\|f(x_{0})-b\|_{\sigma}=0$ and
$\|f(x_{0})-f(a)\|_{\sigma}=\|f(a)+\lambda_{0}e-f(a)\|_{\sigma}=0$, so that
$b=f(a)$). We proved above that if $f$ sends simultaneously diagonalisable
pairs to simultaneously diagonalisable pairs then it is either a Jordan
isomorphism or becomes one after composing with the linear map that can be
written $(x_{0},\lambda)\mapsto(-x_{0},\lambda)$ in the splitting
$J=J_{0}\oplus\mathbf{R}e$. The question is: do one needs to assume that
simultaneously diagonalisable pairs are preserved, or is this always true? We
did not find any evidence that the question was already investigated, even in
the case of symmetric/hermitian matrices.
## References
* [1] E. Andruchow, G. Corach, and D. Stojanoff. Geometrical significance of Löwner-Heinz inequality. Proc. Amer. Math. Soc., 128(4):1031–1037, 2000.
* [2] P. de la Harpe. On hilbert’s metric for simplices. Geometric group theory, 1:97–119, 1993.
* [3] J. Faraut and A. Korányi. Analysis on symmetric cones. Clarendon Press Oxford, 1994.
* [4] J.M. Isidro and A.R. Palacios. Isometries of jb-algebras. manuscripta mathematica, 86(1):337–348, 1995.
* [5] J. Kim and Y. Lim. Jordan automorphic generators of euclidean jordan algebras. J. Korean Math. Soc, 43(3):507–528, 2006.
* [6] L. Molnar and M. Barczy. Linear maps on the space of all bounded observables preserving maximal deviation. Journal of Functional Analysis, 205(2):380–400, 2003.
* [7] Lajos Molnár. Thompson isometries of the space of invertible positive operators. Proc. Amer. Math. Soc., 137(11):3849–3859, 2009.
* [8] JD Wright and MA Youngson. On isometries of jordan algebras. Journal of the London Mathematical Society, 2(2):339, 1978.
|
arxiv-papers
| 2012-07-13T12:18:27 |
2024-09-04T02:49:33.012256
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aur\\'elien Bosch\\'e",
"submitter": "Aur\\'elien Bosch\\'e",
"url": "https://arxiv.org/abs/1207.3214"
}
|
1207.3288
|
Integrating Resource Selection Information with Spatial Capture-Recapture
J. Andrew Royle, U.S. Geological Survey, Patuxent Wildlife Research Center,
Laurel, Maryland, 20708, _email:_ aroyle@usgs.gov
Richard B. Chandler, U.S. Geological Survey, Patuxent Wildlife Research
Center, Laurel, Maryland, 20708, _email:_ rchandler@usgs.gov
Running title. Resource Selection and Spatial Capture-Recapture
Word count. 5670
Summary.
1\. Understanding space usage and resource selection is a primary focus of
many studies of animal populations. Usually, such studies are based on
location data obtained from telemetry, and resource selection functions (RSF)
are used for inference. Another important focus of wildlife research is
estimation and modeling population size and density. Recently developed
spatial capture-recapture (SCR) models accomplish this objective using
individual encounter history data with auxiliary spatial information on
location of capture. SCR models include encounter probability functions that
are intuitively related to RSFs, but to date, no one has extended SCR models
to allow for explicit inference about space usage and resource selection.
2\. In this paper we develop the first statistical framework for jointly
modeling space usage, resource selection, and population density by
integrating SCR data, such as from camera traps, mist-nets, or conventional
catch-traps, with resource selection data from telemetered individuals. We
provide a framework for estimation based on marginal likelihood, wherein we
estimate simultaneously the parameters of the SCR and RSF models.
3\. Our method leads to increases in precision for estimating population
density and parameters of ordinary SCR models. Importantly, we also find that
SCR models alone can estimate parameters of resource selection functions and,
as such, SCR methods can be used as the sole source for studying space-usage;
however, precision will be higher when telemetry data are available.
4\. Finally, we find that SCR models using standard symmetric and stationary
encounter probability models produce biased estimates of density when animal
space usage is related to a landscape covariate. Therefore, it is important
that space usage be taken into consideration, if possible, in studies focused
on estimating density using capture-recapture methods.
Key-words. animal movement, animal sampling, encounter probability,
hierarchical modeling, landscape connectivity, marginal likelihood, resource
selection, space usage, spatial capture-recapture.
## 1 Introduction
Spatial capture-recapture (SCR) models are relatively new methods for
inference about population density from capture-recapture data using auxiliary
information about individual capture locations (Efford, 2004; Borchers and
Efford, 2008; Royle and Young, 2008). SCR models posit that $N$ individuals
are located within a region denoted $\mathcal{S}$. Each individual has a home
range or activity area within which movement occurs during some well-defined
time interval, and the center of the animal’s activity has Cartesian
coordinates $\mathbf{s}_{i}$ for individuals $i=1,\ldots,N$. The population is
sampled using $J$ traps with coordinates ${\bf x}_{j}$ for $j=1,\ldots,J$, and
encounter probability is expressed as a function of the distance between trap
location (${\bf x}_{j}$), and individual activity center (${\bf s}_{i}$).
While SCR models are a relatively recent innovation, their use is already
becoming widespread (Efford et al., 2009; Gardner et al., 2010b, a; Kéry et
al., 2010; Gopalaswamy et al., 2012; Foster and Harmsen, 2012) because they
resolve critical problems with ordinary non-spatial capture-recapture methods
such as ill-defined area sampled and heterogeneity in encounter probability
due to the juxtaposition of individuals with traps (Borchers, 2011).
Furthermore, unlike traditional capture-recapture methods, SCR models allow
for inference about the processes determining spatial variation in population
density.
Despite the increasing popularity of SCR models, every application of them has
been based on encounter probability models, such as the bivariate normal
distribution, that imply symmetric and stationary (invariant to translation)
models for home range. While such simple models might be necessitated in
practice by sparse data, home range size and shape are often not well
represented by stationary distributions because animals select resources that
are unevenly distributed in space. Therefore more complex models are needed to
relate the capture process with the way in which individuals utilize space.
In this paper, we extend SCR capture probability models to accommodate models
of space usage or resource selection, by extending them to include one or more
explicit landscape covariates, which the investigator believes might affect
how individual animals use space within their home range (this is what
(Johnson, 1980) called third-order selection). We do this in a way that is
entirely consistent with the manner in which parameters of classical resource
selection functions (RSF) (Manly et al., 2002) or utilization distributions
(UD) (Worton, 1989; Fieberg and Kochanny, 2005; Fieberg, 2007) are estimated
from animal telemetry data. In fact, we argue that SCR models and RSF/UD
models estimated from telemetry are based on the same basic underlying model
of space usage. The important distinctions between SCR and RSF studies are
that (1) resource selection studies do not result in estimates of population
density and (2) in SCR studies, encounter of individuals is imperfect (i.e.,
“$p<1$”) whereas, with RSF data obtained by telemetry, encounter is perfect.
With respect to the latter point, we can think of the RSF and SCR studies as
being exactly equivalent either if we have a dense array of trapping devices,
or if our telemetry apparatus samples time or space imperfectly. A key concept
that we must confront in order to unify and integrate SCR and RSF data is that
we need to formulate both models in terms of a common latent variable so that
we can make them consistent with respect to some underlying space utilization
process. As we will explain, this latent variable is the number of times that
an individual uses a particular region of the landscape over some period of
time.
The modeling framework we develop here simultaneously resolves three important
problems: (1) it generalizes all existing capture probability models for SCR
data to accommodate realistic patterns of space usage that result in
asymmetric and irregular home ranges; (2) it allows estimation of RSF
parameters directly from SCR data, i.e., absent telemetry data; and (3) it
provides the basis for integrating telemetry data directly into SCR models to
improve estimates of model parameters, including density. Our model greatly
expands the applied relevance of SCR methods for conservation and management,
and for addressing applied and theoretical questions related to animal space
usage and resource selection.
## 2 Spatial Capture-Recapture
A number of distinct observation models have been proposed for spatial
capture-recapture studies (Borchers and Efford, 2008; Royle et al., 2009;
Efford et al., 2009), including Poisson, multinomial, and binomial observation
models. Here we focus on the binomial model in which we suppose that the $J$
traps are operated for $K$ periods (e.g., nights), and the observations are
individual- and trap-specific counts $y_{ij}$, which are binomial with sample
size $K$ and capture probabilities $p_{ij}$ which depend on trap locations
${\bf x}_{j}$ and individual activity centers ${\bf s}_{i}$ as described
subsequently. The vector of trap-specific counts for individual $i$, ${\bf
y}_{i}=(y_{i1},\ldots,y_{iJ})$ is its encounter history. A standard encounter
probability model (Borchers and Efford, 2008) is the Gaussian model in which
$\log(p_{ij})=\alpha_{0}+\alpha_{1}d_{ij}^{2}$ (1)
or, equivalently, $p_{ij}=\lambda_{0}\exp(-d_{ij}^{2}/(2\sigma^{2}))$, where
$d_{ij}$ is the Euclidean distance between points ${\bf s}_{i}$ and ${\bf
x}_{j}$,
$d_{ij}=\|\textbf{s}_{i}-\textbf{x}_{j}\|=\sqrt{(s_{i1}-x_{j1})^{2}+(s_{i2}-x_{j2})^{2}}$,
and $\alpha_{0}=log(\lambda_{0})$ and $\alpha_{1}=-1/(2\sigma^{2})$.
Alternative detection models are used, but all are functions of Euclidean
distance and so we do not consider them further here.
The primary motivation behind our work is that, in all previous applications
of SCR models, simple encounter probability models based only on Euclidean
distance have been used, with estimation based on standard likelihood or
Bayesian methods. These methods regard the activity center for each individual
$i$, ${\bf s}_{i}$, as latent variables and remove them from the likelihood
either under a model of “uniformity” in which ${\bf s}\sim\mbox{Unif}({\cal
S})$ where ${\cal S}$ is a spatial region (the “state-space” of ${\bf s}$), or
a model in which covariates might affect the spatial distribution of
individuals (Borchers and Efford, 2008). The state-space ${\cal S}$ defines
the potential values for any activity center ${\bf s}$, e.g., a polygon
defining available habitat or range of the species under study.
A critical problem with standard SCR models is that the encounter probability
model based on Euclidean distance metric is unaffected by habitat or landscape
structure, and it implies that the space used by individuals is stationary and
symmetric, which may be unreasonable in many applications. For example, if the
common detection model based on a bivariate normal probability distribution
function is used, then the implied space usage by all individuals, no matter
their location in space or local habitat conditions, is symmetric with
circular contours of usage intensity. Subsequently we provide an extension of
this class of SCR models that accommodates asymmetric, irregular and spatially
heterogeneous models of space usage. Thus, “where” an individual lives on the
landscape, and the state of the surrounding landscape, will determine the
character of its usage of space. In particular, we suggest encounter
probability models that imply irregular, asymmetric and non-stationary home
ranges of individuals and that are sensitive to the local landscape being used
by an individual.
## 3 Basic Model of Space Usage
We develop the model here in terms of a discrete landscape purely for
computational expediency. This formulation will accommodate the vast majority
of actual data sets, as almost all habitat or landscape structure data comes
to us in the form of raster data. Let ${\bf x}_{1},\ldots,{\bf x}_{nG}$
identify the center coordinates of a set of $nG$ pixels that define a
landscape. In SCR studies, a subset of the coordinates ${\bf x}$ will
correspond to trap locations where we might observe individuals whereas, in
telemetry studies, animals are observable (by telemetry fixes) at potentially
all coordinates.
Let $z({\bf x})$ denote a covariate measured (or defined) for every pixel
${\bf x}$. For clarity, we develop the basic ideas here in terms of a single
covariate but, in practice, investigators typically have more than 1
covariate, which poses no additional problems. We suppose that a population of
individuals wanders around space in some manner related to the covariate
$z({\bf x})$, and their locations accumulate in pixels by some omnipotent
accounting mechanism. We will define “use of ${\bf x}$” to be the event that
an individual animal appeared in some pixel ${\bf x}$. This is equivalently
stated in the literature in terms of individual having selected ${\bf x}$. As
a biological matter, use is the outcome of individuals moving around their
home range (Hooten et al., 2010), i.e., where an individual is at any point in
time is the result of some movement process. However, to understand space
usage, it is not necessary to entertain explicit models of movement, just to
observe the outcomes, and so we don’t elaborate further on what could be
sensible or useful models of movement.
Suppose that an individual is monitored over some period of time and a fixed
number, say $R$, of use observations are recorded. Let $n({\bf x})$ be the use
frequency of pixel ${\bf x}$ for that individual. i.e., the number of times
that individual used pixel ${\bf x}$ during some period of time. We assume the
following probability distribution for the $nG\times 1$ vector of use
frequencies:
${\bf n}\sim\mbox{Multinom}(R,{\bm{\pi}})$
where ${\bm{\pi}}$ is the $nG\times 1$ vector of use probabilities with
elements (for each pixel):
$\pi({\bf x})=\frac{\exp(\alpha_{2}z({\bf x}))}{\sum_{x}\exp(\alpha_{2}z({\bf
x}))}$
This is the standard RSF model (Manly et al., 2002) used to model telemetry
data. The parameter $\alpha_{2}$ is the effect of the landscape covariate
$z({\bf x})$ on the relative probability of use. Thus, if $\alpha_{2}$ is
positive, the relative probability of use increases as the value of the
covariate increases. In practice, we don’t get to observe $\\{n({\bf x})\\}$
for all individuals but, instead, only for a small subset say
$i=1,2,\ldots,N_{tel}$, which we capture and install telemetry devices on. For
the telemetered individuals, we assume they behave according to the same RSF
model as the population as a whole, which might be justified if individuals
are randomly sampled from the population.
We extend this model slightly to make it more realistic spatially and also
consistent with standard SCR models. Let ${\bf s}$ denote the centroid of an
individuals home range and let $d({\bf s},{\bf x})=||{\bf x}-{\bf s}||$ be the
distance from the home range center ${\bf s}$ of some individual to pixel
${\bf x}$, and let $n({\bf x},{\bf s})$ denote the use frequency of pixel
${\bf x}$ for an individual with activity center ${\bf s}$. We modify the
space usage model to accommodate that space use will be concentrated around an
individual’s home range center (Johnson et al., 2008; Forester et al., 2009):
$\pi({\bf x}|{\bf s})=\frac{\exp(-\alpha_{1}d({\bf x},{\bf
s})^{2}+\alpha_{2}z({\bf x}))}{\sum_{x}\exp(-\alpha_{1}d({\bf x},{\bf
s})^{2}+\alpha_{2}z({\bf x}))}$ (2)
where $\alpha_{1}=1/(2\sigma^{2})$ describes the rate at which encounter
probability declines as a function of distance, $d({\bf x},{\bf s})$. From
ordinary telemetry data, it would be possible to estimate parameters
$\alpha_{1}$, $\alpha_{2}$ and also the activity centers ${\bf s}$ using
standard likelihood methods based on the multinomial likelihood (Johnson et
al., 2008).
Note that Eq. 2 resembles standard encounter models used in spatial capture-
recapture but with an additional covariate $z({\bf x})$. The main difference
between this observation model and the standard SCR model is that the model
here includes the normalizing constant $\sum_{x}\exp(-\alpha_{1}d({\bf x},{\bf
s})^{2}+\alpha_{2}z({\bf x}))$, which ensures that the use distribution is a
proper probability density function. Thus we are able to characterize the
probability of encounter in terms of both distance from activity center and
space use. Note that, under this model for space usage or resource selection,
if there are no covariates, or if $\alpha_{2}=0$, then the probabilities
$\pi({\bf x}|{\bf s})$ are directly proportional to the SCR model for
encounter probability. For example, setting $\alpha_{2}=0$, then this implies
probability of use for pixel ${\bf x}$ is:
$p({\bf x}|{\bf s})\propto\exp(-\alpha_{1}d({\bf x},{\bf s})^{2}).$
Therefore, for whatever model we choose for $p({\bf x},{\bf s})$ in an
ordinary SCR model, we can modify the distance component in the RSF function
in Eq. 2 accordingly to be consistent with that model, by choosing $\pi({\bf
x}|{\bf s})$ according to
$\pi({\bf x}|{\bf s})\propto\exp(log(p({\bf x}|{\bf s}))+\alpha_{2}z({\bf
x}))$
As an illustration of space usage patterns under this model, we simulated a
covariate that represents variation in habitat structure (Fig. 1) such as
might correspond to habitat quality. This was simulated by using a simple
kriging interpolator of spatial noise. Space usage patterns for 8 individuals
in this landscape are shown in Fig. 2, simulated with
$\alpha_{1}=1/(2\sigma^{2})$ with $\sigma=2$ and the coefficient on $z({\bf
x})$ set to $\alpha_{2}=1$. These space usage densities – “home ranges” –
exhibit clear non-stationarity in response to the structure of the underlying
covariate, and they are distinctly asymmetrical. We note that if $\alpha_{2}$
were set to 0, the 8 home ranges shown here would resemble bivariate normal
kernels with $\sigma=2$. Another interesting thing to note is that the
activity centers are not typically located in the pixel of highest use or even
the centroid of usage. That is, the observed “average” location is not an
unbiased estimator of ${\bf s}$ under the model in Eq. 2.
### 3.1 Poisson use model
A natural way to motivate this specific model of space usage is to assume that
individuals make a sequence of random resource selection decisions so that the
outcomes $n({\bf x})$ (for all ${\bf x}$) are marginally independent Poisson
random variables:
$n({\bf x})|{\bf s}\sim\mbox{Poisson}(\lambda({\bf x}|{\bf s}))$
where
$\log(\lambda({\bf x}|{\bf s}))=a_{0}-\alpha_{1}d({\bf x},{\bf
s})^{2}+\alpha_{2}z({\bf x})$
In this case, the number of visits to any particular cell is affected by the
covariate $z({\bf x})$ but has a baseline rate ($\exp(a_{0})$) related to the
amount of movement occurring over some time interval. This is an equivalent
model to the multinomial model given previously in the sense that, if we
condition on the total sample size $R=\sum_{x}n({\bf x})$, then the vector of
use frequencies $\\{n({\bf x})\\}$ for individual with activity center ${\bf
s}$, has a multinomial distribution with probabilities
$\pi({\bf x}|{\bf s})=\frac{\lambda({\bf x}|{\bf s})}{\sum_{x}\lambda({\bf
x}|{\bf s})}$
which is the same as Eq. 2 because $a_{0}$ cancels from the numerator and
denominator of the multinomial cell probabilities and thus this parameter is
not relevant to understanding space usage. Note that if use frequencies are
summarized over $i=1,2,\ldots,N_{tel}$ individuals for each pixel, then a
standard Poisson regression model for the resulting “quadrat counts” is
reasonable. This corresponds to “Design 1” in Manly et al. (2002).
### 3.2 Random Thinning
Suppose our sampling is imperfect so that we only observe a smaller number of
telemetry fixes than actual use frequency, $n({\bf x})$. We express this
“thinning” (or sampling) by assuming the observed number of uses is a binomial
random variable based on a sample of size $n({\bf x})$:
$m({\bf x})\sim\mbox{Bin}(n({\bf x}),\phi_{0}).$
Then, the marginal distribution of the new random variable $m$ is also Poisson
but with mean
$log(\lambda({\bf x}|{\bf s}))=log(\phi_{0})+a_{0}-\alpha_{1}d({\bf x}|{\bf
s})^{2}+\alpha_{2}z({\bf x}).$
Thus, the space-usage model (RSF) for the thinned counts $m$ is the same as
the space-usage model for the original variables $n$. This is because if we
remove $n$ from the conditional model by summing over its possible values,
then the vector of thinned use frequencies ${\bf m}$ (i.e., for all pixels) is
also multinomial with cell probabilities
$\pi({\bf x}|{\bf s})=\frac{\lambda({\bf x}|{\bf s})}{\sum_{x}\lambda({\bf
x}|{\bf s})}$
and so the constants $a_{0}$ and $\phi_{0}$ cancel from both the numerator and
denominator. Thus, the underlying RSF model applies to the true unobserved
count frequencies ${\bf n}$ and also those produced by a random thinning or
sampling process, ${\bf m}$.
In summary, if we conduct a telemetry study of $i=1,2\ldots,N_{tel}$
individuals, the observed data are the $nG\times 1$ vectors of use frequencies
${\bf m}_{i}$ for each individual. We declare these data to be “resource-
selection data” which are typical of the type used to estimate resource-
selection functions (RSFs) (Manly et al., 2002). In fact, the situation we
have described here in which we obtain a random sample of use locations and a
complete census of available locations is referred to as “Design 2” by (Manly
et al., 2002).
### 3.3 Resource Selection in SCR Models
The key to combing RSF data with SCR data is to work with this underlying
resource utilization process and formulate SCR models in terms of that
process. Imagine that we have a sampling device, such as a camera trap, in
every pixel. If the device operates continually then it is no different from a
telemetry instrument. If it operates intermittently or does not expose the
entire area of each pixel then a reasonable model for this imperfect
observation is the “thinned” binomial model given above, where $\phi_{0}$
represents the sampling effectiveness of the device. For data that arise from
SCR studies, the frequency of use for each pixel where a trap is located
serves as an intermediate latent variable that we don’t observe. From a design
standpoint, the main difference between SCR studies and telemetry is that, for
SCR data, we do not have sampling devices in all locations (pixels) in the
landscape. Rather, the data are only recorded at a subsample of them, the trap
locations, which we identify by the specific coordinates ${\bf
x}_{1},\ldots,{\bf x}_{J}$.
So we imagine that the hypothetical perfect data from a camera trapping study
are the counts $m({\bf x})$ only at the specific trap locations ${\bf x}_{j}$,
and for all individuals in the population $i=1,2,\ldots,N$ where $N>N_{tel}$.
We denote the individual- and trap-specific counts by $m_{ij}$ for individual
with activity center ${\bf s}_{i}$ and trap location ${\bf x}_{j}$. In
practice, many (perhaps most) of the $m_{ij}\equiv m({\bf x}_{j},{\bf s}_{i})$
frequencies will be 0, corresponding to individuals not captured in certain
traps. We then construct our SCR encounter probability model based on the view
that these frequencies $m_{ij}$ are latent variables. In particular, under the
SCR model with binary observations, we observe a random variable $y_{ij}=1$ if
the individual $i$ visited the pixel containing trap $j$ and was detected. We
imagine that $y_{ij}$ is related to the latent variable $m_{ij}$ being the
event $m_{ij}>0$, as follows:
$y_{ij}\sim\mbox{Bern}(p_{ij})$
where
$p_{ij}=\Pr(m_{ij}>0)=1-\exp(-\lambda({\bf x}_{j}|{\bf s}_{i}))$
This is the complementary log-log link relating $p_{ij}$ to
$\log(\lambda_{ij})$, setting $\lambda_{ij}\equiv\lambda({\bf x}_{j}|{\bf
s}_{i})$:
$cloglog(p_{ij})=log(\lambda_{ij})$
where
$\log(\lambda_{ij})=\log(\phi_{0})+a_{0}-\alpha_{1}d({\bf x}_{j},{\bf
s}_{i})^{2}+\alpha_{2}z({\bf x}_{j}).$
and we collect the constants so that $\alpha_{0}=log(\phi_{0})+a_{0}$ is the
baseline encounter rate which includes the constant intensity of use by the
individual and also the baseline rate of detection, conditional on use.
## 4 The Joint RSF/SCR Likelihood
To construct the likelihood for SCR data when we have auxiliary covariates on
space usage or direct information on space usage from telemetry data, we
regard the two samples (SCR and RSF) as independent of one another. In
practice, this might not always be the case but (1) the telemetry data often
come from a previous study; (2) Or, the individuals are not the same, or
cannot be reconciled, even if telemetry study occurs simultaneously; (3) In
cases where we can match some individuals between the two samples, regarding
them as independent should only entail a minor loss of efficiency because we
are disregarding more precise information on a small number of activity
centers. Moreover, we believe, it is unlikely in practice to expect the two
samples to be completely reconcilable and that the independence formulation is
the most generally realistic.
Regarding the two data sets as being independent, our approach here is to form
the likelihood for each set of observations as a function of the same
underlying parameters and then combine them. In particular, let ${\cal
L}_{scr}(\alpha_{0},\alpha_{1},\alpha_{2},N;{\bf y}_{scr})$ be the likelihood
for the SCR data in terms of the basic encounter probability parameters and
the total (unknown) population size $N$, and let ${\cal
L}_{rsf}(\alpha_{1},\alpha_{2};{\bf m}_{rsf})$ be the likelihood for the RSF
data based on telemetry which, because the sample size of such individuals is
fixed, does not depend on $N$. Assuming independence of the two datasets, the
joint likelihood is the product of these two pieces:
${\cal L}_{rsf+scr}(\alpha_{0},\alpha_{1},\alpha_{2},N;{\bf y}_{scr},{\bf
m}_{rsf})={\cal L}_{scr}\times{\cal L}_{rsf}$
In what follows, we provide a formulation of each likelihood component. An R
function for obtaining the MLEs of model parameters is given in Appendix 1.
We adopt the notation $f(\cdot)$ to indicate the probability distribution of
whatever observable quantity is in question. e.g., $f(u)$ is the marginal
distribution of $u$ and $f(u|v)$ is the conditional distribution of $u$ given
$v$, etc. We use $g(\cdot)$ to represent the probability distribution of
latent variables. The observation model for the SCR data for individual $i$
and trap $j$, from sampling over $K$ encounter periods, is:
$f(y_{ij}|{\bf s}_{i})=\mbox{Bin}(K,p_{ij}({\bm{\alpha}}))$ (3)
where
$p_{ij}\equiv p(d({\bf x}_{j},{\bf s}_{i}),z({\bf
x}_{j});{\bm{\alpha}})=1-\exp(-\lambda_{ij})$
and
$\lambda_{ij}=\lambda_{0}\exp(-\alpha_{1}d_{ij}^{2}+\alpha_{2}z({\bf x}_{j}))$
We emphasize that this is conditional on the latent variables ${\bf s}_{i}$
(which appear in the distances $d_{ij}$). For these latent variables we adopt
the standard assumption of uniformity, ${\bf s}_{i}\sim\mbox{Unif}({\cal S})$
for each individual $i=1,2,\ldots,N$ (Royle and Young, 2008) where ${\cal S}$
is the state-space of the random variable ${\bf s}$.
The joint distribution of the data for individual $i$, conditional on ${\bf
s}_{i}$, is the product of $J$ binomial terms (i.e., the contributions from
each of $J$ traps):
$f({\bf y}_{i}|{\bf
s}_{i},{\bm{\alpha}})=\prod_{j=1}^{J}\mbox{Bin}(K,p_{ij}({\bm{\alpha}})).$
The marginal likelihood (Borchers and Efford, 2008) is computed by removing
${\bf s}_{i}$, by integration, from the conditional-on-${\bf s}$ likelihood
and regarding the marginal distribution of the data as the likelihood. That
is, we compute:
$f({\bf y}_{i}|{\bm{\alpha}})=\int_{{\cal S}}f({\bf y}_{i}|{\bf
s}_{i},{\bm{\alpha}})g({\bf s}_{i})d{\bf s}_{i}$
where, under the uniformity assumption, we have $g({\bf s})=1/||{\cal S}||$.
The joint likelihood for all $N$ individuals, is the product of $N$ such
terms:
${\cal L}_{scr}({\bm{\alpha}}|{\bf y}_{1},{\bf y}_{2},\ldots,{\bf
y}_{N})=\prod_{i=1}^{N}f({\bf y}_{i}|{\bm{\alpha}})$
In practice, we don’t know $N$ and so we can’t just compute the SCR likelihood
in this manner. Instead, we compute the contributions of the $n$ observed
individuals directly as given above, but then we have to compute the
likelihood contribution for the “all 0” encounter history, i.e., that
corresponding to unobserved individuals. The mechanics of computing that are
the same as for an ordinary observed encounter history, requiring that we
integrate a binomial probability of ${\bf 0}$ over the state-space ${\cal S}$:
$\pi_{0}=\Pr({\bf y}={\bf 0})=\int_{{\cal S}}f({\bf 0}|{\bf
s},{\bm{\alpha}})d{\bf s}.$
We then have to deal with the issue that $n$ itself is a random variable, and
that leads to the combinatorial term in front of the likelihood which involves
the total population size $N$. This produces the conditional-on-$N$ or
“binomial form” of the likelihood (Borchers and Efford, 2008; Royle, 2009):
$\frac{N!}{n!(N-n)!}\left\\{\prod_{i=1}^{n}f({\bf
y}_{i}|{\bm{\alpha}})\right\\}\pi_{0}^{N-n}$
For the RSF data from the sample of individuals with telemetry devices we
adopt the same basic strategy of describing the conditional-on-${\bf s}$
likelihood and then computing the marginal likelihood by averaging over
possible values of ${\bf s}$. We have ${\bf m}_{i}$, the $nG\times 1$ vector
of pixel counts for individual $i$, where these counts are derived from a
telemetry study or similar. We index these elements as $m_{ig}$ for individual
$i$ and grid cell $g$, noting that our index $j$ is reserved only for trap
locations, which are a subset of the $nG$ coordinates ${\bf x}_{1},\ldots,{\bf
x}_{nG}$. The conditional-on-${\bf s}_{i}$ distribution of the telemetry data
from individual $i$ is, omitting the multinomial combinatorial term which does
not depend on parameters,
$f({\bf m}_{i}|{\bf s}_{i},{\bm{\alpha}})\propto\prod_{g=1}^{nG}\pi({\bf
x}_{g}|{\bf s}_{i})^{m_{ig}}$
where
$\pi({\bf x}_{g}|{\bf
s}_{i})=\frac{\exp(-\alpha_{1}d_{ig}^{2}+\alpha_{2}z({\bf
x}_{g}))}{\sum_{g}\exp(-\alpha_{1}d_{ig}^{2}+\alpha_{2}z({\bf x}_{g}))}$
The marginal distribution is
$f({\bf m}_{i}|{\bm{\alpha}})=\int_{{\cal S}}f({\bf m}_{i}|{\bf
s}_{i},{\bm{\alpha}})g({\bf s}_{i})d{\bf s}_{i}$
and therefore the likelihood for the RSF data is
${\cal L}_{rsf}({\bm{\alpha}}|{\bf m}_{1},{\bf m}_{2},\ldots,{\bf
m}_{Ntel})=\prod_{i=1}^{Ntel}f({\bf m}_{i}|{\bm{\alpha}}).$
A key technical aspect of computing these likelihoods is the evaluation of the
2-dimensional integral over the state-space ${\cal S}$, which we approximate
(Appendix 1) by a summation over a fine mesh of points. We note also that the
binomial form of the likelihood here is expressed in terms of the parameter
$N$, the population size for the landscape defined by ${\cal S}$. Given ${\cal
S}$, density is computed as $D({\cal S})=N/\mbox{area}({\cal S})$. In our
simulation study below we report $N$ as the two are equivalent summaries of
the data once ${\cal S}$ is defined. Borchers and Efford (2008) develop a
likelihood based on a further level of marginalization, in which $N$ is
removed from the likelihood by averaging over a Poisson prior for $N$.
## 5 Simulation Analysis
We carried-out a simulation study using the landscape shown in Fig. 1, and
based on populations of size $N=100$ and $N=200$ individuals with activity
centers distributed uniformly over the landscape. This covariate was simulated
by generating a field of spatially correlated noise to emulate a typical
patchy habitat covariate relevant to habitat quality for a species. We
subjected individuals to sampling over $K=10$ sampling periods, using a
$7\times 7$ array of trapping devices located on the the integer coordinates
$(u*5,v*5)$ for $u,v=1,2,3,4,5,6,7$. The SCR encounter model was of the form
$\mbox{
cloglog}(p_{ij})=\alpha_{0}-\frac{1}{2\sigma^{2}}d_{ij}^{2}+\alpha_{2}z({\bf
x}_{j})$
with $\alpha_{0}=-2$, $\sigma=2$ and $\alpha_{2}=1$. In the absence of the
covariate $z$, this corresponds to a RSF that is bivariate normal with
standard deviation 2. These settings yielded an average of about $n=61$
individuals captured for the $N=100$ case and about $n=123$ for the $N=200$
case. The latter case represents what we believe is an extremely large sample
size based on our own experience and thus it should serve to gauge the large
sample bias of the likelihood estimator.
In addition to simulating data from this capture-recapture study, we simulated
2, 4, 8, 12, 16 telemetered individuals to assess the improvement in precision
as sample size increases. For all cases we observed 20 telemetry fixes per
individual, assuming individuals were using space according to a RSF model
with the same parameters as those generating the SCR data. We simulated 500
data sets for each scenario and, for each data set, we fit 3 models: (i) the
SCR only model, in which the telemetry data were not used; (ii) the integrated
SCR/RSF model which combined all of the data for jointly estimating model
parameters; and (iii) the RSF only model which just used the telemetry data
alone (and therefore $\alpha_{0}$ and $N$ are not estimable parameters). The
focus of the simulations was to address the following basic questions: (1) how
much does the root mean-squared error (RMSE) of $\hat{N}$ improve as we add or
increase the number of telemetered individuals? (2) How well does the SCR
model do at estimating the parameter of the RSF with no telemetry data? (3)
How much does the precision of the RSF parameter improve if we add SCR data to
the telemetry data?
Results for $N=100$, $N=200$ and $N_{tel}=(2,4,8,12,16)$ are presented in
Table 1. We note that the first row of each batch (labeled “SCR only”)
represent the same estimator and data configuration. These replicate runs of
the SCR-only situation give us an idea of the inherent MC error in these
simulations, which is roughly about 0.25 and 0.89 on the $N$ scale for the
$N=100$ and $N=200$ cases, respectively. The mean $N$ for the SCR-only
estimator across all 5 simulations for $N=100$ was
$\mbox{mean}(\hat{N})=99.418$, an empirical bias of $0.6\%$. For $N=200$, the
estimated $N$ across all 5 simulations (5 levels of $N_{tel}$) was
$\mbox{mean}(\hat{N})=199.712$, an empirical bias of about $0.15\%$, within
the MC error of the true value of $N=200$. The results suggests a very small
bias of $<1\%$ in the MLE of $N$ for both the SCR-only and combined SCR/RSF
estimators. In practice, we expect a small amount of bias in MLEs as
likelihood theory only guarantees asymptotic unbiasedness.
In terms of RMSE for estimating $N$, we see that (Table 1), generally, there
is about a 5% reduction in RMSE when we have at least 2 telemetered
individuals. And, although there is a lot of MC error in the RMSE quantities,
it might be as much as a 10% reduction as the sample size of captured
individuals increases under the higher $N=200$ setting. This incremental
improvement in RMSE of $\hat{N}$ makes sense because, while the telemetry
provides considerable information about the structural parameters of the
model, it provides no information about mean $p$, i.e. $\alpha_{0}$, which
comes only from the SCR data. Thus estimating $N$ benefits only slightly from
the addition of telemetry data.
The MLE of the RSF parameter $\alpha_{2}$ exhibits negligible or no bias under
both the SCR only and SCR/RSF estimators. It is well-estimated from SCR data
alone and even better than RSF data alone (in terms of RMSE) until we have
more than 200 or so telemetry observations. The biggest improvement from the
use of telemetry data comes in estimating the parameter $\sigma$. We see that
$\hat{\sigma}$ is effectively unbiased, and there is a very large improvement
in RMSE of $\hat{\sigma}$, perhaps as much as 50-60% in some cases, when the
telemetry data are used in the combined estimator (that really doesn’t
translate much into improvements in estimating $N$ as we saw previously).
Improvement due to adding telemetry data diminishes as the expected sample
sizes increases, and so telemetry data does less to improve the precision of
$\hat{\sigma}$ and $\hat{\alpha}_{2}$ for $N=200$ than for $N=100$. This is
because the SCR data along are informative about both of those parameters.
The results as they concern likelihood estimation of $N$ suggest that there is
not a substantial benefit to having telemetry data. Estimators “SCR only” and
“SCR/RSF” both appear approximately unbiased for $N=100$ and $N=200$, and for
any sample size of telemetered individuals. The RMSE is only 5-10% improved
with the addition of telemetry information. However, we find that there is
substantial bias in $\hat{N}$ if we use the misspecified model that contains
no resource selection component. That is if we leave the covariate $z({\bf
x})$ out of the model and incorrectly fit a model with symmetric and spatially
constant encounter model, we see about 20% bias in the estimates of $N$ in a
limited simulation study that we carried-out (Tab. 2). As such, accounting for
resource selection is important, even though, when accounted for, telemetry
data only improves the estimator incrementally. In addition, we find that the
importance of telemetry data is relatively more important for smaller sample
sizes. We carried-out one simulation study for the $N=100$ case but with lower
average encounter probabilities, setting $\alpha_{0}=-3$. This produces
relatively smaller data sets with $E[n]=37$. The results are shown in Tab. 3.
There are some important features evident from this table. First, as a result
of the small samples, the MLE of $N$ is biased for both SCR only and SCR/RSF
estimators although less biased for the SCR/RSF estimator than for SCR only.
The persistent bias in $\hat{N}$ for both models results from the information
about $\alpha_{0}$ coming only from SCR data, and that estimator itself is
intrinsically biased in small samples. Conversely, the estimator of
$\alpha_{2}$, the RSF parameter, appears unbiased for all 3 estimators (SCR
only, SCR/RSF and RSF only), as does the estimator of $\sigma$. We see
relatively larger improvements in RMSE (compared with Tab. 1) of $\hat{N}$,
and those improvements increase substantially as $N_{tel}$ increases.
## 6 Discussion
How animals use space is a fundamental interest to ecologists, and important
in the conservation and management of many species. Normally this is done by
telemetry and models referred to as resource selection functions (Manly et
al., 2002). Conversely, spatial capture-recapture models have grown in
popularity over the last several years (Efford, 2004; Borchers and Efford,
2008; Royle, 2008; Efford et al., 2009; Royle et al., 2009; Gardner et al.,
2010a, b; Kéry et al., 2010; Sollmann et al., 2011; Mollet et al., 2012;
Gopalaswamy et al., 2012). These, and indeed, most, development and
applications of SCR models have focused on density estimation, not
understanding space usage. However, it is intuitive that space usage should
affect encounter probability and thus it should be highly relevant to density
estimation in SCR applications. Despite this, a description of the
relationship between encounter probability and space usage has not been
developed in the literature on spatial capture-recapture models. Essentially
all published applications of SCR models to date have been based on simplistic
encounter probability models that are symmetric and do not vary across space.
One exception is Royle et al. (2012) who developed SCR models that use
ecological distance metrics (“least-cost path”) instead of normal Euclidean
distance. Here we developed an SCR model in terms of a basic underlying model
of space or resource use, that is consistent with existing views of resource
selection functions (RSFs) (Manly et al., 2002).
In developing the SCR model in terms of an underlying model of space usage, we
achieve a number of enormously useful extensions of existing SCR and RSF
methods: (1) We have shown how to integrate classical RSF data from telemetry
with spatial capture-recapture data based on individual encounter histories
obtained by classical arrays of encounter devices or traps. This leads to an
improvement in our ability to estimate density, and also an improvement in our
ability to estimate parameters of the RSF function. Thus, the combined model
is both an extension of standard SCR models and also and extension of standard
RSF models. As many animal population studies have auxiliary telemetry
information, the ability to incorporate such information into SCR studies has
enormous applicability and immediate benefits in many studies. While adding
RSF data to SCR data may increase precision of the MLE of $N$ only
incrementally, the effect can be more substantial in sparse data sets and,
generally, RSF produces relatively huge gains in precision in the MLE of
$\sigma$. (2) We have shown that one can estimate RSF model parameters
directly from SCR data alone. While further exploration of this point is
necessary, it does establish clearly that SCR models are explicit models of
space usage. Because capture-recapture studies are, arguably, more widespread
than telemetry studies alone, this greatly broadens the utility and importance
of data from those studies. (3) It is also now clear that one of the important
parameters of SCR models, that controlling “home range radius”, can be
directly estimated from telemetry data alone. The combined RSF+SCR model does
yield large improvements in estimation of $\sigma$. As a practical matter,
this suggests we could estimate $\sigma$ entirely from data extrinsic to the
SCR study which might provide great freedom in the design of SCR studies. For
example, traps could be spaced far enough apart to generate relatively few
(even no) spatial recaptures, but dramatically increase the coverage of the
population, i.e., the observed sample size of captured individuals relative to
$N$. (4) Finally, we found that an ordinary SCR model with symmetric encounter
probability model produces extremely biased estimates of $N$ when the
population of individuals does exhibit resource selection. As such, it is
important to account for space usage when important covariates are known to
influence space usage patterns.
Use of telemetry data in capture-recapture studies has been suggested
previously. For example, White and Shenk (2001) and Ivan (2012) suggested
using telemetry data to estimate the quantity “probability that an individual
is exposed to sampling” but their estimator requires that individuals are
sampled in proportion to this unknown quantity, which seems impossible to
achieve in many studies. In addition, they do not directly integrate the
telemetry data with the capture-recapture model so that common parameters are
jointly estimated. In fact, they don’t acknowledge shared parameters of the
two models. Sollmann et al. (2012) did recognize this, and used some telemetry
data to estimate directly the parameter $\sigma$ from the bivariate normal SCR
model in order to improve estimates of density. This was an important
conceptual development in the sense that it recognized the relationship
between SCR models and models of space usage, but their model did not include
an explicit resource selection component, and they did not implement a joint
estimation framework.
We developed a formal analysis framework here based on marginal likelihood
(Borchers and Efford, 2008). In principle, Bayesian analysis does not pose any
unique challenges for this new class of models although we expect some loss of
computational efficiency due to the increased number of times the components
of the likelihood would need to be evaluated. We imagine that some problems
would benefit from a Bayesian formulation, however. For example, using an open
population model that allows for recruitment and survival over time (Gardner
et al., 2010a) is convenient to develop in the BUGS language and incorporating
information on unmarked individuals has been done using Bayesian formulations
of SCR models (Chandler and Royle, 2012; Sollmann et al., 2012) but, so far,
not likelihood methods.
In our formulation of the joint likelihood for RSF and SCR data, we assumed
the data from capture-recapture and telemetry studies were independent of one
another. This implies that whether or not an individual enters into one of the
data sets has no effect on whether it enters into the other data set. We
cannot foresee situations in which violation of this assumption should be
problematic or invalidate the estimator under the independence assumption. In
some cases it might so happen that some individuals appear in both the RSF and
SCR data sets. In this case, ignoring that information should entail only an
incremental decrease in precision because a slight bit of information about an
individuals activity center is disregarded. Heuristically, an SCR observation
(encounter in a trap) is like one additional telemetry observation, and so the
misspecification (independence) regards the two pieces of information as
having separate activity centers. Our model pretends that we don’t know
anything about the telemetered individuals in terms of their encounter history
in traps. In principle it shouldn’t be difficult to admit a formal
reconciliation of individuals between the two lists. In that case, we just
combine the two conditional likelihoods before we integrate ${\bf s}$ from the
conditional likelihood. This would be almost trivial to do if all individuals
were reconcilable (or none as in the case we have covered here) but, in
general , we think you will always have an intermediate case – i.e., either
none will be or at most a subset of telemetered individuals will be known.
More likely you have variations of “well, that guy looks telemetered but we
don’t know which guy it is….hmmm” and that case, basically a type of marking
uncertainty or misclassification, is clearly more difficult to deal with.
We conclude that the key benefit of our combined SCR/RSF model is its ability
integrate realistic patterns of space usage directly into SCR models and avoid
extreme bias in estimating $N$ and, secondarily, we are able to obtain RSF
information from SCR alone. Therefore, our new class of integrated SCR/RSF
models allows investigators to model how the landscape and habitat influence
movement and space usage of individuals around their home range, using non-
invasively collected capture-recapture data or capture-recapture data
augmented with telemetry data. This should improve our ability to understand,
and study, aspects of space usage and it might, ultimately, aid in addressing
conservation-related problems such as reserve or corridor design. And, it
should greatly expand the relevance and utility of spatial capture-recapture
beyond simply its use for density estimation.
## Acknowledgments
## References
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## Appendix 1: R script for obtaining MLEs under the SCR+RSF model
### before running this code, put the functions at the end of this script
### into your R workspace
###
## the following block of code makes up a covariate as a spatially correlated
## noise field, with an exponential spatial correlation function
set.seed(1234)
gr<-expand.grid(1:40,1:40)
Dmat<-as.matrix(dist(gr))
V<-exp(-Dmat/5)
z<-t(chol(V))%*%rnorm(1600)
spatial.plot(gr,z)
###
### Set some parameter values
###
alpha0 <- -2
sigma<- 2
beta<- 1
Ntel<-4 # number of individuals with telemeters
nsim<-100
Nfixes<-20 # number of telemetry fixes per individual
N<- 100 # population size
# simulate activity centers of all N individuals
Sid<- sample(1:1600,N,replace=TRUE)
# and coordinates
S<-gr[Sid,]
# now draw centers of telemetered individuals
# have to draw telemetry guys interior or else make up more landscape --
# can’t have truncated telemetry obs
poss.tel<- S[,1]>5 & S[,1]<35 & S[,2]>5 & S[,2]<35
tel.guys<-sample(Sid[poss.tel],Ntel)
sid<-tel.guys
stel<-gr[sid,]
# make up matrix to store RSF data
n<-matrix(NA,nrow=Ntel,ncol=1600)
# for each telemetered guy simulate a number of fixes.
# note that n = 0 for most of the landscape
par(mfrow=c(3,3))
lammat<-matrix(NA,nrow=Ntel,ncol=1600)
for(i in 1:Ntel){
d<- Dmat[sid[i],]
lam<- exp(1 - (1/(2*sigma*sigma))*d*d + beta* z)
n[i,]<-rmultinom(1,Nfixes,lam/sum(lam))
par(mar=c(3,3,3,6))
lammat[i,]<-lam
img<- matrix(lam,nrow=40,ncol=40,byrow=FALSE)
image(1:40,1:40,rot(img),col=terrain.colors(10))
}
## now lets simulate some SCR data on a bunch of guys:
# make a trap array
X<- cbind( sort(rep( seq(5,35,5),7)), rep( seq(5,35,5),7))
ntraps<-nrow(X)
raster.point<-rep(NA,nrow(X))
for(j in 1:nrow(X)){ # which piont in the raster is the trap? must be raster points
raster.point[j]<- (1:1600)[ (X[j,1]==gr[,1]) & (X[j,2] == gr[,2])]
}
points(X,pch=20,cex=2)
D<- e2dist(S,X) ## N x ntraps
Zmat<- matrix(z[raster.point],nrow=N,ncol=ntraps,byrow=TRUE) # note make dims the same
loglam<- alpha0 -(1/(2*sigma*sigma))*D*D + beta*Zmat
p<- 1-exp(-exp(loglam))
## Now simulate SCR data
K<- 10
y<-matrix(NA,nrow=N,ncol=ntraps)
for(i in 1:N){
y[i,]<- rbinom(ntraps,K,p[i,])
}
cap<-apply(y,1,sum)>0
y<-y[cap,]
gr<-as.matrix(gr)
sbar<- (n%*%gr)/as.vector(n%*%rep(1,nrow(gr)))
# Basic SCR model with RSF covariate at trap locations.
tmp1<-nlm(intlik3rsf.v2,c(-3,log(3),1,0),y=y,K=K,X=X,ztrap=z[raster.point],G=gr)
# use telemetry data and activity centers for those are marginalized out of the likelihood
tmp2<-nlm(intlik3rsf.v2,c(-3,log(3),1,0),y=y,K=K,X=X,ztrap=z[raster.point],G=gr,ntel=n,zall=as.vector(z))
# use mean "s" instead of estimating it
tmp3<-nlm(intlik3rsf.v2,c(-3,log(3),1,0),y=y,K=K,X=X,ztrap=z[raster.point],G=gr,ntel=n,zall=as.vector(z),stel=sbar)
# no SCR data, s is random. Here there are 2 extra parameters that are not estimated: start[1] and start[4]
tmp4<-nlm(intlik3rsf.v2,c(-3,log(3),1,0),y=NULL,K=K,X=X,ztrap=z[raster.point],G=gr,ntel=n,zall=as.vector(z))
# Fits SCR model with isotropic Gaussian encounter model
tmp5<- nlm(intlik3rsf.v2,c(-3,log(3),1,0),y=y,K=K,X=X,ztrap=rep(0,ntraps),G=gr)
###
### put all the functions below this line into your R workspace
###
spatial.plot<-
function(x,y){
nc<-as.numeric(cut(y,20))
plot(x,pch=" ")
points(x,pch=20,col=topo.colors(20)[nc],cex=2)
###image.scale(y,col=topo.colors(20))
}
### This is the likelihood function
### It computes several versions of the likelihood depending on the arguments specified
### see the 5 examples above
intlik3rsf.v2 <-function(start=NULL,y=y,K=NULL,X=traplocs,ztrap,G,ntel=NULL,zall=NULL,stel=NULL){
# start = vector of length 5 = starting values
# y = nind x ntraps encounter matrix
# K = how many samples?
# X = trap locations
# ztrap = covariate value at trap locations
# zall = all covariate values for all nG pixels
# ntel = nguys x nG matrix of telemetry fixes in each nG pixels
# stel = home range center of telemetered individuals, IF you wish to estimate it. Not necessary
nG<-nrow(G)
D<- e2dist(X,G)
alpha0<-start[1]
sigma<- exp(start[2])
alpha2<- start[3]
n0<- exp(start[4])
a0<- 1
if(!is.null(y)){
loglam<- alpha0 -(1/(2*sigma*sigma))*D*D + alpha2*ztrap # ztrap recycled over nG
probcap<- 1-exp(-exp(loglam))
#probcap<- (exp(theta0)/(1+exp(theta0)))*exp(-theta1*D*D)
Pm<-matrix(NA,nrow=nrow(probcap),ncol=ncol(probcap))
ymat<-y
ymat<-rbind(y,rep(0,ncol(y)))
lik.marg<-rep(NA,nrow(ymat))
for(i in 1:nrow(ymat)){
Pm[1:length(Pm)]<- (dbinom(rep(ymat[i,],nG),rep(K,nG),probcap[1:length(Pm)],log=TRUE))
lik.cond<- exp(colSums(Pm))
lik.marg[i]<- sum( lik.cond*(1/nG) )
}
nv<-c(rep(1,length(lik.marg)-1),n0)
part1<- lgamma(nrow(y)+n0+1) - lgamma(n0+1)
part2<- sum(nv*log(lik.marg))
out<- -1*(part1+ part2)
}
else{
out<-0
}
if(!is.null(ntel) & !is.null(stel) ){
# this is a tough calculation here
D2<- e2dist(stel,G)^2
# lam is now nG x nG!
lam<- t(exp(a0 - (1/(2*sigma*sigma))*t(D2)+ alpha2*zall)) # recycle zall over all ntel guys
denom<-rowSums(lam)
probs<- lam/denom # each column is the probs for a guy at column [j]
tel.loglik<- -1*sum( ntel*log(probs) )
out<- out + tel.loglik
}
if(!is.null(ntel) & is.null(stel) ){
# this is a tough calculation here
D2<- e2dist(G,G)^2
# lam is now nG x nG!
lam<- t(exp(a0 - (1/(2*sigma*sigma))*t(D2)+ alpha2*zall)) # recycle zall over all ntel guys
denom<-rowSums(lam)
probs<- t(lam/denom) # each column is the probs for a guy at column [j]
marg<- as.vector(rowSums(exp(ntel%*%log(probs))/nG ))
tel.loglik<- -1*sum(log(marg))
out<- out + tel.loglik
}
out
}
Table 1: Mean and RMSE of sampling distribution of the MLE of $N$ and other model parameters under a model of resource selection using only SCR data, SCR combined with RSF data on $N_{tel}$ individuals, and with RSF only data on $N_{tel}$ individuals. Simulations results are based on 500 Monte Carlo simulations of populations containing $N=100$ or $N=200$ individuals. The true parameter values were $\alpha_{2}=1$ and $\sigma=2$. Estimator | N=100 | N=200
---|---|---
$N_{tel}=2$ | $\hat{N}$ | RMSE | $\hat{\alpha}_{2}$ | RMSE | $\hat{\sigma}$ | RMSE | $\hat{N}$ | RMSE | $\hat{\alpha}_{2}$ | RMSE | $\hat{\sigma}$ | RMSE
SCR only: | 99.73 | 9.97 | 0.99 | 0.14 | 2.00 | 0.124 | 198.85 | 14.24 | 0.99 | 0.10 | 2.00 | 0.091
SCR/RSF: | 99.94 | 9.54 | 0.99 | 0.12 | 2.00 | 0.097 | 199.37 | 12.80 | 0.99 | 0.09 | 2.00 | 0.078
RSF only | – | – | 1.03 | 0.33 | 2.00 | 0.160 | – | – | 1.04 | 0.33 | 1.99 | 0.169
$N_{tel}=4$ | | | | | | | | | | | |
SCR only | 99.10 | 9.83 | 0.99 | 0.13 | 2.00 | 0.127 | 200.06 | 15.34 | 1.00 | 0.09 | 2.00 | 0.092
SCR/RSF | 99.17 | 9.47 | 0.99 | 0.11 | 2.00 | 0.086 | 200.25 | 14.36 | 1.00 | 0.08 | 2.01 | 0.073
RSF only | – | – | 0.98 | 0.22 | 2.00 | 0.119 | – | – | 1.02 | 0.21 | 2.01 | 0.122
$N_{tel}=8$ | | | | | | | | | | | |
SCR only | 99.59 | 10.00 | 1.00 | 0.13 | 2.00 | 0.130 | 200.85 | 14.06 | 1.00 | 0.09 | 2.00 | 0.087
SCR/RSF | 98.90 | 10.02 | 0.99 | 0.10 | 2.00 | 0.071 | 200.29 | 13.98 | 1.00 | 0.08 | 2.00 | 0.061
RSF only | – | – | 0.98 | 0.16 | 2.01 | 0.084 | – | – | 0.99 | 0.16 | 2.00 | 0.084
$N_{tel}=12$ | | | | | | | | | | | |
SCR only | 99.44 | 10.73 | 0.98 | 0.13 | 2.02 | 0.128 | 198.76 | 14.47 | 0.99 | 0.10 | 2.00 | 0.091
SCR/RSF | 99.96 | 10.26 | 1.00 | 0.09 | 2.00 | 0.059 | 198.72 | 14.14 | 1.00 | 0.08 | 2.00 | 0.054
RSF only | – | – | 1.01 | 0.12 | 2.00 | 0.069 | – | – | 1.01 | 0.13 | 2.00 | 0.069
$N_{tel}=16$ | | | | | | | | | | | |
SCR only | 99.23 | 10.74 | 0.99 | 0.14 | 2.00 | 0.128 | 200.04 | 14.09 | 0.99 | 0.10 | 2.01 | 0.088
SCR/RSF | 99.20 | 9.79 | 1.00 | 0.09 | 1.99 | 0.057 | 200.25 | 13.40 | 1.00 | 0.07 | 2.00 | 0.047
RSF only | – | – | 1.00 | 0.10 | 1.99 | 0.061 | – | – | 1.00 | 0.11 | 2.00 | 0.055
Table 2: Expected value of $\hat{N}$ and $\hat{\sigma}$ for truth $N=200$ and $\sigma=2$ under a model of resource selection with a single covariate, when the encounter probability model is misspecified by a symmetric and constant model assuming no resource selection; column “bias” is percent bias. | $E[\hat{N}]$ | bias | RMSE | $E[\hat{\sigma}]$ | RMSE
---|---|---|---|---|---
n=2 | 161.48 | -19.2 | 39.98 | 1.84 | 0.180
n=4 | 161.32 | -19.3 | 40.00 | 1.83 | 0.191
n=8 | 161.46 | -19.3 | 40.06 | 1.84 | 0.184
n=12 | 162.40 | -18.8 | 38.95 | 1.84 | 0.185
n=16 | 160.93 | -19.5 | 40.44 | 1.84 | 0.190
Table 3: Mean and RMSE of the sampling distribution of the MLE for model parameters for the $N=100$ and “low $p$” case. For each of 500 simulated data sets, a model was fit using the SCR likelihood only, the joint SCR/RSF likelihood, and the RSF likelihood only. For the latter, the parameter $N$ is not statistically identifiable. Estimator | $E[\hat{N}]$ | RMSE | $E[\hat{\alpha}_{2}]$ | RMSE | $E[\hat{\sigma}]$ | RMSE
---|---|---|---|---|---|---
$N_{tel}=2$ | | | | | |
SCR only | 103.85 | 22.88 | 1.00 | 0.19 | 2.02 | 0.261
SCR/RSF | 102.90 | 20.98 | 1.00 | 0.17 | 2.00 | 0.136
RSF only | – | – | 1.02 | 0.30 | 1.99 | 0.163
$N_{tel}=4$ | | | | | |
SCR only | 105.65 | 26.52 | 1.01 | 0.20 | 2.01 | 0.258
SCR/RSF | 103.55 | 22.92 | 1.01 | 0.14 | 2.00 | 0.104
RSF only | – | – | 1.01 | 0.21 | 1.99 | 0.114
$N_{tel}=8$ | | | | | |
SCR only | 107.41 | 45.05 | 0.99 | 0.19 | 2.01 | 0.254
SCR/RSF | 104.28 | 22.13 | 1.00 | 0.12 | 2.00 | 0.076
RSF only | – | – | 1.01 | 0.15 | 1.99 | 0.081
$N_{tel}=12$ | | | | | |
SCR only | 106.35 | 27.32 | 0.99 | 0.19 | 2.00 | 0.255
SCR/RSF | 104.11 | 21.81 | 1.00 | 0.10 | 2.00 | 0.063
RSF only | – | – | 1.01 | 0.12 | 2.00 | 0.065
$N_{tel}=16$ | | | | | |
SCR only | 104.05 | 31.41 | 0.99 | 0.19 | 2.02 | 0.252
SCR/RSF | 101.98 | 20.78 | 1.00 | 0.09 | 2.00 | 0.055
RSF only | – | – | 1.00 | 0.10 | 2.00 | 0.056
FIGURE CAPTIONS
Figure 1:
A typical habitat covariate reflecting habitat quality or hypothetical utility
of the landscape to a species under study. Home range centers for 8
individuals are shown with black dots.
Figure 2:
Space usage patterns of 8 individuals under a space usage model that contains
a single covariate (shown in Fig. 1). Plotted value is the multinomial
probability $\pi_{ij}$ for pixel $j$ under the model in Eq. 2.
Figure 1: A typical habitat covariate reflecting habitat quality or
hypothetical utility of the landscape to a species under study. Home range
centers for 8 individuals are shown with black dots. Figure 2: Space usage
patterns of 8 individuals under a space usage model that contains a single
covariate (shown in Fig. 1). Plotted value is the multinomial probability
$\pi_{ij}$ for pixel $j$ under the model in Eq. 2.
|
arxiv-papers
| 2012-07-13T16:05:40 |
2024-09-04T02:49:33.031955
|
{
"license": "Public Domain",
"authors": "J. Andrew Royle and Richard B. Chandler",
"submitter": "Andy Royle",
"url": "https://arxiv.org/abs/1207.3288"
}
|
1207.3326
|
# Measuring gravitational lensing of the cosmic microwave background
using cross correlation with large scale structure
Chang Feng Center for Astrophysics and Space Sciences and the Ax Center for
Experimental Cosmology, University of California San Diego, La Jolla, CA 92093
Grigor Aslanyan Department of Physics, University of California at San Diego,
La Jolla, CA 92093 Aneesh V. Manohar Department of Physics, University of
California at San Diego, La Jolla, CA 92093 Brian Keating Center for
Astrophysics and Space Sciences and the Ax Center for Experimental Cosmology,
University of California San Diego, La Jolla, CA 92093 Department of Physics,
University of California at San Diego, La Jolla, CA 92093 Hans P. Paar
Center for Astrophysics and Space Sciences and the Ax Center for Experimental
Cosmology, University of California San Diego, La Jolla, CA 92093 Department
of Physics, University of California at San Diego, La Jolla, CA 92093 Oliver
Zahn Berkeley Center for Cosmological Physics and Lawrence Berkeley
Laboratory, University of California, Berkeley, CA 94720
###### Abstract
We cross correlate the gravitational lensing map extracted from cosmic
microwave background measurements by the Wilkinson Microwave Anisotropy Probe
(WMAP) with the radio galaxy distribution from the NRAO VLA Sky Survey (NVSS)
by using a quadratic estimator technique. We use the full covariance matrix to
filter the data, and calculate the cross-power spectra for the lensing-galaxy
correlation. We explore the impact of changing the values of cosmological
parameters on the lensing reconstruction, and obtain statistical detection
significances at $>3\sigma$. The results of all cross correlations pass the
curl null test as well as a complementary diagnostic test using the NVSS data
in equatorial coordinates. We forecast the potential for Planck and NVSS to
constrain the lensing-galaxy cross correlation as well as the galaxy bias. The
lensing-galaxy cross-power spectra are found to be Gaussian distributed.
###### pacs:
98.70.Vc, 98.62.Sb, 98.80.Es
## I introduction
The cosmic microwave background (CMB) temperature anisotropy contains a wealth
of cosmological information and has played a pivotal role in our understanding
of the Universe. Besides the primordial fluctuations, various secondary
anisotropies, e.g. gravitational lensing, the thermal Sunyaev-Zel’dovich
effect, the kinetic Sunyaev-Zel’dovich effect, as well as the integrated
Sachs-Wolfe effect, are playing an increasingly important role in constraining
cosmological constituents and dynamics.
Among the secondary effects imprinted on the CMB gravitational lensing is of
great importance. The projected gravitational lensing potential is a line-of-
sight probe which contains information about the geometric distance traversed
by CMB photons and time-dependent gravitational potentials. As such it is very
sensitive to late universe parameters, such as the sum of neutrino masses, the
dark energy equation of state and spatial curvature. Since the projected
gravitational lensing potential contains both geometric and structure growth
information, it effectively breaks the angular diameter distance degeneracy
Smith _et al._ (2009a). Gravitational lensing measurements can also be used
to de-lens the $B$-mode polarization of the CMB Smith _et al._ (2010),
enabling us to learn about primordial gravitational waves Kamionkowski _et
al._ (1997) and the energy scale of inflation.
Tentative CMB weak lensing searches have been done with WMAP-7 year data sets
Smidt _et al._ (2011); Feng _et al._ (2012) using non-Gaussian statistics.
However, WMAP-7 alone cannot detect weak lensing of the CMB because WMAP
temperature maps have insufficient sensitivity Feng _et al._ (2012).
Recently, the Atacama Cosmology Telescope Das _et al._ (2011a) and South Pole
Telescope (SPT) van Engelen _et al._ (2012) have performed the first internal
lensing reconstruction detections using non-Gaussianity. In addition, Atacama
Cosmology Telescope and SPT also measured the gravitational lensing signal
from the smoothing effects of the acoustic peaks on the CMB temperature power
spectrum Reichardt _et al._ (2009); Das _et al._ (2011b); Keisler _et al._
(2011). As the experimental sensitivity improves, internal measurements,
either from the power spectrum or the trispectrum, will become more precise in
the near future.
The correlation between lensing and large scale structure arises from large
scale structure, which deflects CMB photons in the late universe. The signal-
to-noise ratio of lensing measurements can be enhanced if the CMB maps are
cross correlated with highly sensitive large scale structure tracers, such as
luminous red galaxies (LRGs) (which cover the redshift range $0.2<z<0.7$),
quasars (which covers the redshift range $z<2.7$) from the Sloan Digital Sky
Survey (SDSS), or the NVSS of radio galaxies which has a higher mean redshift
($z\sim 1$) than the LRGs and quasars. Hirata et al. Hirata _et al._ (2004)
used the cross correlation between WMAP-1 and LRGs and quasars from SDSS
imaging and found no statistically significant signal. Then Smith et al. Smith
_et al._ (2007) used the cross correlation between WMAP-3 and NVSS, and found
a $3.4\sigma$ signal, including systematics. Using a slightly less optimal
estimator than Ref. Smith _et al._ (2007), Hirata et al. Hirata _et al._
(2008) obtained results consistent with, though at slightly lower significance
than, Ref. Smith _et al._ (2007) for WMAP-3 with LRGs (0.95$\sigma$), WMAP-3
with quasars (1.64$\sigma$), and WMAP-3 with NVSS (2.13$\sigma$) respectively.
Recently, SPT found a greater than $4\sigma$ cross correlation between the SPT
convergence field and the galaxy survey from the Blanco Cosmology Survey, the
Wide-field Infrared Survey Explorer, and Spitzer Bleem _et al._ (2012). In
this work, we use WMAP data released in years 1, 3, 5, 7, with NVSS to probe
the lensing-galaxy correlation. We follow the methods developed in Smith et
al. Smith _et al._ (2007) and Hirata et al. Hirata _et al._ (2008) using all
of WMAP’s datasets and compare our results to these earlier analyses.
The structure of the paper is as follows. We introduce the data sets in Sec.
II. Gravitational lensing effects on the CMB as well as the lensing extraction
technique are reviewed in Sec. III. We describe the cross correlation
estimators in Sec. IV, and the forecast for Planck in Sec. V. We discuss our
results in Sec. VI.
Figure 1: WMAP Kp0 mask (left) with $f_{\rm sky}=0.77$ and NVSS mask (right)
with $f_{\rm sky}=0.573$.
## II Data Sets: WMAP and NVSS
The CMB data we use are from WMAP’s Q-, V-, and W-band raw differencing
assemblies (DAs). All of these DAs are masked by the Kp0 mask (Fig. 1) to
remove bright sources and the galactic plane leaving $77\%$ of the sky.
The input for the galaxy distribution is the NVSS of radio galaxies. The NVSS
Condon _et al._ (1998) team provides the software “NVSSlist” to convert its
raw catalog to a deconvolved one which is corrected for known biases and
systematic errors. We use the deconvolved catalog to extract the galaxy count
map. We use this software here, without specifying either a minimum or maximum
flux cut. The NVSS map is pixelized with a HEALPix pixelization scheme with
$N_{\rm side}=256$. We remove the galactic plane ($|b|<10^{\circ}$) and the
part of the sky unobserved by the survey ($\delta<-36.87^{\circ}$). We also
carefully remove bright sources with flux $>1\,\rm Jy$ and mask out a disk of
radius $1^{\circ}$ around them, forming the NVSS mask shown in Fig. 1. The
resulting galaxy count map has $1,224,990$ sources, a sky fraction $f_{\rm
sky}=0.573$, a mean number of sources per pixel $\bar{n}=2.72$, and a surface
density of 170,249 galaxies per steradian. This agrees well with previous
studies Liu _et al._ (2011).
## III Gravitational lensing of the CMB
The effect of lensing on the CMB’s primordial temperature $\tilde{T}$ in
direction ${\bf n}$ can be represented by
$T(\mathbf{n})=\widetilde{T}(\mathbf{n}+\mathbf{d}(\mathbf{n})),$ (1)
where $T$ is the lensed temperature and the deflection angle field
$\mathbf{d}(\mathbf{n})=\mathbf{\nabla}\phi$, with $\phi$ being the lensing
potential. The operator $\nabla$ is the covariance derivative on the sphere
with respect to the angular position ${\bf n}$. We use Gaussian natural units
with $\hbar=c=1$ throughout this paper.
The two-point correlation function of the temperature field is Okamoto and Hu
(2003):
$\displaystyle\left\langle
T_{lm}T_{l^{\prime}m^{\prime}}\right\rangle=\tilde{C}^{TT}_{l}\delta_{ll^{\prime}}\delta_{m-m^{\prime}}(-1)^{m}+\hskip
42.67912pt$
$\displaystyle\sum_{LM}(-1)^{M}\left(\begin{array}[]{ccc}l&l^{\prime}&L\\\
m&m^{\prime}&-M\end{array}\right)f^{TT}_{lLl^{\prime}}\ \phi_{LM},$ (4)
where the second term encodes the effects of lensing with the weighting factor
$f^{TT}_{lLl^{\prime}}$ given by
$\displaystyle
f^{TT}_{lLl^{\prime}}=\tilde{C}_{l}^{TT}{}_{0}F_{l^{\prime}Ll}+\tilde{C}_{l^{\prime}}^{TT}{}_{0}F_{lLl^{\prime}}.$
(5)
We use the standard spherical harmonic decomposition
$T({\bf n})=\displaystyle\sum_{lm}T_{lm}Y_{lm}({\bf n}),$ (6)
which defines the temperature modes $T_{lm}$. We use a similar notation for
all other quantities defined on a sphere, e.g. $\phi_{LM}$ are the modes of
the lensing potential, etc.
Here $\tilde{C}_{l}^{TT}$ is the unlensed temperature power spectrum, and
${}_{0}F_{lLl^{\prime}}=\sqrt{\frac{(2l+1)(2l^{\prime}+1)(2L+1)}{4\pi}}\times\hskip
42.67912pt$
$\displaystyle\frac{1}{2}[L(L+1)+l^{\prime}(l^{\prime}+1)-l(l+1)]\left(\begin{array}[]{ccc}l&L&l^{\prime}\\\
0&0&0\end{array}\right)$ (9)
are proportional to the Wigner $3j$ symbols. Equation (4) provides a way to
extract $\phi_{LM}$ from the $TT$ correlations.
In the late universe, the Poisson equation relates the lensing potential
$\Phi(\mathbf{k})$ to the density contrast $\delta(\mathbf{k})$,
$k^{2}\Phi(\mathbf{k})=\frac{3H_{0}^{2}\Omega_{m}}{2a}\delta(\mathbf{k}),$
(10)
where $\Omega_{m}$ is the matter fraction, $a$ is the scale factor and $H_{0}$
is the Hubble constant. Using the definition $D(\chi)=-2(1/\chi-1/\chi_{*})$,
the projected lensing potential can be expressed as an integral along the
line-of-sight,
$\phi({\bf n})=\int_{0}^{\chi_{*}}d\chi\ \Phi(\chi{\bf n})D(\chi),$ (11)
and it is integrated from 0 to the comoving distance at the last scattering
surface $\chi_{*}$. Here $\chi(z)$ is the comoving distance at redshift $z$.
The galaxy overdensity is also given by a line-of-sight integration as
$g({\bf n})=\frac{\int d\chi\ b_{g}\,\mathcal{N}(\chi)\delta(\chi{\bf
n})}{\int d\chi\ \mathcal{N}(\chi)}.$ (12)
To better understand the projected galaxy overdensity,
$\mathcal{N}(\chi)=dN/d\chi$ is the comoving distance distribution of the
galaxies. For NVSS galaxies, there is a lack of accurate photometric redshifts
so approximations to the redshift distributions are made Xia _et al._ (2011);
Ho _et al._ (2008); Schiavon _et al._ (2012). Following Smith _et al._
(2007), we use a Gaussian distribution
$\frac{dN}{dz}\propto e^{-\frac{(z-z_{0})^{2}}{2\sigma^{2}}}$ (13)
where $\sigma=0.8$ for $z<z_{0}(=1.1)$, and $\sigma=0.3$ for $z>z_{0}$, and
the comoving distribution $dN/d\chi$ is easily derived from Eq. (13).
The parameter $b_{g}(z)$ is the redshift-dependent galaxy bias. To keep the
model as simple as possible we treat the galaxy bias as a constant which can
be determined from a fit to the data shown in Fig. 2. This is different from
Hirata _et al._ (2008); Ho _et al._ (2008) which used the cross correlation
of NVSS with the SDSS and with sources from the 2-Micron All Sky Survey to
determine the galaxy bias. The galaxy bias is of great importance because the
cross correlation can be directly translated into primordial non-Gaussianity
Dalal _et al._ (2008) and may enable general relativity to be tested on
cosmological scales Acquaviva _et al._ (2008).
Equations (11) and (12) give the general definitions for the lensing potential
and galaxy overdensity which are both Gaussian fields, characterized by their
variances, (i.e. the power spectra) $C_{l}^{\phi\phi}$ and $C_{l}^{gg}$. From
Eq. (11) and (12), one sees that the lensing-galaxy cross correlation is built
on the relation described by Eq. (10). Using the Limber approximation $k\sim
l/\chi$ we calculate the theoretical galaxy auto-power spectrum and cross-
power spectrum in Eqs. (14) and (15).
The galaxy-galaxy power spectrum ( $\langle g_{lm}g^{\ast}_{lm}\rangle$ ) is
$\displaystyle C_{l}^{gg}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{1}{\int d\chi\mathcal{N}(\chi)}\right)^{2}\int
d\chi\
b_{g}^{2}\,\frac{\mathcal{N}^{2}(\chi)}{\chi^{2}}P_{\delta}\left(\frac{l}{\chi}\right)$
(14)
and it will be used later for determining the galaxy bias and for simulating
galaxy maps. This power spectrum shows the galaxy clustering strength on
different angular scales. We calculate the power spectrum of the NVSS
overdensity map using two independent methods: a pseudo-$C_{l}$ method and a
spherical harmonic estimation, as described in Refs. Xia _et al._ (2011);
Blake _et al._ (2004) respectively. We find that both methods agree very well
(Fig. 2). As a final check, we have computed the NVSS galaxy power spectrum
using the NVSS galaxy map in both equatorial and galactic coordinates and find
a negligible difference, as expected, since the galaxy clustering is an
intrinsic property of the Universe and does not depend on the choice of
coordinate system. The galaxy power spectrum we use, obtained using the
pixelized map in equatorial coordinates, is plotted in Fig. 2.
The lensing-galaxy cross-power spectrum is
$\displaystyle C_{l}^{\phi g}$ $\displaystyle\simeq$
$\displaystyle\frac{3H_{0}^{2}\Omega_{m}}{2}\frac{1}{\int d\chi\
\mathcal{N}(\chi)}$ (15) $\displaystyle\times$ $\displaystyle\int d\chi\
b_{g}(1+z)D(\chi)\frac{\mathcal{N}(\chi)}{k^{2}\chi^{2}}P_{\delta}\left(\frac{l}{\chi}\right)$
which shows the mutual influence between the gravitational potential and the
galaxy clustering in the late universe on different angular scales.
$P_{\delta}(k)$ is the matter power spectrum defined using the same convention
as Ref. Cooray and Sheth (2002). The primordial scalar curvature perturbations
are evaluated at the pivot scale $k_{0}=0.002\,\textrm{Mpc}^{-1}$. The cross-
power spectrum $C_{l}^{\phi g}$ will be used to simulate the correlated galaxy
maps and will also be fit to data to determine the detection significance.
Figure 2: The NVSS galaxy auto-power spectrum. The $1\sigma$ error bars are
from 1000 Monte Carlo simulations of the NVSS galaxy map with galaxy bias
$b_{g}=1$. For both sets of data points, the red points are from the
pseudo-$C_{l}$ method Xia _et al._ (2011), and the green are from the
spherical harmonic estimation Blake _et al._ (2004). The theoretical galaxy
auto-power spectrum is fit to the red data points derived from the
pseudo-$C_{l}$ method. Both methods show a consistent galaxy auto-power
spectrum from the NVSS data. The first bin of the red data points largely
deviates from the theoretical curve due to systematic effects.
## IV Cross Correlation Estimation
Table 1: The 6-parameter $\Lambda$CDM model used for the simulations of the temperature, galaxy and lensing potential. The derived parameter $\sigma_{8}$, based on the 6-parameter model, is shown in column eight. Using these parameters 1000 galaxy simulations with $b_{g}=1$ were performed to get the reconstructed galaxy biases as well as the $1\sigma$ error bars. From column nine, we see that all the reconstructed galaxy biases are consistent with the input value $b_{g}=1$. Furthermore, the galaxy biases of the real data are calculated based on the simulations and shown in column ten. Two independent methods were used to calculate the galaxy auto-power spectra, as specified in the footnotes. Data set | $\Omega_{b}h^{2}$ | $\Omega_{\rm CDM}h^{2}$ | $H_{0}$ | $A_{s}$ | $n_{s}$ | $\tau$ | $\sigma_{8}$ | $b^{\rm sim}_{g}$ | $b^{\rm data}_{g}$
---|---|---|---|---|---|---|---|---|---
WMAP-1333WMAP1+CBI+ACBAR+2dFGRS+Ly$\alpha$, Spergel _et al._ (2003) | 0.0226 | 0.1104 | 72 | $2.212\times 10^{-9}$ | 0.96 | 0.117 | 0.76 | $0.98\pm 0.12$111Pseudo-$C_{l}$ method Xia _et al._ (2011), | $2.00\pm 0.12$111Pseudo-$C_{l}$ method Xia _et al._ (2011),
| | | | | | | $0.96\pm 0.10$222Spherical harmonic estimation Blake _et al._ (2004) | $1.95\pm 0.10$222Spherical harmonic estimation Blake _et al._ (2004)
WMAP-3444WMAP3 ALL, http://lambda.gsfc.nasa.gov/product/map/dr2/parameters.cfm | 0.02186 | 0.1105 | 70.4 | $2.393\times 10^{-9}$ | 0.947 | 0.073 | 0.77 | $0.98\pm 0.11$111Pseudo-$C_{l}$ method Xia _et al._ (2011), | $1.97\pm 0.11$111Pseudo-$C_{l}$ method Xia _et al._ (2011),
| | | | | | | $0.96\pm 0.09$222Spherical harmonic estimation Blake _et al._ (2004) | $1.92\pm 0.09$222Spherical harmonic estimation Blake _et al._ (2004)
WMAP-5555WMAP5+BAO+SNALL+Ly$\alpha$POST, http://lambda.gsfc.nasa.gov/product/map/dr3/parameters.cfm | 0.02305 | 0.1182 | 69.7 | $2.484\times 10^{-9}$ | 0.969 | 0.094 | 0.85 | $0.98\pm 0.10$111Pseudo-$C_{l}$ method Xia _et al._ (2011), | $1.85\pm 0.10$111Pseudo-$C_{l}$ method Xia _et al._ (2011),
| | | | | | | $0.96\pm 0.08$222Spherical harmonic estimation Blake _et al._ (2004) | $1.81\pm 0.08$222Spherical harmonic estimation Blake _et al._ (2004)
WMAP-7666WMAP7, http://lambda.gsfc.nasa.gov/product/map/dr4/parameters.cfm | 0.02258 | 0.1109 | 71 | $2.43\times 10^{-9}$ | 0.963 | 0.088 | 0.80 | $0.98\pm 0.11$111Pseudo-$C_{l}$ method Xia _et al._ (2011), | $1.91\pm 0.11$111Pseudo-$C_{l}$ method Xia _et al._ (2011),
| | | | | | | $0.96\pm 0.09$222Spherical harmonic estimation Blake _et al._ (2004) | $1.86\pm 0.09$222Spherical harmonic estimation Blake _et al._ (2004)
Monte Carlo simulations are used to estimate the cross correlation between the
CMB and the galaxy distribution. The variances ${\tilde{C}}_{l}^{TT}$,
$C_{l}^{TT}$, $C_{l}^{\phi\phi}$ are computed using CAMB Lewis _et al._
(2000) with the cosmological parameters listed in Table 1. In addition to
these, we derive $C_{l}^{gg}$ and $C_{l}^{\phi g}$ from Eq. (14) and Eq. (15)
with the parameters listed in Table 1. Simulated CMB temperature modes,
${\tilde{a}}_{lm}$, are drawn from Gaussian distributions with zero means and
variances ${\tilde{C}}_{l}^{TT}$. In this work, we will use two sets of
cosmological parameters because we want to check the consistency of our
results with a previous study Smith _et al._ (2007) and also because we want
to explore the impact of using the newest parameters from WMAP-7 on the
lensing-galaxy cross correlations.
We convert these ${\tilde{a}}_{lm}$ to an unlensed temperature map,
${\tilde{T}}({\bf n})$, on which we do a cubic interpolation to precisely
implement Eq. (1). This produces a lensed temperature map $T({\bf n})$ that is
converted back to harmonic space to give the lensed modes $a_{lm}$. Then each
DA’s beam and pixel window transfer function (the pixel window transfer
function has negligible effects on the cross-power spectra) from WMAP are
multiplied by these modes which are subsequently transformed into a
temperature map containing the lensing signal.
We then simulate Gaussian noise in map space where the pixel noise is assumed
to be uncorrelated and Gaussian distributed with zero mean and pixel-
independent variance determined from ${\sigma_{0}}/{\sqrt{N_{\rm obs}}}$.
Here, both $\sigma_{0}$ and $N_{\rm obs}$ are supplied by the WMAP team for
different DAs. We add this noise map to the signal map and apply the Kp0 mask
to get a simulated WMAP DA made in the same way as the real WMAP maps were
produced. The entire procedure can be summarized by Eq. (16) in which $a_{lm}$
is the lensed CMB mode, $n({\bf n})$ the white noise, ${\rm M}^{\rm WMAP}({\bf
n})$ the mask map, $\nu$ the index of the DA channel, $p_{l}$ the pixel window
transfer function, $b_{l}$ the beam transfer function
$\displaystyle T^{(\nu)}({\bf n})$ $\displaystyle=$ $\displaystyle{\rm M}^{\rm
WMAP}({\bf n})\biggl{[}\sum_{lm}p_{l}b^{(\nu)}_{l}a_{lm}Y_{lm}({\bf n})$ (16)
$\displaystyle\qquad+\left(\frac{\sigma_{0}}{\sqrt{N_{\rm obs}({\bf
n})}}\right)^{(\nu)}n({\bf n})\biggr{]}\,.$
To maximize the signal-to-noise ratio, we compute a single harmonic mode
${\hat{a}}_{lm}$ from eight Q, V, W-band DAs. This reduction step is expressed
as Smith _et al._ (2007)
$\displaystyle\mathbf{\hat{a}}$ $\displaystyle=$
$\displaystyle(\mathbf{S}+\mathbf{N})^{-1}\mathbf{a}$ (17) $\displaystyle=$
$\displaystyle\mathbf{S}^{-1/2}\mathbf{A}^{-1}\mathbf{S}^{1/2}\mathbf{N}^{-1}\mathbf{a}.$
Here ${\bf a}$ is the vector of $a_{lm}$s, ${\bf S}$ the signal covariance
matrix, ${\bf N}$ the noise covariance matrix, and ${\bf
A}=\mathbf{I}+\mathbf{S}^{1/2}\mathbf{N}^{-1}\mathbf{S}^{1/2}$. We use the
second form of Eq. (17) and filter the raw CMB modes using a multigrid-
preconditioned-complex conjugate gradient method. The master equation that has
to be solved is
$\mathbf{A}\mathbf{x}=\mathbf{y},$ (18)
where $\mathbf{x}=\mathbf{S}^{1/2}\mathbf{\hat{a}}$, and
$\mathbf{y}=\mathbf{S}^{1/2}\mathbf{N}^{-1}\mathbf{a}$. Equation (18) is
better for numerical computations because ${\bf A}$ is close to the unit
matrix. Appendix A gives details of the numerical calculation of Eq. (18). We
solve Eq. (18) with $\hat{a}_{lm}$ for both the temperature ($\hat{T}_{lm}$)
and the galaxy ($\hat{g}_{lm}$).
We use the standard quadratic estimator to reconstruct a noisy lensing
potential map $\hat{\phi}_{lm}$ in harmonic space Hu (2001a, b),
$\displaystyle\sum_{lm}\hat{\phi}_{lm}Y_{lm}({\bf n})=\nabla^{i}({\
}_{0}A^{T}({\bf n})\nabla_{i}{\ }_{0}B^{T}({\bf n})),$ (19)
where
${}_{0}A^{T}(\mathbf{n})=\displaystyle\sum_{lm}\hat{T}_{lm}Y_{lm}(\mathbf{n})$
(20)
and
${}_{0}B^{T}({\bf{n}})=\displaystyle\sum_{lm}\tilde{C}^{TT}_{l}\hat{T}_{lm}Y_{lm}({\bf{n}}).$
(21)
In the above equations, $\nabla_{i}$ is the gradient operator on a sphere and
$\nabla^{i}=g^{ij}\nabla_{j}$ . Here $g_{ij}$ is the metric of a sphere.
We also use Monte Carlo simulations for the NVSS galaxy maps. The simulated
galaxy modes $g_{lm}$ are drawn from a Gaussian distribution and transformed
into a galaxy overdensity map $g({\bf n})$ at HEALPix resolution $N_{\rm
side}=1024$. The galaxy modes must satisfy the correct galaxy-galaxy auto-
correlation and lensing-galaxy cross correlation. From these two constraints
the simulated galaxy mode must be
$g_{lm}=\frac{C_{l}^{\phi
g}}{C_{l}^{\phi\phi}}\phi_{lm}+\sqrt{C_{l}^{gg}-\frac{(C_{l}^{\phi
g})^{2}}{C_{l}^{\phi\phi}}}G_{lm},$ (22)
where $G_{lm}$ is a complex Gaussian random variable, and $\phi_{lm}$ is
inherited from the deflection field in Eq. (1). From this equation we see that
the lensing-galaxy correlation is encoded in the first term. A NVSS map is
generated where the galaxy number count in each pixel is drawn from a Poisson
distribution with mean
$\lambda({\bf n})=\bar{n}(1+g({\bf n})).$ (23)
The galaxy count map $\lambda({\bf n})$ is used to generate a simulated galaxy
overdensity map $g^{(\text{sim})}({\bf n})$,
$g^{(\text{sim})}({\bf n})={\rm{M}}^{\rm{NVSS}}({\bf
n})\left[\frac{\lambda({\bf n})}{\bar{n}}-1\right],$ (24)
where ${\rm{M}}^{\rm{NVSS}}$ is the NVSS mask shown in Fig. 1.
$g^{(\text{sim})}({\bf n})$ automatically contains the shot-noise with the
variance $N_{l}^{gg}={1}/{\bar{n}}$ for the galaxy overdensity map. We degrade
this map to resolution $N_{\rm side}=256$ i.e. the same as the real NVSS data.
The harmonic mode $g_{lm}^{(\text{sim})}$, which contains the shot-noise, is
obtained from $g^{(\text{sim})}({\bf n})$ and is further filtered using the
same procedure as in Eq. (17),
$\hat{g}_{lm}=({\bf S}+{\bf N})^{-1}g_{lm}^{(\text{sim})}.$ (25)
Here ${\bf S}$ represents the primordial galaxy covariance and ${\bf N}$ is
the shot-noise covariance.
We show the noisy reconstruction of the potential maps and the filtered galaxy
map in Fig. 3, using the measured WMAP and NVSS data.
Figure 3: The noisy reconstruction of the lensing potential map (Eq.(19) from
WMAP-7) band-pass filtered from $20\leq l\leq 40$ (left). The analogous map
from NVSS galaxy data [Eq.(25)] within the band $20\leq l\leq 40$ (right).
The lensing-galaxy cross-power spectrum is the observable which will be
compared with the counterpart from data. The estimator of the lensing-galaxy
cross correlation is expressed as
$C^{\phi
g}_{b}=\frac{1}{\mathcal{F}_{b}}\displaystyle\sum_{\begin{subarray}{c}l\in
b\\\ -l\leq m\leq
l\end{subarray}}(\hat{\phi}_{lm}-\langle\hat{\phi}_{lm}\rangle)^{\ast}\hat{g}_{lm},$
(26)
where $\mathcal{F}_{b}$ is the normalization factor at band $b$. It is shown
in Ref. Smith and Zaldarriaga (2011) that the normalization factor can be
calculated by either a direct or a fast method for the full-sky coverage and
that the fast method converges in $O(10^{2})$ simulations. When there is a
sky-cut these methods account for the sky-cut effect very well and a constant
$f_{\rm sky}$ is often used. The factor $f_{\rm sky}$ is actually a function
of $l$ Verde _et al._ (2003) so a simple constant approximation may
potentially bias the cross-spectra reconstruction. Therefore an end-to-end
simulation Smith _et al._ (2007) is the best way to get the exact
normalization accounting for the sky-cut and that is done here.
Table 2: The two sets of cosmological parameters used in this work: we choose two sets of parameters (labeled “Set-3” and “Set-7”) to do the cross correlation calculations in this work. In order to compare our results with those from the previous studies Smith _et al._ (2007), we use the parameters they used Set-3 from WMAP-3’s cosmological parameters ( row “WMAP-3” in Table 1) combined with the corresponding galaxy bias of Smith et al. Smith _et al._ (2007). Based on the newest cosmological parameters from WMAP-7 ( row “WMAP-7” in Table 1) we construct a new parameter set, Set-7 with the corresponding galaxy bias shown in Table 1. Data set | $\Omega_{b}h^{2}$ | $\Omega_{\rm CDM}h^{2}$ | $H_{0}$ | $A_{s}$ | $n_{s}$ | $\tau$ | $\sigma_{8}$ | $b_{g}$
---|---|---|---|---|---|---|---|---
Set-3 | 0.02186 | 0.1105 | 70.4 | $2.393\times 10^{-9}$ | 0.947 | 0.073 | 0.77 | $1.70$
Set-7 | 0.02258 | 0.1109 | 71 | $2.43\times 10^{-9}$ | 0.963 | 0.088 | 0.80 | $1.91$
Table 3: Measure of lensing-galaxy cross correlation $\mathcal{C}$ and its significance $\mathcal{C}/\Delta\mathcal{C}$ using Set-3. For five columns of this table: the second column shows the simulation results, the third column is the case without gradient stripes removed, the fourth column is the case with gradient stripes removed (this column shows the statistical results of the lensing-galaxy cross correlations). The fifth column is the case by setting the NVSS map in equatorial coordinates as a complementary diagnostic test. Data set | $\mathcal{C}^{\rm{sim}}$ | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$111Without gradient stripes removed. | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$222With gradient stripes removed. | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$333Galaxy map in equatorial coordinate.
---|---|---|---|---|---|---|---
WMAP-1$\times$NVSS | $1.00\pm 0.47$ | $2.13\sigma$ | $1.25\pm 0.47$ | $2.66\sigma$ | $1.24\pm 0.47$ | $2.64\sigma$ | 0.26
WMAP-3$\times$NVSS | $1.00\pm 0.35$ | $2.86\sigma$ | $1.20\pm 0.35$ | $3.43\sigma$ | $1.26\pm 0.35$ | $3.60\sigma$ | 0.17
WMAP-5$\times$NVSS | $1.00\pm 0.31$ | $3.23\sigma$ | $1.24\pm 0.31$ | $4.00\sigma$ | $1.27\pm 0.31$ | $4.10\sigma$ | 0.23
WMAP-7$\times$NVSS | $1.00\pm 0.30$ | $3.33\sigma$ | $1.14\pm 0.30$ | $3.80\sigma$ | $1.16\pm 0.30$ | $3.87\sigma$ | 0.15
Table 4: Measure of lensing-galaxy cross correlation $\mathcal{C}$ and its significance $\mathcal{C}/\Delta\mathcal{C}$ using Set-7. The format of this table is the same as Table 3. Data set | $\mathcal{C}^{\rm{sim}}$ | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$111Without gradient stripes removed. | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$222With gradient stripes removed. | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$333Galaxy map in equatorial coordinate.
---|---|---|---|---|---|---|---
WMAP-1$\times$NVSS | $1.00\pm 0.41$ | $2.44\sigma$ | $1.01\pm 0.41$ | $2.46\sigma$ | $1.00\pm 0.41$ | $2.44\sigma$ | 0.20
WMAP-3$\times$NVSS | $1.00\pm 0.31$ | $3.23\sigma$ | $0.96\pm 0.31$ | $3.10\sigma$ | $1.01\pm 0.31$ | $3.26\sigma$ | 0.13
WMAP-5$\times$NVSS | $1.00\pm 0.28$ | $3.57\sigma$ | $0.98\pm 0.28$ | $3.50\sigma$ | $1.01\pm 0.28$ | $3.61\sigma$ | 0.18
WMAP-7$\times$NVSS | $1.00\pm 0.26$ | $3.85\sigma$ | $0.92\pm 0.26$ | $3.54\sigma$ | $0.93\pm 0.26$ | $3.58\sigma$ | 0.11
Table 5: Fisher matrix analysis for WMAP$\times$NVSS cross correlation. The $1\sigma$ error bars are determined from Eq. (32). We calculate two sets of the optimal bounds for this work, based on two sets of parameters: Set-3 (column two); Set-7 (column three). Data set | $\mathcal{C}^{\textrm{optimal}}$ 111WMAP-3 year cosmological parameters and $b_{g}=1.70$. | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}^{\textrm{optimal}}$ 222WMAP-7 year cosmological parameters and $b_{g}=1.91$. | $\mathcal{C}/\Delta\mathcal{C}$
---|---|---|---|---
WMAP-1$\times$NVSS | $1\pm 0.46$ | $2.17\sigma$ | $1\pm 0.39$ | $2.56\sigma$
WMAP-3$\times$NVSS | $1\pm 0.29$ | $3.45\sigma$ | $1\pm 0.25$ | $4.00\sigma$
WMAP-5$\times$NVSS | $1\pm 0.25$ | $4.00\sigma$ | $1\pm 0.21$ | $4.76\sigma$
WMAP-7$\times$NVSS | $1\pm 0.22$ | $4.55\sigma$ | $1\pm 0.19$ | $5.26\sigma$
Table 6: Results of the curl null tests for WMAP$\times$NVSS cross correlation. The curl null tests are performed based on two sets of parameters: Set-3 (column two); Set-7 (column three). Data set | $\mathcal{C}$111WMAP-3 year cosmological parameters and $b_{g}=1.70$. | $\mathcal{C}/\Delta\mathcal{C}$ | $\mathcal{C}$222WMAP-7 year cosmological parameters and $b_{g}=1.91$. | $\mathcal{C}/\Delta\mathcal{C}$
---|---|---|---|---
WMAP-1$\times$NVSS | $-0.11\pm 0.47$ | $-0.23\sigma$ | $-0.03\pm 0.41$ | $-0.07\sigma$
WMAP-3$\times$NVSS | $0.00\pm 0.35$ | $0.00\sigma$ | $0.04\pm 0.31$ | $0.13\sigma$
WMAP-5$\times$NVSS | $0.05\pm 0.31$ | $0.16\sigma$ | $0.07\pm 0.28$ | $0.25\sigma$
WMAP-7$\times$NVSS | $-0.05\pm 0.30$ | $0.17\sigma$ | $-0.03\pm 0.26$ | $-0.12\sigma$
Table 7: Gaussianity diagnostics for the probability distribution of $\\{\mathcal{C}\\}$ which is constructed from 1000 Monte Carlo simulations. The second column is the Kolmogorov-Smirnov test, and the critical value is 0.04 at 5% confidence level. The Kolmogorov-Smirnov test requires the maximum deviation be $<0.04$ to validate the distribution is Gaussian. The third column is the skewness of $\\{\mathcal{C}\\}$, and the fourth column is the kurtosis of $\\{\mathcal{C}\\}$. The upper values in the cells are the results for Set-3, the lower values for Set-7. For a Gaussian distribution, the skewness should be 0 and the kurtosis should be 3. As can be seen, all the probability distribution functions pass the Kolmogorov-Smirnov test and are consistent with being Gaussian-distributed. Data set | maximum distance[$<$0.04] | skewness[$\sim$0] | kurtosis[$\sim$3]
---|---|---|---
WMAP-1$\times$NVSS | 0.02 | 0.02 | 2.89
0.02 | -0.05 | 2.77
WMAP-3$\times$NVSS | 0.02 | -0.14 | 2.62
0.03 | -0.17 | 2.58
WMAP-5$\times$NVSS | 0.03 | -0.17 | 2.53
0.03 | -0.23 | 2.43
WMAP-7$\times$NVSS | 0.03 | -0.21 | 2.44
0.03 | -0.21 | 2.36
As a systematic check we note that the lensing signal consists of a gradient
and a curl component Cooray _et al._ (2005). The curl component estimator
$\psi_{lm}$ is defined by
$\displaystyle\sum_{lm}\psi_{lm}Y_{lm}({\bf
n})=\displaystyle\sum_{ij}\epsilon^{ij}\nabla_{i}({\ }_{0}A^{T}({\bf
n})\nabla_{j}{\ }_{0}B^{T}({\bf n}))$ (27)
and should vanish because lensing does not generate vorticity. Similar to Eq.
(26), the curl-galaxy cross correlation diagnostic is calculated by
$C^{\psi
g}_{b}=\frac{1}{\mathcal{F}_{b}}\displaystyle\sum_{\begin{subarray}{c}l\in
b\\\ -l\leq m\leq
l\end{subarray}}(\psi_{lm}-\langle\psi_{lm}\rangle)^{\ast}\hat{g}_{lm}$ (28)
which should also vanish.
The amplitude of the cross correlation is determined using
$\mathcal{C}=\frac{\sum_{AB}C^{(\textrm{th})}_{A}\mathbf{C}^{-1}_{AB}C^{(\textrm{obs})}_{B}}{\sum_{AB}C^{(\textrm{th})}_{A}\mathbf{C}^{-1}_{AB}C^{(\textrm{th})}_{B}}.$
(29)
${\bf C}_{AB}$ is the covariance matrix for the band powers and $A$ and $B$
stand for the band power index. We find that the off-diagonal correlations of
${\bf C}_{AB}$ are negligible, and the covariance matrix elements can be
simply replaced by the band power variance $\sigma_{A}^{2}$, i.e. ${\bf
C}_{AB}=\sigma_{A}^{2}\delta_{AB}$.
We have described the procedures used to perform analysis on simulated or
measured data. Now we summarize the analysis of the real WMAP and NVSS data.
We fit the theoretical galaxy auto-power spectrum to the NVSS data in Fig. 2
and determine the galaxy biases (Table 1) using two methods. The error bars
are determined from 1000 simulated galaxy maps with galaxy bias $b_{g}=1$.
Then we choose two sets of parameters (labeled “Set-3” and “Set-7”) in Table 2
to do the cross correlation calculations in this work. In order to compare our
results with those from the previous studies Smith _et al._ (2007), we use
the parameters they used Set-3 from WMAP-3’s cosmological parameters ( row
“WMAP-3” in Table 1) combined with the corresponding galaxy bias of Smith et
al. Smith _et al._ (2007). Based on the newest cosmological parameters from
WMAP-7 ( row “WMAP-7” in Table 1) we construct a new parameter set Set-7 with
the corresponding galaxy bias shown in Table 1. For each of the parameter sets
we calculate four lensing-galaxy cross correlations from WMAP-1 to WMAP-7.
We carefully treat the known systematics of NVSS, i.e. the gradient stripes
which are generated by the declination-dependence of the galaxy overdensity
field due to the low-flux calibration issue Condon _et al._ (1998). We first
make a gradient stripe map only using $m=0$ modes and then subtract it from
the galaxy map. We calculate the lensing-galaxy cross correlations for two
cases: without the gradient stripes removed and with the gradient stripes
removed. We find that this systematic effect does not affect the lensing-
galaxy cross correlations as seen from column $\mathcal{C}^{a}$ and column
$\mathcal{C}^{b}$ in Table 3 and Table 4. The statistical results are those
with the gradient stripes removed which are shown in Figs. 4 and 5 for the
Kp0$+$NVSS mask combination. From the two figures, we find that the lensing-
galaxy cross-power spectra are consistent with the theoretical predictions and
the uncertainty of the cross-power spectrum is decreasing as the year of WMAP
increases. All the error bars are calculated from 1000 Monte Carlo
simulations, which we confirmed to be sufficient for convergences. As a
complementary diagnostic test, we keep the NVSS galaxy overdensity map in
equatorial coordinates and calculate the cross-power spectra and all the
amplitudes are shown in column $\mathcal{C}^{c}$ in Table 3 and Table 4. As
can be seen, they are negligible. All the cross correlation amplitudes are
summarized in Table 3 and Table 4. From the results of WMAP-3$\times$NVSS in
Table 3: for the statistical results, we get lensing detection significance
level of $3.60\sigma$ and Smith _et al._ (2007) got $4.02\sigma$. Both
analyses agree quite well. We find the cross-power spectra from WMAP-5 and
WMAP-7 clearly and firmly show the lensing-galaxy correlation at $>3\sigma$
level for both cases. All the results are within the optimal bounds shown in
Table 5.
Assuming there is no cosmological parity violation the curl-galaxy correlation
should be consistent with zero. We show the results of the curl null tests in
Figs. 6 and 7. As expected, all the correlations are consistent with zero. The
amplitude as well as the significance are given in Table 6.
We pixelized the NVSS catalog with different HEALPix resolutions (e.g. $N_{\rm
side}=512,1024$) to probe the possible pixel artifacts that could afflict the
cross-power spectra and because we want to examine the impact of possible long
range spatial correlations. However, we find that different NVSS pixelization
resolutions do not affect the cross correlation.
We also use the diagonal elements of the covariance matrix to do the analysis
to check the consistency with previous studies. In this case, the estimator
has a larger variance as pointed out by Smith et al. Smith _et al._ (2009b).
This contributes to the difference in significance levels between $4\sigma$
Smith _et al._ (2007) and $2\sigma$ Hirata _et al._ (2008).
Figure 4: (Set-3) The lensing-galaxy cross-power spectra for WMAP$\times$ NVSS
are calculated from Eq. (26). The Kp0 mask is used to remove the contaminated
regions of the WMAP data. WMAP’s data are provided from two Q bands, two V
bands and four W bands. The NVSS mask is applied to the galaxy map to remove
bright sources and unobserved regions. The theoretical cross-power spectra are
shown in blue solid lines, and they are the same for all of the four panels.
The real data are shown in the red scattered points. The statistical amplitude
for WMAP-1$\times$NVSS is $1.24\pm 0.47$, for WMAP-3$\times$NVSS is $1.26\pm
0.35$, for WMAP-5$\times$NVSS is $1.27\pm 0.31$, for WMAP-7$\times$NVSS is
$1.16\pm 0.30$. All the error bars are determined from 1000 Monte Carlo
simulations. We find that the lensing-galaxy cross-power spectra are
consistent with the theoretical predictions and the uncertainty of the cross-
power spectrum is decreasing as the year of WMAP increases.
Figure 5: (Set-7) The lensing-galaxy cross-power spectra for WMAP$\times$ NVSS
are calculated from Eq. (26). See Fig. 4 for detailed descriptions. The
statistical amplitude for WMAP-1$\times$NVSS is $1.00\pm 0.41$, for
WMAP-3$\times$NVSS is $1.01\pm 0.31$, for WMAP-5$\times$NVSS is $1.01\pm
0.28$, for WMAP-7$\times$NVSS is $0.93\pm 0.26$.
Figure 6: (Set-3) The curl null tests for WMAP$\times$ NVSS are calculated
from Eq. (28). The Kp0 mask is used to remove the contaminated regions of the
WMAP data. WMAP data are provided from two Q bands, two V bands and four W
bands. The NVSS mask is applied to the galaxy map to remove bright sources and
unobserved regions. The theoretical lensing-galaxy cross-power spectra with
both WMAP and NVSS in galactic coordinates are shown in blue solid lines for
comparison, and they are the same for all of the four panels. The curl
amplitude for WMAP-1$\times$NVSS is $-0.11\pm 0.47$, for WMAP-3$\times$NVSS is
$0.00\pm 0.35$, for WMAP-5$\times$NVSS is $0.05\pm 0.31$, for
WMAP-7$\times$NVSS is $-0.05\pm 0.30$. As can be seen, all cross-power spectra
for the curl null test are consistent with zero (black dotted line).
Figure 7: (Set-7) The curl null tests for WMAP$\times$ NVSS are calculated
from Eq. (28). See Fig. 6 for detailed descriptions. The curl amplitude for
WMAP-1$\times$NVSS is $-0.03\pm 0.41$, for WMAP-3$\times$NVSS is $0.04\pm
0.31$, for WMAP-5$\times$NVSS is $0.07\pm 0.28$, for WMAP-7$\times$NVSS is
$-0.03\pm 0.26$. As can be seen, all cross-power spectra for the curl null
test are consistent with zero (black dotted line).
## V Forecast for Future Experiments
Figure 8: The signal-to-noise ratio for the lensing-galaxy cross correlation
between Planck and NVSS as a function of the maximum multipole used in the
analysis.
The revealed cross correlation between WMAP and NVSS hints that the detection
significance would be further enhanced if the precision of the CMB data were
improved. The upcoming Planck data will improve upon WMAP, so we expect that
the cross correlation between Planck and NVSS will be more significant. To
predict the optimal bound on the detection signal-to-noise ratio for lensing-
galaxy cross correlation we first calculate the equivalent noise $N_{l}^{\phi
g}$ from the following equation
$N_{l}^{\phi g}=\left[N_{l}^{\phi\phi}N_{l}^{gg}\right]^{1/2}.$ (30)
where $N_{l}^{\phi\phi}$ is the lensing reconstruction noise Okamoto and Hu
(2003) and $N_{l}^{gg}$ is the galaxy shot-noise. The efficient algorithm for
calculating $N_{l}^{\phi\phi}$ is given in Refs. Smith _et al._ (2010); Feng
_et al._ (2012). This reconstruction noise can be minimized by combining
different CMB channels and the minimum noise is
$N_{l}^{\rm
min,\phi\phi}=\frac{1}{\sum_{\nu}\left[N_{l}^{\nu,\phi\phi}\right]^{-1}},$
(31)
Both of the noise spectra effectively propagate the uncertainty $\Delta
C_{l}^{\phi g}$ into the cross-power spectrum $C_{l}^{\phi g}$. Specifically,
we express it as
$\Delta C_{l}^{\phi g}=\sqrt{\frac{2}{(2l+1)f_{\rm sky}}}\ (C_{l}^{\phi
g}+N_{l}^{\phi g}).$ (32)
The optimal bound is then determined from
$\left[\sum_{l}\left(\frac{C_{l}^{\phi g}}{\Delta C_{l}^{\phi
g}}\right)^{2}\right]^{1/2}\,.$
The redshift distribution Eq. (13) was used and the galaxy bias was set equal
to $b_{g}=1$. The instrumental properties for Planck are given in Refs. Miller
_et al._ (2009); The Planck Collaboration (2006). We show the signal-to-noise
ratio for Planck with NVSS as a function of $l_{\text{max}}$ in Fig. 8. We
find that the highest signal-to-noise ratio, i.e. $15\sigma$, saturates at
$l_{\text{max}}=2000$. Since the lensing-galaxy cross-power spectrum scales as
$C_{l}^{\phi g}\propto b_{g}$ as illustrated by Eq. (15), the amplitude of
this cross-power spectra is degenerate with the galaxy bias and the signal-to-
noise for the cross-power spectrum can also serve as a prediction of the
detection significance for the galaxy bias. Thus, we see Planck can detect
$b_{g}$ with high precision which will lead to a better understanding of the
correlation between the baryonic matter distribution and the dark matter
distribution.
## VI Conclusion
We have calculated the lensing-galaxy cross-power spectra using WMAP and NVSS
and the full covariance matrix to filter the data sets. Specifically, we
performed a thorough analysis of WMAP-1, WMAP-3, WMAP-5 and WMAP-7 raw DAs.
The cross correlations between WMAP-5, -7’s 8 DAs
(2Q-bands$+$2V-bands$+$4W-bands) with NVSS clearly and firmly show signals at
$>3\sigma$ level. We took the effects of gradient stripes into account for the
NVSS data, and determined the significance without and with gradient stripes
removed. The major effects caused by the stripes can be seen from the first
bin of either the galaxy auto-power spectrum or the lensing-galaxy cross-power
spectrum; the first bin decreases if the gradient stripes are marginalized
over. However, gradient stripes do not affect the lensing-galaxy correlation
(compare Refs. Schiavon _et al._ (2012); Nolta _et al._ (2004); Vielva _et
al._ (2006)). We have explicitly shown all these results in Tables 3 and 4. In
these two tables, column $\mathcal{C}^{a}$ are the results without the
gradient stripes removed and column $\mathcal{C}^{b}$ are our main results
with the gradient stripes removed. In order to validate the lensing-galaxy
cross correlations, we produced a NVSS galaxy map in equatorial coordinates
directly from the NVSS catalog and cross correlated it with the WMAP DA which
is in galactic coordinates, we find that all the lensing-galaxy cross
correlation amplitudes are negligible.
We investigated the impact of different NVSS pixelization resolutions and
found no effect. We compared the sensitivities of the estimators both with the
full and diagonal covariance matrix and found that the former more effectively
reduces the variance, which is mainly caused by the sky-cut and the
inhomogeneous instrumental noise. However, the former scheme involves the
inversion of a large matrix which is computationally challenging.
We predicted the detection significance for the lensing-galaxy cross
correlation or the galaxy bias for the upcoming Planck data with NVSS and
found the detection significance will be improved by a factor of 5.
The minimum variance of the estimator assumes that the CMB and galaxy
overdensity modes are Gaussian. However, if the CMB contains gravitational
lensing, the bispectrum $\langle TTg\rangle$ is not zero; it induces an
additional variance as indicated in Eq. (64). We analytically and numerically
confirm that this variance is actually not noticeable for WMAP and NVSS as
pointed out in Ref. Smith (2009). Furthermore, being aware of the potential
non-Gaussian shape of the probability distribution function (PDF) Smith and
Kamionkowski (2012), we specifically investigate the PDF of the cross-power
spectrum amplitude $\mathcal{C}$ in terms of Kolmogorov-Smirnov test, the
skewness and the kurtosis. The diagnostic tests are shown in Table 7. All the
PDFs pass the Kolmogorov-Smirnov tests. All the PDFs are consistent with being
Gaussian-distributed (Figs. 9 and 10).
The lensing-galaxy cross correlations effectively link the early universe to
the late universe and the CMB is served as a back light casting the dark
cosmic web (which is formed by the dark matter) throughout the major expansion
history of the universe. The gravitational lensing is a powerful tool to
decode the information of dark matter distribution from the CMB and the
lensing-galaxy cross-correlations further unveil the relationship between
baryonic matter and dark matter.
Figure 9: (Set-3) Probability distribution function for the lensing-galaxy
cross correlation. The likelihood functions are normalized to 1. From 1000
simulations, a set of $\\{\mathcal{C}\\}$ is generated for each one of the
subfigures, then by counting the frequency of $\mathcal{C}$ within a bin, a
step-like function (red) is plotted. For comparison, Gaussian likelihood
(green) is plotted using the mean and the variance of the set
$\\{\mathcal{C}\\}$.
Figure 10: (Set-7) Probability distribution function for the lensing-galaxy
cross correlation. See Fig. 9 for detailed descriptions.
###### Acknowledgements.
We would like to acknowledge helpful discussions with Sudeep Das, Duncan
Hanson, Christian Reichardt, Meir Shimon, and Amit Yadav. We acknowledge the
use of CAMB, HEALPix111http://healpix.jpl.nasa.gov/ Górski _et al._ (2005),
and LAPACK software packages and the LAMBDA archive. The computational
resources required for this work were accessed via the GlideinWMS Sfiligoi
_et al._ (2009) on the Open Science Grid Pordes _et al._ (2007). We are
indebted to Frank Wuerthwein, Igor Sfiligoi, Terrence Martin, and Robert
Konecny for their insight and support.
## Appendix A MULTIGRID-PRECONDITIONED COMPLEX CONJUGATE GRADIENT INVERSION
Given the signal covariance matrix ${\bf S}$ and the noise covariance matrix
${\bf N}$, and an array of the CMB modes ${\bf a}$, we define another
covariance matrix
$\mathbf{A}=\mathbf{I}+\mathbf{S}^{1/2}\mathbf{N}^{-1}\mathbf{S}^{1/2}$, and
two vectors $\mathbf{x}=\mathbf{S}^{1/2}\mathbf{a}$, and
$\mathbf{y}=\mathbf{S}^{1/2}\mathbf{N}^{-1}\mathbf{a}$. For the problem $\bf
A\bf x=\bf y$, we write down the equations for constructing the matrix ${\bf
A}$ and the vector ${\bf y}$,
$\displaystyle N^{-1}_{lml^{\prime}m^{\prime}}$ $\displaystyle=$
$\displaystyle\sum_{\nu}p_{l}b^{(\nu)}_{l}p_{l^{\prime}}b^{(\nu)}_{l^{\prime}}$
(33) $\displaystyle\times\int d\textbf{n}\
Y^{\ast}_{lm}(\textbf{n})Y_{l^{\prime}m^{\prime}}(\textbf{n})\left[\frac{\rm
M({\bf n})}{\sigma^{2}}\right]^{(\nu)}\,,$ $\displaystyle{}[N^{-1}a]_{lm}$
$\displaystyle=$ $\displaystyle\sum_{\nu}p_{l}b^{(\nu)}_{l}$ (34)
$\displaystyle\times\int d\textbf{n}\ Y^{\ast}_{lm}(\textbf{n})\left[\frac{\rm
M({\bf n})\rm H({\bf n})}{\sigma^{2}}\right]^{(\nu)}\,,$ $\displaystyle
w^{(\nu)}_{lm}$ $\displaystyle=$ $\displaystyle\int d\mathbf{n}\
Y_{lm}(\mathbf{n})\left[\frac{\rm M({\bf n})}{\sigma^{2}}\right]^{(\nu)}\,.$
(35)
In the above equations, $p_{l}$ is the window transfer function,
$b_{l}^{(\nu)}$ is the specific beam transfer function corresponding to the DA
of WMAP, and $\rm M({\bf n})$ is the mask map. For WMAP,
$\nu=Q_{1},Q_{2},V_{1},V_{2},W_{1},W_{2},W_{3},W_{4}$, $\rm H({\bf
n})=T(\mathbf{n})$ and $\rm M({\bf n})$ is the Kp0 mask. For NVSS, $\nu=1$ and
$\rm H({\bf n})=g(\mathbf{n})$. Since NVSS has $45$ arc-second FWHM resolution
Condon _et al._ (1998), we set $b_{l}^{(1)}=1$ as NVSS’s beam transfer
function. The correspondence between the continuum and discrete forms of
integration on the sphere is,
$\int
d\mathbf{n}\rightarrow\frac{4\pi}{N_{\textrm{pix}}}\displaystyle\sum_{j},$
(36)
where $j$ denotes the pixel index according to the HEALPix pixelization scheme
and $N_{\textrm{pix}}$ is the total number of pixels.
For comparison, we also use a suboptimal estimator which only takes the
diagonal elements of the inverse noise matrix
$N^{-1}_{lml^{\prime}m^{\prime}}$ shown in Eq. (33).
The filtering using the covariance matrix requires us to solve the linear
equation $\mathbf{A}\mathbf{x}=\mathbf{y}$. We use the preconditioned
conjugate gradient iteration to solve it, and the initial condition is chosen
to be
$\displaystyle{\bf x}^{(0)}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle{\bf r}^{(0)}$ $\displaystyle=$ $\displaystyle{\bf y},$
$\displaystyle{\bf p}^{(1)}$ $\displaystyle=$ $\displaystyle\tilde{{\bf
A}}^{-1}{\bf y},$ (37)
with the preconditioner defined as
$\tilde{{\bf A}}^{-1}=\left(\begin{array}[]{cc}{\bf A}_{0}^{-1}&{\bf 0}\\\
{\bf 0}&{\bf A}^{-1}_{\Delta}\end{array}\right),$ (38)
here ${\bf A}_{\Delta}$ is the diagonal element of the matrix ${\bf A}$.
The iteration procedure is Hirata _et al._ (2004)
$\displaystyle\mathbf{x}^{(i)}$ $\displaystyle=$
$\displaystyle\mathbf{x}^{(i-1)}+\frac{\mathbf{r}^{(i-1)}\tilde{\mathbf{A}}^{-1}\mathbf{r}^{(i-1)}}{\mathbf{p}^{(i)}\mathbf{A}\mathbf{p}^{(i)}}\mathbf{p}^{(i)},$
$\displaystyle\mathbf{r}^{(i)}$ $\displaystyle=$
$\displaystyle\mathbf{y}-\mathbf{A}\mathbf{x}^{(i)},$
$\displaystyle\mathbf{p}^{(i)}$ $\displaystyle=$
$\displaystyle\tilde{\mathbf{A}}^{-1}\mathbf{r}^{(i-1)}+\frac{\mathbf{r}^{(i-1)}\tilde{\mathbf{A}}^{-1}\mathbf{}r^{(i-1)}}{\mathbf{r}^{(i-2)}\tilde{\mathbf{A}}^{-1}\mathbf{r}^{(i-2)}}\mathbf{p}^{(i-1)}.$
(39)
From Eq. (39), we find that two operations $\tilde{\bf A}^{-1}{\bf r}$ and
${\bf A}{\bf p}$ are computationally demanding if we evaluate them directly
because ${\bf A}$ and ${\bf A}_{0}$ are $10^{6}\times 10^{6}$ matrix.
In order to achieve the necessary efficiency, we recursively precondition the
matrix ${\bf A}$ on a much coarser grid. The preconditioner is
$\tilde{{\bf A}}^{-1}=\left(\begin{array}[]{cc}\tilde{\bf A}_{0}^{-1}&{\bf
0}\\\ {\bf 0}&{\bf A}^{-1}_{\Delta}\end{array}\right),$ (40)
and on the coarser grid the preconditioner is $\tilde{\bf A}_{0}^{-1}$. This
multigrid strategy enables us to directly store the matrix $\tilde{\bf A}_{0}$
on the coarsest grid and we can further analytically express the smallest
inversion problem as follows222In the following, we denote subscripts $l_{i}$
or $l_{i},m_{i}$ by $i$ for simplicity, so that $p_{l_{i}}\to p_{i}$,
$Y_{l_{i}m_{i}}\rightarrow Y_{i}$, $N^{-1}_{l_{1}m_{1}l_{2}m_{2}}\rightarrow
N^{-1}_{12}$ etc.
$\displaystyle N_{12}^{-1}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{\nu}\int
p_{1}b^{(\nu)}_{1}Y_{1}^{\ast}p_{2}b^{(\nu)}_{2}Y_{2}\displaystyle\sum_{3}w^{(\nu)}_{3}Y_{3}$
(46) $\displaystyle=$
$\displaystyle\displaystyle\sum_{\nu}\displaystyle\sum_{3}w^{(\nu)}_{3}\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}}$
$\displaystyle\times$
$\displaystyle(-1)^{m_{1}}\left(\begin{array}[]{ccc}l_{1}&l_{2}&l_{3}\\\
0&0&0\end{array}\right)\left(\begin{array}[]{ccc}l_{1}&l_{2}&l_{3}\\\
-m_{1}&m_{2}&m_{3}\end{array}\right)$ $\displaystyle
p_{1}b^{(\nu)}_{1}p_{2}b^{(\nu)}_{2},$
then the problem of preconditioning $\bf A$ with $\tilde{\bf A}_{0}$ on the
finer grid can be iteratively solved by using Eq. (39). For this work we use
three levels of the grids: (1) $N_{\rm side}=512,l_{\rm max}=1000$, (2)
$N_{\rm side}=256,l_{\rm max}=400$, (3) $N_{\rm side}=128,l_{\rm max}=200$. We
split the covariance matrix on the third grid at $l_{\rm split}=30$ to
construct the minimum inversion problem.
For the coarsest grid, we explicitly calculate the inverse noise matrix
$N_{12}^{-1}$ [Eq. (46)] using LAPACK. The iteration process also needs the
multiplication for $\mathbf{A}\mathbf{\lambda}$, and this can be computed
efficiently by doing spherical harmonic transformations:
$\displaystyle\mathbf{A}\mathbf{\lambda}$ $\displaystyle=$
$\displaystyle\displaystyle\sum_{4}(I+S^{1/2}N^{-1}S^{1/2})_{14}\lambda_{4}$
(47) $\displaystyle=$
$\displaystyle\lambda_{1}+\sum_{\nu}p_{1}b^{(\nu)}_{1}S^{1/2}_{1}\biggl{[}\int
d{\bf n}\ Y_{1}^{\ast}\left[\frac{\rm M({\bf n})}{\sigma^{2}}\right]^{(\nu)}$
$\displaystyle\times\left(\sum_{4}p_{4}b^{(\nu)}_{4}S^{1/2}_{4}\lambda_{4}Y_{4}\right)\biggr{]}.$
## Appendix B NON-GAUSSIANITY
There are several possible non-Gaussian effects generated by using a nonzero
bispectrum in the simulation. These can potentially bias our results. We
analytically calculate this non-Gaussian bias in this appendix.
We define the bispectrum by
$\langle a_{1}a_{2}g_{3}\rangle=b_{123}G(123),$ (48)
where $b_{123}=(f_{123}C_{2}^{\rm TT}+f_{213}C_{1}^{\rm TT})C_{l_{3}}^{\phi
g}$, (see footnote 2 for notation) and
$\displaystyle G(123)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(2l_{1}+1)(2l_{2}+1)(2l_{3}+1)}{4\pi}}\left(\begin{array}[]{ccc}l_{1}&l_{2}&l_{3}\\\
0&0&0\end{array}\right)$ (54)
$\displaystyle\left(\begin{array}[]{ccc}l_{1}&l_{2}&l_{3}\\\
m_{1}&m_{2}&m_{3}\end{array}\right).$
The estimator is
$\hat{C}=\frac{1}{\mathcal{F}}(\hat{C}_{A}-\hat{C}_{B})$ (55)
where
$\hat{C}_{A}=\frac{1}{2}\displaystyle\sum_{123}b_{123}G(123)\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{3}$
(56)
and
$\hat{C}_{B}=\frac{1}{2}\displaystyle\sum_{123}b_{123}G(123)\left[C^{\rm
TT}\right]^{-1}_{12}\tilde{g}_{3}.$ (57)
We define
$\displaystyle\tilde{a}$ $\displaystyle=$ $\displaystyle C^{-1}a,$
$\displaystyle f_{k}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum
b_{12k}G(12k)\left[C^{\rm TT}\right]^{-1}_{12},$
$\displaystyle\langle\tilde{a}_{1}\tilde{a}_{2}\rangle$ $\displaystyle=$
$\displaystyle\left[C^{\rm TT}\right]^{-1}_{12},$
$\displaystyle\langle\tilde{g}_{1}\tilde{g}_{2}\rangle$ $\displaystyle=$
$\displaystyle\left[C^{\rm gg}\right]^{-1}_{12}\,.$
The summation index $i$ denotes a sum over $l_{i}m_{i}$.
We define the normalization as
$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\displaystyle\sum_{123456}b_{123}b_{456}G(123)G(456)$
(58) $\displaystyle\times$ $\displaystyle\left[C^{\rm
TT}\right]^{-1}_{14}\left[C^{\rm TT}\right]^{-1}_{25}\left[C^{\rm
gg}\right]^{-1}_{36}.$
The variance of the estimator $\hat{C}$ is $\sigma^{2}(\hat{C})$ which has
contributions from three parts,
$\sigma^{2}(\hat{C})=\sigma^{2}(\hat{C}_{A})+\sigma^{2}(\hat{C}_{B})-2\sigma^{2}(\hat{C}_{A}\hat{C}_{B}).$
(59)
Now we explicitly determine the three variances. For the second term, we have
the relation
$\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{3}\rangle=\displaystyle\sum_{1^{\prime}2^{\prime}3^{\prime}}\left[C^{\rm
TT}\right]^{-1}_{11^{\prime}}\left[C^{\rm
TT}\right]^{-1}_{22^{\prime}}\left[C^{\rm
gg}\right]^{-1}_{33^{\prime}}b_{1^{\prime}2^{\prime}3^{\prime}}G(1^{\prime}2^{\prime}3^{\prime}),$
(60)
so the last two variance terms can be easily expressed as
$\sigma^{2}(\hat{C}_{B})=\sigma^{2}(\hat{C}_{A}\hat{C}_{B})=f^{T}\left[C^{\rm
gg}\right]^{-1}f.$ (61)
For the first term, it is
$\displaystyle\sigma^{2}(\hat{C}_{A})$ $\displaystyle=$
$\displaystyle\frac{1}{4}\displaystyle\sum_{123456}b_{123}b_{456}G(123)G(456)\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{3}\tilde{a}_{4}\tilde{a}_{5}\tilde{g}_{6}\rangle-\frac{1}{4}\displaystyle\sum_{123456}b_{123}b_{456}G(123)G(456)\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{3}\rangle\langle\tilde{a}_{4}\tilde{a}_{5}\tilde{g}_{6}\rangle,$
and can be expanded as
$\displaystyle\frac{1}{4}\displaystyle\sum_{123456}b_{123}b_{456}G(123)G(456)\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{3}\tilde{a}_{4}\tilde{a}_{5}\tilde{g}_{6}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{4}\displaystyle\sum_{123456}b_{123}b_{456}G(123)G(456)\biggl{\\{}\Bigl{[}\underbrace{\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{3}\rangle\langle\tilde{a}_{4}\tilde{a}_{5}\tilde{g}_{6}\rangle}_{\text{second
term}}+\langle\tilde{a}_{4}\tilde{a}_{5}\tilde{g}_{3}\rangle\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{6}\rangle+\langle\tilde{a}_{1}\tilde{a}_{4}\tilde{g}_{3}\rangle\langle\tilde{a}_{2}\tilde{a}_{5}\tilde{g}_{6}\rangle+\langle\tilde{a}_{2}\tilde{a}_{5}\tilde{g}_{3}\rangle\langle\tilde{a}_{1}\tilde{a}_{4}\tilde{g}_{6}\rangle$
$\displaystyle+\langle\tilde{a}_{1}\tilde{a}_{5}\tilde{g}_{3}\rangle\langle\tilde{a}_{2}\tilde{a}_{4}\tilde{g}_{6}\rangle+\langle\tilde{a}_{2}\tilde{a}_{4}\tilde{g}_{3}\rangle\langle\tilde{a}_{1}\tilde{a}_{5}\tilde{g}_{6}\rangle\Big{]}_{3\cdot
3}+\Big{[}\underbrace{\langle\tilde{a}_{1}\tilde{a}_{5}\rangle\langle\tilde{a}_{2}\tilde{a}_{4}\rangle\langle\tilde{g}_{3}\tilde{g}_{6}\rangle+\langle\tilde{a}_{1}\tilde{a}_{4}\rangle\langle\tilde{a}_{2}\tilde{a}_{5}\rangle\langle\tilde{g}_{3}\tilde{g}_{6}\rangle}_{\text{normalization}}+\langle\tilde{a}_{1}\tilde{a}_{2}\rangle\langle\tilde{a}_{4}\tilde{a}_{5}\rangle\langle\tilde{g}_{3}\tilde{g}_{6}\rangle\Bigr{]}_{2\cdot
2\cdot 2}\biggr{\\}}.$ $\displaystyle\sigma^{2}(\hat{C}_{A}-\hat{C}_{B})$
$\displaystyle=$
$\displaystyle\mathcal{F}+\biggl{\\{}\frac{1}{4}\displaystyle\sum_{123456}b_{123}b_{456}G(123)G(456)\Bigl{[}\langle\tilde{a}_{4}\tilde{a}_{5}\tilde{g}_{3}\rangle\langle\tilde{a}_{1}\tilde{a}_{2}\tilde{g}_{6}\rangle+\langle\tilde{a}_{1}\tilde{a}_{4}\tilde{g}_{3}\rangle\langle\tilde{a}_{2}\tilde{a}_{5}\tilde{g}_{6}\rangle+\langle\tilde{a}_{2}\tilde{a}_{5}\tilde{g}_{3}\rangle\langle\tilde{a}_{1}\tilde{a}_{4}\tilde{g}_{6}\rangle$
(64)
$\displaystyle+\langle\tilde{a}_{1}\tilde{a}_{5}\tilde{g}_{3}\rangle\langle\tilde{a}_{2}\tilde{a}_{4}\tilde{g}_{6}\rangle+\langle\tilde{a}_{2}\tilde{a}_{4}\tilde{g}_{3}\rangle\langle\tilde{a}_{1}\tilde{a}_{5}\tilde{g}_{6}\rangle\Bigr{]}\biggr{\\}}_{\rm
nonzero{\ }bispectrum}$ $\displaystyle=$ $\displaystyle O(b^{2})+O(b^{4})$
When all the pieces are put together, we find that the nonvanishing bispectrum
induces an extra term which is $O(b^{4})$. We conclude that for WMAP, this
contribution is very small and we numerically verified that this is indeed the
case.
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|
arxiv-papers
| 2012-07-13T18:42:39 |
2024-09-04T02:49:33.047289
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Chang Feng, Grigor Aslanyan, Aneesh V. Manohar, Brian Keating, Hans P.\n Paar, Oliver Zahn",
"submitter": "Chang Feng",
"url": "https://arxiv.org/abs/1207.3326"
}
|
1207.3597
|
# On Distributability of Petri Nets††thanks: This work was partially supported
by the DFG (German Research Foundation).
An extended abstract of this paper appeared in L. Birkedal, ed.: Proc. 15th
Int. Conf. on Foundations of Software Science and Computation Structures
(FoSSaCS 2012), LNCS 7213, Springer, 2012, pp. 331–345,
doi:http://dx.doi.org/10.1007/978-3-642-28729-9˙22.
Rob van Glabbeek NICTA, Sydney, AustraliaSchool of Computer Science and
Engineering
Univ. of New South Wales, Sydney, Australia rvg@cs.stanford.edu Institute for
Programming and Reactive Systems
TU Braunschweig, Germany Ursula Goltz Jens-Wolfhard Schicke-Uffmann Institute
for Programming and Reactive Systems
TU Braunschweig, Germany goltz@ips.cs.tu-bs.de drahflow@gmx.de
###### Abstract
We formalise a general concept of distributed systems as sequential components
interacting asynchronously. We define a corresponding class of Petri nets,
called LSGA nets, and precisely characterise those system specifications which
can be implemented as LSGA nets up to branching ST-bisimilarity with explicit
divergence.
## 1 Introduction
The aim of this paper is to contribute to a fundamental understanding of the
concept of a distributed reactive system and the paradigms of synchronous and
asynchronous interaction. We start by giving an intuitive characterisation of
the basic features of distributed systems. In particular we assume that
distributed systems consist of components that reside on different locations,
and that any signal from one component to another takes time to travel. Hence
the only interaction mechanism between components is asynchronous
communication.
Our aim is to characterise which system specifications may be implemented as
distributed systems. In many formalisms for system specification or design,
synchronous communication is provided as a basic notion; this happens for
example in process algebras. Hence a particular challenge is that it may be
necessary to simulate synchronous communication by asynchronous communication.
Trivially, any system specification may be implemented distributedly by
locating the whole system on one single component. Hence we need to pose some
additional requirements. One option would be to specify locations for system
activities and then to ask for implementations satisfying this distribution
and still preserving the behaviour of the original specification. This is done
in [2]. Here we pursue a different approach. We add another requirement to our
notion of a distributed system, namely that its components only allow
sequential behaviour. We then ask whether an arbitrary system specification
may be implemented as a distributed system consisting of sequential components
in an optimal way, that is without restricting the concurrency of the original
specification. This is a particular challenge when synchronous communication
interacts with concurrency in the specification of the original system. We
will give a precise characterisation of the class of distributable systems,
which answers in particular under which conditions synchronous communication
may be implemented in a distributed setting.
For our investigations we need a model which is expressive enough to represent
concurrency. It is also useful to have an explicit representation of the
distributed state space of a distributed system, showing in particular the
local control states of components. We choose Petri nets, which offer these
possibilities and additionally allow finite representations of infinite
behaviours. We work within the class of _structural conflict nets_ [8]—a
proper generalisation of the class of one-safe place/transition systems, where
conflict and concurrency are clearly separated.
For comparing the behaviour of systems with their distributed implementation
we need a suitable equivalence notion. Since we think of open systems
interacting with an environment, and since we do not want to restrict
concurrency in applications, we need an equivalence that respects branching
time and concurrency to some degree. Our implementations use transitions which
are invisible to the environment, and this should be reflected in the
equivalence by abstracting from such transitions. However, we do not want
implementations to introduce divergence. In the light of these requirements we
work with two semantic equivalences. _Step readiness equivalence_ is one of
the weakest equivalences that captures branching time, concurrency and
divergence to some degree; whereas _branching ST-bisimilarity with explicit
divergence_ fully captures branching time, divergence, and those aspects of
concurrency that can be represented by concurrent actions overlapping in time.
We obtain the same characterisation for both notions of equivalence, and thus
implicitly for all notions in between these extremes.
We model distributed systems consisting of sequential components as an
appropriate class of Petri nets, called _LSGA nets_. These are obtained by
composing nets with sequential behaviour by means of an asynchronous parallel
composition. We show that this class corresponds exactly to a more abstract
notion of distributed systems, formalised as _distributed nets_ [7].
We then consider distributability of system specifications which are
represented as structural conflict nets. A net $N$ is _distributable_ if there
exists a distributed implementation of $N$, that is a distributed net which is
semantically equivalent to $N$. In the implementation we allow unobservable
transitions, and labellings of transitions, so that single actions of the
original system may be implemented by multiple transitions. However, the
system specifications for which we search distributed implementations are
_plain_ nets without these features.
We give a precise characterisation of distributable nets in terms of a semi-
structural property. This characterisation provides a formal proof that the
interplay between choice and synchronous communication is a key issue for
distributability.
To establish the correctness of our characterisation we develop a new method
for rigorously proving the equivalence of two Petri nets, one of which known
to be plain, up to branching ST-bisimilarity with explicit divergence.
## 2 Basic Notions
In this paper we employ _signed multisets_ , which generalise multisets by
allowing elements to occur in it with a negative multiplicity.
###### Definition 2.1.
multiset Let $X$ be a set.
* –
A signed multiset over $X$ is a function $A\\!:X\rightarrow\mbox{\bbb Z}$,
i.e. $A\in\mbox{\bbb Z}^{X}$.
It is a _multiset_ iff $A\in{\rm Nature}^{X}$, i.e. iff $A(x)\geq 0$ for all
$x\in X$.
* –
$x\in X$ is an _element of_ a signed multiset $A\in{\rm Nature}^{X}$, notation
$x\in A$, iff $A(x)\neq 0$.
* –
For signed multisets $A$ and $B$ over $X$ we write $A\leq B$ iff $A(x)\leq
B(x)$ for all $x\mathbin{\in}X$;
$A\cup B$ denotes the signed multiset over $X$ with $(A\cup
B)(x):=\textrm{max}(A(x),B(x))$,
$A\cap B$ denotes the signed multiset over $X$ with $(A\cap
B)(x):=\textrm{min}(A(x),B(x))$,
$A+B$ denotes the signed multiset over $X$ with $(A+B)(x):=A(x)+B(x)$,
$A-B$ denotes the signed multiset over $X$ with $(A-B)(x):=A(x)-B(x)$, and
for $k\mathbin{\in}{\rm Nature}$ the signed multiset $k\cdot A$ is given by
$(k\cdot A)(x):=k\cdot A(x)$.
* –
The function $\emptyset\\!:X\rightarrow{\rm Nature}$, given by
$\emptyset(x):=0$ for all $x\mathbin{\in}X$, is the _empty_ multiset over $X$.
* –
If $A$ is a signed multiset over $X$ and $Y\subseteq X$ then
$A\mathop{\upharpoonright}Y$ denotes the signed multiset over $Y$ defined by
$(A\mathop{\upharpoonright}Y)(x):=A(x)$ for all $x\mathbin{\in}Y$.
* –
The cardinality $|A|$ of a signed multiset $A$ over $X$ is given by
$|A|:=\sum_{x\in X}|A(x)|$.
* –
A signed multiset $A$ over $X$ is _finite_ iff $|A|<\infty$, i.e., iff the set
$\\{x\mid x\mathbin{\in}A\\}$ is finite.
We write $A\in_{f}\mbox{\bbb Z}^{X}$ or $A\in_{f}{\rm Nature}^{X}$ to indicate
that $A$ is a finite (signed) multiset over $X$.
* –
Any function $f:X\rightarrow\mbox{\bbb Z}$ or $f:X\rightarrow\mbox{\bbb
Z}^{Y}$ from $X$ to either the integers or the signed multisets over some set
$Y$ extends to the finite signed multisets $A$ over $X$ by $f(A)=\sum_{x\in
X}A(x)\cdot f(x)$.
Two signed multisets $A\\!:X\rightarrow\mbox{\bbb Z}$ and
$B\\!:Y\rightarrow\mbox{\bbb Z}$ are _extensionally equivalent_ iff
$A\mathop{\upharpoonright}(X\cap Y)=B\mathop{\upharpoonright}(X\cap Y)$,
$A\mathop{\upharpoonright}(X\setminus Y)=\emptyset$, and
$B\mathop{\upharpoonright}(Y\setminus X)=\emptyset$. In this paper we often do
not distinguish extensionally equivalent signed multisets. This enables us,
for instance, to use $A+B$ even when $A$ and $B$ have different underlying
domains. A multiset $A$ with $A(x)\in\\{0,1\\}$ for all $x$ is identified with
the set $\\{x\mid A(x)=1\\}$. A signed multiset with elements $x$ and $y$,
having multiplicities $-2$ and $3$, is denoted as
$-2\cdot\\{x\\}+3\cdot\\{y\\}$.
We consider here general labelled place/transition systems with arc weights.
Arc weights are not necessary for the results of the paper, but are included
for the sake of generality.
###### Definition 2.2.
Petri net Let Act be a set of _visible actions_ and $\tau\mathbin{\not\in}{\rm
Act}$ be an _invisible action_. Let ${\rm Act}_{\tau}:={\rm
Act}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$. A (_labelled_)
_Petri net_ (_over ${\rm Act}_{\tau}$_) is a tuple $N=(S,T,F,M_{0},\ell)$
where
* –
$S$ and $T$ are disjoint sets (of _places_ and _transitions_),
* –
$F:(S\times T\cup T\times S)\rightarrow{\rm Nature}$ (the _flow relation_
including _arc weights_),
* –
$M_{0}:S\rightarrow{\rm Nature}$ (the _initial marking_), and
* –
$\ell:T\rightarrow{\rm Act}_{\tau}$ (the _labelling function_).
Petri nets are depicted by drawing the places as circles and the transitions
as boxes, containing their label. Identities of places and transitions are
displayed next to the net element. When $F(x,y)>0$ for $x,y\mathbin{\in}S\cup
T$ there is an arrow (_arc_) from $x$ to $y$, labelled with the _arc weight_
$F(x,y)$. Weights 1 are elided. When a Petri net represents a concurrent
system, a global state of this system is given as a _marking_ , a multiset $M$
of places, depicted by placing $M(s)$ dots (_tokens_) in each place $s$. The
initial state is $M_{0}$.
To compress the graphical notation, we also allow universal quantifiers of the
form $\forall x.\phi(x)$ to appear in the drawing (cf. Figure 3). A quantifier
replaces occurrences of $x$ in element identities with all concrete values for
which $\phi(x)$ holds, possibly creating a set of elements instead of the
depicted single one. An arc of which only one end is replicated by a given
quantifier results in a fan of arcs, one for each replicated element. If both
ends of an arc are affected by the same quantifier, an arc is created between
pairs of elements corresponding to the same $x$, but not between elements
created due to differing values of $x$.
The behaviour of a Petri net is defined by the possible moves between markings
$M$ and $M^{\prime}$, which take place when a finite multiset $G$ of
transitions _fires_. In that case, each occurrence of a transition $t$ in $G$
consumes $F(s,t)$ tokens from each place $s$. Naturally, this can happen only
if $M$ makes all these tokens available in the first place. Next, each $t$
produces $F(t,s)$ tokens in each $s$. Definition LABEL:df-firing formalises
this notion of behaviour.
###### Definition 2.3.
preset Let $N=(S,T,F,M_{0},\ell)$ be a Petri net and $x\mathbin{\in}S\cup T$.
The multisets ${\vphantom{x}}^{\bullet}x,~{}{x}^{\bullet}:S\cup
T\rightarrow{\rm Nature}$ are given by ${\vphantom{x}}^{\bullet}x(y)=F(y,x)$
and ${x}^{\bullet}(y)=F(x,y)$ for all $y\mathbin{\in}S\cup T$. If $x\in T$,
the elements of ${\vphantom{x}}^{\bullet}x$ and ${x}^{\bullet}$ are called
_pre-_ and _postplaces_ of $x$, respectively, and if $x\in S$ we speak of
_pre-_ and _posttransitions_. The _token replacement function_
$\llbracket\\_\\!\\_\rrbracket:T\rightarrow\mbox{\bbb Z}^{S}$ is given by
$\llbracket t\rrbracket={t}^{\bullet}-{\vphantom{t}}^{\bullet}t$ for all $t\in
T$. These functions extend to finite signed multisets as usual (see Definition
LABEL:df-multiset).
###### Definition 2.4.
firing Let $N\mathbin{=}(S,T,F,M_{0},\ell)$ be a Petri net,
$G\mathbin{\in}{\rm Nature}^{T}\\!$, $G$ non-empty and finite, and
$M,M^{\prime}\in{\rm Nature}^{S}$.
$G$ is a _step_ from $M$ to $M^{\prime}$, written
$M~{}[G\rangle_{N}~{}M^{\prime}$, iff
* –
${\vphantom{G}}^{\bullet}G\leq M$ ($G$ is _enabled_) and
* –
$M^{\prime}=(M-{\vphantom{G}}^{\bullet}G)+{G}^{\bullet}=M+\llbracket
G\rrbracket$.
Note that steps are (finite) multisets, thus allowing self-concurrency, i.e.
the same transition can occur multiple times in a single step. We write
$M~{}[t\rangle_{N}~{}M^{\prime}$ for
$M\mathrel{[\\{t\\}\rangle_{N}}M^{\prime}$, whereas $M[G\rangle_{N}$
abbreviates $\exists M^{\prime}.~{}M\mathrel{[G\rangle_{N}}M^{\prime}$. We may
omit the subscript $N$ if clear from context.
In our nets transitions are labelled with _actions_ drawn from a set ${\rm
Act}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$. This makes it
possible to see these nets as models of _reactive systems_ that interact with
their environment. A transition $t$ can be thought of as the occurrence of the
action $\ell(t)$. If $\ell(t)\mathbin{\in}{\rm Act}$, this occurrence can be
observed and influenced by the environment, but if $\ell(t)\mathbin{=}\tau$,
it cannot and $t$ is an _internal_ or _silent_ transition. Transitions whose
occurrences cannot be distinguished by the environment carry the same label.
In particular, since the environment cannot observe the occurrence of internal
transitions at all, they are all labelled $\tau$.
The labelling function $\ell$ extends to finite multisets of transitions
$G\in\mbox{\bbb Z}^{T}$ by $\ell(G):=\sum_{t\in T}G(t)\cdot\\{\ell(t)\\}$. For
$A,B\in\mbox{\bbb Z}^{{\rm Act}_{\tau}}$ we write $A\equiv B$ iff
$\ell(A)(a)=\ell(B)(a)$ for all $a\in{\rm Act}$, i.e. iff $A$ and $B$ contain
the same (numbers of) visible actions, allowing
$\ell(A)(\tau)\neq\ell(B)(\tau)$. Hence $\ell(G)\equiv\emptyset$ indicates
that $\ell(t)=\tau$ for all transitions $t\in T$ with $G(t)\neq 0$.
###### Definition 2.5.
onesafe Let $N=(S,T,F,M_{0},\ell)$ be a Petri net.
* –
The set $[M_{0}\rangle_{N}$ of _reachable markings of $N$_ is defined as the
smallest set containing $M_{0}$ that is closed under $[G\rangle_{N}$, meaning
that if $M\in[M_{0}\rangle_{N}$ and $M\mathrel{[G\rangle_{N}}M^{\prime}$ then
$M^{\prime}\in[M_{0}\rangle_{N}$.
* –
$N$ is _one-safe_ iff $M\in[M_{0}\rangle_{N}\Rightarrow\forall s\in
S.~{}M(s)\leq 1$.
* –
The _concurrency relation_ $\mathord{\smile}\subseteq T^{2}$ is given by
$t\smile u\Leftrightarrow\exists
M\mathbin{\in}[M_{0}\rangle.~{}M[\\{t\\}\mathord{+}\\{u\\}\rangle$.
* –
$N$ is a _structural conflict net_ iff for all $t,u\in T$ with $t\smile u$ we
have ${\vphantom{t}}^{\bullet}t\cap{\vphantom{u}}^{\bullet}u=\emptyset$.
We use the term _plain nets_ for Petri nets where $\ell$ is injective and no
transition has the label $\tau$, i.e. essentially unlabelled nets.
This paper first of all aims at studying finite Petri nets: nets with finitely
many places and transitions. However, our work also applies to infinite nets
with the properties that ${\vphantom{t}}^{\bullet}t\neq\emptyset$ for all
transitions $t\in T$, and any reachable marking (a) is finite, and (b) enables
only finitely many transitions. Henceforth, we call such nets _finitary_.
Finitariness can be ensured by requiring
$|M_{0}|\mathbin{<}\infty\wedge\forall t\in
T.\,{\vphantom{t}}^{\bullet}t\neq\emptyset\wedge\forall x\in S\cup
T.\,|{x}^{\bullet}|<\infty$, i.e. that the initial marking is finite, no
transition has an empty set of preplaces, and each place and transition has
only finitely many outgoing arcs.
## 3 Semantic Equivalences
In this section, we give an overview on some semantic equivalences for
reactive systems. Most of these may be defined formally for Petri nets in a
uniform way, by first defining equivalences for transition systems and then
associating different transition systems with a Petri net. This yields in
particular different non-interleaving equivalences for Petri nets.
###### Definition 3.1.
LTS Let $\mathfrak{Act}$ be a set of _visible actions_ and
$\tau\mathbin{\not\in}\mathfrak{Act}$ be an _invisible action_. Let
$\mathfrak{Act}_{\tau}:=\mathfrak{Act}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$.
A _labelled transition system_ (LTS) (_over $\mathfrak{Act}_{\tau}$_) is a
triple $\mathfrak{L}=(\mathfrak{S},\mathfrak{T},\mathfrak{M_{0}})$ with
* –
$\mathfrak{S}$ a set of _states_ ,
* –
$\mathfrak{T}\subseteq\mathfrak{S}\times\mathfrak{Act}_{\tau}\times\mathfrak{S}$
a _transition relation_
* –
and $\mathfrak{M_{0}}\in\mathfrak{S}$ the _initial state_.
Given an LTS $(\mathfrak{S},\mathfrak{T},\mathfrak{M_{0}})$ with
$\mathfrak{M},\mathfrak{M}^{\prime}\in\mathfrak{S}$ and
$\alpha\in\mathfrak{Act}_{\tau}$, we write
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ for
$(\mathfrak{M},\alpha,\mathfrak{M}^{\prime})\in\mathfrak{T}$. We write
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}$ for
$\exists\mathfrak{M}^{\prime}.~{}\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ and
$\mathfrak{M}\arrownot\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}$ for
$\nexists\mathfrak{M}^{\prime}.~{}\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$.
Furthermore, $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\alpha\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ denotes
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}\vee(\alpha\mathbin{=}\tau\wedge\mathfrak{M}\mathbin{=}\mathfrak{M}^{\prime})$,
meaning that in case $\alpha\mathbin{=}\tau$ performing a $\tau$-transition is
optional. For $\,a_{1}a_{2}\cdots a_{n}\in\mathfrak{Act}^{*}$ we write
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to31.74887pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\,a_{1}a_{2}\cdots a_{n}~{}$\hskip
2.5pt}}$}}\mathfrak{M}^{\prime}$ when
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{2}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\cdots\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{n}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}\vspace{-4pt}$
where $\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}$ denotes the reflexive and transitive closure of
$\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}$. A state
$\mathfrak{M}\in\mathfrak{S}$ is said to be _reachable_ iff there is a
$\sigma\in\mathfrak{Act}^{*}$ such that
$\mathfrak{M_{0}}\stackrel{{\scriptstyle\sigma}}{{\Longrightarrow}}\mathfrak{M}$.
The set of all reachable states is denoted by $[\mathfrak{M_{0}}\rangle$. In
case there are $\mathfrak{M}_{i}\in[\mathfrak{M_{0}}\rangle$ for all $i\geq 1$
with $\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$ the LTS is said to
display _divergence_.
Many semantic equivalences on LTSs that in some way abstract from internal
transitions are defined in the literature; an overview can be found in [5]. On
divergence-free LTSs, the most discriminating semantics in the spectrum of
equivalences of [5], and the only one that fully respects the branching
structure of related systems, is _branching bisimilarity_ , proposed in [11].
###### Definition 3.2.
branching LTS Two LTSs
$(\mathfrak{S}_{1},\mathfrak{T}_{1},\mathfrak{M_{0}}_{1})$ and
$(\mathfrak{S}_{2},\mathfrak{T}_{2},\mathfrak{M_{0}}_{2})$ are _branching
bisimilar_ iff there exists a relation
$\mathcal{B}\,\subseteq\mathfrak{S}_{1}\times\mathfrak{S}_{2}$—a _branching
bisimulation_ —such that, for all $\alpha\mathbin{\in}\mathfrak{Act}_{\tau}$:
1. 1.
$\mathfrak{M_{0}}_{1}\mathcal{B}\,\mathfrak{M_{0}}_{2}$;
2. 2.
if $\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}_{1}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$
then $\exists\mathfrak{M}^{\dagger}_{2},\mathfrak{M}^{\prime}_{2}$ such that
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}_{2}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\alpha\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}$,
$\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}^{\dagger}_{2}$ and
$\mathfrak{M}^{\prime}_{1}\mathcal{B}\,\mathfrak{M}^{\prime}_{2}$;
3. 3.
if $\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}_{2}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}$
then $\exists\mathfrak{M}^{\dagger}_{1},\mathfrak{M}^{\prime}_{1}$ such that
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}_{1}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\alpha\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$,
$\mathfrak{M}^{\dagger}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}^{\prime}_{1}\mathcal{B}\,\mathfrak{M}^{\prime}_{2}$.
Branching bisimilarity _with explicit divergence_ [11, 9], is a variant of
branching bisimilarity that fully respects the diverging behaviour of related
systems. Since in this paper we mainly compare systems of which one admits no
divergence at all, the definition simplifies to the requirement that the other
system may not diverge either.
One of the semantics reviewed in [5] that respects branching time and
divergence only to a small extent, is _readiness equivalence_ , proposed in
[14].
###### Definition 3.3.
readiness Let $\mathfrak{L}=(\mathfrak{S},\mathfrak{T},\mathfrak{M_{0}})$ be
an LTS, $\sigma\in\mathfrak{Act}^{*}$ and $X\subseteq\mathfrak{Act}$.
$\langle\sigma,X\rangle$ is a _ready pair_ of $\mathfrak{L}$ iff
$\exists\mathfrak{M}.~{}\mathfrak{M_{0}}\stackrel{{\scriptstyle\sigma}}{{\Longrightarrow}}\mathfrak{M}\wedge\mathfrak{M}\arrownot\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\wedge\,X=\\{a\mathbin{\in}\mathfrak{Act}\mid\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\\}.$
We write $\mathfrak{R}(\mathfrak{L})$ for the set of all ready pairs of
$\mathfrak{L}$.
Two LTSs $\mathfrak{L}_{1}$ and $\mathfrak{L}_{2}$ are _readiness equivalent_
iff $\mathfrak{R}(\mathfrak{L}_{1})=\mathfrak{R}(\mathfrak{L}_{2})$.
As indicated in [6], see in particular the diagram on Page 317 (or 88),
equivalences on LTSs have been ported to Petri nets and other causality
respecting models of concurrency chiefly in five ways: we distinguish
_interleaving semantics_ , _step semantics_ , _split semantics_ , _ST-
semantics_ and _causal semantics_. Causal semantics fully respect the causal
relationships between the actions of related systems, whereas interleaving
semantics fully abstract from this information. Step semantics differ from
interleaving semantics by taking into account the possibility of multiple
actions to occur simultaneously (in _one step_); this carries a minimal amount
of causal information. ST-semantics respect causality to the extent that it
can be expressed in terms of the possibility of durational actions to overlap
in time. They are formalised by executing a visible action $a$ in two phases:
its start $a^{+}$ and its termination $a^{-}$. Moreover, terminating actions
are properly matched with their starts. Split semantics are a simplification
of ST-semantics in which the matching of starts and terminations is dropped.
Interleaving semantics on Petri nets can be formalised by associating to each
net $N=(S,T,F,M_{0},\ell)$ the LTS $(\mathfrak{S},\mathfrak{T},M_{0})$ with
$\mathfrak{S}$ the set of markings of $N$ and $\mathfrak{T}$ given by
$M_{1}\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}M_{2}:\Leftrightarrow\exists\,t\mathbin{\in}T.~{}\alpha\mathbin{=}\ell(t)\wedge
M_{1}~{}[t\rangle~{}M_{2}.$
Here we take $\mathfrak{Act}:={\rm Act}$. Now each equivalence on LTSs from
[5] induces a corresponding interleaving equivalence on nets by declaring two
nets equivalent iff the associated LTSs are. For example, _interleaving
branching bisimilarity_ is the relation of Definition LABEL:df-branching_LTS
with the $\mathfrak{M}$’s denoting markings, and the $\alpha$’s actions from
${\rm Act}_{\tau}$.
Step semantics on Petri nets can be formalised by associating another LTS to
each net. Again we take $\mathfrak{S}$ to be the markings of the net, and
$\mathfrak{M_{0}}$ the initial marking, but this time $\mathfrak{Act}$
consists of the _steps_ over ${\rm Act}$, the non-empty, finite multisets $A$
of visible actions from ${\rm Act}$, and the transition relation
$\mathfrak{T}$ is given by
$M_{1}\stackrel{{\scriptstyle
A}}{{\longrightarrow}}M_{2}:\Leftrightarrow\exists\,G\in_{f}{\rm
Nature}^{T}.~{}A=\ell(G)\wedge M_{1}~{}[G\rangle~{}M_{2}$
with $\tau$-transitions defined just as in the interleaving case. In
particular, the step version of readiness equivalence would be the relation of
Definition LABEL:df-readiness with the $\mathfrak{M}$’s denoting markings, the
$a$’s steps over ${\rm Act}$, and the $\sigma$’s sequences of steps. However,
variations in this type of definition are possible. In this paper, following
[7], we employ a form of step readiness semantics that is a bit closer to
interleaving semantics: $\sigma$ is a sequence of single actions, whereas the
menu $X$ of possible continuations after $\sigma$ is a set of steps.
###### Definition 3.4.
step readiness Let $N=(S,T,F,M_{0},\ell)$ be a Petri net, $\sigma\in{\rm
Act}^{*}$ and $X\subseteq{\rm Nature}^{{\rm Act}}$. $\langle\sigma,X\rangle$
is a _step ready pair_ of $N$ iff
$\exists M.M_{0}\stackrel{{\scriptstyle\sigma}}{{\Longrightarrow}}M\wedge
M\arrownot\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\wedge\,X=\\{A\mathbin{\in}{\rm
Nature}^{\rm Act}\mid M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle A$\hskip
2.5pt}\hfil}}$}}\\}.$
We write $\mathcal{R}(N)$ for the set of all step ready pairs of $N$.
Two Petri nets $N_{1}$ and $N_{2}$ are _step readiness equivalent_ ,
$N_{1}\approx_{\mathscr{R}}N_{2}$, iff
$\mathcal{R}(N_{1})=\mathcal{R}(N_{2})$.
Next we propose a general definition on Petri nets of ST-versions of each of
the semantics of [5]. Again we do this through a mapping from nets to a
suitable LTS. An _ST-marking_ of a net $(S,T,F,M_{0},\ell)$ is a pair
$(M,U)\mathbin{\in}{\rm Nature}^{S}\mathord{\times}T^{*}$ of a normal marking,
together with a sequence of transitions _currently firing_. The _initial_ ST-
marking is $\mathfrak{M_{0}}:=(M_{0},\varepsilon)$. The elements of ${\rm
Act}^{\pm}:=\\{a^{+},\,a^{-n}\mid a\mathbin{\in}{\rm Act},~{}n\mathbin{>}0\\}$
are called _visible action phases_ , and $Act^{\pm}_{\tau}:={\rm
Act}^{\pm}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$. For $U\in
T^{*}$, we write $t\in^{(n)}U$ if $t$ is the $n^{\it th}$ element of $U$.
Furthermore $U^{-n}$ denotes $U$ after removal of the $n^{\it th}$ transition.
###### Definition 3.5.
ST-marking Let $N=(S,T,F,M_{0},\ell)$ be a Petri net, labelled over ${\rm
Act}_{\tau}$.
The _ST-transition relations_ $\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}$ for $\eta\mathbin{\in}{\rm
Act}^{\pm}_{\tau}$ between ST-markings are given by
$(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{+}$\hskip 2.5pt}\hfil}}$}}(M^{\prime},U^{\prime})$ iff $\exists
t\mathbin{\in}T.~{}\ell(t)=a\wedge M[t\rangle\wedge
M^{\prime}=M-{\vphantom{t}}^{\bullet}t\wedge U^{\prime}=Ut$.
$(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-n}$\hskip 2.5pt}\hfil}}$}}(M^{\prime},U^{\prime})$ iff $\exists
t\in^{(n)}U.~{}\ell(t)=a\wedge U^{\prime}=U^{-n}\wedge
M^{\prime}=M+{t}^{\bullet}$.
$(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}(M^{\prime},U^{\prime})$ iff
$M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}M^{\prime}\wedge U^{\prime}=U$.
Now the ST-LTS associated to a net $N$ is
$(\mathfrak{S},\mathfrak{T},\mathfrak{M_{0}})$ with $\mathfrak{S}$ the set of
ST-markings of $N$, $\mathfrak{Act}:={\rm Act}^{\pm}$, $\mathfrak{T}$ as
defined in Definition LABEL:df-ST-marking, and $\mathfrak{M_{0}}$ the initial
ST-marking. Again, each equivalence on LTSs from [5] induces a corresponding
ST-equivalence on nets by declaring two nets equivalent iff their associated
LTSs are. In particular, _branching ST-bisimilarity_ is the relation of
Definition LABEL:df-branching_LTS with the $\mathfrak{M}$’s denoting ST-
markings, and the $\alpha$’s action phases from ${\rm Act}^{\pm}_{\tau}$. We
write $N_{1}\approx^{\Delta}_{bSTb}N_{2}$ iff $N_{1}$ and $N_{2}$ are
branching ST-bisimilar with explicit divergence.
_ST-bisimilarity_ was originally proposed in [10]. It was extended to a
setting with internal actions in [18], based on the notion of _weak
bisimilarity_ of [13], which is a bit less discriminating than branching
bisimilarity. The above can be regarded as a reformulation of the same idea;
the notion of weak ST-bisimilarity defined according to the recipe above
agrees with the ST-bisimilarity of [18].
The next proposition says that branching ST-bisimilarity with explicit
divergence is more discriminating than (i.e. _stronger_ than, _finer_ than,
or included in) step readiness equivalence.
###### Proposition 3.6.
step ready ST Let $N_{1}$ and $N_{2}$ be Petri nets. If
$N_{1}\approx^{\Delta}_{bSTb}N_{2}$ then $N_{1}\approx_{\mathscr{R}}N_{2}$.
###### Proof 3.7.
Suppose $N_{1}\approx^{\Delta}_{bSTb}N_{2}$ and
$\langle\sigma,X\rangle\in\mathcal{R}(N_{1})$. By symmetry it suffices to show
that $\langle\sigma,X\rangle\in\mathcal{R}(N_{2})$.
There must be a branching bisimulation $\mathcal{B}\,$ between the ST-markings
of $N_{1}=(S_{1},T_{1},F_{1},{M_{0}}_{1},\ell_{1})$ and
$N_{2}=(S_{2},T_{2},F_{2},{M_{0}}_{2},\ell_{2})$. In particular,
$({M_{0}}_{1},\epsilon)\mathcal{B}\,({M_{0}}_{2},\epsilon)$. Let
$\sigma:=a_{1}a_{2}\cdots a_{n}\in{\rm Act}^{*}$. Then
${M_{0}}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{2}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\cdots\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{n}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}M^{\prime}_{1}$ for a marking $M^{\prime}_{1}\mathbin{\in}{\rm
Nature}^{S_{1}}$ with $X=\\{A\mathbin{\in}{\rm Nature}^{\rm Act}\mid
M^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle A$\hskip
2.5pt}\hfil}}$}}\\}$ and
$M^{\prime}_{1}\arrownot\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}$.
Hence $({M_{0}}_{1},\epsilon)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{1}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
a_{1}^{-1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{2}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
a_{2}^{-1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\cdots\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{n}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
a_{n}^{-1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}(M^{\prime}_{1},\epsilon)$. Thus, using the properties of a
branching bisimulation on the ST-LTSs associated to $N_{1}$ and $N_{2}$, there
must be a marking $M^{\prime}_{2}\mathbin{\in}{\rm Nature}^{S_{2}}$ such that
$({M_{0}}_{2},\epsilon)\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{1}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
a_{1}^{-1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{2}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
a_{2}^{-1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\cdots\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\stackrel{{\scriptstyle
a_{n}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
a_{n}^{-1}}}{{\longrightarrow}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\\!(M^{\prime}_{2},\epsilon)$ and
$(M^{\prime}_{1},\epsilon)\mathcal{B}\,(M^{\prime}_{2},\epsilon)$. Since
$(M^{\prime}_{1},\epsilon)\arrownot\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}$,
the ST-marking $(M^{\prime}_{1},\epsilon)$ admits no divergence. As
$\approx^{\Delta}_{bSTb}$ respects this property, also
$(M^{\prime}_{2},\epsilon)$ admits no divergence, and there must be an
$M^{\prime\prime}_{2}\mathbin{\in}{\rm Nature}^{S_{2}}$ with
$M^{\prime\prime}_{2}\arrownot\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}$
and $(M^{\prime}_{2},\epsilon)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}(M^{\prime\prime}_{2},\epsilon)$. Clause 3. of a branching
bisimulation gives
$(M^{\prime}_{1},\epsilon)\mathcal{B}\,(M^{\prime\prime}_{2},\epsilon)$, and
Definition LABEL:df-ST-marking yields
${M_{0}}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\sigma$\hskip 2.5pt}\hfil}}$}}M^{\prime\prime}_{2}$.
Now let $B=\\{b_{1},\ldots,b_{n}\\}\in X$. Then
$M^{\prime}_{1}\stackrel{{\scriptstyle B}}{{\longrightarrow}}$, so
$(M^{\prime}_{1},\epsilon)\stackrel{{\scriptstyle
b_{1}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
b_{2}^{+}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
b_{m}^{+}}}{{\longrightarrow}}$. Property 2. of a branching bisimulation
implies $(M^{\prime\prime}_{2},\epsilon)\stackrel{{\scriptstyle
b_{1}^{+}}}{{\longrightarrow}}\stackrel{{\scriptstyle
b_{2}^{+}}}{{\longrightarrow}}\cdots\stackrel{{\scriptstyle
b_{m}^{+}}}{{\longrightarrow}}$ and hence
$M^{\prime\prime}_{2}\stackrel{{\scriptstyle B}}{{\longrightarrow}}$.
Likewise, with Property 3., $M^{\prime\prime}_{2}\stackrel{{\scriptstyle
B}}{{\longrightarrow}}$ implies $M^{\prime}_{1}\stackrel{{\scriptstyle
B}}{{\longrightarrow}}$ for all $B\in{\rm Nature}^{\rm Act}$. It follows that
$\langle\sigma,X\rangle\in\mathcal{R}(N_{2})$.
In this paper we employ both step readiness equivalence and branching ST-
bisimilarity with explicit divergence. Fortunately it will turn out that for
our purposes the latter equivalence coincides with its split version (since
always one of the compared nets is plain, see Proposition LABEL:pr-split).
A _split marking_ of a net $N=(S,T,F,M_{0},\ell)$ is a pair $(M,U)\in{\rm
Nature}^{S}\times{\rm Nature}^{T}$ of a normal marking $M$, together with a
multiset of transitions currently firing. The _initial_ split marking is
$\mathfrak{M_{o}}:=(M_{0},\emptyset)$. A split marking can be regarded as an
abstraction from an ST-marking, in which the total order on the (finite)
multiset of transitions that are currently firing has been dropped. Let ${\rm
Act}^{\pm}_{\rm split}:=\\{a^{+},\,a^{-}\mid a\in{\rm Act}\\}$.
###### Definition 3.8.
split marking Let $N=(S,T,F,M_{0},\ell)$ be a Petri net, labelled over ${\rm
Act}_{\tau}$.
The _split transition relations_ $\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\zeta$\hskip 2.5pt}\hfil}}$}}$ for $\zeta\mathbin{\in}{\rm
Act}^{\pm}_{\rm
split}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$ between split
markings are given by
$(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{+}$\hskip 2.5pt}\hfil}}$}}(M^{\prime},U^{\prime})$ iff $\exists
t\mathbin{\in}T.~{}\ell(t)=a\wedge M[t\rangle\wedge
M^{\prime}=M-{\vphantom{t}}^{\bullet}t\wedge U^{\prime}=U+\\{t\\}$.
$(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-}$\hskip 2.5pt}\hfil}}$}}(M^{\prime},U^{\prime})$ iff $\exists
t\mathbin{\in}U.~{}\ell(t)=a\wedge U^{\prime}=U-\\{t\\}\wedge
M^{\prime}=M+{t}^{\bullet}$.
$(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}(M^{\prime},U^{\prime})$ iff
$M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}M^{\prime}\wedge U^{\prime}=U$.
Note that $(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{+}$\hskip 2.5pt}\hfil}}$}}$ iff $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$, whereas $(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-}$\hskip 2.5pt}\hfil}}$}}$ iff $a\in\ell(U)$. With induction on
reachability of markings it is furthermore easy to check that
$(M,U)\in[\mathfrak{M_{0}}\rangle$ iff $\ell(U)\in{\rm Nature}^{\rm Act}$ and
$M+\\!{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle$.
The split LTS associated to a net $N$ is
$(\mathfrak{S},\mathfrak{T},\mathfrak{M_{0}})$ with $\mathfrak{S}$ the set of
split markings of $N$, $\mathfrak{Act}:={\rm Act}^{\pm}$, $\mathfrak{T}$ as
defined in Definition LABEL:df-split_marking, and $\mathfrak{M_{0}}$ the
initial split marking. Again, each equivalence on LTSs from [5] induces a
corresponding split equivalence on nets by declaring two nets equivalent iff
their associated LTSs are. In particular, _branching split bisimilarity_ is
the relation of Definition LABEL:df-branching_LTS with the $\mathfrak{M}$’s
denoting split markings, and the $\alpha$’s action phases from ${\rm
Act}^{\pm}_{\rm
split}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$.
For $\mathfrak{M}=(M,U)\in{\rm Nature}^{S}\times T^{*}$ an ST-marking, let
$\overline{\mathfrak{M}}=(M,\overline{U})\in{\rm Nature}^{S}\times{\rm
Nature}^{T}$ be the split marking obtained by converting the sequence $U$ into
the multiset $\overline{U}$, where $\overline{U}(t)$ is the number of
occurrences of the transition $t\in T$ in $U$. Moreover, define
$\ell(\mathfrak{M})$ by $\ell(M,U):=\ell(U)$ and $\ell(t_{1}t_{2}\cdots
t_{k}):=\ell(t_{1})\ell(t_{2})\cdots\ell(t_{k})$. Furthermore, for
$\eta\in{\rm Act}^{\pm}_{\tau}$, let $\overline{\eta}\in{\rm Act}^{\pm}_{\rm
split}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$ be given by
$\overline{a^{+}}:=a^{+}$, $\overline{a^{-n}}:=a^{-}$ and
$\overline{\tau}:=\tau$.
###### Observation 1
Let $\mathfrak{M},\mathfrak{M}^{\prime}$ be ST-markings,
$\mathfrak{M}^{\dagger}$ a split marking, $\eta\mathbin{\in}{\rm
Act}^{\pm}_{\tau}$ and $\zeta\in{\rm Act}^{\pm}_{\rm split}\cup\\{\tau\\}$.
Then
* –
$\mathfrak{M}\in{\rm Nature}^{S}\times T^{*}$ is the initial ST-marking of $N$
iff $\overline{\mathfrak{M}}\in{\rm Nature}^{S}\times{\rm Nature}^{T}$ is the
initial split marking of $N$;
* –
if $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ then
$\overline{\mathfrak{M}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\overline{\eta}$\hskip
2.5pt}\hfil}}$}}\overline{\mathfrak{M}^{\prime}}$;
* –
if $\overline{\mathfrak{M}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\zeta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}$ then
there is a $\mathfrak{M}^{\prime}\in{\rm Nature}^{S}\times T^{*}$ and
$\eta\in{\rm Act}^{\pm}_{\tau}$ such that
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$,
$\overline{\eta}=\zeta$ and
$\overline{\mathfrak{M}^{\prime}}=\mathfrak{M}^{\dagger}$;
* –
if $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ then
$\overline{\mathfrak{M}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\overline{\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}}$\hskip
2.5pt}\hfil}}$}}\overline{\mathfrak{M}^{\prime}}$;
* –
if $\overline{\mathfrak{M}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\zeta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}$ then there is a
$\mathfrak{M}^{\prime}\in{\rm Nature}^{S}\times T^{*}$ and $\eta\in{\rm
Act}^{\pm}_{\tau}$ such that $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$, $\overline{\eta}=\zeta$ and
$\overline{\mathfrak{M}^{\prime}}=\mathfrak{M}^{\dagger}$;
* –
if $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ then
$\overline{\mathfrak{M}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\overline{\mathfrak{M}^{\prime}}$;
* –
if $\overline{\mathfrak{M}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}$ then there is a
$\mathfrak{M}^{\prime}\in{\rm Nature}^{S}\times T^{*}$ such that
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ and
$\overline{\mathfrak{M}^{\prime}}=\mathfrak{M}^{\dagger}$. $\Box$
###### Lemma 3.9.
label sequence Let $N_{1}=(S_{1},T_{1},F_{1},{M_{0}}_{1},\ell)$ and
$N_{2}=(S_{2},T_{2},F_{2},{M_{0}}_{2},\ell_{2})$ be two nets, $N_{2}$ being
plain; let $\mathfrak{M}_{1},\mathfrak{M}^{\prime}_{1}$ be ST-markings of
$N_{1}$, and $\mathfrak{M}_{2},\mathfrak{M}^{\prime}_{2}$ ST-markings of
$N_{2}$. If $\ell(\mathfrak{M}_{2})=\ell(\mathfrak{M}_{1})$,
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$ and
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta^{\prime}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}$ with
$\overline{\eta^{\prime}}=\overline{\eta}$, then there is an
$\mathfrak{M}^{\prime\prime}_{2}$ with
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}_{2}$,
$\ell(\mathfrak{M}^{\prime\prime}_{2})=\ell(\mathfrak{M}^{\prime}_{1})$, and
$\overline{\mathfrak{M}^{\prime\prime}_{2}}=\overline{\mathfrak{M}^{\prime}_{2}}$.
###### Proof 3.10.
If $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ or
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ then $\ell(\mathfrak{M}^{\prime})$ is
completely determined by $\ell(\mathfrak{M})$ and $\eta$. For this reason the
requirement
$\ell(\mathfrak{M}^{\prime\prime}_{2})=\ell(\mathfrak{M}^{\prime}_{1})$ will
hold as soon as the other requirements are met.
First suppose $\eta$ is of the form $\tau$ or $a^{+}$. Then
$\overline{\eta}=\eta$ and moreover $\overline{\eta^{\prime}}=\overline{\eta}$
implies $\eta^{\prime}=\eta$. Thus we can take
$\mathfrak{M}^{\prime\prime}_{2}:=\mathfrak{M}^{\prime}_{2}$.
Now suppose $\eta:=a^{-n}$ for some $n>0$. Then $\eta^{\prime}=a^{-m}$ for
some $m>0$. As $\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}$, the $n^{\it th}$ element of
$\ell(\mathfrak{M}_{1})$ must (exist and) be $a$. Since
$\ell(\mathfrak{M}_{2})=\ell(\mathfrak{M}_{1})$, also the $n^{\it th}$ element
of $\ell(\mathfrak{M}_{2})$ must be $a$, so there is an
$\mathfrak{M}^{\prime\prime}_{2}$ with
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}_{2}$. Let
$\mathfrak{M}_{2}:=(M_{2},U_{2})$. Then $U_{2}$ is a sequence of transitions
of which the $n^{\it th}$ and the $m^{\it th}$ elements are both labelled $a$.
Since the net $N_{2}$ is plain, those two transitions must be equal. Let
$\mathfrak{M}^{\prime}_{2}:=(M^{\prime}_{2},U^{\prime}_{2})$ and
$\mathfrak{M^{\prime\prime}}_{2}:=(M^{\prime\prime}_{2},U^{\prime\prime}_{2})$.
We find that $M^{\prime\prime}_{2}\mathbin{=}M^{\prime}_{2}$ and
$\overline{U^{\prime\prime}_{2}}\mathbin{=}\overline{U^{\prime}_{2}}$. It
follows that
$\overline{\mathfrak{M}^{\prime\prime}_{2}}=\overline{\mathfrak{M}^{\prime}_{2}}$.
###### Observation 2
If $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ for ST-markings
$\mathfrak{M},\mathfrak{M}^{\prime}$ then
$\ell(\mathfrak{M}^{\prime})=\ell(\mathfrak{M})$.
###### Observation 3
If $\ell(\mathfrak{M}_{1})=\ell(\mathfrak{M}_{2})$ and
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-n}$\hskip 2.5pt}\hfil}}$}}$ for some $a\in{\rm Act}$ and $n>0$, then
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-n}$\hskip 2.5pt}\hfil}}$}}$.
###### Observation 4
If $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-n}$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ and
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-n}$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}$ for some $a\in{\rm
Act}$ and $n>0$, then $\mathfrak{M}^{\prime}_{1}=\mathfrak{M}^{\prime}_{2}$.
###### Proposition 3.11.
split Let $N_{1}=(S_{1},T_{1},F_{1},{M_{0}}_{1},\ell)$ and
$N_{2}=(S_{2},T_{2},F_{2},{M_{0}}_{2},\ell_{2})$ be two nets, $N_{2}$ being
plain. Then $N_{1}$ and $N_{2}$ are branching ST-bisimilar (with explicit
divergence) iff they are branching split bisimilar (with explicit divergence).
###### Proof 3.12.
Suppose $\mathcal{B}\,$ is a branching ST-bisimulation between $N_{1}$ and
$N_{2}$. Then, by Observation 1, the relation $\mathcal{B}\,_{\rm
split}:=\\{(\overline{\mathfrak{M}_{1}},\overline{\mathfrak{M}_{2}})\mid(\mathfrak{M}_{1},\mathfrak{M}_{2})\in\mathcal{B}\,\\}$
is a branching split bisimulation between $N_{1}$ and $N_{2}$.
Now let $\mathcal{B}\,$ be a branching split bisimulation between $N_{1}$ and
$N_{2}$. Then, using Observation 1, the relation $\mathcal{B}\,_{\rm
ST}:=\\{(\mathfrak{M}_{1},\mathfrak{M}_{2})\mid\ell_{1}(\mathfrak{M}_{1})=\ell_{2}(\mathfrak{M}_{2})\wedge(\overline{\mathfrak{M}_{1}},\overline{\mathfrak{M}_{2}})\in\mathcal{B}\,\\}$
turns out to be a branching ST-bisimulation between $N_{1}$ and $N_{2}$:
1. 1.
$\mathfrak{M_{0}}_{1}\mathcal{B}\,_{\rm ST}\mathfrak{M_{0}}_{2}$ follows from
Observation 1, using that
$\overline{\mathfrak{M_{0}}_{1}}\mathcal{B}\,\overline{\mathfrak{M_{0}}_{2}}$
and
$\ell(\mathfrak{M_{0}}_{1})\mathbin{=}\ell(\mathfrak{M_{0}}_{2})\mathbin{=}\epsilon$.
2. 2.
Suppose $\mathfrak{M}_{1}\mathcal{B}\,_{\rm ST}\mathfrak{M}_{2}$ and
$\mathfrak{M}_{1}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$. Then
$\overline{\mathfrak{M}_{1}}\mathcal{B}\,\overline{\mathfrak{M}_{2}}$ and
$\overline{\mathfrak{M}_{1}}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\overline{\eta}$\hskip
2.5pt}\hfil}}$}}\overline{\mathfrak{M}^{\prime}_{1}}$. Hence
$\exists\mathfrak{M}^{\dagger}_{2},\mathfrak{M}^{\ddagger}_{2}$ such that
$\overline{\mathfrak{M}_{2}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}_{2}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\overline{\eta}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\ddagger}_{2}$,
$\overline{\mathfrak{M}_{1}}\mathcal{B}\,\mathfrak{M}^{\dagger}_{2}$ and
$\overline{\mathfrak{M}^{\prime}_{1}}\mathcal{B}\,\mathfrak{M}^{\ddagger}_{2}$.
As $N_{2}$ is plain, $\mathfrak{M}^{\dagger}_{2}=\overline{\mathfrak{M}_{2}}$.
By Observation 1, using that
$\overline{\mathfrak{M}_{2}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\overline{\eta}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\ddagger}_{2}$,
$\exists\mathfrak{M}^{\prime}_{2},\,\eta^{\prime}$ such that
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta^{\prime}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}$,
$\overline{\eta^{\prime}}=\overline{\eta}$ and
$\overline{\mathfrak{M}^{\prime}_{2}}=\mathfrak{M}^{\ddagger}_{2}$. By Lemma
LABEL:lem-label_sequence, there is an ST-marking
$\mathfrak{M}^{\prime\prime}_{2}$ such that
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}_{2}$,
$\ell(\mathfrak{M}^{\prime\prime}_{2})=\ell(\mathfrak{M}^{\prime}_{1})$, and
$\overline{\mathfrak{M}^{\prime\prime}_{2}}=\overline{\mathfrak{M}^{\prime}_{2}}=\mathfrak{M}^{\ddagger}_{2}$.
It follows that $\mathfrak{M}^{\prime}_{1}\mathcal{B}\,_{\rm
ST}\mathfrak{M}^{\prime\prime}_{2}$.
3. 3.
Suppose $\mathfrak{M}_{1}\mathcal{B}\,_{\rm ST}\mathfrak{M}_{2}$ and
$\mathfrak{M}_{2}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}$. Then
$\overline{\mathfrak{M}_{1}}\mathcal{B}\,\overline{\mathfrak{M}_{2}}$ and
$\overline{\mathfrak{M}_{2}}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\overline{\eta}$\hskip
2.5pt}\hfil}}$}}\overline{\mathfrak{M}^{\prime}_{2}}$. Hence
$\exists\mathfrak{M}^{\dagger}_{1},\mathfrak{M}^{\ddagger}_{1}$ such that
$\overline{\mathfrak{M}_{1}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\dagger}_{1}\\!\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\overline{\eta}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\ddagger}_{1}$,
$\mathfrak{M}^{\dagger}_{1}\mathcal{B}\,\overline{\mathfrak{M}_{2}}$ and
$\mathfrak{M}^{\ddagger}_{1}\mathcal{B}\,\overline{\mathfrak{M}^{\prime}_{2}}$.
By Observation 1, $\exists\mathfrak{M}^{*}_{1}$ such that
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{*}_{1}$ and
$\overline{\mathfrak{M}^{*}_{1}}=\mathfrak{M}^{\dagger}_{1}$. By Observation
2, $\ell(\mathfrak{M}^{*}_{1})=\ell(\mathfrak{M}_{1})=\ell(\mathfrak{M}_{2})$,
so $\mathfrak{M}^{*}_{1}\mathcal{B}\,_{\rm ST}\mathfrak{M}_{2}$. Since $N_{2}$
is plain, $\eta\neq\tau$.
* •
Let $\eta=a^{+}$ for some $a\in{\rm Act}$. Using that
$\overline{\mathfrak{M}^{*}_{1}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\overline{\eta}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\ddagger}_{1}$, by Observation 1
$\exists\mathfrak{M}^{\prime}_{1},\,\eta^{\prime}$ such that
$\mathfrak{M}^{*}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta^{\prime}\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$,
$\overline{\eta^{\prime}}=\overline{\eta}$ and
$\overline{\mathfrak{M}^{\prime}_{1}}=\mathfrak{M}^{\ddagger}_{1}$. It must be
that $\eta^{\prime}=\eta=a^{+}$ and
$\ell(\mathfrak{M}^{\prime}_{1})=\ell(\mathfrak{M}^{*}_{1})a=\ell(\mathfrak{M}_{2})a=\ell(\mathfrak{M}^{\prime}_{2})$.
Hence $\mathfrak{M}^{\prime}_{1}\mathcal{B}\,_{\rm
ST}\mathfrak{M}^{\prime}_{2}$.
* •
Let $\eta=a^{-n}$ for some $a\in{\rm Act}$ and $n>0$. By Observation 3,
$\exists\mathfrak{M}^{\prime}_{1}$ with
$\mathfrak{M}^{*}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$. By
Part 2. of this proof, $\exists\mathfrak{M}^{\prime\prime}_{2}$ such that
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\mbox{\tiny\rm(}\eta\mbox{\tiny\rm)}$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}_{2}$ and
$\mathfrak{M}^{\prime}_{1}\mathcal{B}\,_{\rm
ST}\mathfrak{M}^{\prime\prime}_{2}$. By Observation 4
$\mathfrak{M}^{\prime\prime}_{2}=\mathfrak{M}^{\prime}_{2}$.
Since the net $N_{2}$ is plain, it has no divergence. In such a case, the
requirement “with explicit divergence” requires $N_{1}$ to be free of
divergence as well, regardless of whether split or ST-semantics is in used.
In this paper we will not consider causal semantics. The reason is that our
distributed implementations will not fully preserve the causal behaviour of
nets. We will further comment on this in the conclusion.
## 4 Distributed Systems
In this section, we stipulate what we understand by a distributed system, and
subsequently formalise a model of distributed systems in terms of Petri nets.
* –
A distributed system consists of components residing on different locations.
* –
Components work concurrently.
* –
Interactions between components are only possible by explicit communications.
* –
Communication between components is time consuming and asynchronous.
Asynchronous communication is the only interaction mechanism in a distributed
system for exchanging signals or information.
* –
The sending of a message happens always strictly before its receipt (there is
a causal relation between sending and receiving a message).
* –
A sending component sends without regarding the state of the receiver; in
particular there is no need to synchronise with a receiving component. After
sending the sender continues its behaviour independently of receipt of the
message.
As explained in the introduction, we will add another requirement to our
notion of a distributed system, namely that its components only allow
sequential behaviour.
Formally, we model distributed systems as nets consisting of component nets
with sequential behaviour and interfaces in terms of input and output places.
###### Definition 4.1.
component Let $N\mathbin{=}(S,T,F,M_{0},\ell)$ be a Petri net,
$I,O\mathbin{\subseteq}S$, $I\mathop{\cap}O\mathbin{=}\emptyset$ and
${O}^{\bullet}=\emptyset$.
* 1.
$(N,I,O)$ is a _component with interface $(I,O)$_.
* 2.
$(N,I,O)$ is a _sequential_ component with interface $(I,O)$ iff
$\exists Q\mathbin{\subseteq}S\mathord{\setminus}(I\cup O)$ with $\forall t\in
T.|{\vphantom{t}}^{\bullet}t\mathop{\upharpoonright}Q|=1\wedge|{t}^{\bullet}\\!\mathop{\upharpoonright}Q|=1$
and $|M_{0}\mathop{\upharpoonright}Q|=1$.
An input place $i\mathbin{\in}I$ of a component
$\mathcal{C}\mathbin{=}(N,I,O)$ can be regarded as a mailbox of $\mathcal{C}$
for a specific type of messages. An output place $o\mathbin{\in}O$, on the
other hand, is an address outside $\mathcal{C}$ to which $\mathcal{C}$ can
send messages. Moving a token into $o$ is like posting a letter. The condition
${o}^{\bullet}=\emptyset$ says that a message, once posted, cannot be
retrieved by the component.
A set of places like $Q$ above is called an _$S$ -invariant_. The requirements
guarantee that the number of tokens in these places remains constant, in this
case $1$. It follows that no two transitions can ever fire concurrently (in
one step). Conversely, whenever a net is sequential, in the sense that no two
transitions can fire in one step, it is easily converted into a behaviourally
equivalent net with the required $S$-invariant, namely by adding a single
marked place with a self-loop to all transitions. This modification preserves
virtually all semantic equivalences on Petri nets from the literature,
including $\approx^{\Delta}_{bSTb}$.
Next we define an operator for combining components with asynchronous
communication by fusing input and output places.
###### Definition 4.2.
parcomp Let $\mathfrak{K}$ be an index set.
Let $((S_{k},T_{k},F_{k},{M_{0}}_{k},\ell_{k}),I_{k},O_{k})$ with
$k\in\mathfrak{K}$ be components with interface such that $(S_{k}\cup
T_{k})\cap(S_{l}\cup T_{l})=(I_{k}\cup O_{k})\cap(I_{l}\cup O_{l})$ for all
$k,l\in\mathfrak{K}$ with $k\neq l$ (components are disjoint except for
interface places) and $I_{k}\cap I_{l}=\emptyset$ for all $k,l\in\mathfrak{K}$
with $k\neq l$ (mailboxes cannot be shared; any message has a unique
recipient).
Then the _asynchronous parallel composition_ of these components is defined by
$\Big{\|}_{i\in\mathfrak{K}}((S_{k},T_{k},F_{k},{M_{0}}_{k},\ell_{k}),I_{k},O_{k})=((S,T,F,{M_{0}},\ell),I,O)\vspace{-.5ex}$
with
$S\mathord{=}\bigcup_{k\in\mathfrak{K}}S_{k},~{}T\mathord{=}\bigcup_{k\in\mathfrak{K}}\\!T_{k},~{}F\mathord{=}\bigcup_{k\in\mathfrak{K}}F_{k},~{}M_{0}\mathord{=}\sum_{k\in\mathfrak{K}}{M_{0}}_{k},~{}\ell\mathord{=}\bigcup_{k\in\mathfrak{K}}\ell_{k}$
(componentwise union of all nets),
$I\mathord{=}\bigcup_{k\in\mathfrak{K}}I_{k}$ (we accept additional inputs
from outside), and
$O\mathord{=}\bigcup_{k\in\mathfrak{K}}O_{k}\setminus\bigcup_{k\in\mathfrak{K}}I_{k}$
(once fused with an input, $o\mathbin{\in}O_{I}$ is no longer an output).
###### Observation 5
$\|$ is associative.
This follows directly from the associativity of the (multi)set union operator.
$\Box$
We are now ready to define the class of nets representing systems of
asynchronously communicating sequential components.
###### Definition 4.3.
LSGA A Petri net $N$ is an _LSGA net_ (a _locally sequential globally
asynchronous net_) iff there exists an index set $\mathfrak{K}$ and sequential
components with interface $\mathcal{C}_{k},~{}k\mathbin{\in}\mathfrak{K}$,
such that $(N,I,O)=\|_{k\in\mathfrak{K}}\mathcal{C}_{k}$ for some $I$ and $O$.
Up to $\approx^{\Delta}_{bSTb}$—or any reasonable equivalence preserving
causality and branching time but abstracting from internal activity—the same
class of LSGA systems would have been obtained if we had imposed, in
Definition LABEL:df-component, that $I$, $O$ and $Q$ form a partition of $S$
and that ${\vphantom{I}}^{\bullet}I=\emptyset$. However, it is essential that
our definition allows multiple transitions of a component to read from the
same input place.
In the remainder of this section we give a more abstract characterisation of
Petri nets representing distributed systems, namely as _distributed_ Petri
nets, which we introduced in [7]. This will be useful in Section 5, where we
investigate distributability using this more semantic characterisation. We
show below that the concrete characterisation of distributed systems as LSGA
nets and this abstract characterisation agree.
Following [2], to arrive at a class of nets representing distributed systems,
we associate _localities_ to the elements of a net $N=(S,T,F,M_{0},\ell)$. We
model this by a function $D:S\cup T\rightarrow\textrm{Loc}$, with Loc a set of
possible locations. We refer to such a function as a _distribution_ of $N$.
Since the identity of the locations is irrelevant for our purposes, we can
just as well abstract from Loc and represent $D$ by the equivalence relation
$\equiv_{D}$ on $S\cup T$ given by $x\equiv_{D}y$ iff $D(x)=D(y)$.
Following [7], we impose a fundamental restriction on distributions, namely
that when two transitions can occur in one step, they cannot be co-located.
This reflects our assumption that at a given location actions can only occur
sequentially.
In [7] we observed that Petri nets incorporate a notion of synchronous
interaction, in that a transition can fire only by synchronously taking the
tokens from all of its preplaces. In general the behaviour of a net would
change radically if a transition would take its input tokens one by one—in
particular deadlocks may be introduced. Therefore we insist that in a
distributed Petri net, a transition and all its input places reside on the
same location. There is no reason to require the same for the output places of
a transition, for the behaviour of a net would not change significantly if
transitions were to deposit their output tokens one by one [7].
This leads to the following definition of a distributed Petri net.
###### Definition 4.4.
distributed[7] A Petri net $N=(S,T,F,M_{0},\ell)$ is _distributed_ iff there
exists a distribution $D$ such that
* (1)
$\forall s\in S,~{}t\in T.~{}\hskip
1.0pts\in{\vphantom{t}}^{\bullet}t\Rightarrow t\equiv_{D}s$,
* (2)
$\forall t,u\in T.~{}t\smile u\Rightarrow t\not\equiv_{D}u$.
A typical example of a net which is not distributed is shown in Figure 2 on
Page 2. Transitions $t$ and $v$ are concurrently executable and hence should
be placed on different locations. However, both have preplaces in common with
$u$ which would enforce putting all three transitions on the same location. In
fact, distributed nets can be characterised in the following semi-structural
way.
###### Observation 6
A Petri net is distributed iff there is no sequence $t_{0},\ldots,t_{n}$ of
transitions with $t_{0}\smile t_{n}$ and
${\vphantom{t_{i-1}}}^{\bullet}t_{i-1}\cap{\vphantom{t_{i}}}^{\bullet}t_{i}\neq\emptyset$
for $i=1,\ldots,n$. $\Box$
We proceed to show that the classes of LSGA nets and distributable nets
essentially coincide. That every LSGA net is distributed follows because we
can place each sequential component on a separate location. The following two
lemmas constitute a formal argument. Here we call a component with interface
$(N,I,O)$ distributed iff $N$ is distributed.
###### Lemma 4.5.
sequential component distributed Any sequential component with interface is
distributed.
###### Proof 4.6.
As a sequential component displays no concurrency, it suffices to co-locate
all places and transitions.
Lemma LABEL:lem-parcompdistributed states that the class of distributed nets
is closed under asynchronous parallel composition.
###### Lemma 4.7.
parcompdistributed Let $\mathcal{C}_{k}=(N_{k},I_{k},O_{k})$,
$k\mathbin{\in}\mathfrak{K}$, be components with interface, satisfying the
requirements of Definition LABEL:df-parcomp, which are all distributed. Then
$\|_{k\in\mathfrak{K}}\mathcal{C}_{k}$ is distributed.
###### Proof 4.8.
We need to find a distribution $D$ satisfying the requirements of Definition
LABEL:df-distributed.
Every component $\mathcal{C}_{k}$ is distributed and hence comes with a
distribution $D_{k}$. Without loss of generality the codomains of all $D_{k}$
can be assumed disjoint.
Considering each $D_{k}$ as a function from net elements onto locations, a
partial function $D_{k}^{\prime}$ can be defined which does not map any places
in $O_{k}$, denoting that the element may be located arbitrarily, and behaves
as $D_{k}$ for all other elements. As an output place has no posttransitions
within a component, any total function larger than (i.e. a superset of)
$D_{k}^{\prime}$ is still a valid distribution for $N_{k}$.
Now $D^{\prime}=\bigcup_{k\in\mathfrak{K}}D_{k}^{\prime}$ is a (partial)
function, as every place shared between components is an input place of at
most one. The required distribution $D$ can be chosen as any total function
extending $D^{\prime}$; it satisfies the requirements of Definition LABEL:df-
distributed since the $D_{k}$’s do.
###### Corollary 4.9.
LSGA distributed Every LSGA net is distributed. $\Box$
Conversely, any distributed net $N$ can be transformed in an LSGA net by
choosing co-located transitions with their pre- and postplaces as sequential
components and declaring any place that belongs to multiple components to be
an input place of component $N_{k}$ if it is a preplace of a transition in
$N_{k}$, and an output place of component $N_{l}$ if it is a postplace of a
transition in $N_{l}$ and not an input place of $N_{l}$. Furthermore, in order
to guarantee that the components are sequential in the sense of Definition
LABEL:df-component, an explicit control place is added to each
component—without changing behaviour—as explained below Definition LABEL:df-
component. It is straightforward to check that the asynchronous parallel
composition of all so-obtained components is an LSGA net, and that it is
equivalent to $N$ (using $\approx_{\mathscr{R}}$, $\approx^{\Delta}_{bSTb}$,
or any other reasonable equivalence).
###### Theorem 4.10.
bothdistributedequal For any distributed net $N$ there is an LSGA net
$N^{\prime}$ with $N^{\prime}\approx^{\Delta}_{bSTb}N$.
###### Proof 4.11.
Let $N=(S,T,F,M_{0},\ell)$ be a distributed net with a distribution $D$. Then
an equivalent LSGA net $N^{\prime}$ can be constructed by composing sequential
components with interfaces as follows.
For each equivalence class $[x]$ of net elements according to $D$ a sequential
component $(N_{[x]},I_{[x]},O_{[x]})$ is created. Each such component contains
one new and initially marked place $p_{[x]}$ which is connected via self-loops
to all transitions in $[x]$. The interface of the component is formed by
$I_{[x]}:=(S\cap[x])$111Alternatively, we could take
$I_{[x]}:={(T\backslash[x])}^{\bullet}\cap[x]$. and $O_{[x]}:={([x]\cap
T)}^{\bullet}\setminus[x]$. Formally,
$N_{[x]}:=(S_{[x]},T_{[x]},F_{[x]},{M_{0}}_{[x]},\ell_{[x]})$ with
* •
$S_{[x]}=((S\cap[x])\cup O_{[x]}\cup\\{p_{[x]}\\}$,
* •
$T_{[x]}=T\cap[x]$,
* •
$F_{[x]}=F\mathop{\upharpoonright}(S_{[x]}\cup
T_{[x]})^{2}\cup\\{(p_{[x]},t),(t,p_{[x]})\mid t\in T_{[x]}\\}$,
* •
${M_{0}}_{[x]}=(M_{0}\mathop{\upharpoonright}[x])\cup\\{p_{[x]}\\}$, and
* •
$\ell_{[x]}=\ell\mathop{\upharpoonright}[x]$.
All components overlap at interfaces only, as the sole places not in an
interface are the newly created $p_{[x]}$. The $I_{[x]}$ are disjoint as the
equivalence classes $[x]$ are, so
$(N^{\prime},I^{\prime},O^{\prime}):=\|_{[x]\in(S\cup
T)/D}(N_{[x]},O_{[x]},I_{[x]})$ is well-defined. It remains to be shown that
$N^{\prime}\approx^{\Delta}_{bSTb}N$. The elements of $N^{\prime}$ are exactly
those of $N$ plus the new places $p_{[x]}$, which stay marked continuously
except when a transition from $[x]$ is firing, and never connect two
concurrently enabled transitions. Hence there exists a bijection between the
ST-markings of $N^{\prime}$ and $N$ that preserves the ST-transition relations
between them, i.e. the associated ST-LTSs are isomorphic. From this it follows
that $N^{\prime}\approx^{\Delta}_{bSTb}N$.
###### Observation 7
Every distributed Petri net is a structural conflict net. $\Box$
###### Corollary 4.12.
LSGA-structuralconflict Every LSGA net is a structural conflict net. $\Box$
Further on, we use a more liberal definition of a distributed net, called
_essentially distributed_. We will show that up to $\approx^{\Delta}_{bSTb}$
any essentially distributed net can be converted into a distributed net. In
[7] we employed an even more liberal definition of a distributed net, which we
call here _externally distributed_. Although we showed that up to step
readiness equivalence any externally distributed net can be converted into a
distributed net, this does not hold for $\approx^{\Delta}_{bSTb}$.
###### Definition 4.13.
externally distributed A net $N=(S,T,F,M_{0},\ell)$ is _essentially
distributed_ iff there exists a distribution $D$ satisfying (1) of Definition
LABEL:df-distributed and
* ($2^{\prime}$)
$\forall t,u\in T.~{}t\smile u\wedge\ell(t)\neq\tau\Rightarrow
t\not\equiv_{D}u$.
It is _externally distributed_ iff there exists a distribution $D$ satisfying
(1) and
* ($2^{\prime\prime}$)
$\forall t,u\in T.~{}t\smile u\wedge\ell(t),\ell(u)\neq\tau\Rightarrow
t\not\equiv_{D}u$.
Instead of ruling out co-location of concurrent transitions in general,
essentially distributed nets permit concurrency of internal
transitions—labelled $\tau$—at the same location. Externally distributed nets
even allow concurrency between external and internal transitions at the same
location. If the transitions $t$ and $v$ in the net of Figure 2 would both be
labelled $\tau$, the net would be essentially distributed, although not
distributed; in case only $v$ would be labelled $\tau$ the net would be
externally distributed but not essentially distributed. Essentially
distributed nets need not be structural conflict nets; in fact, _any_ net
without external transitions is essentially distributed.
The following proposition says that up to $\approx^{\Delta}_{bSTb}$ any
essentially distributed net can be converted into a distributed net.
###### Proposition 4.14.
essentiallydistributedequal For any essentially distributed net $N$ there is a
distributed net $N^{\prime}$ with $N^{\prime}\approx^{\Delta}_{bSTb}N$.
###### Proof 4.15.
The same construction as in the proof of Theorem LABEL:thm-
bothdistributedequal applies: $N^{\prime}$ differs from $N$ by the addition,
for each location $[x]$, of a marked place $p_{[x]}$ that is connected through
self-loops to all transitions at that location. This time there exists a
bijection between the _reachable_ ST-markings of $N^{\prime}$ and $N$ that
preserves the ST-transition relations between them. This bijection exists
because a reachable ST-marking is a pair $(M,U)$ with $U$ a sequence of
_external_ transitions only; this follows by a straightforward induction on
reachability by ST-transitions. From this it follows that
$N^{\prime}\approx^{\Delta}_{bSTb}N$.
Likewise, up to $\approx_{\mathscr{R}}$ any externally distributed net can be
converted into a distributed net.
###### Proposition 4.16.
externallydistributedequal[7] For any externally distributed net $N$ there is
a distributed net $N^{\prime}$ with $N^{\prime}\approx_{\mathscr{R}}N$.
###### Proof 4.17.
Again the same construction applies. This time there exists a bijection
between the markings of $N^{\prime}$ and $N$ that preserves the step
transition relations between them, i.e. the associated step transition systems
are isomorphic. Here we use that the transitions in the associated LTS involve
either a multiset of concurrently firing _external_ transitions, or a single
internal one. From this, step readiness equivalence follows.
The counterexample in Figure 2 shows that up to
$N^{\prime}\approx^{\Delta}_{bSTb}N$ not any externally distributed net can be
converted into a distributed net. Sequentialising the component with actions
$a$, $b$ and $\tau$ would disable the execution $\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{+}$\hskip 2.5pt}\hfil}}$}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
c^{+}$\hskip 2.5pt}\hfil}}$}}$.
0.8$p$$q$
$a$
$t$
$b$
$u$
$c$
$v$
Figure 1: A fully marked M.
0.8$p$$q$
$a$
$t$
$b$
$u$
$\tau$
$v$$r$
$c$
$w$
Figure 2: Externally distributed, but not distributable.
###### Definition 4.18.
canonical Given any Petri net $N$, the _canonical co-location relation_
$\equiv_{C}$ on $N$ is the equivalence relation on the places and transitions
of $N$ _generated_ by Condition (1) of Definition LABEL:df-distributed, i.e.
the smallest equivalence relation $\equiv_{D}$ satisfying (1). The _canonical
distribution_ of $N$ is the distribution $C$ that maps each place or
transition to its $\equiv_{C}$-equivalence class.
###### Observation 8
A Petri net that is distributed (resp. essentially or externally distributed)
w.r.t. any distribution $D$, is distributed (resp. essentially or externally
distributed) w.r.t. its canonical distribution.
Hence a net is distributed (resp. essentially or externally distributed) iff
its canonical distribution $D$ satisfies Condition (2) of Definition LABEL:df-
distributed (resp. Condition ($2^{\prime}$) or ($2^{\prime\prime}$) of
Definition LABEL:df-externally_distributed).
## 5 Distributable Systems
We now consider Petri nets as specifications of concurrent systems and ask the
question which of those specifications can be implemented as distributed
systems. This question can be formalised as
> Which Petri nets are semantically equivalent to distributed nets?
Of course the answer depends on the choice of a suitable semantic equivalence.
Here we will answer this question using the two equivalences discussed in the
introduction. We will give a precise characterisation of those nets for which
we can find semantically equivalent distributed nets. For the negative part of
this characterisation, stating that certain nets are not distributable, we
will use step readiness equivalence, which is one of the simplest and least
discriminating equivalences imaginable that abstracts from internal actions,
but preserves branching time, concurrency and divergence to some small degree.
As explained in [7], giving up on any of these latter three properties would
make any Petri net distributable, but in a rather trivial and unsatisfactory
way. For the positive part, namely that all other nets are indeed
distributable, we will use the most discriminating equivalence for which our
implementation works, namely branching ST-bisimilarity with explicit
divergence, which is finer than step readiness equivalence. Hence we will
obtain the strongest possible results for both directions and it turns out
that the concept of distributability is fairly robust w.r.t. the choice of a
suitable equivalence: any equivalence notion between step readiness
equivalence and branching ST-bisimilarity with explicit divergence will yield
the same characterisation.
###### Definition 5.1.
distributable A Petri net $N$ is _distributable_ up to an equivalence
$\approx$ iff there exists a distributed net $N^{\prime}$ with
$N^{\prime}\approx N$.
Formally we give our characterisation of distributability by classifying which
finitary plain structural conflict nets can be implemented as distributed
nets, and hence as LSGA nets. In such implementations, we use invisible
transitions. We study the concept “distributable” for plain nets only, but in
order to get the largest class possible we allow non-plain implementations,
where a given transition may be split into multiple transitions carrying the
same label.
It is well known that sometimes a global protocol is necessary to implement
synchronous interaction present in system specifications. In particular, this
may be needed for deciding choices in a coherent way, when these choices
require agreement of multiple components. The simple net in Figure 2 shows a
typical situation of this kind. Independent decisions of the two choices might
lead to a deadlock. As remarked in [7], for this particular net there exists
no satisfactory distributed implementation that fully respects the reactive
behaviour of the original system. Indeed such M-structures, representing
interference between concurrency and choice, turn out to play a crucial rôle
for characterising distributability.
###### Definition 5.2.
fullM Let $N=(S,T,F,M_{0},\ell)$ be a Petri net. $N$ has a _fully reachable
pure M_ iff
$\exists t,u,v\in
T.{\vphantom{t}}^{\bullet}t\cap{\vphantom{u}}^{\bullet}u\neq\emptyset\wedge{\vphantom{u}}^{\bullet}u\cap{\vphantom{v}}^{\bullet}v\neq\emptyset\wedge{\vphantom{t}}^{\bullet}t\cap{\vphantom{v}}^{\bullet}v=\emptyset\wedge\exists
M\in[M_{0}\rangle.{\vphantom{t}}^{\bullet}t\cup{\vphantom{u}}^{\bullet}u\cup{\vphantom{v}}^{\bullet}v\subseteq
M$.
Note that Definition LABEL:df-fullM implies that $t\neq u$, $u\neq v$ and
$t\neq v$.
We now give an upper bound on the class of distributable nets by adopting a
result from [7].
###### Theorem 5.3.
trulysyngltfullm Let $N$ be a plain structural conflict Petri net. If $N$ has
a fully reachable pure M, then $N$ is not distributable up to step readiness
equivalence.
###### Proof 5.4.
In [7] this theorem was obtained for plain one-safe nets.222In [7] the theorem
was claimed and proven only for plain nets with a fully reachable _visible_
pure M; however, for plain nets the requirement of visibility is irrelevant.
The proof applies verbatim to plain structural conflict nets as well.
Since $\approx^{\Delta}_{bSTb}$ is finer than $\approx_{\mathscr{R}}$, this
result holds also for distributability up to $\approx^{\Delta}_{bSTb}$ (and
any equivalence between $\approx_{\mathscr{R}}$ and
$\approx^{\Delta}_{bSTb}$).
In the following, we establish that this upper bound is tight, and hence a
finitary plain structural conflict net is distributable iff it has no fully
reachable pure M. For this, it is helpful to first introduce macros in Petri
nets for reversibility of transitions.
### 5.1 Petri nets with reversible transitions
A _Petri net with reversible transitions_ generalises the notion of a Petri
net; its semantics is given by a translation to an ordinary Petri net, thereby
interpreting the reversible transitions as syntactic sugar for certain net
fragments. It is defined as a tuple $(S,T,\Omega,\mbox{\it\i},F,M_{0},\ell)$
with $S$ a set of places, $T$ a set of (reversible) transitions, labelled by
$\ell:T\rightarrow{\rm
Act}\stackrel{{\scriptstyle\mbox{\huge.}}}{{\cup}}\\{\tau\\}$, $\Omega$ a set
of _undo interfaces_ with the relation $\mbox{\it\i}\subseteq\Omega\times T$
linking interfaces to transitions, $M_{0}\mathbin{\in}{\rm Nature}^{S}$ an
initial marking, and
$F\\!:(S\times T\times\\{{\scriptstyle\it
in,~{}early,~{}late,~{}out,~{}far}\\}\rightarrow{\rm Nature})$
the flow relation. When $F(s,t,{\scriptstyle\it type})>0$ for
${\scriptstyle\it type}\in\\{{\scriptstyle\it
in,~{}early,~{}late,~{}out,~{}far}\\}$, this is depicted by drawing an arc
from $s$ to $t$, labelled with its arc weight $F(s,t,{\scriptstyle\it type})$,
of the form 0.7, 0.7, 0.7, 0.7, 0.7, respectively. For $t\mathbin{\in}T$ and
${\scriptstyle\it type}\in\\{{\scriptstyle\it
in,~{}early,~{}late,~{}out,~{}far}\\}$, the multiset of places $t^{\it
type}\mathbin{\in}{\rm Nature}^{S}$ is given by $t^{\it
type}(s)=F(s,t,{\scriptstyle\it type})$. When $s\mathbin{\in}t^{\it type}$ for
${\scriptstyle\it type}\in\\{{\scriptstyle\it in,~{}early,~{}late}\\}$, the
place $s$ is called a _preplace_ of $t$ of type type; when
$s\mathbin{\in}t^{\it type}$ for ${\scriptstyle\it type}\in\\{{\scriptstyle\it
out,~{}far}\\}$, $s$ is called a _postplace_ of $t$ of type type. For each
undo interface $\omega\mathbin{\in}\Omega$ and transition $t$ with
$\mbox{\it\i}(\omega,t)$ there must be places $\textsf{undo}_{\omega}(t)$,
$\textsf{reset}_{\omega}(t)$ and $\textsf{ack}_{\omega}(t)$ in $S$. A
transition with a nonempty set of interfaces is called _reversible_ ; the
other (_standard_) transitions may have pre- and postplaces of types in and
out only—for these transitions $t^{\it
in}\mathbin{=}{\vphantom{t}}^{\bullet}t$ and $t^{\it
out}\mathbin{=}{t}^{\bullet}$. In case $\Omega=\emptyset$, the net is just a
normal Petri net.
A global state of a Petri net with reversible transitions is given by a
marking $M\mathbin{\in}{\rm Nature}^{S}$, together with the state of each
reversible transition “currently in progress”. Each transition in the net can
fire as usual. A reversible transition can moreover take back (some of) its
output tokens, and be _undone_ and _reset_. When a transition $t$ fires, it
consumes $\sum_{{\scriptstyle\it type}\in\\{{\scriptstyle\it
in,~{}early,~{}late}\\}}F(s,t,{\scriptstyle\it type})$ tokens from each of its
preplaces $s$ and produces $\sum_{{\scriptstyle\it type}\in\\{{\scriptstyle\it
out,~{}far}\\}}F(s,t,{\scriptstyle\it type})$ tokens in each of its postplaces
$s$. A reversible transition $t$ that has fired can start its reversal by
consuming a token from $\textsf{undo}_{\omega}(t)$ for one of its interfaces
$\omega$. Subsequently, it can take back one by one a token from its
postplaces of type far. After it has retrieved all its output of type far, the
transition is undone, thereby returning $F(s,t,{\scriptstyle\it early})$
tokens in each of its preplaces $s$ of type early. Afterwards, by consuming a
token from $\textsf{reset}_{\omega}(t)$, for the same interface $\omega$ that
started the undo-process, the transition terminates its chain of activities by
returning $F(s,t,{\scriptstyle\it late})$ tokens in each of its late preplaces
$s$. At that occasion it also produces a token in $\textsf{ack}_{\omega}(t)$.
Alternatively, two tokens in $\textsf{undo}_{\omega}(t)$ and
$\textsf{reset}_{\omega}(t)$ can annihilate each other without involving the
transition $t$; this also produces a token in $\textsf{ack}_{\omega}(t)$. The
latter mechanism comes in action when trying to undo a transition that has not
yet fired.
Figure LABEL:fig-reversible shows the translation of a reversible transition
$t$ with $\ell(t)\mathbin{=}a$ into an ordinary net fragment.
0.8$(in)$$(late)$$(early)$$\textsf{undo}_{\omega}(t)$$\textsf{reset}_{\omega}(t)$$\textsf{ack}_{\omega}(t)$$(far)$$(out)$
$a$
$t$
$\omega$
0.8$f$$o$$i$$l$$e$$\textsf{take}(f,t)$
$\tau$
$t\cdot\textsf{undo}(f)$$\textsf{took}(f,t)$
$\tau$
$t\cdot\textsf{undo}_{\omega}$
$\tau$
$t\cdot\textsf{undone}$$\textsf{fired}(t)$$\rho(t)\\!$
$a$
$t\cdot\textsf{fire}$
$\tau$
$t\cdot\textsf{reset}_{\omega}$$\textsf{undo}_{\omega}(t)$$\rho_{\omega}(t)$
$\tau$
$t\cdot\textsf{elide}_{\omega}$$\textsf{ack}_{\omega}(t)$$\textsf{reset}_{\omega}(t)$$\forall
f\in t^{\,\it far}$$\forall o\in t^{out}$$\forall i\in t^{in}$$\forall l\in
t^{late}$$\forall e\in
t^{early}$$\forall\omega.\,\mbox{\it\i}(\omega,t)$$withreversibletransitionstranslatesintothePetrinetcontainingallplaces$S$,initiallymarkedasindicatedby$M_0$,allstandardtransitionsin$T$,labelledaccordingto$ℓ$,alongwiththeirpre-
andpostplaces,andfurthermoreallnetelementsmentionedinTable~{}\ref{tab-
reversible}.Here\hypertarget{Tback}{}$T^←$denotesthesetofreversibletransitionsin$T.
### 5.2 The conflict replicating implementation
Now we establish that a finitary plain structural conflict net that has no
fully reachable pure M is distributable. We do this by proposing the _conflict
replicating implementation_ of any such net, and show that this implementation
is always (a) essentially distributed, and (b) equivalent to the original net.
In order to get the strongest possible result, for (b) we use branching ST-
bisimilarity with explicit divergence.
0.8$\begin{array}[]{l}\displaystyle\forall j\in T^{\prime}\\\ \forall
p\in{\vphantom{j}}^{\bullet}j\\\ \forall h<^{\\#}j\\\ \forall i\leq^{\\#}j\\\
\forall k\geq^{\\#}j\\\ \forall l>^{\\#}j\\\ \forall
q\in{\vphantom{i}}^{\bullet}i\\\ \forall c\in{q}^{\bullet}\\\ \forall
r\in{i\,}^{\bullet}\\\ \forall
t\in\Omega_{i}:=\begin{array}[t]{@{}l@{}}\\{\textsf{initialise}_{c}\mid
c\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i\\}+\mbox{}\\\
\\{\textsf{transfer}^{b}_{c}\mid
b<^{\\#}c\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i\\}\end{array}\\\
\forall
u\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j\\\
\end{array}$$F(p,j)$$F(i,r)$$F(q,i)$$p$
$\tau$
$\textsf{distribute}_{p}$$p_{j}$$\textsf{pre}^{j}_{k}$$\pi_{j}$
$\tau$
$\textsf{initialise}_{j}$
$u$$\textsf{undo}_{u}(\textsf{initialise}_{j})$$\textsf{reset}_{u}(\textsf{initialise}_{j})$$\textsf{ack}_{u}(\textsf{initialise}_{j})$$\textsf{trans}^{h}_{j}\textsf{-in}$$\pi_{h\\#j}$
$\tau$
$\textsf{transfer}^{h}_{j}$
$u$$\textsf{undo}_{u}(\textsf{transfer}^{h}_{j})$$\textsf{reset}_{u}(\textsf{transfer}^{h}_{j})$$\textsf{ack}_{u}(\textsf{transfer}^{h}_{j})$$\textsf{trans}^{h}_{j}\textsf{-out}$$\textsf{pre}^{i}_{j}$$\pi_{j\\#l}$
$\ell(i)$
$\textsf{execute}^{i}_{j}$$\textsf{undo}_{i}(t)$$\textsf{fetch}_{i,j}^{q,c}\textsf{-in}$$q_{c}$
$\tau$
$\textsf{fetch}_{i,j}^{q,c}$$\textsf{fetch}_{i,j}^{q,c}\textsf{-out}$
$\tau$
$\textsf{fetched}^{i}_{j}$$\textsf{ack}_{i}(t)$$\textsf{reset}_{i}(t)$
$\tau$
$\textsf{finalise}^{i}$$r$
Figure 3: The conflict replicating implementation
To define the conflict replicating implementation of a net
$N=(S,T,F,M_{0},\ell)$ we fix an arbitrary well-ordering $<$ on its
transitions. We let $b,c,g,h,i,j,k,l$ range over these ordered transitions,
and write
* –
$i\mathbin{\\#}j$ iff $i\neq
j\wedge{\vphantom{i}}^{\bullet}i\cap{\vphantom{j}}^{\bullet}j\neq\emptyset$
(transitions $i$ and $j$ are _in conflict_), and
$i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j$
iff $i\mathbin{\\#}j\vee i\mathbin{=}j$,
* –
$i<^{\\#}\\!j$ iff $i<j\wedge i\mathbin{\\#}j$, and $i\leq^{\\#}\\!j$ iff
$i<^{\\#}\\!j\vee i=j$.
Figure 3 shows the conflict replicating implementation of $N$. It is presented
as a Petri net
$\mathcal{I}(N)=(S^{\prime},T^{\prime},F^{\prime},\Omega,\mbox{\it\i},M^{\prime}_{0},\ell^{\prime})$
with reversible transitions. The set $\Omega$ of undo interfaces is $T$, and
for $i\mathbin{\in}\Omega$ we have $\mbox{\it\i}(i,t)$ iff
$t\mathbin{\in}\Omega_{i}$, where the sets of transitions
$\Omega_{i}\mathbin{\in}{\rm Nature}^{T^{\prime}}$ are specified in Figure 3.
The implementation $\mathcal{I}(N)$ inherits the places of $N$ (i.e.
$S^{\prime}\supseteq S$), and we postulate that
$M^{\prime}_{0}\mathord{\upharpoonright}S=M_{0}$. Given this, Figure 3 is not
merely an illustration of $\mathcal{I}(N)$—it provides a complete and accurate
description of it, thereby defining the conflict replicating implementation of
any net. In interpreting this figure it is important to realise that net
elements are completely determined by their name (identity), and exist only
once, even if they show up multiple times in the figure. For instance, the
place $\pi_{h\\#j}$ with $h\mathord{=}2$ and $j\mathord{=}5$ (when using
natural numbers for the transitions in $T$) is the same as the place
$\pi_{j\\#l}$ with $j\mathord{=}2$ and $l\mathord{=}5$; it is a standard
preplace of $\textsf{execute}^{i}_{2}$ (for all $i\leq^{\\#}\\!2$), a standard
postplace of $\textsf{fetched}^{i}_{2}$, as well as a late preplace of
$\textsf{transfer}^{2}_{5}$. A description of this net after expanding the
macros for reversible transitions appears in Table 6 on Page 6. The rôle of
the transitions $\textsf{distribute}_{p}$ for $p\mathbin{\in}S$ is to
distribute a token in $p$ to copies $p_{j}$ of $p$ in the localities of all
transitions $j\mathbin{\in}T$ with $p\mathbin{\in}{\vphantom{j}}^{\bullet}j$.
In case $j$ is enabled in $N$, the transition $\textsf{initialise}_{j}$ will
become enabled in $\mathcal{I}(N)$. These transitions put tokens in the places
$\textsf{pre}^{j}_{k}$, which are preconditions for all transitions
$\textsf{execute}^{j}_{k}$, which model the execution of $j$ at the location
of $k$. When two conflicting transitions $h$ and $j$ are both enabled in $N$,
the first steps $\textsf{initialise}_{h}$ and $\textsf{initialise}_{j}$
towards their execution in $\mathcal{I}(N)$ can happen in parallel. To prevent
them from executing both, $\textsf{execute}^{j}_{j}$ (of $j$ at its own
location) is only possible after $\textsf{transfer}^{h}_{j}$, which disables
$\textsf{execute}^{h}_{h}$. The main idea behind the conflict replicating
implementation is that a transition $h\mathbin{\in}T$ is primarily executed by
a sequential component of its own, but when a conflicting transition $j$ gets
enabled, the sequential component implementing $j$ may “steal” the possibility
to execute $h$ from the home component of $h$, and keep the options to do $h$
and $j$ open until one of them occurs. To prevent $h$ and $j$ from stealing
each other’s initiative, which would result in deadlock, a global asymmetry is
built in by ordering the transitions. Transition $j$ can steal the initiative
from $h$ only when $h<j$. In case $j$ is also in conflict with a transition
$l$, with $j<l$, the initiative to perform $j$ may subsequently be stolen by
$l$. In that case either $h$ and $l$ are in conflict too—then $l$ takes
responsibility for the execution of $h$ as well—or $h$ and $l$ are
concurrent—in that case $h$ will not be enabled, due to the absence of fully
reachable pure Ms in $N$. The absence of fully reachable pure Ms also
guarantees that it cannot happen that two concurrent transitions $j$ and $k$
both steal the initiative from an enabled transition $h$. After the firing of
$\textsf{execute}^{i}_{j}$ all tokens that were left behind in the process of
carefully orchestrating this firing will have to be cleaned up, in order to
prepare the net for the next activity in the same neighbourhood. This is the
reason for the reversibility of the transitions preparing the firing of
$\textsf{execute}^{i}_{j}$. Hence there is an undo interface for each
transition $i\in T^{\prime}$, cleaning up the mess made in preparation of
firing $\textsf{execute}^{i}_{j}$ for some $j\geq^{\\#}i$. $\Omega_{i}$ is the
multiset of all transitions $t$ that could possibly have contributed to this.
For each of them the undo interface $i$ is activated, by
$\textsf{execute}^{i}_{j}$ depositing a token in $\textsf{undo}_{i}(t)$. After
all preparatory transitions that have fired are undone, tokens appear in the
places $p_{c}$ for all $p\mathbin{\in}{\vphantom{i}}^{\bullet}i$ and
$c\mathbin{\in}{p}^{\bullet}$. These are collected by
$\textsf{fetch}_{i,j}^{p,c}$, after which all transitions in $\Omega_{i}$ get
a reset signal. Those that have fired and were undone are reset, and those
that never fired perform $\textsf{elide}_{i}(t)$. In either case a token
appears in $\textsf{ack}_{i}(t)$. These are collected by
$\textsf{finalise}^{i}$, which finishes the process of executing $i$ by
depositing tokens in its postplaces.
0.8
$p$$q$$r$$s$$v$$x$$y$$z$
$a$
$1$
$b$
$2$
$c$
$3$
$d$
$4$
$e$
$5$and more powerful. By means of a simplification a similar method can be
obtained, also in three steps, for establishing the equivalence of two Petri
nets up to interleaving branching bisimilarity with explicit divergence. This
is elaborated at the end of this section.
###### Definition 5.5.
deterministic A labelled transition system
$(\mathfrak{S},\mathfrak{T},\mathfrak{M_{0}})$ is called _deterministic_ if
for all reachable states $\mathfrak{M}\in[\mathfrak{M_{0}}\rangle$ we have
$\mathfrak{M}\arrownot\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}$ and if
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ and
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}$ for some $a\in\mathfrak{Act}$
then $\mathfrak{M}^{\prime}=\mathfrak{M}^{\prime\prime}$.
Deterministic systems may not have reachable $\tau$-transitions at all; this
way, if $\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\sigma$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}$ and
$\mathfrak{M}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\sigma$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}$
for some $\sigma\in\mathfrak{Act}^{*}$ then
$\mathfrak{M}^{\prime}=\mathfrak{M}^{\prime\prime}$. Note that the labelled
transition system associated to a plain Petri net is deterministic; the same
applies to the ST-LTS, the split LTS or the step LTS associated to such a net.
###### Lemma 5.6.
plain branching bisimilarity Let
$(\mathfrak{S}_{1},\mathfrak{T}_{1},\mathfrak{M_{0}}_{1})$ and
$(\mathfrak{S}_{2},\mathfrak{T}_{2},\mathfrak{M_{0}}_{2})$ be two labelled
transition systems, the latter being deterministic. Suppose there is a
relation $\mathcal{B}\,\subseteq\mathfrak{S}_{1}\times\mathfrak{S}_{2}$ such
that
1. (a)
$\mathfrak{M_{0}}_{1}\mathcal{B}\,\mathfrak{M_{0}}_{2}$,
2. (b)
if $\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$ then
$\mathfrak{M}^{\prime}_{1}\mathcal{B}\,\mathfrak{M}_{2}$,
3. (c)
if $\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}$ for some $a\in\mathfrak{Act}$ then
$\exists\mathfrak{M}^{\prime}_{2}.~{}\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}\wedge\mathfrak{M}^{\prime}_{1}\mathcal{B}\,\mathfrak{M}^{\prime}_{2}$,
4. (d)
if $\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ for some $a\in\mathfrak{Act}$ then either
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ or $\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}$
5. (e)
and there is no infinite sequence
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$ with
$\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ for some $\mathfrak{M}_{2}$.
Then $\mathcal{B}\,$ is a branching bisimulation, and the two LTSs are
branching bisimilar with explicit divergence.
* Proof:
It suffices to show that $\mathcal{B}\,$ satisfies Conditions 1–3 of
Definition LABEL:df-branching_LTS; the condition on explicit divergence
follows immediately from (e), using that a deterministic LTS admits no
divergence at all.
1. 1.
By (a).
2. 2.
In case $\alpha=\tau$ this follows directly from (b), and otherwise from (c).
In both cases $\mathfrak{M}^{\dagger}_{2}:=\mathfrak{M}_{2}$ and when
$\alpha=\tau$ also $\mathfrak{M}^{\prime}_{2}:=\mathfrak{M}_{2}$.
3. 3.
Suppose $\mathfrak{M}_{1}\mathcal{B}\,\mathfrak{M}_{2}$ and
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\alpha$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{\prime}_{2}$.
Since $(\mathfrak{S}_{2},\mathfrak{T}_{2},\mathfrak{M_{0}}_{2})$ is
deterministic, $\alpha=a\in{\rm Act}$. By (d) we have either
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{1}_{1}$ or
$\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{1}_{1}$ for some
$\mathfrak{M}^{1}_{1}\in\mathfrak{S}_{1}$. In the latter case (b) yields
$\mathfrak{M}^{1}_{1}\mathcal{B}\,\mathfrak{M}_{2}$, and using (d) again,
either $\mathfrak{M}^{1}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{2}_{1}$ or
$\mathfrak{M}^{1}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{2}_{1}$ for some
$\mathfrak{M}^{2}_{1}\in\mathfrak{S}_{1}$. Repeating this argument, if the
choice between $a$ and $\tau$ is made $k$ times in favour of $\tau$ (with
$k\geq 0$), we obtain $\mathfrak{M}^{k}_{1}\mathcal{B}\,\mathfrak{M}_{2}$
(where $\mathfrak{M}^{0}_{1}:=\mathfrak{M}_{1}$) and either
$\mathfrak{M}^{k}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{k+1}_{1}$ or
$\mathfrak{M}^{k}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\mathfrak{M}^{k+1}_{1}$. By (e),
at some point the choice must be made in favour of $a$, say at
$\mathfrak{M}^{k}_{1}$. Thus $\mathfrak{M}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{k}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{k+1}_{1}$, with
$\mathfrak{M}^{k}_{1}\mathcal{B}\,\mathfrak{M}_{2}$. We take
$\mathfrak{M}^{\dagger}_{1}$ and $\mathfrak{M}^{\prime}_{1}$ from Definition
LABEL:df-branching_LTS to be $\mathfrak{M}^{k}_{1}$ and
$\mathfrak{M}^{k+1}_{1}$. It remains to show that
$\mathfrak{M}^{k+1}_{1}\mathcal{B}\,\mathfrak{M}^{\prime}_{2}$. By (c) there
is an $\mathfrak{M}^{\prime\prime}_{2}\in\mathfrak{S}_{2}$ with
$\mathfrak{M}_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}\mathfrak{M}^{\prime\prime}_{2}$ and
$\mathfrak{M}^{k+1}_{1}\mathcal{B}\,\mathfrak{M}^{\prime\prime}_{2}$. Since
$(\mathfrak{S}_{2},\mathfrak{T}_{2},\mathfrak{M_{0}}_{2})$ is deterministic,
$\mathfrak{M}^{\prime}_{2}=\mathfrak{M}^{\prime\prime}_{2}$. $\Box$
###### Lemma 5.7.
1ST Let $N=(S,T,F,M_{0},\ell)$ and
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
be two nets, $N^{\prime}$ being plain. Suppose there is a relation
$\mathcal{B}\,\subseteq({\rm Nature}^{S}\times{\rm Nature}^{T})\times({\rm
Nature}^{S^{\prime}}\times{\rm Nature}^{T^{\prime}})$ such that
1. (a)
$(M_{0},\emptyset)\mathcal{B}\,(M^{\prime}_{0},\emptyset)$,
2. (b)
if $(M_{1},U_{1})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$ and
$(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}(M_{2},U_{2})$ then
$(M_{2},U_{2})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$,
3. (c)
if $(M_{1},U_{1})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$ and
$(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}(M_{2},U_{2})$ for some
$\eta\in{\rm Act}^{\pm}$
then
$\exists(M^{\prime}_{2},U^{\prime}_{2}).~{}(M^{\prime}_{1},U^{\prime}_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip
2.5pt}\hfil}}$}}(M^{\prime}_{2},U^{\prime}_{2})\wedge(M_{2},U_{2})\mathcal{B}\,(M_{2}^{\prime},U^{\prime}_{2})$,
4. (d)
if $(M_{1},U_{1})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$ and
$(M^{\prime}_{1},U^{\prime}_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}$ with $\eta\in{\rm Act}^{\pm}$
then either $\mathord{(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}}$ or
$\mathord{(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}}$
5. (e)
and there is no infinite sequence $(M,U)\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}(M_{2},U_{2})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$ with
$(M,U)\mathcal{B}\,(M^{\prime},U^{\prime})$ for some
$(M^{\prime},U^{\prime})$.
Then $\mathcal{B}\,$ is a branching split bisimulation, and
$N\approx^{\Delta}_{bSTb}N^{\prime}$.
###### Proof 5.8.
That $N$ and $N^{\prime}$ are branching split bisimilar with explicit
divergence follows directly from Lemma LABEL:lem-plain_branching_bisimilarity
by taking $(\mathfrak{S}_{1},\mathfrak{T}_{1},\mathfrak{M_{0}}_{1})$ and
$(\mathfrak{S}_{2},\mathfrak{T}_{2},\mathfrak{M_{0}}_{2})$ to be the split
LTSs associated to $N$ and $N^{\prime}$ respectively. Here we use that the
split LTS associated to a plain net is deterministic. The final conclusion
follows by Proposition LABEL:pr-split.
Lemma LABEL:lem-1ST provides a method for proving
$N\approx^{\Delta}_{bSTb}N^{\prime}$ that can be more efficient than directly
checking the definition. In particular, the intermediate states
$\mathfrak{M}^{\dagger}$ and the sequence of $\tau$-transitions
$\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{=}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{=}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\Rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle$\hskip
2.5pt}\hfil}}$}}$ from Definition LABEL:df-branching_LTS do not occur in Lemma
LABEL:lem-plain_branching_bisimilarity, and hence not in Lemma LABEL:lem-1ST.
Moreover, in Condition (d) one no longer has the match the targets of
corresponding transitions. Lemma LABEL:lem-2ST below, when applicable,
provides an even more efficient method: it is no longer needed to specify the
branching split bisimulation $\mathcal{B}\,$, and the targets have disappeared
from the transitions in Condition 2c as well. Instead, we have acquired
Condition 1, but this is structural property, which is relatively easy to
check.
###### Lemma 5.9.
2ST Let $N=(S,T,F,M_{0},\ell)$ be a net and
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
be a plain net with $S^{\prime}\subseteq S$ and
$M^{\prime}_{0}=M_{0}\upharpoonright S^{\prime}$. Suppose:
1. 1.
$\forall t\mathbin{\in}T,~{}\ell(t)\neq\tau.~{}\exists
t^{\prime}\mathbin{\in}T^{\prime},~{}\ell(t^{\prime})=\ell(t).~{}\exists
G\in_{f}{\rm Nature}^{T},~{}\ell(G)\equiv\emptyset.~{}\llbracket
t^{\prime}\rrbracket=\llbracket t+G\rrbracket$.
2. 2.
For any $G\in_{f}\mbox{\bbb Z}^{T}$ with $\ell(G)\equiv\emptyset$,
$M^{\prime}\mathbin{\in}{\rm Nature}^{S^{\prime}}$,
$U^{\prime}\mathbin{\in}{\rm Nature}^{T^{\prime}}$ and $U\mathbin{\in}{\rm
Nature}^{T}$ with $\ell^{\prime}(U^{\prime})\mathbin{=}\ell(U)$,
$M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$
and
$M:=M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket-{\vphantom{U}}^{\bullet}U\in{\rm Nature}^{S}$ with
$M+{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle_{N}$, it holds that:
1. (a)
there is no infinite sequence $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$
2. (b)
if $M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$ then $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ or $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}$
3. (c)
and if $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\mathbin{\in}{\rm Act}$ then
$M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$.
Then $N\approx^{\Delta}_{bSTb}N^{\prime}$.
* Proof:
Define $\mathcal{B}\,\subseteq({\rm Nature}^{S}\times{\rm
Nature}^{T})\times({\rm Nature}^{S^{\prime}}\times{\rm Nature}^{T^{\prime}})$
by
$(M,U)\mathcal{B}\,(M^{\prime},U^{\prime}):\Leftrightarrow\ell^{\prime}(U^{\prime})\mathbin{=}\ell(U)\wedge
M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}\linebreak[3]\wedge\exists
G\in_{f}\mbox{\bbb Z}^{T}.~{}\ell(G)\equiv\emptyset\wedge
M+{\vphantom{U}}^{\bullet}U=M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket\in[M_{0}\rangle_{N}$. It suffices to show that $\mathcal{B}\,$
satisfies Conditions (a)–(e) of Lemma LABEL:lem-1ST.
1. (a)
Take $G=\emptyset$.
2. (b)
Suppose $(M_{1},U_{1})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$ and
$(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}(M_{2},U_{2})$. Then
$\ell^{\prime}(U^{\prime}_{1})\mathbin{=}\ell(U_{1})\wedge
M^{\prime}_{1}+\\!{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}\linebreak[2]\wedge\exists
G\in_{f}\mbox{\bbb Z}^{T}.~{}\ell(G)\mathbin{\equiv}\emptyset\wedge
M_{1}=M^{\prime}_{1}+\\!{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket-\\!{\vphantom{U_{1}}}^{\bullet}U_{1}\wedge
M_{1}+{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle_{N}$ and moreover
$M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}M_{2}\wedge U_{2}=U_{1}$. So
$M_{1}[t\rangle M_{2}$ for some $t\mathbin{\in}T$ with
$\ell(t)\mathbin{=}\tau$. Hence $M_{2}=M_{1}+\llbracket
t\rrbracket=M^{\prime}_{1}+\\!{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}+(M_{0}\mathord{-}M^{\prime}_{0})+\llbracket
G+t\rrbracket\linebreak[2]-\\!{\vphantom{U_{1}}}^{\bullet}U_{1}$. Since
$(M_{1}+{\vphantom{U_{1}}}^{\bullet}U_{1})[t\rangle(M_{2}+{\vphantom{U_{1}}}^{\bullet}U_{1})$,
we have $M_{2}+{\vphantom{U_{1}}}^{\bullet}U_{1}\in[M_{0}\rangle_{N}$. Since
also $\ell(G+t)\equiv\emptyset$ it follows that
$(M_{2},U_{1})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$.
3. (c)
Suppose $(M_{1},U_{1})\mathcal{B}\,(M_{1}^{\prime},U^{\prime}_{1})$ and
$(M_{1},U_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\eta$\hskip 2.5pt}\hfil}}$}}(M_{2},U_{2})$, with
$\eta\in{\rm Act}^{\pm}$. Then
$\ell^{\prime}(U^{\prime}_{1})\mathbin{=}\ell(U_{1})$,
$M^{\prime}_{1}+\\!{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$
and
$\exists G\in_{f}\mbox{\bbb Z}^{T}.~{}\ell(G)\mathbin{\equiv}\emptyset\wedge
M_{1}+\\!{\vphantom{U_{1}}}^{\bullet}U_{1}=M^{\prime}_{1}+\\!{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket\in[M_{0}\rangle_{N}.$ (1)
First suppose $\eta=a^{+}$. Then $\exists
t\mathbin{\in}T.~{}\ell(t)\mathbin{=}a\wedge M_{1}[t\rangle\wedge
M_{2}=M_{1}-{\vphantom{t}}^{\bullet}t\wedge U_{2}=U_{1}+\\{t\\}$. Using that
$M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$, by Condition 2c we have
$M^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$, i.e. $M^{\prime}_{1}[t^{\prime}\rangle$ for some
$t^{\prime}\in T$ with $\ell^{\prime}(t^{\prime})=a$. Let
$M^{\prime}_{2}:=M^{\prime}_{1}-{\vphantom{t}}^{\bullet}t$ and
$U^{\prime}_{2}:=U^{\prime}_{1}+\\{t^{\prime}\\}$. Then
$(M^{\prime}_{1},U^{\prime}_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{+}$\hskip 2.5pt}\hfil}}$}}(M^{\prime}_{2},U^{\prime}_{2})$. Moreover,
$\ell(U_{2})=\ell(U^{\prime}_{2})$,
$M^{\prime}_{2}+{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2}=M^{\prime}_{1}+{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$
and
$M_{2}+{\vphantom{U_{2}}}^{\bullet}U_{2}=M_{1}+{\vphantom{U_{1}}}^{\bullet}U_{1}$.
In combination with (1) this yields
$M_{2}+\\!{\vphantom{U_{2}}}^{\bullet}U_{2}=M_{1}+{\vphantom{U_{1}}}^{\bullet}U_{1}=M^{\prime}_{1}+\\!{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket=M^{\prime}_{2}+\\!{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket,$
so $(M_{2},U_{2})\mathcal{B}\,(M_{2}^{\prime},U^{\prime}_{2})$.
Now suppose $\eta=a^{-}$. Then $\exists t\mathbin{\in}U_{1}.\linebreak[3]\
\ell(t)\mathbin{=}a\wedge U_{2}\mathbin{=}U_{1}\mathord{-}\\{t\\}\wedge
M_{2}=M_{1}+{t}^{\bullet}$. Since
$\ell^{\prime}(U^{\prime}_{1})\mathbin{=}\ell(U_{1})$ there is a
$t^{\prime}\mathbin{\in}U^{\prime}_{1}$ with $\ell(t^{\prime})\mathbin{=}a$.
Let $M^{\prime}_{2}:=M^{\prime}_{1}+{t^{\prime}}^{\bullet}$ and
$U^{\prime}_{2}:=U^{\prime}_{1}-\\{t^{\prime}\\}$. Then
$(M^{\prime}_{1},U^{\prime}_{1})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle
a^{-}$\hskip 2.5pt}\hfil}}$}}(M^{\prime}_{2},U^{\prime}_{2})$. By
construction, $\ell(U_{2})=\ell(U^{\prime}_{2})$. Moreover,
$M_{2}+{\vphantom{U_{2}}}^{\bullet}U_{2}=M_{1}+{t}^{\bullet}+{\vphantom{U_{1}}}^{\bullet}U_{1}-{\vphantom{t}}^{\bullet}t=(M_{1}+{\vphantom{U_{1}}}^{\bullet}U_{1})+\llbracket
t\rrbracket$, and likewise
$M^{\prime}_{2}+{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2}=(M^{\prime}_{1}+{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1})+\llbracket
t^{\prime}\rrbracket$ (2)
so
$(M^{\prime}_{1}+{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1})[t^{\prime}\rangle(M^{\prime}_{2}+{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2})$.
Since
$M^{\prime}_{1}+{\vphantom{U^{\prime}_{1}}}^{\bullet}U^{\prime}_{1}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$,
this yields
$M^{\prime}_{2}+{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$.
Moreover,
$M_{2}+\\!{\vphantom{U_{2}}}^{\bullet}U_{2}=M_{1}+{t}^{\bullet}+\\!{\vphantom{U_{1}}}^{\bullet}U_{1}-\\!{\vphantom{t}}^{\bullet}t=M_{1}+\\!{\vphantom{U_{1}}}^{\bullet}U_{1}+\llbracket
t\rrbracket\in[M_{0}\rangle_{N}$. Furthermore, combining (1) and (2) gives
$\exists G\in_{f}\mbox{\bbb Z}^{T}.~{}\ell(G)\mathbin{\equiv}\emptyset\wedge
M_{2}+\\!{\vphantom{U_{2}}}^{\bullet}U_{2}-\llbracket
t\rrbracket=M^{\prime}_{2}+\\!{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2}-\llbracket
t^{\prime}\rrbracket+(M_{0}-M^{\prime}_{0})+\llbracket G\rrbracket.$ (3)
By Condition 1 of Lemma LABEL:lem-2ST, $\exists
t^{\prime\prime}\mathbin{\in}T^{\prime},~{}\ell(t^{\prime\prime})=\ell(t).~{}\exists
G_{t}\in_{f}{\rm Nature}^{T},~{}\ell(G_{t})\equiv\emptyset.~{}\llbracket
t\rrbracket=\llbracket t^{\prime\prime}-G_{t}\rrbracket$. Since $N^{\prime}$
is a plain net, it has only one transition $t^{\dagger}$ with
$\ell(t^{\dagger})\mathbin{=}a$, so $t^{\prime\prime}\mathbin{=}t^{\prime}$.
Substitution of $\llbracket t^{\prime}-G_{t}\rrbracket$ for $t$ in (3) yields
$\qquad\exists G\in_{f}\mbox{\bbb
Z}^{T}.~{}\ell(G)\mathbin{\equiv}\emptyset\wedge
M_{2}+\\!{\vphantom{U_{2}}}^{\bullet}U_{2}=M^{\prime}_{2}+\\!{\vphantom{U^{\prime}_{2}}}^{\bullet}U^{\prime}_{2}+(M_{0}-M^{\prime}_{0})+\llbracket
G-G_{t}\rrbracket.$
Since $\ell(G-G_{t})\equiv\emptyset$ we obtain
$(M_{2},U_{2})\mathcal{B}\,(M_{2}^{\prime},U^{\prime}_{2})$.
4. (d)
Follows directly from Condition 2b and Definition LABEL:df-split_marking.
5. (e)
Follows directly from Condition 2a and Definition LABEL:df-split_marking.
$\Box$
In Lemma LABEL:lem-2ST a relation is explored between markings $M$ and
$M+\llbracket H\rrbracket$ (where $M$ is
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})$
of Lemma LABEL:lem-2ST, $H:=G$, and $M+\llbracket H\rrbracket$ is
$M+\\!{\vphantom{U}}^{\bullet}U$ of Lemma LABEL:lem-2ST). In such a case, we
can think of $M$ as an “original marking”, and of $M+\llbracket H\rrbracket$
as a modification of this marking by the token replacement $\llbracket
H\rrbracket$. The next lemma provides a method to trace certain places $s$
marked by $M+\llbracket H\rrbracket$ (or transitions $t$ that are enabled
under $M+\llbracket H\rrbracket$) back to places that must have been marked by
$M$ before taking into account the token replacement $\llbracket H\rrbracket$.
Such places are called _faithful origins_ of $s$ (or $t$). In tracking the
faithful origins of places and transitions, we assume that the places marked
by $M$ are taken from a set $S^{+}$ and the transitions in $H$ from a set
$T^{+}$. In Lemma LABEL:lem-origin we furthermore assume that the flow
relation restricted to $S\cup T^{+}$ is acyclic. We will need this lemma in
proving the correctness of our final method of proving
$N\approx^{\Delta}_{bSTb}N^{\prime}$.
###### Definition 5.10.
faithful Let $N=(S,T,F,M_{0},\ell)$ be a Petri net, $T^{+}\subseteq T$ a set
of transitions and $S^{+}\subseteq S$ a set of places.
* •
A _path_ in $N$ is an alternating sequence $\pi=x_{0}x_{1}x_{2}\cdots
x_{n}\in(S\cup T)^{*}$ of places and transitions, such that
$F(x_{i},x_{i+1})>0$ for $0\mathbin{\leq}i\mathbin{<}n$. The _arc weight_
$F(\pi)$ of such a path is the product $\Pi_{0}^{n-1}F(x_{i},x_{i+1})$.
* •
A place $s\in S$ is called _faithful_ w.r.t. $T^{+}$ and $S^{+}$ iff
$|\\{s\\}\cap S^{+}|+\sum_{t\in T^{+}}F(t,s)=1$.
* •
A path $x_{0}x_{1}x_{2}\cdots x_{n}\in(S\cup T)^{*}$ from $x_{0}$ to $x_{n}$
is _faithful_ w.r.t. $T^{+}$ and $S^{+}$ iff all intermediate nodes $x_{i}$
for $0\leq i<n$ are either transitions in $T^{+}$ or faithful places w.r.t.
$T^{+}$ and $S^{+}$.
* •
For $x\in S\cup T$, the _infinitary multiset_ ${}^{*}x\in({\rm
Nature}\cup\\{\infty\\})^{S^{+}}$ of _faithful origins_ of $x$ is given by
${}^{*}x(s)=\sup\\{F(\pi)\mid\pi\mbox{ is a faithful path from $s\in S^{+}$ to
$x$}\\}$. (So ${}^{*}x(s)=0$ if no such path exists.)
Suppose a marking $M_{2}$ is reachable from a marking $M_{1}\in{\rm
Nature}^{S^{+}}$ by firing transitions from $T^{+}$ only. Then, if a faithful
place $s$ bears a token under $M_{2}$—i.e. $M_{2}(s)>0$—this token has a
unique source: if $s\in S^{+}$ it must stem from $M_{1}$ and otherwise it must
be produced by the unique transition $t\mathbin{\in}T^{+}$ with
$F(t,s)\mathbin{=}1$. In a net without arc weights, ${}^{*}x$ is always a set,
namely the set of places $s$ in $S^{+}$ from which the flow relation of the
net admits a path to $x$ that passes only through faithful places and
transitions from $T^{+}$ (with the possible exception of $x$ itself). For nets
with arc weights, the underlying set of ${}^{*}x$ is the same, and the
multiplicity of $s\in\mbox{}^{*}x$ is obtained by multiplying all arc weights
on the qualifying path from $s$ to $x$; in case of multiple such paths, we
take the upper bound over all such paths (which could yield the value
$\infty$).
###### Observation 9
Let $(S,T,F,M_{0},\ell)$ be a Petri net, $T^{+}\subseteq T$ a set of
transitions and ${S^{+}}\subseteq S$ a set of places. For faithful places $s$
and transitions $t\in T$ we have
${}^{*}s=\left\\{\begin{array}[]{@{}ll@{}}\\{s\\}&\mbox{if}~{}s\in{S^{+}}\\\
{}^{*}t&\mbox{if}~{}t\in T^{+}\wedge
F(t,s)=1\end{array}\right.\qquad\qquad\vspace{-1.5pt}^{*}t=\bigcup\\{F(s,t)\cdot\mbox{}^{*}s\mid
s\in{\vphantom{t}}^{\bullet}t\wedge s{\rm~{}faithful}\\}.$
###### Lemma 5.11.
origin Let $(S,T,F,M_{0},\ell)$ be a Petri net, $T^{+}\subseteq T$ a set of
transitions such that $F\upharpoonright(S\cup T^{+})$ is acyclic, and
${S^{+}}\subseteq S$ a set of places. Let $M\in{\rm Nature}^{S^{+}}$ and
$H\in_{f}{\rm Nature}^{T^{+}}$, such that $M+\llbracket H\rrbracket\in{\rm
Nature}^{S}$. Then
1. (a)
for any faithful place $s$ w.r.t. $T^{+}$ and ${S^{+}}$ we have $(M+\llbracket
H\rrbracket)(s)\cdot\mbox{}^{*}s\leq M$;
2. (b)
for any $k\in{\rm Nature}$, and any transition $t$ with $(M+\llbracket
H\rrbracket)[k\cdot\\{t\\}\rangle$, we have $k\cdot\mbox{}^{*}t\leq M$.
###### Proof 5.12.
We apply induction on $|H|$.
(a). When $(M+\llbracket H\rrbracket)(s)=0$ it trivially follows that
$(M+\llbracket H\rrbracket)(s)\cdot\mbox{}^{*}s\leq M$. So suppose
$(M+\llbracket H\rrbracket)(s)>0$. Then either $s\in{S^{+}}$ or there is a
unique $t\in T^{+}$ with $H(t)>0$ and $F(t,s)=1$. In the first case, using
that $s\in{u}^{\bullet}$ for no $u\in T^{+}$, we have $(M+\llbracket
H\rrbracket)(s)\leq M(s)$, so $(M+\llbracket
H\rrbracket)(s)\cdot\mbox{}^{*}s\leq M(s)\cdot\\{s\\}\leq M$. In the latter
case, $(M+\llbracket H\rrbracket)(s)\leq M(s)+\sum_{u\in T^{+}}H(u)\cdot
F(u,s)=H(t)$ and ${}^{*}s=\mbox{}^{*}t$. Let $U:=\\{u\in T^{+}\mid
H(u)>0\wedge uF^{+}t\\}$ be the set of transitions occurring in $H$ from which
the flow relation of the net offers a non-empty path to $t$. As
$F\upharpoonright(S\cup T^{+})$ is acyclic, $t\notin U$, so
$H\\!\upharpoonright\\!U<H$. Let $s^{\prime}$ be any place with
$s^{\prime}\in{\vphantom{u}}^{\bullet}u$ for some transition $u\in U$. Then,
by construction of $U$, it cannot happen that $s^{\prime}\in{v}^{\bullet}$ for
some transition $v\notin U$ with $H(v)>0$. Hence $(M+\llbracket
H\\!\upharpoonright\\!U\rrbracket)(s^{\prime})\geq(M+\llbracket
H\rrbracket)(s^{\prime})\geq 0$.Moreover, for any other place
$s^{\prime\prime}$ we have
${\vphantom{(H\\!\upharpoonright\\!U)}}^{\bullet}(H\\!\upharpoonright\\!U)(s^{\prime\prime})=0$
and thus $(M+\llbracket
H\\!\upharpoonright\\!U\rrbracket)(s^{\prime\prime})\geq
M(s^{\prime\prime})\geq 0$. It follows that $M+\llbracket
H\\!\upharpoonright\\!U\rrbracket\in{\rm Nature}^{S}$. For each
$s^{\prime\prime\prime}\in{\vphantom{t}}^{\bullet}t$ we have
${(H-H\\!\upharpoonright\\!U)}^{\bullet}(s^{\prime\prime\prime})=0$ and
${\vphantom{(H-H\\!\upharpoonright\\!U)}}^{\bullet}(H-H\\!\upharpoonright\\!U)(s^{\prime\prime\prime})\geq
H(t)\cdot{\vphantom{t}}^{\bullet}t(s^{\prime\prime\prime})$ and therefore
$0\leq(M+\llbracket H\rrbracket)(s^{\prime\prime\prime})\leq(M+\llbracket
H\\!\upharpoonright\\!U\rrbracket)(s^{\prime\prime\prime})-H(t)\cdot{\vphantom{t}}^{\bullet}t(s^{\prime\prime\prime})$,
and hence $H(t)\cdot{\vphantom{t}}^{\bullet}t\linebreak[3]\leq M+\llbracket
H\\!\upharpoonright\\!U\rrbracket$. It follows that $(M+\llbracket
H\\!\upharpoonright\\!U\rrbracket)[H(t)\cdot\\{t\\}\rangle$. Thus, by
induction, $(M+\llbracket H\rrbracket)(s)\cdot\mbox{}^{*}s\leq
H(t)\cdot\mbox{}^{*}t\leq M$. (b). Let $(M+\llbracket
H\rrbracket)[k\cdot\\{t\\}\rangle$. For any faithful
$s\in{\vphantom{t}}^{\bullet}t$ we have $(M+\llbracket H\rrbracket)(s)\geq
k\cdot F(s,t)$, and thus, using (a),
$k\cdot F(s,t)\cdot\mbox{}^{*}s\leq(M+\llbracket
H\rrbracket)(s)\cdot\mbox{}^{*}s\leq M\;.\vspace{-1.5pt}$
Therefore, by Observation 9, $k\cdot\mbox{}^{*}t=\bigcup\\{k\cdot
F(s,t)\cdot\mbox{}^{*}s\mid s\in{\vphantom{t}}^{\bullet}t\wedge
s{\rm~{}faithful}\\}\leq M$.
The following theorem is the main result of this section. It presents a method
for proving $N\approx^{\Delta}_{bSTb}N^{\prime}$ for $N$ a net and
$N^{\prime}$ a plain net. Its main advantage w.r.t. directly using the
definition, or w.r.t. application of Lemma LABEL:lem-1ST or LABEL:lem-2ST, is
the replacement of requirements on the dynamic behaviour of nets by structural
requirements. Such requirements are typically easier to check. Replacing the
requirement “$M+{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle_{N}$” in Condition 5
by “$M+{\vphantom{U}}^{\bullet}U\in{\rm Nature}^{S}$” would have yielded an
even more structural version of this theorem; however, that version turned out
not to be strong enough for the verification task performed in Section 6.
###### Theorem 5.13.
3ST Let $N=(S,T,F,M_{0},\ell)$ be a net and
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
be a plain net with $S^{\prime}\subseteq S$ and
$M^{\prime}_{0}=M_{0}\upharpoonright S^{\prime}$. Suppose there exist sets
$T^{+}\subseteq T$ and $T^{-}\subseteq T$ and a class $\mbox{\it
NF}\subseteq\mbox{\bbb Z}^{T}$, such that
1. 1.
$F\upharpoonright(S\cup T^{+})$ is acyclic.
2. 2.
$F\upharpoonright(S\cup T^{-})$ is acyclic.
3. 3.
$\forall t\mathbin{\in}T,~{}\ell(t)\neq\tau.~{}\exists
t^{\prime}\mathbin{\in}T^{\prime},~{}\ell(t^{\prime})=\ell(t).~{}\left({\vphantom{t^{\prime}}}^{\bullet}t^{\prime}\leq\mbox{}^{*}t\wedge\exists
G\in_{f}{\rm Nature}^{T},~{}\ell(G)\equiv\emptyset.~{}\llbracket
t^{\prime}\rrbracket=\llbracket t+G\rrbracket\right)$.
Here $\mbox{}^{*}t$ is the multiset of faithful origins of $t$ w.r.t. $T^{+}$
and $S^{\prime}\cup\\{s\in S\mid M_{0}(s)>0\\}$.
4. 4.
There exists a function $f:T\rightarrow{\rm Nature}$ with $f(t)>0$ for all
$t\mathbin{\in}T$, extended to $\mbox{\bbb Z}^{T}$ as in Definition LABEL:df-
multiset, such that for each $G\in_{f}\mbox{\bbb Z}^{T}$ with
$\ell(G)\equiv\emptyset$ there is an $H\in_{f}\mbox{\it NF}$ with
$\ell(H)\equiv\emptyset$, $\llbracket H\rrbracket=\llbracket G\rrbracket$ and
$f(H)=f(G)$.
5. 5.
For every $M^{\prime}\in{\rm Nature}^{S^{\prime}}$, $U^{\prime}\in{\rm
Nature}^{T^{\prime}}$ and $U\in{\rm Nature}^{T}$ with
$\ell(U)=\ell^{\prime}(U^{\prime})$ and
$M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$,
there is an $H_{M^{\prime},U}\in_{f}{\rm Nature}^{T^{+}}$ with
$\ell(H_{M^{\prime},U})\equiv\emptyset$, such that for each
$H\mathbin{\in_{f}}\mbox{\it NF}$ with
$M:=M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H\rrbracket-{\vphantom{U}}^{\bullet}U\in{\rm Nature}^{S}$ and
$M+{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle_{N}$:
1. (a)
$M_{M^{\prime},U}:=M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H_{M^{\prime},U}\rrbracket-{\vphantom{U}}^{\bullet}U\in{\rm Nature}^{S}$,
2. (b)
if $M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$ then
$M_{M^{\prime},U}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$,
3. (c)
$H\leq H_{M^{\prime},U}$.
4. (d)
if $H(u)<0$ then $u\in T^{-}$,
5. (e)
if $H(u)<0$ and $H(t)>0$ then
${\vphantom{u}}^{\bullet}u\cap{\vphantom{t}}^{\bullet}t=\emptyset$,
6. (f)
if $H(u)<0$ and $(M+\\!{\vphantom{U}}^{\bullet}U)[t\rangle$ with
$\ell(t)\neq\tau$ then
${\vphantom{u}}^{\bullet}u\cap{\vphantom{t}}^{\bullet}t=\emptyset$,
7. (g)
if $(M+\\!{\vphantom{U}}^{\bullet}U)[\\{t\\}\mathord{+}\\{u\\}\rangle$ and and
$t^{\prime},u^{\prime}\in T^{\prime}$ with $\ell^{\prime}(t^{\prime})=\ell(t)$
and $\ell^{\prime}(u^{\prime})=\ell(u)$, then
${\vphantom{t^{\prime}}}^{\bullet}t^{\prime}\cap{\vphantom{u^{\prime}}}^{\bullet}u^{\prime}=\emptyset$.
Then $N\approx^{\Delta}_{bSTb}N^{\prime}$.
* Proof:
It suffices to show that Condition 2 of Lemma LABEL:lem-2ST holds (for
Condition 1 of Lemma LABEL:lem-2ST is part of Condition 3 above). So let
$G\in_{f}\mbox{\bbb Z}^{T}$ with $\ell(G)\equiv\emptyset$,
$M^{\prime}\mathbin{\in}{\rm Nature}^{S^{\prime}}$,
$U^{\prime}\mathbin{\in}{\rm Nature}^{T^{\prime}}\\!$ and $U\mathbin{\in}{\rm
Nature}^{T}\\!$ with $\ell^{\prime}(U^{\prime})\mathbin{=}\ell(U)$,
$M^{\prime}\mathord{+}\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$,
$M:=M^{\prime}\mathord{+}\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\mathord{+}(M_{0}\mathord{-}M^{\prime}_{0})\mathord{+}\llbracket
G\rrbracket\mathord{-}\\!{\vphantom{U}}^{\bullet}U\mathbin{\in}{\rm
Nature}^{S}$ and $M+{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle_{N}$.
1. (a)
Suppose $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$. Then there are
transitions $t_{i}\in T$ with $\ell(t_{i})=\tau$, for all $i\mathbin{\geq}1$,
such that $M[t_{1}\rangle M_{1}[t_{2}\rangle M_{2}[t_{3}\rangle\cdots$. As
also
$(M+\\!{\vphantom{U}}^{\bullet}U)[t_{1}\rangle(M_{1}+\\!{\vphantom{U}}^{\bullet}U)[t_{2}\rangle(M_{2}+\\!{\vphantom{U}}^{\bullet}U)[t_{3}\rangle\cdots$,
it follows that
$(M_{i}+\\!{\vphantom{U}}^{\bullet}U)\mathbin{\in}[M_{0}\rangle_{N}$ for all
$i\geq 1$. Let $G_{0}:=G$ and for all $i\geq 1$ let
$G_{i+1}:=G_{i}+\\{t_{i}\\}$. Then $\ell(G_{i})\equiv\emptyset$ and
$M_{i}=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
G_{i}\rrbracket-\\!{\vphantom{U}}^{\bullet}U$. Moreover,
$f(G_{i+1})=f(G_{i})+f(t_{i})>f(G_{i})$. For all $i\geq 1$, using Condition 4,
let $H_{i}\mathbin{\in_{f}}\mbox{\it NF}$ be so that $\llbracket
H_{i}\rrbracket\mathbin{=}\llbracket G_{i}\rrbracket$ and
$f(H_{i})\mathbin{=}f(G_{i})$. Then
$M_{i}=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H_{i}\rrbracket-\\!{\vphantom{U}}^{\bullet}U$ and
$f(H_{0})<f(H_{1})<f(H_{2})<\cdots$. However, from Condition 5c we get
$f(H_{i})\leq f(H_{M^{\prime}})$ for all $i\geq 1$. The sequence
$M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$ therefore must be
finite.
2. (b)
Now suppose $M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$. By Condition 4 above there exists an
$H\in_{f}\mbox{\it NF}$ such that $\ell(H)\equiv\emptyset$ and $\llbracket
H\rrbracket=\llbracket G\rrbracket$, and hence
$M=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H\rrbracket-\\!{\vphantom{U}}^{\bullet}U$. Let $H^{-}:=\\{u\in T\mid
H(u)<0\\}$.
* –
First suppose $H^{-}\neq\emptyset$. By Condition 5d, $H^{-}\subseteq T^{-}$.
By Condition 2, $<^{-}:=(F\upharpoonright(S\cup T^{-}))^{+}$ is a partial
order on $S\cup T^{-}$, and hence on $H^{-}$. Let $u$ be a minimal transition
in $H^{-}$ w.r.t. $<^{-}$. By definition, for all $s\in S$,
$M(s)=M^{\prime}(s)+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}(s)+(M_{0}-M^{\prime}_{0})(s)+\\!\sum_{t\in
T}H(t)\cdot F(t,s)+\\!\sum_{t\in T}\\!-H(t)\cdot F(s,t)+\\!\sum_{t\in
U}\\!-U(t)\cdot F(t,s).\vspace{-2ex}$ (4)
As $M^{\prime}_{0}=M_{0}\upharpoonright S^{\prime}$, we have
$M^{\prime}_{0}\leq M_{0}$. Hence the first three summands in this equation
are always positive (or $0$). Now assume $s\in{\vphantom{u}}^{\bullet}u$.
Since $u$ is minimal w.r.t. $<^{-}$, there is no $t\in T$ with $H(t)<0$ and
$F(t,s)\neq 0$. Hence also all summands $H(t)\cdot F(t,s)$ are positive. By
Condition 5e, there is no $t\in T$ with $H(t)>0$ and $F(s,t)\neq 0$, so all
summands $-H(t)\cdot F(s,t)$ are positive as well. By Condition 5f, there is
no $t\in T$ with $U(t)>0$ and $F(s,t)\neq 0$, for this would imply that
$\ell(t)\neq\tau$ and $(M+\\!{\vphantom{U}}^{\bullet}U)[t\rangle$, so no
summands in (4) are negative. Thus $0\leq-H(u)\cdot F(s,u)\leq M(s)$. Since
$H(u)\leq-1$, this implies $M(s)\geq F(s,u)$. Hence $u$ is enabled in $M$. As
$\ell(u)=\tau$, we have $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}$.
* –
Next suppose $H^{-}\\!=\emptyset$ but $H\neq H_{M^{\prime},U}$. Let
$H^{\smile}:=\\{u\in T\mid H_{M^{\prime},U}(u)-H(u)>0\\}$. Then
$H^{\smile}\neq\emptyset$ by Condition 5c. Since $H_{M^{\prime},U}\in_{f}{\rm
Nature}^{T^{+}}\\!\\!$, $H^{\smile}\subseteq T^{+}$. By Condition 1,
$<^{+}:=(F\upharpoonright(S\cup T^{+}))^{+}$ is a partial order on $S\cup
T^{+}$, and hence on $H^{\smile}$. Let $u$ be a minimal transition in
$H^{\smile}$ w.r.t. $<^{+}$. We have
$M=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H_{M^{\prime},U}+(H-H_{M^{\prime},U})\rrbracket-\\!{\vphantom{U}}^{\bullet}U=M_{M^{\prime},U}+\llbracket
H-H_{M^{\prime},U}\rrbracket$. Hence, for all $s\in S$,
$M(s)=M_{M^{\prime},U}(s)+\sum_{t\in T}(H-H_{M^{\prime},U})(t)\cdot
F(t,s)+\sum_{t\in T}-(H-H_{M^{\prime},U})(t)\cdot F(s,t)\;.$ (5)
By Condition 5a, $M_{M^{\prime},U}\in{\rm Nature}^{S}$. By Condition 5c,
$H-H_{M^{\prime},U}\leq 0$. For $s\in{\vphantom{u}}^{\bullet}u$ there is
moreover no $t\in H^{\smile}$ with $s\in{t}^{\bullet}$, so no $t\in T$ with
$(H-H_{M^{\prime},U})(t)<0$ and $F(t,s)\neq 0$. Hence no summands in (5) are
negative. It follows that $0\leq-(H\mathord{-}M_{M^{\prime},U})(u)\cdot
F(s,t)\leq M(s)$. Since $(H\mathord{-}H_{M^{\prime},U})(u)\leq-1$, this
implies $M(s)\geq F(s,u)$. Hence $u$ is enabled in $M$. As $\ell(u)=\tau$, we
have $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}$.
* –
Finally suppose $H=H_{M^{\prime},U}$. Then $M=M_{M^{\prime},U}$ and
$M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ follows by Condition 5b.
3. (c)
Next suppose $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$. Then there is a $t\in T$ with
$\ell(t)=a\neq\tau$ and $M[t\rangle$. So
$(M+\\!{\vphantom{U}}^{\bullet}U)[t\rangle$. We will first show that
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$. By Condition 4 there exists an $H_{0}\in_{f}\mbox{\it
NF}\subseteq{\rm Nature}^{T}$ such that $\ell(H_{0})\equiv\emptyset$ and
$\llbracket H_{0}\rrbracket=\llbracket G\rrbracket$, and hence
$M+\\!{\vphantom{U}}^{\bullet}U=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H_{0}\rrbracket\in[M_{0}\rangle_{N}$. For our first step, it suffices to show
that whenever $H\mathbin{\in_{f}}\mbox{\it NF}$ with
$M_{H}:=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H\rrbracket\mathbin{\in}[M_{0}\rangle$and $M_{H}[t\rangle$, then
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$. We show this by induction on $f(H_{M^{\prime},U}-H)$,
observing that $f(H_{M^{\prime},U}-H)\in{\rm Nature}$ by Conditions 5c (with
empty $U$) and 4.
We consider two cases, depending on the emptiness of $H^{-}:=\\{u\in T\mid
H(u)<0\\}$.
First assume $H^{-}\\!\mathbin{=}\emptyset$. Then $H\mathbin{\in_{f}}{\rm
Nature}^{T}\\!$. By Condition 5c (with empty $U$) we even have
$H\mathbin{\in_{f}}{\rm Nature}^{T^{+}}\\!\\!$. Let $\mbox{}^{*}t$ denote the
multiset of faithful origins of $t$ w.r.t. $T^{+}$ and
$S^{+}:=S^{\prime}\cup\\{s\in S\mid M_{0}(s)>0\\}$. By Lemma 5.11(b), taking
$k\mathbin{=}1$, substituting
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})$
for the “$M$” of that lemma, and using Condition 1 of Theorem LABEL:thm-3ST,
${}^{*}t\leq
M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})$.
So by Condition 3 of Theorem LABEL:thm-3ST there is a $t^{\prime}\in
T^{\prime}$ with $\ell(t^{\prime})=\ell(t)$ and
${\vphantom{t^{\prime}}}^{\bullet}t^{\prime}\leq
M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})$.
Since ${\vphantom{t^{\prime}}}^{\bullet}t^{\prime}\in{\rm
Nature}^{S^{\prime}}$ and
$M^{\prime}_{0}=M_{0}\\!\upharpoonright\\!S^{\prime}$, this implies
${\vphantom{t^{\prime}}}^{\bullet}t^{\prime}\leq
M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$. It follows that
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[t^{\prime}\rangle_{N^{\prime}}$
and hence
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$.
Now assume $H^{-}\neq\emptyset$. By the same proof as for (b) above, case
$H^{-}\neq\emptyset$, there is a transition $u\in H^{-}$ that is enabled in
$M_{H}$. So $M_{H}[u\rangle M_{1}$ for some $M_{1}\in[M_{0}\rangle_{N}$, and
$M_{1}=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H+u\rrbracket$. By Condition 5f of Theorem LABEL:thm-3ST (still with empty
$U$), ${\vphantom{u}}^{\bullet}u\cap{\vphantom{t}}^{\bullet}t=\emptyset$, and
thus $M_{1}[t\rangle$. By Condition 4 of Theorem LABEL:thm-3ST there exists an
$H_{1}\mathbin{\in_{f}}\mbox{\it NF}$ such that
$\ell(H_{1})\mathbin{\equiv}\emptyset$, $\llbracket
H_{1}\rrbracket\mathbin{=}\llbracket H+u\rrbracket$, and
$f(H_{1})\mathbin{=}f(H+u)\mathbin{>}f(H)$. Thus $M_{1}=M_{H_{1}}$ and
$f(H_{M^{\prime},U}-H_{1})<f(H_{M^{\prime},U}-H)$. By induction we obtain
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$.
By the above reasoning, there is a $t^{\prime}\in T^{\prime}$ such that
$\ell^{\prime}(t^{\prime})=\ell(t)$ and
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[t^{\prime}\rangle$.
Now take any $u^{\prime}\in U^{\prime}$. Then there must be an $u\in U$ with
$\ell^{\prime}(u^{\prime})=\ell(u)$. Since $M[t\rangle$, we have
$(M+\\!{\vphantom{U}}^{\bullet}U)[\\{t\\}\mathord{+}\\{u\\}\rangle$ and by
Condition 5g we obtain
${\vphantom{t^{\prime}}}^{\bullet}t^{\prime}\cap{\vphantom{u^{\prime}}}^{\bullet}u^{\prime}=\emptyset$.
It follows that $M^{\prime}[t^{\prime}\rangle$, and hence
$M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$. $\Box$
### Digression: Interleaving semantics
Above, a method is presented for establishing the equivalence of two Petri
nets, one of which known to be plain, up to branching ST-bisimilarity with
explicit divergence. Here, we simplify this result into a method for
establishing the equivalence of the two nets up interleaving branching
bisimilarity with explicit divergence. This result is not applied in the
current paper.
###### Lemma 5.14.
1 Let $N=(S,T,F,M_{0},\ell)$ and
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
be two nets, $N^{\prime}$ being plain. Suppose there is a relation
$\mathcal{B}\,\subseteq{\rm Nature}^{S}\times{\rm Nature}^{S^{\prime}}$ such
that
1. (a)
$M_{0}\mathcal{B}\,M^{\prime}_{0}$,
2. (b)
if $M_{1}\mathcal{B}\,M_{1}^{\prime}$ and $M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}M_{2}$ then
$M_{2}\mathcal{B}\,M_{1}^{\prime}$,
3. (c)
if $M_{1}\mathcal{B}\,M_{1}^{\prime}$ and $M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}M_{2}$ for some $a\in{\rm Act}$ then $\exists
M^{\prime}_{2}.~{}M^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}M^{\prime}_{2}\wedge M_{2}\mathcal{B}\,M^{\prime}_{2}$,
4. (d)
if $M_{1}\mathcal{B}\,M_{1}^{\prime}$ and
$M^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ for some $a\in{\rm Act}$ then either
$\mathord{M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}}$ or $\mathord{M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}}$
5. (e)
and there is no infinite sequence $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$ with
$M\mathcal{B}\,M^{\prime}$ for some $M^{\prime}$.
Then $N$ and $N^{\prime}$ are interleaving branching bisimilar with explicit
divergence.
###### Proof 5.15.
This follows directly from Lemma LABEL:lem-plain_branching_bisimilarity by
taking $(\mathfrak{S}_{1},\mathfrak{T}_{1},\mathfrak{M_{0}}_{1})$ and
$(\mathfrak{S}_{2},\mathfrak{T}_{2},\mathfrak{M_{0}}_{2})$ to be the
interleaving LTSs associated to $N$ and $N^{\prime}$ respectively. Here we use
that the LTS associated to a plain net is deterministic.
###### Lemma 5.16.
2 Let $N=(S,T,F,M_{0},\ell)$ be a net and
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
be a plain net with $S^{\prime}\subseteq S$ and
$M^{\prime}_{0}=M_{0}\upharpoonright S^{\prime}$. Suppose:
1. 1.
$\forall t\mathbin{\in}T,~{}\ell(t)\neq\tau.~{}\exists
t^{\prime}\mathbin{\in}T^{\prime},~{}\ell(t^{\prime})=\ell(t).~{}\exists
G\in_{f}{\rm Nature}^{T},~{}\ell(G)\equiv\emptyset.~{}\llbracket
t^{\prime}\rrbracket=\llbracket t+G\rrbracket$.
2. 2.
For any $G\mathbin{\in_{f}}\mbox{\bbb Z}^{T}$ with
$\ell(G)\mathbin{\equiv}\emptyset$,
$M^{\prime}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$ and
$M:=M^{\prime}\mathord{+}(M_{0}\mathord{-}M^{\prime}_{0})\mathord{+}\llbracket
G\rrbracket\mathbin{\in}[M_{0}\rangle_{N}$, it holds that:
1. (a)
there is no infinite sequence $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip
2.5pt}\hfil}}$}}M_{2}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}\cdots$,
2. (b)
if $M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$ then
$\mathord{M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}}$ or $\mathord{M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}}$
3. (c)
and if $M\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$ then
$M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$.
Then $N$ and $N^{\prime}$ are interleaving branching bisimilar with explicit
divergence.
* Proof:
Define $\mathcal{B}\,\subseteq{\rm Nature}^{S}\times{\rm Nature}^{S^{\prime}}$
by $M\mathcal{B}\,M^{\prime}:\Leftrightarrow
M^{\prime}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}\wedge\exists
G\mathbin{\in_{f}}\mbox{\bbb
Z}^{T}.~{}M=M^{\prime}\mathord{+}(M_{0}\mathord{-}M^{\prime}_{0})\mathord{+}\llbracket
G\rrbracket\mathbin{\in}[M_{0}\rangle_{N}\linebreak[3]\wedge\ell(G)\equiv\emptyset$.
It suffices to show that $\mathcal{B}\,$ satisfies Conditions (a)–(e) of Lemma
LABEL:lem-1.
1. (a)
Take $G=\emptyset$.
2. (b)
Suppose $M_{1}\mathcal{B}\,M_{1}^{\prime}$ and
$M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip
2.5pt$\scriptstyle\tau$\hskip 2.5pt}\hfil}}$}}M_{2}$. Then $\exists
G\in_{f}\mbox{\bbb
Z}^{T}\\!\\!.~{}M_{1}=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket\wedge\ell(G)\equiv\emptyset$ and $\exists
t\mathbin{\in}T.~{}\ell(t)=\tau\wedge M_{2}=M_{1}+\llbracket
t\rrbracket=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket G+t\rrbracket$.
Moreover, $M_{1}\in[M_{0}\rangle_{N}$ and hence $M_{2}\in[M_{0}\rangle_{N}$.
Furthermore, $M_{1}^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$ and
$\ell(G+t)\equiv\emptyset$, so $M_{2}\mathcal{B}\,M^{\prime}_{1}$.
3. (c)
Suppose $M_{1}\mathcal{B}\,M_{1}^{\prime}$ and
$M_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}M_{2}$. Then $\exists G\in_{f}\mbox{\bbb
Z}^{T}\\!\\!.~{}M_{1}=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G\rrbracket\wedge\ell(G)\equiv\emptyset$ and $\exists
t\mathbin{\in}T.~{}\ell(t)=a\neq\tau\wedge M_{2}=M_{1}+\llbracket
t\rrbracket=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket G+t\rrbracket$.
Moreover, $M_{1}\in[M_{0}\rangle_{N}$ and hence $M_{2}\in[M_{0}\rangle_{N}$.
Furthermore, $M_{1}^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$. By
Condition 1 of Lemma LABEL:lem-2, $\exists
t^{\prime}\mathbin{\in}T^{\prime},~{}\ell(t^{\prime})\mathbin{=}\ell(t).\linebreak[3]\
\exists G_{t}\mathbin{\in_{f}}{\rm
Nature}^{T},~{}\ell(G_{t})\equiv\emptyset.~{}\llbracket t\rrbracket=\llbracket
t^{\prime}-G_{t}\rrbracket$. Substitution of $\llbracket
t^{\prime}-G_{t}\rrbracket$ for $t$ yields $M_{2}=M^{\prime}_{1}+\llbracket
t^{\prime}\rrbracket+(M_{0}\mathord{-}M^{\prime}_{0})+\llbracket
G-G_{t}\rrbracket$. By Condition 2c,
$M^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$, so $M^{\prime}_{1}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}M^{\prime}_{2}$ for some
$M^{\prime}_{2}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$. As $t^{\prime}$ is the
only transition in $T^{\prime}$ with $\ell^{\prime}(t^{\prime})=a$, we must
have $M^{\prime}_{1}[t^{\prime}\rangle M^{\prime}_{2}$. So
$M^{\prime}_{1}+\llbracket t^{\prime}\rrbracket=M^{\prime}_{2}$. Since
$\ell(G-G_{t})\equiv\emptyset$ it follows that
$M_{2}\mathcal{B}\,M^{\prime}_{2}$.
4. (d)
Follows directly from Condition 2b.
5. (e)
Follows directly from Condition 2a. $\Box$
The above is a variant of this Lemma LABEL:lem-2ST that requires Condition 2
only for $U=U^{\prime}=\emptyset$, and allows to conclude that $N$ and
$N^{\prime}$ are interleaving branching bisimilar (instead of branching ST-
bisimilar) with explicit divergence. Likewise, the below is a variant of
Theorem LABEL:thm-3ST that requires Condition 5 only for
$U=U^{\prime}=\emptyset$, and misses Condition 5g.
###### Theorem 5.17.
3 Let $N=(S,T,F,M_{0},\ell)$ be a net and
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
be a plain net with $S^{\prime}\subseteq S$ and
$M^{\prime}_{0}=M_{0}\upharpoonright S^{\prime}$. Suppose there exist sets
$T^{+}\subseteq T$ and $T^{-}\subseteq T$ and a class $\mbox{\it
NF}\subseteq\mbox{\bbb Z}^{T}$, such that
1. 1–4.
Conditions 1–4 from Theorem LABEL:thm-3ST hold, and
2. 5.
For every reachable marking $M^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$
there is an $H_{M^{\prime}}\in_{f}{\rm Nature}^{T^{+}}$ with
$\ell(H_{M^{\prime}})\equiv\emptyset$, such that for each $H\in_{f}\mbox{\it
NF}$ with $M:=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H\rrbracket\in[M_{0}\rangle_{N}$ one has:
1. (a)
$M_{M^{\prime}}:=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H_{M^{\prime}}\rrbracket\in{\rm Nature}^{S}$,
2. (b)
if $M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$ with $a\in{\rm Act}$ then
$M_{M^{\prime}}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$,
3. (c)
$H\leq H_{M^{\prime}}$,
4. (d)
if $H(u)<0$ then $u\in T^{-}$,
5. (e)
if $H(u)<0$ and $H(t)>0$ then
${\vphantom{u}}^{\bullet}u\cap{\vphantom{t}}^{\bullet}t=\emptyset$,
6. (f)
if $H(u)<0$ and $M[t\rangle$ with $\ell(t)\neq\tau$ then
${\vphantom{u}}^{\bullet}u\cap{\vphantom{t}}^{\bullet}t=\emptyset$.
Then $N$ and $N^{\prime}$ are interleaving branching bisimilar with explicit
divergence.
###### Proof 5.18.
A straightforward simplification of the proof of Theorem LABEL:thm-3ST.
## 6 The Correctness Proof
We now apply the preceding theory to prove the correctness of the conflict
replicating implementation.
###### Theorem 6.1.
correctness Let $N$ be a finitary plain structural conflict net without a
fully reachable pure M.
Then $\mathcal{I}(N)\approx^{\Delta}_{bSTb}N$.
* Proof:
In this proof the given finitary plain structural conflict net without a fully
reachable pure M will be
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$,
and its conflict replicated implementation $\mathcal{I}(N^{\prime})$ is called
$N=(S,T,F,M_{0},\ell)$. This convention matches the one of Section LABEL:sec-
method, but is the reverse of the one used in Section 5; it pays off in terms
of a significant reduction in the number of primes in this paper.
For future reference, Table 6 provides a place-oriented representation of the
conflict replicating implementation of a given net
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$,
with the macros for reversible transitions expanded. Here
$\mbox{\hyperlink{Tback}{$T^{\leftarrow}$}}=\\{\textsf{initialise}_{j}\mid
j\mathbin{\in}T^{\prime}\\}\cup\\{\textsf{transfer}^{h}_{j}\mid
h<^{\\#}j\mathbin{\in}T^{\prime}\\}$, whereas
$(\textsf{transfer}^{h}_{j})^{\,\it
far}=\\{\textsf{trans}^{h}_{j}\textsf{-out}\\}$ and
$(\textsf{initialise}_{j})^{\,\it far}=\\{\textsf{pre}^{j}_{k}\mid
k\geq^{\\#}j\\}\cup\\{\textsf{trans}^{h}_{j}\textsf{-in}\mid h<^{\\#}j\\}$.
We will obtain Theorem LABEL:thm-correctness as an application of Theorem
LABEL:thm-3ST. Following the construction of $N$ described in Section 5.2, we
indeed have $S^{\prime}\subseteq S$ and $M^{\prime}_{0}=M_{0}\upharpoonright
S^{\prime}$. Let $T^{+}\subseteq T$ be the set of transitions
$\textsf{distribute}_{p}\qquad\textsf{initialise}_{j}\cdot\textsf{fire}\qquad\textsf{transfer}^{h}_{j}\cdot\textsf{fire}\qquad$
(6)
for any applicable values of $p\mathbin{\in}S^{\prime}$ and
$h,j\mathbin{\in}T^{\prime}\\!$. Furthermore,
$T^{-}:=(T\setminus(T^{+}\cup\\{\textsf{execute}^{i}_{j}\mid
i\leq^{\\#}\\!j\in T^{\prime}\\}))$. We start with checking Conditions 1, 2
and 3 of Theorem LABEL:thm-3ST.
1. 1.
Let $<^{+}$ be the partial order on $T^{+}$ given by the order of listing in
(6)—so
$\textsf{initialise}_{i}\cdot\textsf{fire}<^{+}\textsf{transfer}^{h}_{j}\\!\cdot\textsf{fire}$,
for any $i\in T^{\prime}$ and $h<^{\\#}j\in T^{\prime}$, but the transitions
$\textsf{transfer}^{h}_{j}\cdot\textsf{fire}$ and
$\textsf{transfer}^{k}_{l}\cdot\textsf{fire}$ for $(i,j)\neq(k,l)$ are
unordered. By examining Table 6 we see that for any place with a pretransition
$t$ in $T^{+}$, all its posttransitions $u$ in $T^{+}$ appear higher in the
$<^{+}$-ordering: $t<^{+}u$. From this it follows that $F\upharpoonright(S\cup
T^{+})$ is acyclic.
2. 2.
Let $<^{-}\\!$ be the partial order on $T^{-}\\!$ given by the row-wise order
of the following enumeration of $T^{-}\\!$:
$\begin{array}[]{l@{\quad}l@{\quad}l@{\quad}ll}t\cdot\textsf{undo}_{i}&\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(f)&\textsf{transfer}^{h}_{j}\cdot\textsf{undone}&\textsf{initialise}_{j}\cdot\textsf{undo}(f)&\textsf{initialise}_{j}\cdot\textsf{undone}\\\
\textsf{fetch}_{i,j}^{p,c}&\textsf{fetched}^{i}_{j}&t\cdot\textsf{reset}_{i}&t\cdot\textsf{elide}_{i}&\textsf{finalise}^{i}\end{array}$
for any $t\in\\{\textsf{initialise}_{j},~{}\textsf{transfer}^{h}_{j}\\}$ and
any applicable values of $f\mathbin{\in}S$, $p\mathbin{\in}S^{\prime}$, and
$h,i,j,c\mathbin{\in}T^{\prime}\\!$. By examining Table 6 we see that for any
place with a pretransition $t$ in $T^{-}$, all its posttransitions $u$ in
$T^{-}$ appear higher in the $<^{-}$-ordering: $t<^{-}u$. From this it follows
that $F\upharpoonright(S\cup T^{-})$ is acyclic.
$\begin{array}[]{@{}llll@{}}{}\\\\[-30.1388pt]
\textbf{Place}&\textrm{Pretransitions}\hfill\scriptstyle\rm{arc~{}weights}&\textrm{Posttransitions}\hfill\scriptstyle\rm{arc~{}weights}&\textrm{for
all}\\\ \hline\cr
p&\textsf{finalise}^{i}\hfill\mbox{\scriptsize$F^{\prime}(i,p)$}&\textsf{distribute}_{p}~{}~{}~{}\mbox{\scriptsize(if
${p}^{\bullet}\mathbin{\neq}\emptyset$)}&p\mathbin{\in}S^{\prime},~{}i\in{\vphantom{p}}^{\bullet}p\\\
p_{c}&\left\\{\begin{array}[]{@{}l@{}}\textsf{distribute}_{p}\\\
\textsf{initialise}_{c}\cdot\textsf{undone}~{}~{}\hfill\mbox{\scriptsize$F^{\prime}(p,c)$}\\!\\!\end{array}\right.&\begin{array}[]{@{}l@{}}\textsf{initialise}_{c}\cdot\textsf{fire}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\hfill\mbox{\scriptsize$F^{\prime}(p,c)$}\\\
\textsf{fetch}_{i,j}^{p,c}\hfill\mbox{\scriptsize$F^{\prime}(p,i)$}\end{array}&\begin{array}[]{@{}l@{}}p\mathbin{\in}S^{\prime},~{}c\in{p}^{\bullet}\\\
j\geq^{\\#}i\in{p}^{\bullet}\end{array}\\\
\pi_{c}~{}\hfill\mbox{(marked)}&\textsf{initialise}_{c}\cdot\textsf{reset}_{i}&\textsf{initialise}_{c}\cdot\textsf{fire}&i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}c\in
T^{\prime}\\\
\textsf{pre}^{i}_{j}&\left\\{\begin{array}[]{@{}l@{}}\textsf{initialise}_{i}\cdot\textsf{fire}\\\
\textsf{execute}^{i}_{j}\end{array}\right.&\begin{array}[]{@{}l@{}}\textsf{execute}^{i}_{j}\\\
\textsf{initialise}_{i}\cdot\textsf{undo}(\textsf{pre}^{i}_{j})\end{array}&\begin{array}[]{@{}l@{}}j\geq^{\\#}i\in
T^{\prime}\end{array}\\\
\textsf{trans}^{h}_{j}\textsf{-in}&\left\\{\begin{array}[]{@{}l@{}}\textsf{initialise}_{j}\cdot\textsf{fire}\\\
\textsf{transfer}^{h}_{j}\cdot\textsf{undone}\end{array}\right.&\begin{array}[]{@{}l@{}}\textsf{transfer}^{h}_{j}\cdot\textsf{fire}\\\
\textsf{initialise}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-in})\end{array}&h<^{\\#}j\in
T^{\prime}\\\
\textsf{trans}^{h}_{j}\textsf{-out}&\left\\{\begin{array}[]{@{}l@{}}\textsf{transfer}^{h}_{j}\cdot\textsf{fire}\\\
\textsf{execute}^{i}_{j}\end{array}\right.&\begin{array}[]{@{}l@{}}\textsf{execute}^{i}_{j}\\\
\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out})\\!\\!\\!\end{array}&h<^{\\#}j\in
T^{\prime},~{}i\leq^{\\#}\\!j\\\
\pi_{j\\#l}~{}\hfill\mbox{(marked)}&\left\\{\begin{array}[]{@{}l@{}}\textsf{fetched}^{i}_{j}\\\
\textsf{transfer}^{j}_{l}\cdot\textsf{reset}_{c}\end{array}\right.&\begin{array}[]{@{}l@{}}\textsf{execute}^{i}_{j}\\\
\textsf{transfer}^{j}_{l}\cdot\textsf{fire}\end{array}&\begin{array}[]{@{}l@{}}i\leq^{\\#}\\!j<^{\\#}l\in
T^{\prime},~{}c\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}l\end{array}\\\
\textsf{fetch}_{i,j}^{p,c}\textsf{-in}&\textsf{execute}^{i}_{j}&\textsf{fetch}_{i,j}^{p,c}&j\geq^{\\#}i\mathbin{\in}T^{\prime},~{}p\mathbin{\in}{\vphantom{i}}^{\bullet}i,~{}c\mathbin{\in}{p}^{\bullet}\\\
\textsf{fetch}_{i,j}^{p,c}\textsf{-out}&\textsf{fetch}_{i,j}^{p,c}&\textsf{fetched}^{i}_{j}&j\geq^{\\#}i\mathbin{\in}T^{\prime},~{}p\mathbin{\in}{\vphantom{i}}^{\bullet}i,~{}c\mathbin{\in}{p}^{\bullet}\\\
\hline\cr\rule{0.0pt}{12.0pt}\textsf{undo}_{i}(t)&\textsf{execute}^{i}_{j}\cdot\textsf{fire}&t\cdot\textsf{undo}_{i},\quad
t\cdot\textsf{elide}_{i}&j\geq^{\\#}i\in T^{\prime},~{}t\in\Omega_{i}\\\
\textsf{reset}_{i}(t)&\textsf{fetched}^{i}_{j}&t\cdot\textsf{reset}_{i},\quad
t\cdot\textsf{elide}_{i}&j\geq^{\\#}i\in T^{\prime},~{}t\in\Omega_{i}\\\
\textsf{ack}_{i}(t)&t\cdot\textsf{reset}_{i},\quad
t\cdot\textsf{elide}_{i}&\textsf{finalise}^{i}&i\in
T^{\prime},~{}t\in\Omega_{i}\\\
\textsf{fired}(t)&t\cdot\textsf{fire}&t\cdot\textsf{undo}_{i}&t\in
T^{\leftarrow},~{}\Omega_{i}\ni t\\\
\rho_{i}(t)&t\cdot\textsf{undo}_{i}&t\cdot\textsf{reset}_{i}&t\in
T^{\leftarrow},~{}\Omega_{i}\ni t\\\
\textsf{take}(f,t)&t\cdot\textsf{undo}_{i}&t\cdot\textsf{undo}(f)&t\in
T^{\leftarrow},~{}\Omega_{i}\ni t,~{}f\mathbin{\in}t^{\,\it far}\\\
\textsf{took}(f,t)&t\cdot\textsf{undo}(f)&t\cdot\textsf{undone}&t\in
T^{\leftarrow},~{}f\in t^{\,\it far}\\\
\rho(t)&t\cdot\textsf{undone}&t\cdot\textsf{reset}_{i}&t\in
T^{\leftarrow},~{}\Omega_{i}\ni t\\\ \end{array}$
Table 2: The conflict replicating implementation.
1. 3.
The only transitions $t\in T$ with $\ell(t)\neq\tau$ are
$\textsf{execute}^{i}_{j}$, with $i\leq^{\\#}\\!j\in T^{\prime}$. So take
$i\leq^{\\#}\\!j\in T^{\prime}$. Then the only transition
$t^{\prime}\mathbin{\in}T^{\prime}$ with
$\ell^{\prime}(t^{\prime})\mathbin{=}\ell(\textsf{execute}^{i}_{j})$ is $i$.
Now two statements regarding $i$ and $\textsf{execute}^{i}_{j}$ need to be
proven. For the first, note that, for any $p\in{\vphantom{i}}^{\bullet}i$, the
places $p$, $p_{i}$ and $\textsf{pre}^{i}_{j}$ are faithful w.r.t. $T^{+}$ and
$S^{\prime}\cup\\{s\in S\mid M_{0}(s)>0\\}$. Hence
$~{}p~{}~{}\textsf{distribute}_{p}~{}~{}p_{i}~{}~{}\textsf{initialise}_{i}\cdot\textsf{fire}~{}~{}\textsf{pre}^{i}_{j}~{}~{}\textsf{execute}^{i}_{j}~{}$
is a faithful path from $p$ to $\textsf{execute}^{i}_{j}$. The arc weight of
this path is $F^{\prime}(p,i)$. Thus
${\vphantom{i}}^{\bullet}i\leq\mbox{}^{*}\textsf{execute}^{i}_{j}$.
The second statement holds because, for all $i\leq^{\\#}\\!j\in T^{\prime}$,
$\llbracket
i\rrbracket=\llbracket\textsf{execute}^{i}_{j}+\\!\\!\sum_{p\in{\vphantom{i}}^{\bullet}i}\big{(}F^{\prime}(p,i)\cdot\textsf{distribute}_{p}+\\!\\!\sum_{c\in{p}^{\bullet}}\textsf{fetch}_{i,j}^{p,c}\big{)}+\textsf{fetched}^{i}_{j}+\textsf{finalise}^{i}+\sum_{t\in\Omega_{i}}t\cdot\textsf{elide}_{i}\rrbracket.$
(7)
To check that these equations hold, note that
$\begin{array}[]{@{}l@{~}c@{~}l@{}}\llbracket\textsf{distribute}_{p}\rrbracket\hfil~{}&=\hfil~{}&-\\{p\\}+\\{p_{c}\mid
c\in{p}^{\bullet}\\},\\\
\llbracket\textsf{execute}^{i}_{j}\rrbracket\hfil~{}&=\hfil~{}&-\\{\pi_{j\\#l}\mid
l\geq^{\\#}j\\}+\\{\textsf{fetch}_{i,j}^{p,c}\textsf{-in}\mid
p\mathbin{\in}{\vphantom{i}}^{\bullet}i,~{}c\mathbin{\in}{p}^{\bullet}\\}+\\{\textsf{undo}_{i}(t)\mid
t\in\Omega_{i}\\},\\\
\llbracket\textsf{fetch}_{i,j}^{p,c}\rrbracket\hfil~{}&=\hfil~{}&-\\{\textsf{fetch}_{i,j}^{p,c}\textsf{-in}\\}-F^{\prime}(p,i)\cdot\\{p_{c}\\}+\\{\textsf{fetch}_{i,j}^{p,c}\textsf{-out}\\},\\\
\llbracket\textsf{fetched}^{i}_{j}\rrbracket\hfil~{}&=\hfil~{}&-\\{\textsf{fetch}_{i,j}^{p,c}\textsf{-out}\mid
p\mathbin{\in}{\vphantom{i}}^{\bullet}i,~{}c\mathbin{\in}{p}^{\bullet}\\}+\\{\pi_{j\\#l}\mid
l\geq^{\\#}j\\}+\\{\textsf{reset}_{i}(t)\mid t\in\Omega_{i}\\},\\\ \llbracket
t\cdot\textsf{elide}_{i}\rrbracket\hfil~{}&=\hfil~{}&-\\{\textsf{undo}_{i}(t),~{}\textsf{reset}_{i}(t)\mid
t\in\Omega_{i}\\}+\\{\textsf{ack}_{i}(t)\mid t\in\Omega_{i}\\},\\\
\llbracket\textsf{finalise}^{i}\rrbracket\hfil~{}&=\hfil~{}&-\\{\textsf{ack}_{i}(t)\mid
t\in\Omega_{i}\\}+\raisebox{0.0pt}[0.0pt][0.0pt]{$\displaystyle\sum_{r\in{i}^{\bullet}}F^{\prime}(i,r)\cdot\\{r\\}$}.\\\\[4.30554pt]
\end{array}$
Before we define the class $\mbox{\it NF}\subseteq\mbox{\bbb Z}^{T}$ of signed
multisets of transitions in normal form, and verify conditions 4 and 5, we
derive some properties of the conflict replicating implementation
$N=\mathcal{I}(N^{\prime})$.
###### Claim 1.
G-properties For any $M^{\prime}\in\mbox{\bbb Z}^{S^{\prime}}$ and
$G\in_{f}\mbox{\bbb Z}^{T}$ such that
$M:=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket G\rrbracket\in{\rm
Nature}^{S}$ we have
$\displaystyle G(t\cdot\textsf{elide}_{i})+G(t\cdot\textsf{undo}_{i})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$
$\displaystyle\sum_{j\geq^{\\#}i}G(\textsf{execute}^{i}_{j})$ (8)
$\displaystyle G(\textsf{finalise}^{i})\leq
G(t\cdot\textsf{elide}_{i})+G(t\cdot\textsf{reset}_{i})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$
$\displaystyle\sum_{j\geq^{\\#}i}G(\textsf{fetched}^{i}_{j})$ (9)
$\displaystyle G(t\cdot\textsf{reset}_{i})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
G(t\cdot\textsf{undo}_{i})$ (10)
for each $i\in T^{\prime}$ and $t\mathbin{\in}\Omega_{i}$. Moreover, for each
$t\in T^{\leftarrow}$ and $f\in t^{\,\it far}$,
$\sum_{\\{\omega\mid
t\in\Omega_{\omega}\\}}\\!\\!\\!\\!\\!\\!\\!\\!G(t\cdot\textsf{reset}_{\omega})\leq
G(t\cdot\textsf{undone})\leq
G(t\cdot\textsf{undo}(f))\leq\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\{\omega\mid
t\in\Omega_{\omega}\\}}\\!\\!\\!\\!\\!\\!\\!\\!G(t\cdot\textsf{undo}_{\omega})\leq
G(t\cdot\textsf{fire})$ (11)
and for each appropriate $c,h,i,j,l\in T^{\prime}$ and $p\in S^{\prime}$:
$\displaystyle G(\textsf{fetched}^{i}_{j})\leq G(\textsf{fetch}_{i,j}^{p,c})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
G(\textsf{execute}^{i}_{j})$ (12) $\displaystyle
G(\textsf{initialise}_{j}\cdot\textsf{fire})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
1+\sum_{\omega}G(\textsf{initialise}_{j}\cdot\textsf{reset}_{\omega})$ (13)
$\displaystyle
G(\textsf{transfer}^{h}_{j}\cdot\textsf{fire})-G(\textsf{transfer}^{h}_{j}\cdot\textsf{undone})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
G(\textsf{initialise}_{j}\cdot\textsf{fire})-G(\textsf{initialise}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-in}))$
(14) $\displaystyle
G(\textsf{transfer}^{j}_{l}\\!\cdot\textsf{fire})+\sum_{i\leq^{\\#}\\!j}G(\textsf{execute}^{i}_{j})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
1+\sum_{\omega}G(\textsf{transfer}^{j}_{l}\\!\cdot\textsf{reset}_{\omega})+\sum_{i\leq^{\\#}\\!j}G(\textsf{fetched}^{i}_{j})$
(15) $\displaystyle\mbox{if ~{}$M[\textsf{execute}^{i}_{j}\rangle$~{}
then}\quad 1$ $\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
G(\textsf{initialise}_{i}\cdot\textsf{fire})-G(\textsf{initialise}_{i}\cdot\textsf{undo}(\textsf{pre}^{i}_{j}))$
(16) $\displaystyle\mbox{if ~{}$\exists
i.~{}M[\textsf{execute}^{i}_{j}\rangle$~{} then}\quad 1$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
G(\textsf{transfer}^{h}_{j}\cdot\textsf{fire})-G(\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out}))$
(17)
$F^{\prime}(p,c)\mathord{\cdot}\big{(}G(\textsf{initialise}_{c}\\!\cdot\textsf{fire})\mathord{-}G(\textsf{initialise}_{c}\\!\cdot\textsf{undone})\big{)}+\sum_{j\geq^{\\#}i\in{p}^{\bullet}}F^{\prime}(p,i)\cdot
G(\textsf{fetch}_{i,j}^{p,c})\leq G(\textsf{distribute}_{p})\vspace{-15pt}$
(18) $\displaystyle G(\textsf{distribute}_{p})$
$\displaystyle\\!\\!\\!\\!\leq\\!\\!\\!\\!$ $\displaystyle
M^{\prime}(p)+\sum_{\\{i\in T^{\prime}\mid
p\in{i}^{\bullet}\\}}G(\textsf{finalise}^{i}).$ (19)
* Proof:
For any $i\in T^{\prime}$ and $t\in\Omega_{i}$, we have
$M(\textsf{undo}_{i}(t))=\big{(}\sum_{j\geq^{\\#}i}G(\textsf{execute}^{i}_{j})\big{)}-G(t\cdot\textsf{elide}_{i})-G(t\cdot\textsf{undo}_{i})\geq
0,$
given that
$M^{\prime}(\textsf{undo}_{i}(t))=(M_{0}-M^{\prime}_{0})(\textsf{undo}_{i}(t))=\emptyset$.
In this way, the place $\textsf{undo}_{i}(t)$ gives rise to the inequation (8)
about $G$. Likewise, the places $\textsf{ack}_{i}(t)$, $\textsf{reset}_{i}(t)$
and $\rho_{i}(t)$, respectively, contribute (9) and (10), whereas $\rho(t)$,
$\textsf{took}(t)$, $\textsf{take}(t)$ and $\textsf{fired}(t)$ yield (11). The
remaining inequations arise from $\textsf{fetch}_{i,j}^{p,c}\textsf{-out}$,
$\textsf{fetch}_{i,j}^{p,c}\textsf{-in}$, $\pi_{j}$,
$\textsf{trans}^{h}_{j}\textsf{-in}$, $\pi_{j\\#l}$, $\textsf{pre}^{i}_{j}$,
$\textsf{trans}^{h}_{j}\textsf{-out}$, $p_{c}$ and $p$, respectively.
(15) can be rewritten as $T^{j}_{l}+\sum_{i\leq^{\\#}\\!j}E^{i}_{j}\leq 1$,
where
$T^{j}_{l}:=G(\textsf{transfer}^{j}_{l}\cdot\textsf{fire})-\sum_{\omega}G(\textsf{transfer}^{j}_{l}\cdot\textsf{reset}_{\omega})$
and $E^{i}_{j}:=G(\textsf{execute}^{i}_{j})-G(\textsf{fetched}^{i}_{j})$. By
(11) $\sum_{\omega}G(\textsf{transfer}^{j}_{l}\cdot\textsf{reset}_{i})\leq
G(\textsf{transfer}^{j}_{l}\cdot\textsf{fire})$, so $T^{j}_{l}\geq 0$, and
likewise, by (12), $E^{i}_{j}\geq 0$ for all $i\leq^{\\#}\\!j$. Hence, for all
$i\leq^{\\#}\\!j<^{\\#}l\in T^{\prime}$,
$0\leq T^{j}_{l}\leq 1\qquad 0\leq E^{i}_{j}\leq 1\qquad
T^{j}_{l}+\sum_{i\leq^{\\#}\\!j}E^{i}_{j}\leq 1.\vspace{-2ex}$ (20)
In our next claim we study triples $(M,M^{\prime},G)$ with
1. (A)
$M\in[M_{0}\rangle_{N}$, $M^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$
and $G\in_{f}\mbox{\bbb Z}^{T}$,
2. (B)
$M=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket G\rrbracket$,
3. (C)
$G(\textsf{finalise}^{i})=0$ for all $i\in T^{\prime}$,
4. (D)
$G(\textsf{distribute}_{p})\leq M^{\prime}(p)$ for all $p\in S^{\prime}$,
5. (E)
$G(\textsf{fetched}^{k}_{l})\geq 0$ for all $k\leq^{\\#}\\!l\in T^{\prime}$,
6. (F)
$\displaystyle G(\textsf{distribute}_{p})\geq F^{\prime}(p,i)\cdot
G(\textsf{execute}^{i}_{j})$ for all $i\leq^{\\#}\\!j\in T^{\prime}$ and
$p\in{\vphantom{i}}^{\bullet}i$,
7. (G)
$0\leq G(\textsf{execute}^{i}_{j})\leq 1$ for all $i\leq^{\\#}\\!j\in
T^{\prime}$,
8. (H)
$\displaystyle G(\textsf{distribute}_{p})\geq F^{\prime}(p,j)\cdot
G(\textsf{execute}^{i}_{j})$ for all $i\leq^{\\#}\\!j\in T^{\prime}$ and
$p\in{\vphantom{j}}^{\bullet}j$,
9. (I)
(in the notation of (20)) if $E^{i}_{j}=1$ with $i\leq^{\\#}\\!j\in
T^{\prime}$ then $T^{h}_{j}=1$ for all $h<^{\\#}j$,
10. (J)
there are no
$j\geq^{\\#}i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\leq^{\\#}l\in
T^{\prime}$ with $(i,j)\neq(k,\ell)$, $G(\textsf{execute}^{i}_{j})>0$ and
$G(\textsf{execute}^{k}_{l})>0$,
11. (K)
there are no
$i\leq^{\\#}j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\leq^{\\#}l\in
T^{\prime}$ with $(i,j)\neq(k,\ell)$, $G(\textsf{execute}^{i}_{j})>0$ and
$G(\textsf{execute}^{k}_{l})>0$.
Given such a triple $(M_{1},M^{\prime}_{1},G_{1})$ and a transition $t\in T$,
we define $\textit{next}(M_{1},M^{\prime}_{1},G_{1},t)=:(M,M^{\prime},G)$ as
follows: Let $G_{2}:=G_{1}+\\{t\\}$. Take $M:=M_{1}+\llbracket
t\rrbracket=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket G_{2}\rrbracket$.
In case $t$ is not of the form $\textsf{finalise}^{i}$ we take
$M^{\prime}:=M^{\prime}_{1}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$ and
$G:=G_{2}\in_{f}\mbox{\bbb Z}^{T}$. In case
$t\mathbin{=}\textsf{finalise}^{i}$ for some $i\in T^{\prime}$ we have
$1=G_{2}(\textsf{finalise}^{i})\leq\sum_{j\geq^{\\#}i}G_{2}(\textsf{execute}^{i}_{j})=\sum_{j\geq^{\\#}i}G_{1}(\textsf{execute}^{i}_{j})$
by (C), (9) and (12), so by (G) and (J) there is a unique $j\geq^{\\#}i$ with
$G_{1}(\textsf{execute}^{i}_{j})=1$. We take
$M^{\prime}:=M^{\prime}_{1}+\llbracket i\rrbracket$ and
$G:=G_{2}-G^{i}_{\\!\\!j}$, where $G^{i}_{\\!\\!j}$ is the right-hand side of
(7).
###### Claim 2.
reachable
1. (1)
If $M_{1}[t\rangle$ and $(M_{1},M^{\prime}_{1},G_{1})$ satisfies (A)-(K), then
so does $\textit{next}(M_{1},M^{\prime}_{1},G_{1},t)$.
2. (2)
For any $M\in[M_{0}\rangle_{N}$ there exist $M^{\prime}$ and $G$ such that
(A)-(K) hold.
* Proof:
(2) follows from (1) via induction on the reachability of $M$. In case
$M=M_{0}$ we take $M^{\prime}:=M^{\prime}_{0}$ and $G:=\emptyset$. Clearly,
(A)–(K) are satisfied.
Hence we now show (1). Let
$(M,M^{\prime},G):=\textit{next}(M_{1},M^{\prime}_{1},G_{1},t)$. We check that
$(M,M^{\prime},G)$ satisfies the requirements (A)–(K).
1. (A)
By construction, $M\in[M_{0}\rangle_{N}$ and $G\in_{f}\mbox{\bbb Z}^{T}$. If
$t$ is not of the form $\textsf{finalise}^{i}$ we have
$M^{\prime}\mathbin{=}M_{1}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$.
Otherwise, by (D) and (F) we have $M^{\prime}_{1}(p)\geq
G_{1}(\textsf{distribute}_{p})\geq F^{\prime}(p,i)$ for all
$p\in{\vphantom{i}}^{\bullet}i$, and hence $M^{\prime}_{1}[i\rangle$. This in
turn implies that $M^{\prime}=M^{\prime}_{1}+\llbracket
i\rrbracket\in[M^{\prime}_{0}\rangle_{N^{\prime}}$.
2. (B)
In case $t$ is not of the form $\textsf{finalise}^{i}$ we have
$M=M_{1}+\llbracket
t\rrbracket=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G_{1}+t\rrbracket=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket G\rrbracket.$
In case $t=\textsf{finalise}^{i}$ we have
$M=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket
G_{2}\rrbracket=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket G\rrbracket$,
using that $\llbracket i\rrbracket=\llbracket G^{i}_{\\!\\!j}\rrbracket$.
3. (C)
In case $t=\textsf{finalise}^{i}$ we have
$G(\textsf{finalise}^{i})=G_{1}(\textsf{finalise}^{i})+1-G^{i}_{\\!\\!j}(\textsf{finalise}^{i})=0+1-1=0$.
Otherwise $G(\textsf{finalise}^{i})=G_{1}(\textsf{finalise}^{i})+0=0+0=0$.
4. (D)
This follows immediately from (C) and (19).
5. (E)
The only time that this invariant is in danger is when
$t=\textsf{finalise}^{i}$. Then
$G=G_{1}+\\{\textsf{finalise}^{i}\\}-G^{i}_{\\!\\!j}$ for a certain
$j\geq^{\\#}i$ with $G_{1}(\textsf{execute}^{i}_{j})=1$. By (J)333We use (J)
and (E) for $G_{1}$ only, making use of the induction hypothesis.
$G_{1}(\textsf{execute}^{i}_{l})\leq 0$ for all $l\geq^{\\#}i$ with $l\neq j$.
Hence by (12) $G_{1}(\textsf{fetched}^{i}_{l})\leq 0$ for all such $l$. By (C)
$G_{2}(\textsf{finalise}^{i})=G_{1}(\textsf{finalise}^{i})+1=1$, so by (9)
$\sum_{l\geq^{\\#}i}G_{1}(\textsf{fetched}^{i}_{l})=\sum_{l\geq^{\\#}i}G_{2}(\textsf{fetched}^{i}_{l})>0$;
hence it must be that $G_{1}(\textsf{fetched}^{i}_{j})>0$. By (E)3
$G_{1}(\textsf{fetched}^{k}_{l})\geq 0$ for all $k\leq^{\\#}\\!l\in
T^{\prime}$. Given that $G^{i}_{\\!\\!j}(\textsf{fetched}^{i}_{j})=1$ and
$G^{i}_{\\!\\!j}(\textsf{fetched}^{k}_{l})=0$ for all $(k,l)\neq(i,j)$, we
obtain $G(\textsf{fetched}^{k}_{l})\geq 0$ for all $k\leq^{\\#}\\!l\in
T^{\prime}$.
6. (F)
Take $i\mathbin{\leq^{\\#}\\!}j\mathbin{\in}T^{\prime}$ and
$p\mathbin{\in}{\vphantom{i}}^{\bullet}i$. There are two occasions where the
invariant is in danger: when $t=\textsf{execute}^{i}_{j}$ and when
$t=\textsf{finalise}^{k}$ with $k\in T^{\prime}$. First let
$t=\textsf{execute}^{i}_{j}$. Then $M_{1}[\textsf{execute}^{i}_{j}\rangle$.
Thus,
Now let $t=\textsf{finalise}^{k}$ with $k\in T^{\prime}$. By (11)
$G(\textsf{initialise}_{i}\cdot\textsf{fire})-G(\textsf{initialise}_{i}\cdot\textsf{undone})\geq
0$. So by (18), (E), and (12) $G(\textsf{distribute}_{p})\geq 0$. For this
reason we may assume, w.l.o.g., that $G(\textsf{execute}^{i}_{j})\geq 1$.
We have $G=G_{1}+\\{\textsf{finalise}^{k}\\}-G^{k}_{l}$ for certain
$l\geq^{\\#}k$ with $G_{1}(\textsf{execute}^{k}_{l})\mathbin{=}1$. Since
$G^{i}_{\\!\\!j}(\textsf{execute}^{i}_{j})\mathbin{\geq}0$, we also have
$G_{1}(\textsf{execute}^{i}_{j})\geq 1$. By (J) this implies that
$\neg(i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$
or $(i,j)=(k,l)$. In the latter case we have
$G(\textsf{execute}^{i}_{j})=\mbox{}$
$G_{1}(\textsf{execute}^{i}_{j})-G^{i}_{\\!\\!j}(\textsf{execute}^{i}_{j})=1-1=0$,
contradicting our assumption. In the former case
$p\notin{\vphantom{k}}^{\bullet}k$, so $G^{k}_{l}(\textsf{distribute}_{p})=0$
and hence $G(\textsf{distribute}_{p})=G_{1}(\textsf{distribute}_{p})\geq
F^{\prime}(p,i)\cdot G_{1}(\textsf{execute}^{i}_{j})=F^{\prime}(p,i)\cdot
G(\textsf{execute}^{i}_{j})$.
7. (G)
That $G(\textsf{execute}^{i}_{j})\geq 0$ follows from (E) and (12). If
$G(\textsf{execute}^{i}_{j})\geq 2$ for some $i\leq^{\\#}\\!j\in T^{\prime}$
then $M^{\prime}(p)\geq G(\textsf{distribute}_{p})\geq 2\cdot F^{\prime}(p,i)$
for all $p\in{\vphantom{i}}^{\bullet}i$, using (D) and (F), so
$M^{\prime}[2\cdot\\{i\\}\rangle_{N^{\prime}}$. Since $N^{\prime}$ is a
finitary structural conflict net, it has no self-concurrency, so this is
impossible.
8. (H)
Take $i\mathbin{\leq^{\\#}\\!}j\mathbin{\in}T^{\prime}$ and
$p\mathbin{\in}{\vphantom{j}}^{\bullet}j$. The case $i=j$ follows from (F), so
assume $i<^{\\#}j$. By (11) we have
$G(\textsf{initialise}_{i}\cdot\textsf{fire})-G(\textsf{initialise}_{i}\cdot\textsf{undone})\geq
0$. So by (18), (E), and (12) $G(\textsf{distribute}_{p})\geq 0$. Hence, using
(G), we may assume, w.l.o.g., that $G(\textsf{execute}^{i}_{j})=1$. We need to
investigate the same two cases as in the proof of (F) above. First let
$t=\textsf{execute}^{i}_{j}$. Then $M_{1}[\textsf{execute}^{i}_{j}\rangle$.
Thus,
Now let $t=\textsf{finalise}^{k}$ with $k\in T^{\prime}$. We have
$G=G_{1}+\\{\textsf{finalise}^{k}\\}-G^{k}_{l}$ for certain $l\geq^{\\#}k$
with $G_{1}(\textsf{execute}^{k}_{l})=1$. Since
$G^{i}_{\\!\\!j}(\textsf{execute}^{i}_{j})\mathbin{\geq}0$, we also have
$G_{1}(\textsf{execute}^{i}_{j})\geq 1$. By (K) this implies that
$\neg(j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$
or $(i,j)=(k,l)$. In the latter case $G(\textsf{execute}^{i}_{j})=\mbox{}$
$G_{1}(\textsf{execute}^{i}_{j})-G^{i}_{\\!\\!j}(\textsf{execute}^{i}_{j})=1-1=0$,
contradicting our assumption. In the former case
$p\notin{\vphantom{k}}^{\bullet}k$, so $G^{k}_{l}(\textsf{distribute}_{p})=0$
and hence $G(\textsf{distribute}_{p})=G_{1}(\textsf{distribute}_{p})\geq
F^{\prime}(p,j)\cdot G_{1}(\textsf{execute}^{i}_{j})=F^{\prime}(p,j)\cdot
G(\textsf{execute}^{i}_{j})$.
9. (I)
Let $i\mathbin{\leq^{\\#}\\!}j\mathbin{\in}T^{\prime}$ and $h<^{\\#}j$. Since,
for all $k\mathbin{\leq^{\\#}\\!}l\mathbin{\in}T^{\prime}$,
$G^{k}_{l}(\textsf{transfer}^{h}_{j}\\!\cdot\textsf{fire})\mathbin{=}\sum_{\omega}G^{k}_{l}(\textsf{transfer}^{h}_{j}\\!\cdot\textsf{reset}_{\omega})\mathbin{=}0$
and $G^{k}_{l}(\textsf{execute}^{i}_{j})=G^{k}_{l}(\textsf{fetched}^{i}_{j})$,
the invariant is preserved when $t$ has the form $\textsf{finalise}^{b}\\!$.
Using (20), it is in danger only when $t=\textsf{execute}^{i}_{j}$ or
$t=\textsf{transfer}^{h}_{j}\\!\cdot\textsf{reset}_{\omega}$ for some $\omega$
with $\textsf{transfer}^{h}_{j}\mathbin{\in}\Omega_{\omega}$.
First assume $M_{1}[\textsf{execute}^{i}_{j}\rangle$ and
$T^{h}_{j}=G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{fire})-\sum_{\omega}G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{reset}_{\omega})=0$.
Then
$\begin{array}[b]{r@{~\leq~}ll}\lx@intercol\hfil
1~{}\leq~{}&G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{fire})-G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\mbox{$\textsf{trans}^{h}_{j}\textsf{-out}$}))&\mbox{(by
(\ref{transout}))}\\\
~{}\leq~{}&G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{fire})-\sum_{\omega}G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{reset}_{\omega})=0&\mbox{(by
(\ref{took}))},\\\ \end{array}$
which is a contradiction.
Next assume $t=\textsf{transfer}^{h}_{j}\\!\cdot\textsf{reset}_{k}$ with
$k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j$,
and $E^{i}_{j}=1$. By (E) and (G) the latter implies that
$G_{1}(\textsf{execute}^{i}_{j})=1$ and $G_{1}(\textsf{fetched}^{i}_{j})=0$.
Then
$\begin{array}[b]{r@{~\leq~}ll}\lx@intercol\hfil
0~{}=~{}&G_{1}(\textsf{finalise}^{k})&\mbox{(by (\ref{r2}))}\\\
~{}\leq~{}&G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{elide}_{k})+G_{1}(\textsf{transfer}^{h}_{j}\cdot\textsf{reset}_{k})&\mbox{(by
(\ref{reset}))}\\\
\lx@intercol\hfil~{}<~{}&G(\textsf{transfer}^{h}_{j}\cdot\textsf{elide}_{k})+G(\textsf{transfer}^{h}_{j}\cdot\textsf{reset}_{k})\\\
~{}\leq~{}&\sum_{l\geq^{\\#}k}G(\textsf{fetched}^{k}_{l})&\mbox{(by
(\ref{reset}))}.\end{array}$
Hence $G_{1}(\textsf{fetched}^{k}_{l})=G(\textsf{fetched}^{k}_{l})>0$ for some
$l\geq^{\\#}k$, and by (12) also $G_{1}(\textsf{execute}^{k}_{l})>0$. Using
(K) we obtain $(i,\\!j)\mathop{=}(k,l)$, thereby obtaining a contradiction
($0\mathbin{=}G_{1}(\textsf{fetched}^{i}_{j})\mathbin{=}G_{1}(\textsf{fetched}^{k}_{l})\mathbin{>}0$).
10. (J)
Let
$j\geq^{\\#}i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\leq^{\\#}l\in
T^{\prime}$ with $(i,j)\neq(k,\ell)$. The invariant is in danger only when
$t\mathbin{=}\textsf{execute}^{i}_{j}$ or
$t\mathbin{=}\textsf{execute}^{k}_{l}$. W.l.o.g. let
$t\mathbin{=}\textsf{execute}^{k}_{l}$, with
$G_{1}(\textsf{execute}^{k}_{l})\mathbin{=}0$ and
$G_{1}(\textsf{execute}^{i}_{j})\mathbin{\geq}1$.
Making a case distinction, first assume
$G(\textsf{fetched}^{i}_{j})\mathbin{\geq}1$. Using (D), (F) and that
$G(\textsf{execute}^{k}_{l})=1$, $M^{\prime}(p)\geq
G(\textsf{distribute}_{p})\geq F^{\prime}(p,k)$ for all
$p\in{\vphantom{k}}^{\bullet}k$. Likewise, $M^{\prime}(p)\geq
G(\textsf{distribute}_{p})\geq F^{\prime}(p,i)$ for all
$p\in{\vphantom{i}}^{\bullet}i$. Moreover, just as in the proof of (F), we
derive, for all $p\in{\vphantom{i}}^{\bullet}i\cap{\vphantom{k}}^{\bullet}k$,
$\begin{array}[b]{@{}r@{~\geq~}ll@{}}\lx@intercol M^{\prime}(p)\geq
G(\textsf{distribute}_{p})\hfil\lx@intercol&\mbox{(by (\ref{r3}))}\\\
\mbox{}~{}\geq~{}&\displaystyle
F^{\prime}(p,k)\cdot\big{(}G(\textsf{initialise}_{k}\cdot\textsf{fire})-G(\textsf{initialise}_{k}\cdot\textsf{undone})\big{)}+\sum_{h\geq^{\\#}g\in{p}^{\bullet}}F^{\prime}(p,g)\cdot
G(\textsf{fetch}_{g,h}^{p,k})&\mbox{(by (\ref{p_j}))}\\\
~{}\geq~{}&\displaystyle
F^{\prime}(p,k)\cdot\big{(}G(\textsf{initialise}_{k}\cdot\textsf{fire})-G(\textsf{initialise}_{k}\cdot\textsf{undone})\big{)}+\sum_{h\geq^{\\#}g\in{p}^{\bullet}}F^{\prime}(p,g)\cdot
G(\textsf{fetched}^{g}_{h})&\mbox{(by (\ref{fetch}))}\\\
~{}\geq~{}&F^{\prime}(p,k)\cdot\big{(}G(\textsf{initialise}_{k}\cdot\textsf{fire})-G(\textsf{initialise}_{k}\cdot\textsf{undone})\big{)}+F^{\prime}(p,i)\cdot
G(\textsf{fetched}^{i}_{j})&\mbox{(by (\ref{rFp}))}\\\
~{}\geq~{}&F^{\prime}(p,k)\cdot\big{(}G(\textsf{initialise}_{k}\cdot\textsf{fire})-G(\textsf{initialise}_{k}\cdot\textsf{undo}(\textsf{pre}^{k}_{l}))\big{)}+F^{\prime}(p,i)\cdot
G(\textsf{fetched}^{i}_{j})&\mbox{(by (\ref{took}))}\\\
~{}\geq~{}&F^{\prime}(p,k)+F^{\prime}(p,i)&\mbox{(by
(\ref{pre}))}.\end{array}$
It follows that $M^{\prime}[\\{k\\}\mathord{+}\\{i\\}\rangle$. As
$i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k$
and $N^{\prime}$ is a finitary structural conflict net, this is impossible.
(Note that this argument holds regardless whether $i=k$.)
Now assume $G(\textsf{fetched}^{i}_{j})\leq 0$. Then, in the notation of (20),
$E^{i}_{j}=1$. Since $G_{1}(\textsf{execute}^{k}_{l})=0$, (E) and (12) yield
$G_{1}(\textsf{fetched}^{k}_{l})=0$. Hence $G(\textsf{execute}^{k}_{l})=1$ and
$G(\textsf{fetched}^{k}_{l})=0$, so $E^{k}_{l}=1$. We will conclude the proof
by deriving a contradiction from $E^{i}_{j}=E^{k}_{l}=1$. In case $j=l$ this
contradiction emerges immediately from (20). By symmetry it hence suffices to
consider the case $j<l$.
By (D) and (H) we have $M^{\prime}(p)\geq G(\textsf{distribute}_{p})\geq
F^{\prime}(p,j)$ for all $p\in{\vphantom{j}}^{\bullet}j$, so
$M^{\prime}[j\rangle$. Likewise $M^{\prime}[l\rangle$ and, using (F),
$M^{\prime}[i\rangle$ and $M^{\prime}[k\rangle$. Since
$j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k$
and $N^{\prime}$ has no fully reachable pure M,
$j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k$.
Since
$j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}l$
and $N^{\prime}$ has no fully reachable pure M,
$j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}l$.
So $j<^{\\#}l$. By (20), using that $E^{i}_{j}=1$, $T^{j}_{l}=0$. This is in
contradiction with $E^{k}_{l}=1$ and (I).
11. (K)
Suppose that $G(\textsf{execute}^{i}_{j})>0$ and
$G(\textsf{execute}^{k}_{l})>0$, with
$i\leq^{\\#}j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\leq^{\\#}l\in
T^{\prime}$. By (D) and (H) we have
$M^{\prime}(p)\mathbin{\geq}G(\textsf{distribute}_{p})\mathbin{\geq}F^{\prime}(p,j)$
for all $p\mathbin{\in}{\vphantom{j}}^{\bullet}j$, so $M^{\prime}[j\rangle$.
Likewise, using (F), $M^{\prime}[i\rangle$ and $M^{\prime}[k\rangle$. Since
$i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k$
and $N^{\prime}$ has no fully reachable pure M,
$i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k$.
Using this, the result follows from (J).
###### Claim 3.
extra For any $M\in[M_{0}\rangle_{N}$ there exist
$M^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$ and $G\in_{f}\mbox{\bbb
Z}^{T}$ satisfying (A)–(K) from Claim LABEL:cl-reachable, and
1. (L)
there are no
$j\geq^{\\#}i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\leq^{\\#}l\in
T^{\prime}$ with $M[\textsf{execute}^{i}_{j}\rangle$ and
$G(\textsf{execute}^{k}_{l})>0$,
2. (M)
there are no
$i\leq^{\\#}j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k\leq^{\\#}l\in
T^{\prime}$ with $M[\textsf{execute}^{i}_{j}\rangle$ and
$G(\textsf{execute}^{k}_{l})>0$,
3. (N)
if $M[\textsf{execute}^{i}_{j}\rangle$ for $i\leq^{\\#}\\!j\in T^{\prime}$
then $M^{\prime}[j\rangle$.
* Proof:
Given $M$, by Claim LABEL:cl-reachable(2) there are $M^{\prime}$ and $G$ so
that the triple $(M,M^{\prime},G)$ satisfies (A)–(K). Assume
$M[\textsf{execute}^{i}_{j}\rangle$ for some $i\leq^{\\#}\\!j\in T^{\prime}$.
Let $M_{1}:=M+\llbracket\textsf{execute}^{i}_{j}\rrbracket$ and
$G_{1}:=G+\\{\textsf{execute}^{i}_{j}\\}$. By (G)
$G(\textsf{execute}^{i}_{j})\geq 0$, so $G_{1}(\textsf{execute}^{i}_{j})>0$.
By Claim LABEL:cl-reachable(1) the triple ($M_{1},M^{\prime},G_{1}$) satisfies
(A)–(K).
1. (L)
Suppose $G(\textsf{execute}^{k}_{l})>0$ for certain
$l\geq^{\\#}k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i$.
In case $(i,j)=(k,\ell)$ we have $G_{1}(\textsf{execute}^{i}_{j})\geq 2$,
contradicting (G). In case $(i,j)\neq(k,\ell)$, $G_{1}$ fails (J), also a
contradiction.
2. (M)
Suppose $G(\textsf{execute}^{k}_{l})>0$ for certain
$l\geq^{\\#}k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j$.
Then $G_{1}$ fails (G) or (K), a contradiction.
3. (N)
By (D) and (H) $M^{\prime}(p)\geq G_{1}(\textsf{distribute}_{p})\geq F(p,j)$
for all $p\in{\vphantom{j}}^{\bullet}j$, so $M^{\prime}[j\rangle$.
###### Claim 4.
concurrency If
$M[\\{\textsf{execute}^{i}_{j}\\}\mathord{+}\\{\textsf{execute}^{k}_{l}\\}\rangle$
for some $M\in[M_{0}\rangle_{N}$ then
$\neg(i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$.
* Proof:
Suppose
$M[\\{\textsf{execute}^{i}_{j}\\}\mathord{+}\\{\textsf{execute}^{k}_{l}\\}\rangle$
for some $M\in[M_{0}\rangle_{N}$. By Claim LABEL:cl-reachable(2) there exist
$M^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$ and $G\in_{f}\mbox{\bbb
Z}^{T}$ satisfying (A)–(K). Let
$M_{1}:=M+\llbracket\textsf{execute}^{k}_{l}\rrbracket$ and
$G_{1}:=G\mathbin{+}\\{\textsf{execute}^{k}_{l}\\}$. By Claim LABEL:cl-
reachable(1) the triple $(M_{1},M^{\prime},G_{1})$ satisfies (A)–(K). Let
$M_{2}:=M_{1}+\llbracket\textsf{execute}^{i}_{j}\rrbracket$ and
$G_{2}:=G_{1}\mathbin{+}\\{\textsf{execute}^{i}_{j}\\}$. Again by Claim
LABEL:cl-reachable(1), also the triple $(M_{2},M^{\prime},G_{2})$ satisfies
(A)–(K). By (G) $G(\textsf{execute}^{i}_{j})\mathbin{\geq}0$, so in case
$(i,j)\mathbin{=}(k,l)$ we obtain
$G_{2}(\textsf{execute}^{i}_{j})\mathbin{\geq}2$, contradicting (G). Hence
$(i,j)\mathbin{\neq}(k,l)$. Moreover, $G_{2}(\textsf{execute}^{k}_{l})>0$ and
$G_{2}(\textsf{execute}^{i}_{j})>0$. Now (J) implies
$\neg(i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$.
For any $t\in\\{\textsf{initialise}_{j},~{}\textsf{transfer}^{h}_{j}\\}$ with
$h,j\mathbin{\in}T^{\prime}$, and any $\omega\mathbin{\in}\Omega$ with
$t\in\Omega_{\omega}$, we write
$t(\omega):=t\cdot\textsf{fire}+t\cdot\textsf{undo}_{\omega}+\big{(}\sum_{f\in
t^{\,\it
far}}t\cdot\textsf{undo}(f)\big{)}+t\cdot\textsf{undone}+t\cdot\textsf{reset}_{\omega}\;.\vspace{-1ex}$
The transition $t$ has no preplaces of type in, nor postplaces of type out. By
checking in Table LABEL:tab-reversible or Figure LABEL:fig-reversible that
each other place occurs as often in
${\vphantom{u(\omega)}}^{\bullet}u(\omega)+{(u\cdot\textsf{elide}_{\omega})}^{\bullet}$
as in
${u(\omega)}^{\bullet}+{\vphantom{(u\cdot\textsf{elide}_{\omega})}}^{\bullet}(u\cdot\textsf{elide}_{\omega})$,
one verifies, for any $\omega\in\Omega$ with $t\in\Omega_{\omega}$, that
$\llbracket t(\omega)\rrbracket=\llbracket
t\cdot\textsf{elide}_{\omega}\rrbracket.$ (21)
Let $\equiv$ be the congruence relation on finite signed multisets of
transitions generated by
$\displaystyle t(\omega)$ $\displaystyle\equiv$ $\displaystyle
t\cdot\textsf{elide}_{\omega}$ (22)
for all $t\in\\{\textsf{initialise}_{j},~{}\textsf{transfer}^{h}_{j}\mid
h,j\mathbin{\in}T^{\prime}\\}$ and $\omega\in\Omega$ with $\Omega_{\omega}\ni
t$. Here _congruence_ means that $G_{1}\mathbin{\equiv}G_{2}$ implies $k\cdot
G_{1}\mathbin{\equiv}k\cdot G_{2}$ and $G_{1}+H\mathbin{\equiv}G_{2}+H$ for
all $k\mathbin{\in}\mbox{\bbb Z}$ and $H\in_{f}\mbox{\bbb Z}^{T}$. Using (21)
$G_{1}\equiv G_{2}$ implies $\llbracket G_{1}\rrbracket=\llbracket
G_{2}\rrbracket$.
###### Claim 5.
0 If $M^{\prime}=\llbracket G\rrbracket$ for $M^{\prime}\in\mbox{\bbb
Z}^{S^{\prime}}$ and $G\in_{f}\mbox{\bbb Z}^{T}$ such that for all $i\in
T^{\prime}$ we have $G(\textsf{finalise}^{i})=0$ and either $\forall
j\geq^{\\#}i.~{}G(\textsf{execute}^{i}_{j})\geq 0$ or $\forall
j\geq^{\\#}i.~{}G(\textsf{execute}^{i}_{j})\leq 0$, then $G\equiv\emptyset$.
* Proof:
Let $M^{\prime}$ and $G$ be as above. W.l.o.g. we assume
$G(t\cdot\textsf{elide}_{\omega})=0$ for all
$t\in\\{\textsf{initialise}_{j},~{}\textsf{transfer}^{h}_{j}\\}$ and all
$\omega\in\Omega$ with $t\in\Omega_{\omega}$, for any $G$ can be brought into
that form by applying (22). For each $s\in S\setminus S^{\prime}$ we have
$M^{\prime}(s)=0$, and using this the inequations (8)–(12) and (18) of Claim
LABEL:cl-G-properties turn into equations. For each $i\in T^{\prime}$ we have
$G(\sum_{j\geq^{\\#}i}\textsf{execute}^{i}_{j})=0$, using (the equational form
of) (8)–(10), and that $G(\textsf{finalise}^{i})=0$. Since
$G(\textsf{execute}^{i}_{j})\geq 0$ (or $\mbox{}\leq 0$) for all
$j\geq^{\\#}i$, this implies that $G(\textsf{execute}^{i}_{j})=0$ for each
$i\leq^{\\#}\\!j\in T^{\prime}$. With (12) we obtain
$G(\textsf{fetched}^{i}_{j})=G(\textsf{fetch}_{i,j}^{p,c})=0$ for each
applicable $p,c,i,j$. Using that $G(t\cdot\textsf{elide}_{\omega})=0$ for each
applicable $t$ and $\omega$, with (9)–(11) and (18) we find $G(t)=0$ for all
$t\in T$.
###### Claim 6.
D Let $M:=M^{\prime}+(M_{0}\mathord{-}M^{\prime}_{0})+\llbracket
H\rrbracket\in[M_{0}\rangle_{N}$ for
$M^{\prime}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$ and
$H\in_{f}\mbox{\bbb Z}^{T}$ with $H(\textsf{execute}^{i}_{j})\mathbin{=}0$ for
all $i\leq^{\\#}\\!j\in T^{\prime}$.
1. (a)
If $H(\textsf{finalise}^{i})<0$ and $H(\textsf{finalise}^{k})<0$ for certain
$i,k\in T^{\prime}$ then $\neg(i\mathrel{\\#}k)$.
2. (b)
If $M[\textsf{execute}^{i}_{j}\rangle$ and $H(\textsf{finalise}^{k})<0$ for
certain $i,k\in T^{\prime}$ then
$\neg(i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$
and
$\neg(j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$.
3. (c)
$H(\textsf{distribute}_{p})\geq 0$ for all $p\in S^{\prime}$ (with
${p}^{\bullet}\neq\emptyset$).
4. (d)
Let
$c\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i\in
T^{\prime}$. If $H(\textsf{distribute}_{p})\geq F^{\prime}(p,c)$ for all
$p\in{\vphantom{c}}^{\bullet}c$, then $H(\textsf{finalise}^{i})=0$.
5. (e)
If $M[\textsf{execute}^{i}_{j}\rangle$ with $i\leq^{\\#}\\!j\in T^{\prime}$
then $M^{\prime}[j\rangle$.
* Proof:
By Claim LABEL:cl-extra there exist
$M^{\prime}_{1}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$ and
$G_{1}\in_{f}\mbox{\bbb Z}^{T}$ satisfying (B)–(N) (with $M$, $M^{\prime}_{1}$
and $G_{1}$ playing the rôles of $M$, $M^{\prime}$ and $G$). In particular,
$M=M^{\prime}_{1}+(M_{0}-M^{\prime}_{0})+\llbracket G_{1}\rrbracket$,
$G_{1}(\textsf{finalise}^{i})=0$ for all $i\in T^{\prime}$, and
$G_{1}(\textsf{execute}^{i}_{j})\geq 0$ for all $i\leq^{\\#}\\!j\in
T^{\prime}$. Using (J), for each $i\in T^{\prime}$ there is at most one
$j\geq^{\\#}i$ with $G_{1}(\textsf{execute}^{i}_{j})>0$; we denote this $j$ by
$f(i)$, and let $f(i):=i$ when there is no such $j$. This makes
$f:T^{\prime}\rightarrow T^{\prime}$ a function, satisfying
$G_{1}(\textsf{execute}^{i}_{j})=0$ for all $j\geq^{\\#}i$ with $j\neq f(i)$.
Given that $H(\textsf{execute}^{i}_{j})\mathbin{=}0$ for all
$i\leq^{\\#}\\!j\in T^{\prime}$, (8)–(10) (or (9) and (12)) imply
$H(\textsf{finalise}^{i})\leq 0$ for all $i\in T^{\prime}$. Let
$M^{\prime}_{2}:=M^{\prime}+\sum_{i\mathbin{\in}T^{\prime}}H(\textsf{finalise}^{i})\cdot\llbracket
i\rrbracket$ and
$G_{2}:=H-\sum_{i\mathbin{\in}T^{\prime}}H(\textsf{finalise}^{i})\cdot
G^{i}_{\\!f(i)}$, where $G^{i}_{\\!\\!j}$ is the right-hand side of (7). Then
$M=M^{\prime}+(M_{0}-M^{\prime}_{0})+\llbracket
H\rrbracket=M^{\prime}_{2}+(M_{0}-M^{\prime}_{0})+\llbracket G_{2}\rrbracket$,
using that $\llbracket i\rrbracket=\llbracket G^{i}_{\\!f(i)}\rrbracket$.
Moreover, $G_{2}(\textsf{finalise}^{i})=0$ for all $i\mathbin{\in}T^{\prime}$,
using that $G^{i}_{\\!f(i)}(\textsf{finalise}^{i})=1$.
It follows that $M^{\prime}_{1}-M^{\prime}_{2}=\llbracket
G_{2}-G_{1}\rrbracket$. Moreover, we have
$(G_{2}-G_{1})(\textsf{finalise}^{i})=0$ for all $i\in T^{\prime}$. We proceed
to show that $G_{2}-G_{1}$ satisfies the remaining precondition of Claim
LABEL:cl-0. So let $i\in T^{\prime}$. In case $H(\textsf{finalise}^{i})=0$,
for all $j\geq^{\\#}i$ we have $G_{2}(\textsf{execute}^{i}_{j})=0$, and
$G_{1}(\textsf{execute}^{i}_{j})\geq 0$ by (G). Hence
$(G_{2}-G_{1})(\textsf{execute}^{i}_{j})\leq 0$. In case
$H(\textsf{finalise}^{i})<0$, we have $G_{2}(\textsf{execute}^{i}_{f(i)})\geq
1$, and hence, using (G), $(G_{2}-G_{1})(\textsf{execute}^{i}_{f(i)})\geq 0$.
Furthermore, for all $j\neq f(i)$, $G_{2}(\textsf{execute}^{i}_{j})\geq 0$ and
$G_{1}(\textsf{execute}^{i}_{j})=0$, so again
$(G_{2}-G_{1})(\textsf{execute}^{i}_{j})\geq 0$.
Thus we may apply Claim LABEL:cl-0, which yields $G_{2}\equiv G_{1}$. It
follows that
$M^{\prime}_{2}=M^{\prime}_{1}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$.
1. (a)
Suppose that $H(\textsf{finalise}^{i})<0$ and $H(\textsf{finalise}^{k})<0$ for
certain $i\mathrel{\\#}k\in T^{\prime}$. Then
$G_{2}(\textsf{execute}^{i}_{f(i)})>0$ and
$G_{2}(\textsf{execute}^{k}_{f(k)})>0$, so
$G_{1}(\textsf{execute}^{i}_{f(i)})>0$ and
$G_{1}(\textsf{execute}^{k}_{f(k)})>0$, contradicting (J).
2. (b)
Suppose that $M[\textsf{execute}^{i}_{j}\rangle$ and
$H(\textsf{finalise}^{k})<0$ for certain
$k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i$
or
$k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j$.
Then
$G_{1}(\textsf{execute}^{k}_{f(k)})=\mbox{}$$G_{2}(\textsf{execute}^{k}_{f(k)})>0$,
contradicting (L) or (M).
3. (c)
By (a), for any given $p\in S^{\prime}$ there is at most one
$i\in{p}^{\bullet}$ with $H(\textsf{finalise}^{i})<0$. For all $i\in
T^{\prime}$ with $i\notin{p}^{\bullet}$ we have
$G^{i}_{\\!f(i)}(\textsf{distribute}_{p})=0$. First suppose
$k\in{p}^{\bullet}$ satisfies $H(\textsf{finalise}^{k})<0$. Then
$G_{1}(\textsf{execute}^{k}_{f(k)})\begin{array}[t]{@{~=~}l}{}=~{}\lx@intercol
G_{2}(\textsf{execute}^{k}_{f(k)})\\\ {}=~{}\lx@intercol
H(\textsf{execute}^{k}_{f(k)})-\sum_{i\in
T^{\prime}}H(\textsf{finalise}^{i})\cdot
G^{i}_{\\!f(i)}(\textsf{execute}^{k}_{f(k)})\\\ {}=~{}\lx@intercol
0-H(\textsf{finalise}^{k}),\end{array}$
so by (F) $G_{1}(\textsf{distribute}_{p})\geq-F^{\prime}(p,k)\cdot
H(\textsf{finalise}^{k})$. Hence
$H(\textsf{distribute}_{p})~{}\begin{array}[t]{@{}l}=~{}G_{2}(\textsf{distribute}_{p})+\sum_{i\in
T^{\prime}}H(\textsf{finalise}^{i})\cdot
G^{i}_{\\!f(i)}(\textsf{distribute}_{p})\\\
=~{}G_{1}(\textsf{distribute}_{p})+H(\textsf{finalise}^{k})\cdot
G^{k}_{f(k)}(\textsf{distribute}_{p})\\\ \geq~{}-F^{\prime}(p,k)\cdot
H(\textsf{finalise}^{k})+H(\textsf{finalise}^{k})\cdot
F^{\prime}(p,k)=0.\end{array}$
In case there is no $i\in{p}^{\bullet}$ with $H(\textsf{finalise}^{i})<0$ we
have
$H(\textsf{distribute}_{p})=G_{2}(\textsf{distribute}_{p})+\sum_{i\in
T^{\prime}}H(\textsf{finalise}^{i})\cdot
G^{i}_{\\!f(i)}(\textsf{distribute}_{p})=G_{1}(\textsf{distribute}_{p})\geq
0\vspace{-2ex}$
by (F) and (G).
4. (d)
Since $H(\textsf{finalise}^{i})\leq 0$ and
$G^{i}_{\\!f(i)}(\textsf{distribute}_{p})\geq 0$ for all
$i\mathbin{\in}T^{\prime}$, also using (c), all summands in
$H(\textsf{distribute}_{p})+\sum_{i\in
T^{\prime}}-H(\textsf{finalise}^{i})\cdot
G^{i}_{\\!f(i)}(\textsf{distribute}_{p})$ are positive. Now suppose
$H(\textsf{finalise}^{i})<0$ for certain $i\mathbin{\in}T^{\prime}$. Then,
using (D), for all $p\in{\vphantom{i}}^{\bullet}i$,
$M^{\prime}_{1}(p)\geq
G_{1}(\textsf{distribute}_{p})=G_{2}(\textsf{distribute}_{p})\geq
G^{i}_{\\!f(i)}(\textsf{distribute}_{p})=F^{\prime}(p,i).$
Furthermore, let
$c\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i$
and suppose $H(\textsf{distribute}_{p})\geq F^{\prime}(p,c)$ for all
$p\in{\vphantom{c}}^{\bullet}c$. Then, using (D),
$M^{\prime}_{1}(p)\geq
G_{1}(\textsf{distribute}_{p})=G_{2}(\textsf{distribute}_{p})\geq
H(\textsf{distribute}_{p})\geq F^{\prime}(p,c)$
for all $p\in{\vphantom{c}}^{\bullet}c$. Moreover, if
$p\in{\vphantom{c}}^{\bullet}c\cap{\vphantom{i}}^{\bullet}i$ then
$M^{\prime}_{1}(p)\geq G_{2}(\textsf{distribute}_{p})\geq
H(\textsf{distribute}_{p})+G^{i}_{\\!f(i)}(\textsf{distribute}_{p})\geq
F^{\prime}(p,c)+F^{\prime}(p,i).$
Hence $M^{\prime}_{2}[\\{c\\}\mathord{+}\\{i\\}\rangle$. However, since
$c\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i$
and $N^{\prime}$ is a structural conflict net, this is impossible.
5. (e)
Suppose $M[\textsf{execute}^{i}_{j}\rangle$ with $i\leq^{\\#}\\!j\in
T^{\prime}$. Then $M^{\prime}_{1}[j\rangle$ by (N). Now
$M^{\prime}=M^{\prime}_{1}+\sum_{k\mathbin{\in}T^{\prime}}-H(\textsf{finalise}^{k})\cdot\llbracket
k\rrbracket$, with $-H(\textsf{finalise}^{k})\geq 0$ for all $k\in
T^{\prime}$. Whenever $-H(\textsf{finalise}^{k})>0$ then
$\neg(j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$
by (b). Hence $M^{\prime}[j\rangle$.
We now define the class $\mbox{\it NF}\subseteq\mbox{\bbb Z}^{T}$ of signed
multisets of transitions in _normal form_ by $H\in\mbox{\it NF}$ iff
$\ell(H)\equiv\emptyset$ and, for all
$t\in\\{\textsf{initialise}_{j},~{}\textsf{transfer}^{h}_{j}\mid
h,j\mathbin{\in}T^{\prime}\\}$:
1. (NF-1)
$H(t\cdot\textsf{elide}_{\omega})\leq 0$ for each $\omega\mathbin{\in}\Omega$,
2. (NF-2)
$H(t\cdot\textsf{undo}_{\omega})\geq 0$ for each $\omega\mathbin{\in}\Omega$,
or $H(t\cdot\textsf{fire})\geq 0$,
3. (NF-3)
and if $H(t\cdot\textsf{elide}_{\omega})<0$ for any
$\omega\mathbin{\in}\Omega$, then $H(t\cdot\textsf{undo}_{\omega})\leq 0$ and
$H(t\cdot\textsf{fire})\leq 0$.
We proceed verifying the remaining conditions of Theorem LABEL:thm-3ST.
1. 4.
By applying (22), each signed multiset $G\in_{f}\mbox{\bbb Z}^{T}$ with
$\ell(G)\equiv\emptyset$ can be converted into a signed multiset
$H\in_{f}\mbox{\it NF}$ with $\ell(H)\equiv\emptyset$, such that $\llbracket
H\rrbracket=\llbracket G\rrbracket$. Namely, for any
$t\in\\{\textsf{initialise}_{j},~{}\textsf{transfer}^{h}_{j}\mid
h,j\mathbin{\in}T^{\prime}\\}$, first of all perform the following three
transformations, until none is applicable:
1. (i)
correct a positive count of a transition $t\cdot\textsf{elide}_{\omega}$ in
$G$ by adding $t(\omega)-t\cdot\textsf{elide}_{\omega}$ to $G$;
2. (ii)
if both $H(t\cdot\textsf{undo}_{\omega})<0$ for some $\omega$ and
$H(t\cdot\textsf{fire})<0$, correct this in the same way;
3. (iii)
and if, for some $\omega$, $t\mathord{\cdot}\textsf{elide}_{\omega}$ has a
negative and $t\mathord{\cdot}\textsf{undo}_{\omega}$ a positive count, add
$t\cdot\textsf{elide}_{\omega}-t(\omega)$.
Note that transformation (iii) will never be applied to the same $\omega$ as
(i) or (ii), so termination is ensured. Properties (NF-1) and (NF-2) then hold
for $t$. After termination of (i)–(iii), perform
1. (iv)
if, for some $\omega$, $H(t\cdot\textsf{elide}_{\omega})<0$ and
$H(t\cdot\textsf{fire})>0$, add $t\cdot\textsf{elide}_{\omega}-t(\omega)$.
This will ensure that also (NF-3) is satisfied, while preserving (NF-1) and
(NF-2).
Define the function $f:T\rightarrow{\rm Nature}$ by $f(u):=1$ for all $u\in T$
not of the form $u=t\cdot\textsf{elide}_{\omega}$, and
$f(t\cdot\textsf{elide}_{\omega}):=f(t(\omega))$ (applying the last item of
Definition LABEL:df-multiset). Then surely $f(G)=f(H)$.
2. 5.
Let $M^{\prime}\in{\rm Nature}^{S^{\prime}}$, $U^{\prime}\in{\rm
Nature}^{T^{\prime}}$ and $U\in{\rm Nature}^{T}$ with
$\ell(U)=\ell^{\prime}(U^{\prime})$ and
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$.
Since $N^{\prime}$ is a finitary structural conflict net, it admits no self-
concurrency, so, as ${\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\leq
M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$,
the multiset $U^{\prime}$ must be a set. As $N^{\prime}$ is plain, this
implies that the multiset $\ell^{\prime}(U^{\prime})$ is a set. Since
$\ell(U)=\ell^{\prime}(U^{\prime})$, also $\ell(U)$, and hence $U$, must be a
set. All its elements have the form $\textsf{execute}^{i}_{j}$ for
$i\leq^{\\#}\\!j\in T^{\prime}$, since these are the only transitions in $T$
with visible labels. Note that $U^{\prime}$ is completely determined by $U$,
namely by $U^{\prime}=\\{i\mid\exists j.~{}\textsf{execute}^{i}_{j}\in U\\}$.
We take
$H_{M^{\prime},U}:=\sum_{p\in
S^{\prime}}(M^{\prime}\mathord{+}\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})(p)\cdot\\{\textsf{distribute}_{p}\\}+\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle}\\!\\!\\!\\!\left(\\{\textsf{initialise}_{j}\cdot\textsf{fire}\\}+\sum_{h<^{\\#}j,~{}\nexists\textsf{execute}^{g}_{h}\in
U}\\{\textsf{transfer}^{h}_{j}\cdot\textsf{fire}\\}\right)$
Since $N^{\prime}$ is finitary, $H_{M^{\prime},U}\in_{f}{\rm Nature}^{T^{+}}$.
Moreover, $\ell(H_{M^{\prime},U})\equiv\emptyset$.
Let $H\mathbin{\in_{f}}\mbox{\it NF}$ with
$M:=M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}+(M_{0}\mathord{-}M^{\prime}_{0})+\llbracket
H\rrbracket-{\vphantom{U}}^{\bullet}U\in{\rm Nature}^{S}$ and
$M+{\vphantom{U}}^{\bullet}U\in[M_{0}\rangle_{N}$. Since
$H\mathbin{\in}\mbox{\it NF}$, and thus $\ell(H)\equiv\emptyset$,
$H(\textsf{execute}^{i}_{j})=0$. From here on we apply Claim LABEL:cl-G-
properties and Claim LABEL:cl-D with $M+{\vphantom{U}}^{\bullet}U$ and
$M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ playing the rôles of
$M$ and $M^{\prime}$. Note that the preconditions of these claims are met.
That $H(\textsf{execute}^{i}_{j})=0$ for all
$i\leq^{\\#}\\!j\mathbin{\in}T^{\prime}$, together with (8) and the
requirements (NF-1) and (NF-3) for normal forms, yields
$H(t\cdot\textsf{elide}_{i})\leq 0$ as well as $H(t\cdot\textsf{undo}_{i})\leq
0$. Using this, (9)–(12) imply that
$H(u)\leq 0~{}\mbox{ for each }~{}u\in T^{-}.\vspace{-4ex}$ (23)
###### Claim 7.
C Let $c\mathbin{\in}T^{\prime}$ and $p\in{\vphantom{c}}^{\bullet}c$. Then
* –
if $H(\textsf{initialise}_{c}\cdot\textsf{fire})>0$ then
$H(\textsf{fetch}_{i,j}^{p,c})=0$ for all $i\in{p}^{\bullet}$ and
$j\geq^{\\#}i$, and
* –
if $H(\textsf{transfer}^{b}_{c}\cdot\textsf{fire})>0$ for some $b<^{\\#}c$
then $H(\textsf{fetch}_{i,j}^{p,c})=0$ for all $i\in{p}^{\bullet}$ and
$j\geq^{\\#}i$.
* Proof:
Suppose that $H(t\cdot\textsf{fire})>0$, for $t=\textsf{initialise}_{c}$ or
$t=\textsf{transfer}^{b}_{c}$. Then (13) resp. (20) together with (23) implies
that $H(t\cdot\textsf{reset}_{\omega})=0$ for each $\omega$ with
$t\in\Omega_{\omega}$. In order words, $H(t\cdot\textsf{reset}_{i})=0$ for
each
$i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}c$,
so in particular for each $i\in{p}^{\bullet}$. Furthermore,
$H(t\cdot\textsf{elide}_{i})\geq 0$, by requirement (NF-3) of normal forms.
With (9), this yields $\sum_{j\geq^{\\#}i}H(\textsf{fetched}^{i}_{j})\geq 0$,
and (23) implies $H(\textsf{fetched}^{i}_{j})=0$ for each $j\geq^{\\#}i$. Now
(12, 23) gives $H(\textsf{fetch}_{i,j}^{p,c})=0$ for each
$j\geq^{\\#}i\in{p}^{\bullet}$.
We proceed to verify the requirements (5a)–(5g) of Theorem LABEL:thm-3ST.
1. (5a)
To show that $M_{M^{\prime},U}\in{\rm Nature}^{S}$, it suffices to apply it to
the preplaces of transitions in $H_{M^{\prime},U}+U$:
$\begin{array}[]{@{}l@{~=~}ll}M_{M^{\prime},U}(p)\hfil~{}=~{}&0&\mbox{for all
}p\in S^{\prime}\;;\\\
M_{M^{\prime},U}(p_{j})\hfil~{}=~{}&\left\\{\begin{array}[]{@{}ll@{}}(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})(p)-F^{\prime}(p,j)&\mbox{if
}(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\\\
(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})(p)&\mbox{otherwise}\end{array}\right.&\mbox{for
}p\mathbin{\in}S^{\prime},~{}j\mathbin{\in}{p}^{\bullet};\\\
M_{M^{\prime},U}(\pi_{j})\hfil~{}=~{}&\left\\{\begin{array}[]{@{}l@{\qquad\;\quad}l@{}}\phantom{-}0&\mbox{if
}(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\\\
\phantom{-}1&\mbox{otherwise}\end{array}\right.&\mbox{for }j\in T^{\prime};\\\
M_{M^{\prime},U}(\textsf{pre}^{j}_{k})\hfil~{}=~{}&\left\\{\begin{array}[]{@{}l@{\qquad\;\quad}l@{}}\phantom{-}1&\mbox{if
}(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\wedge\textsf{execute}^{j}_{k}\notin
U\\\ -1&\mbox{if
}\neg(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\wedge\textsf{execute}^{j}_{k}\in
U\\\ \phantom{-}0&\mbox{otherwise}\end{array}\right.&\mbox{for
}j\leq^{\\#}\\!k\in T^{\prime};\\\
M_{M^{\prime},U}(\pi_{h\\#j})\hfil~{}=~{}&\left\\{\begin{array}[]{@{}l@{\qquad\;\quad}l@{}}\phantom{-}0&\mbox{if
}\exists\textsf{execute}^{g}_{h}\in
U\vee(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\\\
\phantom{-}1&\mbox{otherwise}\end{array}\right.&\mbox{for }h<^{\\#}j\in
T^{\prime}\\\
M_{M^{\prime},U}(\textsf{trans}^{h}_{j}\textsf{-in})\hfil~{}=~{}&\left\\{\begin{array}[]{@{}l@{\qquad\;\quad}l@{}}\phantom{-}1&\mbox{if
}(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\wedge\exists\textsf{execute}^{g}_{h}\in
U\\\ \phantom{-}0&\mbox{otherwise}\end{array}\right.&\mbox{for }h<^{\\#}j\in
T^{\prime};\\\
M_{M^{\prime},U}(\textsf{trans}^{h}_{j}\textsf{-out})\hfil~{}=~{}&\left\\{\begin{array}[]{@{}l@{\qquad\;\quad}l@{}}\phantom{-}1&\makebox[0.0pt][l]{if
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\wedge\nexists\textsf{execute}^{g}_{h}\in
U\wedge\nexists\textsf{execute}^{i}_{j}\in U$}\\\ -1&\makebox[0.0pt][l]{if
$\big{(}\neg(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle\vee\exists\textsf{execute}^{g}_{h}\in
U\big{)}\wedge\exists\textsf{execute}^{i}_{j}\in U$}\\\
\phantom{-}0&\mbox{otherwise}\end{array}\right.&\begin{array}[]{@{}l@{}}\mbox{}\\\
\mbox{}\\\ \mbox{for }h<^{\\#}j\in T^{\prime}.\end{array}\\\ \end{array}$
For all these places $s$ we indeed have that $M_{M^{\prime},U}(s)\geq 0$, for
the circumstances yielding the two exceptions above cannot occur:
* –
Suppose $\textsf{execute}^{j}_{k}\in U$ with $j\leq^{\\#}\\!k\in T^{\prime}$.
Then $j\in U^{\prime}$, so ${\vphantom{j}}^{\bullet}j\subseteq
M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ and
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle$.
Consequently, $M_{M^{\prime},U}(\textsf{pre}^{j}_{k})\neq-1$ for all
$j\leq^{\\#}\\!k\in T^{\prime}$.
* –
Suppose $\textsf{execute}^{i}_{j}\in U$ with $i\leq^{\\#}\\!j\in T^{\prime}$.
Then
${\vphantom{\textsf{execute}^{i}_{j}}}^{\bullet}\textsf{execute}^{i}_{j}\leq{\vphantom{U}}^{\bullet}U$,
so $(M+\\!{\vphantom{U}}^{\bullet}U)[\textsf{execute}^{i}_{j}\rangle$. Claim
LABEL:cl-D(e) with $M+\\!{\vphantom{U}}^{\bullet}U$ and
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ in the rôles of
$M$ and $M^{\prime}$ yields
$(M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle$.
If moreover $\textsf{execute}^{g}_{h}\mathbin{\in}U$ with
$g\mathbin{\leq^{\\#}\\!}h\mathbin{<^{\\#}}\\!j$, then
$\\{g\\}\mathord{+}\\{i\\}\subseteq U^{\prime}$, so
${\vphantom{\\{}}^{\bullet}\\{g\\}\mathord{+}\\!{\vphantom{\\{}}^{\bullet}\\{i\\}\subseteq
M^{\prime}\mathord{+}\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ and
$(M^{\prime}+{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[\\{g\\}\mathord{+}\\{i\\}\rangle$.
In particular, $g\smile i$, and since $N^{\prime}$ is a structural conflict
net, ${\vphantom{g}}^{\bullet}g\cap{\vphantom{i}}^{\bullet}i=\emptyset$. By
Claim LABEL:cl-D(e)—as
above—$(M^{\prime}\mathord{+}\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[h\rangle$,
so
${\vphantom{g}}^{\bullet}g\cup{\vphantom{h}}^{\bullet}h\cup{\vphantom{j}}^{\bullet}j\cup{\vphantom{i}}^{\bullet}i\subseteq
M^{\prime}\mathord{+}\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}\mathbin{\in}[M^{\prime}_{0}\rangle_{N^{\prime}}$.
Moreover, since $g\leq^{\\#}\\!h<^{\\#}j\geq^{\\#}i$, we have
${\vphantom{g}}^{\bullet}g\cap{\vphantom{h}}^{\bullet}h\neq\emptyset$,
${\vphantom{h}}^{\bullet}h\cap{\vphantom{i}}^{\bullet}i\neq\emptyset$ and
${\vphantom{i}}^{\bullet}i\cap{\vphantom{j}}^{\bullet}j\neq\emptyset$. Now in
case also
${\vphantom{h}}^{\bullet}h\cap{\vphantom{i}}^{\bullet}i\neq\emptyset$, the
transitions $g$, $h$ and $i$ constitute a fully reachable pure M; otherwise
$h\smile i$ and $h$, $j$ and $i$ constitute a fully reachable pure M. Either
way, we obtain a contradiction. Consequently,
$M_{M^{\prime},U}(\textsf{trans}^{h}_{j}\textsf{-out})\neq-1$ for all
$h<^{\\#}j\in T^{\prime}$.
2. (5b)
Suppose $M^{\prime}\mathrel{\hbox{$\mathop{\hbox
to15.00002pt{$\mathord{-}\mkern-6.0mu\cleaders\hbox{$\mkern-2.0mu\mathord{-}\mkern-2.0mu$}\hfill\mkern-6.0mu\mathord{\rightarrow}$}}\limits^{\hbox
to15.00002pt{\hfil\hbox{\vrule
height=6.45831pt,depth=4.30554pt,width=0.0pt\hskip 2.5pt$\scriptstyle a$\hskip
2.5pt}\hfil}}$}}$; say $M^{\prime}[i\rangle$ with $\ell^{\prime}(i)=a$. Let
$j$ be the largest transition in $T^{\prime}$ w.r.t. the well-ordering $<$ on
$T$ such that $i\leq^{\\#}\\!j$ and
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle$. It
suffices to show that $M_{M^{\prime},U}[\textsf{execute}^{i}_{j}\rangle$, i.e.
that $M_{M^{\prime},U}(\textsf{pre}^{i}_{j})\mathord{=}1$,
$M_{M^{\prime},U}(\textsf{trans}^{h}_{j}\textsf{-out})\mathord{=}1$ for all
$h\mathbin{<^{\\#}}\\!j$, and $M_{M^{\prime},U}(\pi_{j\\#l})\mathord{=}1$ for
all $l\mathbin{>^{\\#}}\\!j$.
If $\textsf{execute}^{i}_{j}\in U$ we would have $i\in U^{\prime}$ and hence
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[2\cdot\\{i\\}\rangle$.
Since $N^{\prime}$ is a finitary structural conflict net, this is impossible.
Therefore $\textsf{execute}^{i}_{j}\not\in U$ and, using the calculations from
(a) above, $M_{M^{\prime},U}(\textsf{pre}^{i}_{j})=1$.
Let $h<^{\\#}j$. To establish that
$M_{M^{\prime},U}(\textsf{trans}^{h}_{j}\textsf{-out})=1$ we need to show that
there is no $k\leq^{\\#}\\!j$ with $\textsf{execute}^{k}_{j}\in U$ and no
$g\leq^{\\#}\\!h$ with $\textsf{execute}^{g}_{h}\in U$. First suppose
$\textsf{execute}^{k}_{j}\in U$ for some $k\leq^{\\#}\\!j$. Then $k\in
U^{\prime}$ and hence
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[\\{i\\}\mathord{+}\\{k\\}\rangle$.
This implies $i\smile k$, and, as $N^{\prime}$ is a structural conflict net,
${\vphantom{i}}^{\bullet}i\cap{\vphantom{k}}^{\bullet}k=\emptyset$. Hence the
transitions $i$, $j$ and $k$ are all different, with
${\vphantom{i}}^{\bullet}i\cap{\vphantom{j}}^{\bullet}j\neq\emptyset$ and
${\vphantom{j}}^{\bullet}j\cap{\vphantom{k}}^{\bullet}k\neq\emptyset$ but
${\vphantom{i}}^{\bullet}i\cap{\vphantom{k}}^{\bullet}k=\emptyset$. Moreover,
the reachable marking
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ enables all three
of them. Hence $N^{\prime}$ contains a fully reachable pure M, which
contradicts the assumptions of Theorem LABEL:thm-correctness.
Next suppose $\textsf{execute}^{g}_{h}\in U$ for some $g\leq^{\\#}\\!h$. Then
$(M+\\!{\vphantom{U}}^{\bullet}U)[\textsf{execute}^{g}_{h}\rangle$, so
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[h\rangle$ by
Claim LABEL:cl-D(e). Moreover, $g\in U^{\prime}$, so
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[\\{i\\}\mathord{+}\\{g\\}\rangle$.
This implies $g\smile i$, and
${\vphantom{g}}^{\bullet}g\cap{\vphantom{i}}^{\bullet}i=\emptyset$. Moreover,
${\vphantom{g}}^{\bullet}g\cap{\vphantom{h}}^{\bullet}h\neq\emptyset$,
${\vphantom{h}}^{\bullet}h\cap{\vphantom{j}}^{\bullet}j\neq\emptyset$ and
${\vphantom{j}}^{\bullet}j\cap{\vphantom{i}}^{\bullet}i\neq\emptyset$, while
the reachable marking
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ enables all these
transitions. Depending on whether
${\vphantom{h}}^{\bullet}h\cap{\vphantom{i}}^{\bullet}i=\emptyset$, either
$h$, $j$ and $i$, or $g$, $h$ and $i$ constitute a fully reachable pure M,
contradicting the assumptions of Theorem LABEL:thm-correctness.
Let $l>^{\\#}j$. To establish that $M_{M^{\prime},U}(\pi_{j\\#l})=1$ we need
to show that there is no $k\leq^{\\#}\\!j$ with $\textsf{execute}^{k}_{j}\in
U$—already done above—and that
$\neg(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[l\rangle$.
Suppose
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[l\rangle$.
Considering that $j$ was the largest transition with $i\leq^{\\#}\\!j$ and
$(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[j\rangle$, we
cannot have $i<^{\\#}l$. Hence the transitions $i$, $j$ and $l$ are all
different, with
${\vphantom{i}}^{\bullet}i\cap{\vphantom{j}}^{\bullet}j\neq\emptyset$ and
${\vphantom{j}}^{\bullet}j\cap{\vphantom{l}}^{\bullet}l\neq\emptyset$ but
${\vphantom{i}}^{\bullet}i\cap{\vphantom{l}}^{\bullet}l=\emptyset$. Moreover,
the reachable marking
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$ enables all three
of them. Hence $N^{\prime}$ contains a fully reachable pure M, which
contradicts the assumptions of Theorem LABEL:thm-correctness.
3. (5c)
We have to show that $H(t)\leq H_{M^{\prime},U}(t)$ for each $t\in T$.
1. $\bullet$
In case $t\in T^{-}$ this follows from (23) and $H_{M^{\prime},U}\in{\rm
Nature}^{T^{+}}\\!\\!$.
2. $\bullet$
In case $t=\textsf{execute}^{i}_{j}$ it follows since
$\ell(H)\equiv\emptyset$.
3. $\bullet$
In case $t=\textsf{distribute}_{p}$ it follows from (19) and (23).
4. $\bullet$
Next let $t=\textsf{initialise}_{c}\cdot\textsf{fire}$ for some $c\in
T^{\prime}$. In case $H(\textsf{initialise}_{c}\cdot\textsf{fire})\leq 0$
surely we have $H(\textsf{initialise}_{c}\cdot\textsf{fire})\leq
H_{M^{\prime},U}(\textsf{initialise}_{c}\cdot\textsf{fire})$. So without
limitation of generality we may assume that
$H(\textsf{initialise}_{c}\cdot\textsf{fire})>0$. By (13, 23) we have
$H(\textsf{initialise}_{c}\\!\cdot\textsf{fire})=1$. Using (18), Claim
LABEL:cl-C, (23) and (19) we obtain, for all $p\in{\vphantom{c}}^{\bullet}c$,
$F^{\prime}(p,c)\cdot H(\textsf{initialise}_{c}\cdot\textsf{fire})\leq
H(\textsf{distribute}_{p})\leq(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})(p).$
Hence $c$ is enabled under
$M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime}$, which implies
$H_{M^{\prime},U}(\textsf{initialise}_{c}\cdot\textsf{fire})=1$.
5. $\bullet$
Let $t\mathbin{=}\textsf{transfer}^{b}_{c}\cdot\textsf{fire}$ for some
$b\mathbin{<^{\\#}}\\!c\mathbin{\in}T^{\prime}\\!$. As above, we may assume
$H(\textsf{transfer}^{b}_{c}\\!\cdot\textsf{fire})\mathbin{>}0$. By (20, 23)
we have $H(\textsf{transfer}^{b}_{c}\\!\cdot\textsf{fire})=1$. Using (23) and
that $H(\textsf{execute}^{g}_{b})=0$ for all $g\leq^{\\#}\\!b$, it follows
that $(M+\\!{\vphantom{U}}^{\bullet}U)(\pi_{b\\#c})=0$. Hence
$\neg(M+\\!{\vphantom{U}}^{\bullet}U)[\textsf{execute}^{g}_{b}\rangle$ for all
$g\leq^{\\#}\\!b$, and thus $\nexists\textsf{execute}^{g}_{b}\in U$. For all
$p\in{\vphantom{c}}^{\bullet}c$ we derive
$\begin{array}[]{@{}r@{~\leq~}ll}\lx@intercol F^{\prime}(p,c)\cdot
H(\textsf{transfer}^{b}_{c}\cdot\textsf{fire})\hfil\\\
\mbox{}~{}\leq~{}&F^{\prime}(p,c)\cdot\big{(}H(\textsf{transfer}^{b}_{c}\cdot\textsf{fire})-H(\textsf{transfer}^{b}_{c}\cdot\textsf{undone})\big{)}&(\ref{T-negative})\\\
~{}\leq~{}&F^{\prime}(p,c)\cdot\big{(}H(\textsf{initialise}_{c}\cdot\textsf{fire})-H(\textsf{initialise}_{c}\cdot\textsf{undo}(\mbox{$\textsf{trans}^{b}_{c}\textsf{-in}$}))\big{)}&(\ref{transin})\\\
~{}\leq~{}&F^{\prime}(p,c)\cdot\big{(}H(\textsf{initialise}_{c}\cdot\textsf{fire})-H(\textsf{initialise}_{c}\cdot\textsf{undone})\big{)}&(\ref{took})\\\
\lx@intercol\hfil\mbox{}~{}=~{}&\displaystyle\mbox{[the same as
above]}+\sum_{j\geq^{\\#}i\in{p}^{\bullet}}F^{\prime}(p,i)\cdot
H(\textsf{fetch}_{i,j}^{p,c})&(\mbox{Claim~{}\ref{cl-C}})\\\\[-10.0pt]
~{}\leq~{}&H(\textsf{distribute}_{p})&(\ref{p_j})\\\
~{}\leq~{}&\displaystyle(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})(p)+\sum_{\\{i\in
T^{\prime}\mid
p\in{i}^{\bullet}\\}}H(\textsf{finalise}^{i})&(\ref{p})\\\\[-10.0pt]
~{}\leq~{}&(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})(p)&(\ref{T-negative}).\end{array}$
Hence $(M^{\prime}+\\!{\vphantom{U^{\prime}}}^{\bullet}U^{\prime})[c\rangle$,
and thus $H_{M^{\prime},U}(\textsf{transfer}^{b}_{c})=1$.
4. (5d)
If $u\notin T^{-}$, yet $H(u)\neq 0$, then $u$ is either
$\textsf{distribute}_{p}$, $\textsf{initialise}_{j}\cdot\textsf{fire}$ or
$\textsf{transfer}^{h}_{j}\cdot\textsf{fire}$ for suitable $p\in S^{\prime}$
or $h,j\in T^{\prime}$. For $u=\textsf{distribute}_{p}$ the requirement
follows from Claim LABEL:cl-D(c); otherwise Property (NF-2), together with
(11), guarantees that $H(u)\geq 0$.
5. (5e)
If $H(t)\mathbin{>}0$ and $H(u)\mathbin{<}0$, then $t\mathbin{\in}T^{+}$ and
$u\mathbin{\in}T^{-}$. The only candidates for
${\vphantom{t}}^{\bullet}t\cap{\vphantom{u}}^{\bullet}u\neq\emptyset$ are
* –
$p_{c}\in{\vphantom{(\textsf{initialise}_{c}\cdot\textsf{fire})}}^{\bullet}(\textsf{initialise}_{c}\cdot\textsf{fire})\cap{\vphantom{(\textsf{fetch}_{i,j}^{p,c})}}^{\bullet}(\textsf{fetch}_{i,j}^{p,c})$
for $p\in S^{\prime}$, $c,i\in{p}^{\bullet}$ and $j\geq^{\\#}i$,
* –
$\textsf{trans}^{b}_{c}\textsf{-in}\in{\vphantom{(\textsf{transfer}^{b}_{c}\cdot\textsf{fire})}}^{\bullet}(\textsf{transfer}^{b}_{c}\cdot\textsf{fire})\cap{\vphantom{(\textsf{initialise}_{c}\cdot\textsf{undo}(\textsf{trans}^{b}_{c}\textsf{-in}))}}^{\bullet}(\textsf{initialise}_{c}\cdot\textsf{undo}(\textsf{trans}^{b}_{c}\textsf{-in}))$
for $b\leq^{\\#}\\!c\in T^{\prime}$.
We investigate these possibilities one by one.
* –
$H(\textsf{initialise}_{c}\cdot\textsf{fire})>0\wedge
H(\textsf{fetch}_{i,j}^{p,c})<0$ cannot occur by Claim LABEL:cl-C.
* –
Suppose $H(\textsf{transfer}^{b}_{c}\cdot\textsf{fire})>0$. By (20, 23) we
have $H(\textsf{transfer}^{b}_{c}\\!\cdot\textsf{fire})=1$. Through the
derivation above, in the proof of requirement (c), using (23, 14, 11), Claim
LABEL:cl-C and (18), we obtain $H(\textsf{distribute}_{p})\geq
F^{\prime}(p,c)$ for all $p\in{\vphantom{c}}^{\bullet}c$. Now Claim
LABEL:cl-D(d) yields $H(\textsf{finalise}^{i})=0$ for all
$i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}c$.
By (9) and (23) we obtain
$H(\textsf{initialise}_{c}\\!\cdot\textsf{reset}_{i})\mathbin{=}0$ for each
such $i$. Hence
$\sum_{i\stackrel{{\scriptstyle\\#}}{{=}}c}H(\textsf{initialise}_{c}\\!\cdot\textsf{reset}_{i})\mathbin{=}0$,
and thus
$H(\textsf{initialise}_{c}\cdot\textsf{undo}(\textsf{trans}^{b}_{c}\textsf{-in}))=0$
by (11, 23).
6. (5f)
If $H(u)<0$ and $(M+\\!{\vphantom{U}}^{\bullet}U)[t\rangle$ with
$\ell(t)\neq\tau$, then $t=\textsf{execute}^{i}_{j}$ for some
$i\leq^{\\#}\\!j\in T^{\prime}$ and $u\mathbin{\in}T^{-}$. The only candidates
for ${\vphantom{t}}^{\bullet}t\cap{\vphantom{u}}^{\bullet}u\neq\emptyset$ are
* –
$\textsf{pre}^{i}_{j}\in{\vphantom{(\textsf{execute}^{i}_{j})}}^{\bullet}(\textsf{execute}^{i}_{j})\cap{\vphantom{(\textsf{initialise}_{j}\cdot\textsf{undo}(\textsf{pre}^{i}_{j}))}}^{\bullet}(\textsf{initialise}_{j}\cdot\textsf{undo}(\textsf{pre}^{i}_{j}))$
and
* –
$\textsf{trans}^{h}_{j}\textsf{-out}\in{\vphantom{(\textsf{execute}^{i}_{j})}}^{\bullet}(\textsf{execute}^{i}_{j})\cap{\vphantom{(\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out}))}}^{\bullet}(\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out}))$
for $h<^{\\#}j$.
We investigate these possibilities one by one.
* –
Suppose $(M+\\!{\vphantom{U}}^{\bullet}U)[\textsf{execute}^{i}_{j}\rangle$. By
Claim LABEL:cl-D(b), $H(\textsf{finalise}^{k})\geq 0$ for each
$k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}i$.
By (9) and (23) we obtain
$H(\textsf{initialise}_{i}\\!\cdot\textsf{reset}_{k})\mathbin{=}0$ for each
such $k$. Hence
$\displaystyle\sum_{k\stackrel{{\scriptstyle\\#}}{{=}}i}H(\textsf{initialise}_{i}\\!\cdot\textsf{reset}_{k})\mathbin{=}0$,
and thus
$H(\textsf{initialise}_{i}\cdot\textsf{undo}(\textsf{pre}^{i}_{j}))=0$ by (11,
23).
* –
Suppose $(M+\\!{\vphantom{U}}^{\bullet}U)[\textsf{execute}^{i}_{j}\rangle$ and
$h<^{\\#}j$. By Claim LABEL:cl-D(b), $H(\textsf{finalise}^{k})\geq 0$ for each
$k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}j$.
By (9) and (23)
$H(\textsf{transfer}^{h}_{j}\\!\cdot\textsf{reset}_{k})\mathbin{=}0$ for each
such $k$. So
$\displaystyle\sum_{k\stackrel{{\scriptstyle\\#}}{{=}}j}H(\textsf{transfer}^{h}_{j}\\!\cdot\textsf{reset}_{k})\mathbin{=}0$,
and
$H(\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out}))=0$
by (11, 23).
7. (5g)
Suppose
$(M+\\!{\vphantom{U}}^{\bullet}U)[\\{t\\}\mathord{+}\\{u\\}\rangle_{N}$, and
$i,k\in T^{\prime}$ with $\ell^{\prime}(i)=\ell(t)$ and
$\ell^{\prime}(k)=\ell(u)$. Since the net $N^{\prime}$ is plain, $t$ and $u$
must have the form $\textsf{execute}^{i}_{j}$ and $\textsf{execute}^{k}_{j}$
for some $j>^{\\#}i$ and $l>^{\\#}k$. Claim LABEL:cl-concurrency yields
$\neg(i\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$
and hence ${\vphantom{i}}^{\bullet}i\cap{\vphantom{k}}^{\bullet}k=\emptyset$.
$\Box$
Thus, we have established that the conflict replicating implementation
$\mathcal{I}(N^{\prime})$ of a finitary plain structural conflict net
$N^{\prime}$ without a fully reachable pure M is branching ST-bisimilar with
explicit divergence to $N^{\prime}$. It remains to be shown that
$\mathcal{I}(N^{\prime})$ is essentially distributed.
###### Lemma 6.2.
S-invariant Let $N$ be the conflict replicating implementation of a finitary
net
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$;
let $j,l\in T^{\prime}\\!$, with $l\mathbin{>^{\\#}}j$. Then no two
transitions from the set $\\{\textsf{execute}^{i}_{j}\mid
i\leq^{\\#}\\!j\\}\cup\\{\textsf{transfer}^{j}_{l}\cdot\textsf{fire}\\}\cup\mbox{}$
$\\{\textsf{transfer}^{j}_{l}\cdot\textsf{undo}(\mbox{$\textsf{trans}^{j}_{l}\textsf{-out}$})\\}\cup\\{\textsf{execute}^{k}_{l}\mid
k\leq^{\\#}\\!l\\}$ can fire concurrently.
###### Proof 6.3.
For each $i\mathbin{\leq^{\\#}\\!}j$ pick an arbitrary preplace $q_{i}$ of
$i$. The set
$\\{\textsf{fetch}^{q_{i},i}_{i,j}\textsf{-in},~{}\textsf{fetch}^{q_{i},i}_{i,j}\textsf{-out}\mid
i\leq^{\\#}\\!j\\}\cup\mbox{}$
$\\{\pi_{j\\#l},~{}\textsf{trans}^{j}_{l}\textsf{-out},~{}\textsf{took}(\textsf{trans}^{j}_{l}\textsf{-out},\textsf{transfer}^{j}_{l}),~{}\rho(\textsf{transfer}^{j}_{l}\\}$
is an _S-invariant_ : there is always exactly one token in this set. This is
the case because each transition from $N$ has as many preplaces as postplaces
in this set. The transitions from $\\{\textsf{execute}^{i}_{j}\mid
i\leq^{\\#}\\!j\\}\cup\\{\textsf{transfer}^{j}_{l}\cdot\textsf{fire}\\}\cup\\{\textsf{transfer}^{j}_{l}\cdot\textsf{undo}(\mbox{$\textsf{trans}^{j}_{l}\textsf{-out}$})\\}\vspace{-2pt}\cup\\{\textsf{execute}^{k}_{l}\mid
k\leq^{\\#}\\!l\\}$ each have a preplace in this set. Hence no two of them can
fire concurrently.
###### Lemma 6.4.
essentially distributed Let $N$ be the conflict replicating implementation
$\mathcal{I}(N^{\prime})$ of a finitary plain structural conflict net
$N^{\prime}=(S^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0},\ell^{\prime})$
without a fully reachable pure M. Then for any
$i\leq^{\\#}\\!j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}c\in
T^{\prime}$ and $f\in(\textsf{initialise}_{c})^{\,\it far}$, the transitions
$\textsf{execute}^{i}_{j}$ and $\textsf{initialise}_{c}\cdot\textsf{undo}(f)$
cannot fire concurrently.
###### Proof 6.5.
Suppose these transitions can fire concurrently, say from the marking
$M\in[M_{0}\rangle_{N}$. By Claim LABEL:cl-extra, there are
$M^{\prime}\in[M^{\prime}_{0}\rangle_{N^{\prime}}$ and $G\in_{f}\mbox{\bbb
Z}^{T}$ such that (B)–(N) hold. Let $t:=\textsf{initialise}_{c}$,
$G_{1}:=G+\\{t\cdot\textsf{undo}(f)\\}$ and $M_{1}\mathbin{:=}M+\llbracket
t\mathord{\cdot}\textsf{undo}(f)\rrbracket$. Then (11), applied to the triples
$(M,M^{\prime},G)$ and $(M_{1},M^{\prime},G_{1})$, yields
$\sum_{\makebox[7.00002pt][l]{$\scriptstyle\\{\omega\mid
t\in\Omega_{\omega}\\}$}}G(t\cdot\textsf{reset}_{\omega})\leq
G(t\cdot\textsf{undo}(f))<G_{1}(t\cdot\textsf{undo}(f))\leq\sum_{\makebox[7.00002pt]{$\scriptstyle\\{\omega\mid
t\in\Omega_{\omega}\\}$}}G_{1}(t\cdot\textsf{undo}_{\omega})=\sum_{\makebox[7.00002pt]{$\scriptstyle\\{\omega\mid
t\in\Omega_{\omega}\\}$}}G(t\cdot\textsf{undo}_{\omega}).$
Hence, there is an $\omega$ with $t\in\Omega_{\omega}$ and
$G(t\cdot\textsf{reset}_{\omega})<G(t\cdot\textsf{undo}_{\omega})$. This
$\omega$ must have the form $k\in T^{\prime}$ with
$k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}c$.
We now obtain
$\begin{array}[b]{r@{~\leq~}ll}\lx@intercol\hfil
0~{}=~{}&G(\textsf{finalise}^{k})&\mbox{(by (\ref{r2}))}\\\
~{}\leq~{}&G(t\cdot\textsf{elide}_{k})+G(t\cdot\textsf{reset}_{k})&\mbox{(by
(\ref{reset}))}\\\
\lx@intercol\hfil~{}<~{}&G(t\cdot\textsf{elide}_{k})+G(t\cdot\textsf{undo}_{k})\\\
~{}\leq~{}&\sum_{l\geq^{\\#}k}G(\textsf{execute}^{k}_{l})&\mbox{(by
(\ref{undo}))}.\end{array}$
Hence, there is an
$l\geq^{\\#}k\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}c$
with $G(\textsf{execute}^{k}_{l})>0$. By (M) we obtain
$\neg(j\mathbin{\raisebox{0.0pt}[0.0pt][0.0pt]{$\stackrel{{\scriptstyle\\#}}{{=}}$}}k)$,
so ${\vphantom{j}}^{\bullet}j\cap{\vphantom{k}}^{\bullet}k=\emptyset$.
Additionally, we have
${\vphantom{j}}^{\bullet}j\cap{\vphantom{c}}^{\bullet}c\neq\emptyset$ and
${\vphantom{c}}^{\bullet}c\cap{\vphantom{k}}^{\bullet}k\neq\emptyset$. By (N)
we obtain $M^{\prime}[j\rangle$, and by (D) and (F) $M^{\prime}[k\rangle$.
Furthermore, by (11),
$G(t\cdot\textsf{undo}(f))<G_{1}(t\cdot\textsf{undo}(f))\leq
G_{1}(t\cdot\textsf{fire})=G(t\cdot\textsf{fire})$, so, for all
$p\mathbin{\in}{\vphantom{c}}^{\bullet}c$,
$\begin{array}[b]{r@{~\leq~}ll}F^{\prime}(p,c)~{}\leq~{}&F^{\prime}(p,c)\cdot\big{(}G(t\cdot\textsf{fire})-G(t\cdot\textsf{undo}(f))\big{)}\\\
~{}\leq~{}&F^{\prime}(p,c)\cdot\big{(}G(t\cdot\textsf{fire})-G(t\cdot\textsf{undone})\big{)}&\mbox{(by
(\ref{took}))}\\\
~{}\leq~{}&G(\textsf{distribute}_{p})-\sum_{j\geq^{\\#}i\in{p}^{\bullet}}F^{\prime}(p,i)\cdot
G(\textsf{fetch}_{i,j}^{p,c})&\mbox{(by (\ref{p_j}))}\\\
~{}\leq~{}&G(\textsf{distribute}_{p})&\mbox{(by (\ref{rFp}) and
(\ref{fetch}))}\\\ ~{}\leq~{}&M^{\prime}(p)&\mbox{(by (\ref{r3}).}\\\
\end{array}$
It follows that $M^{\prime}[c\rangle$. Thus $N^{\prime}$ contains a fully
reachable pure M, which contradicts the assumptions of Lemma LABEL:lem-
essentially_distributed.
###### Theorem 6.6.
cri-distributed Let $N$ be the conflict replicating implementation
$\mathcal{I}(N^{\prime})$ of a finitary plain structural conflict net
$N^{\prime}$ without a fully reachable pure M. Then $N$ is essentially
distributed.
###### Proof 6.7.
We take the canonical distribution $D$ of $N$, in which $\equiv_{D}$ is the
equivalence relation on places and transitions generated by Condition (1) of
Definition LABEL:df-distributed. We need to show that this distribution
satisfies Condition ($2^{\prime}$) of Definition LABEL:df-
externally_distributed. A given transition $t$ with $\ell(t)\neq\tau$ must
have the form $\textsf{execute}^{i}_{j}$ for some $i\leq^{\\#}\\!j\in
T^{\prime}$. By following the flow relation of $N$ one finds the places and
transitions that, under the canonical distribution, are co-located with
$\textsf{execute}^{i}_{j}$:
$\begin{array}[]{@{}l@{}}\pi_{j\\#l}\rightarrow\textsf{transfer}^{j}_{l}\cdot\textsf{fire}\leftarrow\textsf{trans}^{j}_{l}\textsf{-in}\rightarrow\textsf{initialise}_{l}\cdot\textsf{undo}(\mbox{$\textsf{trans}^{j}_{l}\textsf{-in}$})\leftarrow\textsf{take}(\textsf{trans}^{j}_{l}\textsf{-in},\textsf{initialise}_{l})\\\
~{}~{}\downarrow\\\ \textsf{execute}^{i}_{j}\\\ ~{}~{}\uparrow\\\
\textsf{trans}^{h}_{j}\textsf{-out}\rightarrow\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out})\leftarrow\textsf{take}(\textsf{trans}^{h}_{j}\textsf{-out},\textsf{transfer}^{h}_{j})\\\
~{}~{}\downarrow\\\ \textsf{execute}^{g}_{j}\\\ ~{}~{}\uparrow\\\
\textsf{pre}^{g}_{j}\rightarrow\textsf{initialise}_{g}\cdot\textsf{undo}(\textsf{pre}^{g}_{j})\leftarrow\textsf{take}(\textsf{pre}^{g}_{j},\textsf{initialise}_{g})\end{array}$
for all $l\mathbin{>^{\\#}}j$, $h\mathbin{<^{\\#}}j$ and $g\leq^{\\#}\\!j$. We
need to show that none of these transitions can happen concurrently with
$\textsf{execute}^{i}_{j}$. For transitions
$\textsf{transfer}^{j}_{l}\cdot\textsf{fire}$ and $\textsf{execute}^{g}_{j}$
this follows directly from Lemma LABEL:lem-S-invariant. For
$\textsf{transfer}^{h}_{j}\cdot\textsf{undo}(\textsf{trans}^{h}_{j}\textsf{-out})$
this also follows from Lemma LABEL:lem-S-invariant, in which $j$, $k$ and $l$
play the rôle of the current $h$, $i$ and $j$. For the transitions
$\textsf{initialise}_{l}\cdot\textsf{undo}(\mbox{$\textsf{trans}^{j}_{l}\textsf{-in}$})$
and $\textsf{initialise}_{g}\cdot\textsf{undo}(\textsf{pre}^{g}_{j})$ this has
been established in Lemma LABEL:lem-essentially_distributed.
Our main result follows by combining Theorems LABEL:thm-correctness and
LABEL:thm-cri-distributed and Proposition LABEL:pr-
essentiallydistributedequal:
###### Theorem 6.8.
fullmgttrulysync Let $N$ be a finitary plain structural conflict net without a
fully reachable pure M. Then $N$ is distributable up to
$\approx^{\Delta}_{bSTb}$.
###### Corollary 6.9.
fullmeqtrulysync Let $N$ be a finitary plain structural conflict net. Then $N$
is distributable iff it has no fully reachable pure M.
## 7 Conclusion
In this paper, we have given a precise characterisation of distributable Petri
nets in terms of a semi-structural property. Moreover, we have shown that our
notion of distributability corresponds to an intuitive notion of a distributed
system by establishing that any distributable net may be implemented as a
network of asynchronously communicating components. In order to formalise what
qualifies as a valid implementation, we needed a suitable equivalence
relation. We have chosen step readiness equivalence for showing the
impossibility part of our characterisation, since it is one of the simplest
and least discriminating semantic equivalences imaginable that abstracts from
internal actions but preserves branching time, concurrency and divergence to
some small degree. For the positive part, stating that all other nets are
implementable, we have introduced a combination of several well known rather
discriminating equivalences, namely a divergence sensitive version of
branching bisimulation adapted to ST-semantics. Hence our characterisation is
rather robust against the chosen equivalence; it holds in fact for all
equivalences between these two notions. However, ST-equivalence (and our
version of it) preserves the causal structure between action occurrences only
as far as it can be expressed in terms of the possibility of durational
actions to overlap in time. Hence a natural question is whether we could have
chosen an even stronger causality sensitive equivalence for our
implementability result, respecting e.g. pomset equivalence or history
preserving bisimulation. Our conflict replicating implementation does not
fully preserve the causal behaviour of nets; we are convinced that we have
chosen the strongest possible equivalence for which our implementation works.
It is an open problem to find a class of nets that can be implemented
distributedly while preserving divergence, branching time and causality in
full. Another line of research is to investigate which Petri nets can be
implemented as distributed nets when relaxing the requirement of preserving
the branching structure. If we allow linear time correct implementations
(using a step trace equivalence), we conjecture that all Petri nets become
distributable. However, also in this case it is problematic, in fact even
impossible in our setting, to preserve the causal structure, as has been shown
in [17]. A similar impossibility result has been obtained in the world of the
$\pi$-calculus in [15]. The interplay between choice and synchronous
communication has already been investigated in quite a number of approaches in
different frameworks. We refer to [7] for a rather comprehensive overview and
concentrate here on recent and closely related work. The idea of modelling
asynchronously communicating sequential components by sequential Petri nets
interacting though buffer places has already been considered in [16]. There
Wolfgang Reisig introduces a class of systems, represented as Petri nets,
where the relative speeds of different components are guaranteed to be
irrelevant. His class is a strict subset of our LSGA nets, requiring
additionally, amongst others, that all choices in sequential components are
free, i.e. do not depend upon the existence of buffer tokens, and that places
are output buffers of only one component. Another quite similar approach was
taken in [4], where transition labels are classified as being either input or
output. There, asynchrony is introduced by adding new buffer places during net
composition. This framework does not allow multiple senders for a single
receiver. Other notions of distributed and distributable Petri nets are
proposed in [12, 2, 3]. In these works, given a distribution of the
transitions of a net, the net is distributable iff it can be implemented by a
net that is distributed w.r.t. that distribution. The requirement that
concurrent transitions may not be co-located is absent; given the fixed
distribution, there is no need for such a requirement. These papers differ
from each other, and from ours, in what counts as a valid implementation. A
comparison of our criterion with that of Hopkins [12] is provided in [7]. In
[7] we have obtained a characterisation similar to Corollary LABEL:cor-
fullmeqtrulysync, but for a much more restricted notion of distributed
implementation (_plain distributability_), disallowing nontrivial transition
labellings in distributed implementations. We also proved that fully reachable
pure Ms are not implementable in a distributed way, even when using transition
labels (Theorem LABEL:thm-trulysyngltfullm). However, we were not able to show
that this upper bound on the class of distributable systems was tight. Our
current work implies the validity of Conjecture 1 of [7]. While in [7] we
considered only one-safe place/transition systems, the present paper employs a
more general class of place/transition systems, namely structural conflict
nets. This enables us to give a concrete characterisation of distributed nets
as systems of sequential components interacting via non-safe buffer places.
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* [17] J.-W. Schicke, K. Peters & U. Goltz (2011): _Synchrony vs. Causality in Asynchronous Petri Nets_. In B. Luttik & F. Valencia, editors: Proceedings 18th International Workshop on Expressiveness in Concurrency, Aachen, Germany, 5th September 2011, Electronic Proceedings in Theoretical Computer Science 64, pp. 119–131, 10.4204/EPTCS.64.9.
* [18] W. Vogler (1993): _Bisimulation and Action Refinement_. Theoretical Computer Science 114(1), pp. 173–200, 10.1016/0304-3975(93)90157-O.
|
arxiv-papers
| 2012-07-16T08:18:17 |
2024-09-04T02:49:33.074056
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rob van Glabbeek, Ursula Goltz and Jens-Wolfhard Schicke-Uffmann",
"submitter": "Rob van Glabbeek",
"url": "https://arxiv.org/abs/1207.3597"
}
|
1207.3634
|
Further author information: (Send correspondence to G. Agapito)
G. Agapito: E-mail: agapito@arcetri.astro.it
# Infinite impulse response modal filtering in visible adaptive optics
G. Agapitoa C. Arcidiaconob F. Quirós-Pachecoa A. Puglisia S. Espositoa
aOsservatorio Astrofisico di Arcetri Largo E. Fermi 5 Firenze Italy;
bOsservatorio Astronomico di Bologna Via Ranzani 1 Bologna Italy
###### Abstract
Diffraction limited resolution adaptive optics (AO) correction in visible
wavelengths requires a high performance control. In this paper we investigate
infinite impulse response filters that optimize the wavefront correction: we
tested these algorithms through full numerical simulations of a single-
conjugate AO system comprising an adaptive secondary mirror with 1127
actuators and a pyramid wavefront sensor (WFS). The actual practicability of
the algorithms depends on both robustness and knowledge of the real system:
errors in the system model may even worsen the performance. In particular we
checked the robustness of the algorithms in different conditions, proving that
the proposed method can reject both disturbance and calibration errors.
###### keywords:
Adaptive optics, Pyramid wavefront sensor, telescope vibrations, adaptive
secondary mirror, modal control optimization
## 1 Introduction
Modern 8m telescopes are able to provide up to 90% Strehl Ratio (SR) on high
flux regime in H band (Esposito _et al._[1]), and reach a maximum resolution
of about 30mas (FLAO@LBT at the J band or the Hubble Space Telescope at the U
band or using lucky imaging technique on large telescope at R band Law _et
al._[2]). Since the maximum achievable resolution is fixed by diffraction
retrieving Point Spread Function (PSF), which Full Width at Half Maximum
(FWHM) is $\lambda/D$, where $\lambda$ is the imaged wavelength and $D$ the
telescope diameter. 8-m class telescope could improve the maximum achievable
resolution making images at shorter wavelenghts, such as V band, which could
provide a 12mas resolution. However behind these numbers is hidden a
limitation: an Adaptive Optics (AO) system for visible wavelengths has tighter
requirements compared to an near infra-red one. In fact shorter wavelengths
correspond to smaller Fried’s parameters, $r_{0}$, hence more degrees of
correction and low measurement noise values are required to obtain a good
correction.
Optimized control techniques such as a Modal Control Optimization[3],
observer-based and model-based[4, 5, 6, 7, 8, 9, 10] approaches can increase
the performance of an AO system in comparison to an integrator controller.
These methods need to work in realistic conditions, _i.e._ without an accurate
knowledge of the real system and with complex disturbances like calibration
errors and telescope vibrations. In particular, Modal Control Optimization has
only one degree of freedom per mode, that is the integrator gain, so it cannot
efficiently reject disturbance condition such as telescope vibrations, while
the other methods rely on the accuracy of the system model.
For these reasons, starting from the work of Dessenne _et al._[11], we have
chosen to design a method that is data driven, so it relies as little as
possible on the system model, and can have many degrees of freedoms as to
adapt itself to the disturbances.
The remainder of this paper is as follows. Sec.2 introduces the Infinite
Impulse Response (IIR) filter based control along with other controllers that
we will use in the performance evaluation as comparison. Sec.3 briefly
describes the AO system considered in our simulations. Finally in Sec.4 the
controllers are analyzed through numerical simulations from both performance
and robustness viewpoint.
## 2 AO Control Systems
Figure 1: Adaptive Optics Control loop.
The AO system control loop considered in this work is represented in Fig. 1.
The controller gets the modal measurements as input and gives the modal
commands as output. The most used controller in AO systems is the integrator,
so we have chosen it to make a comparison that will be presented in the next
sections.
### 2.1 IIR filter based control
To determine the optimal control expressed as an IIR filter we realized an
algorithm that minimized the residual variance for each mode $i$ as a function
of the IIR filter parameters:
$[\hat{a}_{i},\hat{b}_{i}]=\min_{[a_{i},b_{i}]}J_{i}\,,$ (1)
where $a_{i}$ and $b_{i}$ are two vectors of the IIR filter denominator and
numerator coefficients:
$C_{i}(z)=\frac{\sum\limits_{j=1}^{N_{b}}b_{i}(j)z^{-j}}{\sum\limits_{l=1}^{N_{a}}a_{i}(l)z^{-l}}\,,$
(2)
where $N_{a}$ and $N_{b}$ are the order of respectively the denominator and
numerator. When the modes are orthonormal, minimizing the sum of the residual
modal variances is equivalent to minimizing the residual phase variance in the
direction of the Natural Guide Star (NGS), which is the goal of the control
system studied in this paper. The cost, that is the measured residual modal
variance of the i-_th_ mode $J_{i}$, is:
$J_{i}=\sum\limits_{f=\frac{T}{n}}^{\frac{T}{2}}\Phi_{i}^{meas}(\omega)=\sum\limits_{f=\frac{T}{n}}^{\frac{T}{2}}\|W(z)\|^{2}\Phi_{i}^{ol}(\omega)\,,$
(3)
where:
$W(z)=\frac{1}{1+\mathrm{C}(z)\mathrm{WFS}(z)\mathrm{DM}(z)}.$ (4)
$\omega$ is the discrete frequency, $n$ is the number of points of the
discrete signals, $T$ is the sampling period, $\Phi_{i}^{meas}(\omega)$ is the
Power Spectral Density (PSD) of the closed loop i-_th_ modal coefficient
determined from the WFS measurements, and $\Phi_{i}^{ol}(\omega)$ is the PSD
of the open loop i-_th_ modal coefficient. The open loop measurement $o_{i}$
would be the input signal of the AO system without the control feedback, and
we reconstruct it as the sum of the closed loop measurement $s_{i}$ and
command $c_{i}$. At iteration $k$, taking into account the delay $d$, it can
be written as:
$o_{i}(k)=s_{i}(k)+c_{i}(k-d)\,.$ (5)
It can be shown (see Dessenne _et al._[11]) that:
$s_{i}(k)=W(z)o_{i}(k)\,,$ (6)
so that the relationship shown in Eq.3 is demonstrated.
This algorithm must verify the stability of the loop during the minimization
of the cost $J_{i}$, and it must discard the controller that do not stabilize
the loop. In fact, an unstable controller could produce the minimum cost, but
it could not be implemented.
To take into account the modelling errors, like pupil shifts or approximations
of DM and WFS models, a stability constrain is not enough. Therefore the
controller must guarantee some robustness, so small variations of the model
parameters would not drive the controller to instability. Thus we impose
$\max\|{\mu}\left(\chi_{i}(z)\right)\|^{2}\leq\eta$, where $\mu(\cdot)$ gives
the roots of the polynomial, $0<\eta<1$, and $\chi_{i}(z)=1+C_{i}(z)P(z)$ is
the characteristic polynomial, where $P(z)$ is the AO system model:
$P(z)=\mathrm{WFS}(z)\mathrm{DM}(z)\,,$ (7)
we have simply assumed that $\mathrm{WFS}(z)=z^{-1}$ and
$\mathrm{DM}(z)=z^{-1}$, in order to model a simple AO loop with two frames
delay.
Note that $\eta$ gives a measurement of the stability margin since if $\eta$
is equal to 1 only stability is guaranteed, while smaller values of $\eta$
correspond to a greater distance from instability, _i.e._ a greater
robustness.
In this work we choose $\eta=0.8$, which corresponds to a gain margin of 0.2.
Hence the algorithm can be summarized as:
* •
Acquire with a closed loop the mode measurement $s_{i}$ and commands $c_{i}$.
* •
Determine $\Phi_{i}^{ol}(f)$.
* •
Choose $N_{a}$ and $N_{b}$ and the starting IIR filter parameters $a_{i}(0)$,
$b_{i}(0)$.
* •
Search the combination of IIR filter parameters $\hat{a}_{i}$, $\hat{b}_{i}$
which minimizes $J_{i}$ and satisfies
$\max\|{\mu}\left(\chi_{i}(z)\right)\|^{2}\leq\eta$.
In this work, the minimization of $J_{i}$ is based on the downhill simplex
method of Nelder & Mead [12], and we opted for $N_{a}=N_{b}=3$ in case of
turbulence and 2 coefficients more for each vibration. So, in the next
simulations, Tip and Tilt have $N_{a}=N_{b}=7$ while the other modes have
$N_{a}=N_{b}=3$, because, as presented in Sec.3, Tip and Tilt are affected by
two vibrations.
### 2.2 Modal Gain Integrator
The Transfer Function of the modal integrator is:
$C(z)=\frac{g_{i}}{1-z^{-1}}$ (8)
where $g_{i}$ is the integrator gain of the _i_ -th mode, and $z$ is the
variable of the Z-transform.
In this work, we refer as integrator the particular case in which all the
gains are equal, and as Optimized Modal Gain Integrator[3] (OMGI) the
particular case in which each modal gain is optimized. For both cases we
optimize the gain: in the integrator case we optimized the global gain by a
trial and error procedure _i.e._ we ran many simulations with different gains
and then we chose the one that produced the best results in terms of Strehl
Ratio (SR); instead, in case of OMGI, we determine the gain following the same
algorithm presented in the previous section, but instead of minimizing the
cost $J_{i}$ as a function of the IIR filter parameters, we minimized it as
function of the integrator gain $g_{i}$:
$\hat{g}_{i}=\min_{g_{i}}J_{i}\,,$ (9)
since the integrator controller is an IIR filter with $N_{a}=2$ $N_{b}=1$ and
a single variable parameter, the gain $g$.
An example of IIR and OMGI Rejection Transfer Functions (RTF) $W$ is shown in
Fig.2. Note that the OMGI give better rejection at the lowest frequencies, but
the bandwidth rapidly decreases with the mode order. The integrator controller
bandwidth is in direct proportion to the peak at high frequencies, and this
peak increase the residual phase due to the measurement noise. Hence, the high
order modes, that are more affected by noise, have a lower bandwidth. Instead
the IIR filter gives a bandwidth greater than $20\,\mathrm{Hz}$ for all the
modes with low peaks at high frequencies.
Figure 2: RTFs for various modes with the IIR filter based control with
$N_{a}=N_{b}=3$ (left), and the OMGI (right). This RTFs are determined for the
case magnitude 12.5 without vibrations nor pupil shifts. All the other
parameters may be found in Tab.1.
## 3 AO system and simulator
The AO system analyzed in this work is based on a pyramid wavefront sensor
(WFS) with tilt modulation (Ragazzoni[14]) and on an adaptive secondary mirror
(ASM) (Salinari _et al._ [15]). We chose to work with an 8m class telescope,
an ASM with 1127 actuators, and a WFS with 40 $\times$ 40 sub-apertures. As
detector of the WFS we considered an ANDOR iXon X3 897 camera[16] with Read
Out Noise (RON), Excess Noise, and Clock Induced Charge (CIC) (Daigle _et
al._[17]) values shown in Tab.1. All simulations reported hereafter rely on
the End-to-End numerical simulator used to perform the performance analysis
and optimization of the First Light Adaptive Optics (FLAO) system of the LBT
telescope [18].
The atmospheric turbulence was simulated with a set of phase screens with Von-
Karman statistics that were displaced in front of the telescope pupil to
emulate the time-evolving turbulence according to the Taylor hypothesis.
Pupil shifts due to the sensor optical misalignments are a common calibration
error. We will evaluate in our simulations the robustness of the controller to
this kind of errors. The pupil shifts considered in the simulations have an
amplitude of $0.1\,\mathrm{pixel}$ in both _x_ and _y_ directions.
We add in the simulations, as external disturbance, telescope vibrations.
Vibrations may arise from many different situations, e.g. telescope
orientation, telescope tracking errors, and wind shaking. In particular, since
vibrations cause displacements of the image they have a major impact on so-
called tip/tilt modes. Hence, we chose to introduce Tip/Tilt vibrations with
central frequencies of $13\,\mathrm{Hz}$ and $22\,\mathrm{Hz}$, an RMS of
$20\,\mathrm{mas}$ each, and a damping ratio of $0.01$.
Telescope
---
Effective diameter ($D$) | $8\,\mathrm{m}$
Central obstruction | $0.138\,\mathrm{D}$
Pyramid WFS
Sensing wavelength ($\lambda$) | $0.75\,\mathrm{\mu m}$
Number of sub-apertures | $40\times 40$
Tilt modulation radius | $2.0\frac{\lambda}{D}$
Total average transmission | $0.41$
RON (electrons per pixel) | $0.06\,\mathrm{e^{-}RMS}$
Clock Induced Charge | $0.005\,\mathrm{e^{-}/pixel/frame}$
Excess Noise | $\sqrt{2}$
Guide star
Mag. zero point (Johnson [13]) | $1.76\,10^{-8}\,\mathrm{J/s/m^{2}/\mu m}$
ASM
---
Modes (Karhunen-Loève) | $1127$
Turbulence
Outer scale ($L_{0}$) | $25\,\mathrm{m}$
Mean wind speed | $10.6\,\mathrm{m/s}$
Telescope Vibration
Frequencies | $13,22\,\mathrm{Hz}$
Standard deviation | $20\,\mathrm{mas}$
Damping ratio | $0.01$
Loop paramenters
Sampling frequency | $1000\,\mathrm{Hz}$
Delay | 2 frames
Table 1: Summary of the simulation parameters.
## 4 Simulations
Firstly we will check the performance of the integrator with turbulence and
without pupil shifts nor vibrations. We chose to run all the simulations in
between magnitude 8.5 and 12.5. All the simulation parameters are listed in
Tab.1. The gain is determined with the method described in Sec.2.2. In Fig.3
and Tab.2 are shown the results. As we expect the performance of the system
rapidly decreases at the shortest wavelengths. Moreover the decrease is
greater in the visible wavelengths, in fact, while at K band from magnitude
$8.5$ to $12.5$ less than $8\%$ of SR is lost, at V band the difference is
about $40\%$.
Figure 3: Summary of simulation results for the integrator controller in case of turbulence without pupil shifts nor vibrations. Integrator gain is determined as described in Sec.2.2. Seeing 0.8′′, see Table 1 for the other parameters. band | V | R | H | K
---|---|---|---|---
magnitude | Strehl Ratio $\%$
8.5 | 54.8 | 73.4 | 94.5 | 96.8
9.5 | 50.1 | 70.1 | 93.7 | 96.4
10.5 | 42.4 | 64.3 | 92.2 | 95.5
11.5 | 30.7 | 54.4 | 89.5 | 93.9
12.5 | 14.9 | 37.1 | 83.3 | 90.1
Table 2: Summary of simulation results for the integrator controller.
Integrator gain is determined as described in Sec.2.2. Seeing 0.8′′, see Table
1 for the other parameters.
In the next simulations we considered only these two magnitudes, $8.5$ and
$12.5$, to make a comparison between all the controllers. We will compare the
controller in 4 difference observation conditions: the first one is with
turbulence and without pupil shifts nor vibrations, then second and third case
with turbulence and with pupil shifts or vibrations, respectively, and the
last one with all the disturbances included.
The controllers parameters are optimized for each condition. The gain found
for the integrator and the minimum, maximum and mean gain of the OMGI are
shown in Tab.3. Regarding the IIR filter Fig.4 shown, as an example, two RTFs
for the tip/tilt modes in two different cases: on the left, with only
turbulence and, on the right, with turbulence, pupil shifts, and vibrations.
Note that in the first case we design a filter with $N_{a}=N_{b}=3$, while
with all the disturbances $N_{a}=N_{b}=7$.
Controller | integrator | OMGI
---|---|---
magnitude | g | g min | g max | g mean
Turbulence
8.5 | 0.6 | 0.23 | 0.88 | 0.50
12.5 | 0.3 | 0.01 | 0.85 | 0.20
Turbulence + Pupil shifts
8.5 | 0.6 | 0.08 | 0.92 | 0.33
12.5 | 0.3 | 0.03 | 0.85 | 0.23
Turbulence + Vibrations
8.5 | 0.8 | 0.23 | 0.92 | 0.50
12.5 | 0.4 | 0.01 | 0.93 | 0.22
Turbulence + Pupil shifts + Vibration
8.5 | 0.7 | 0.08 | 0.92 | 0.33
12.5 | 0.4 | 0.01 | 0.92 | 0.23
Table 3: Summary of simulation optimal gain. Comparison between Integrator and
OMGI. Seeing 0.8′′.
Figure 4: Rejection Transfer Function (RTF) for the Tip mode with an IIR
filter based control with $N_{a}=N_{b}=3$ (left), and with $N_{a}=N_{b}=7$
(right). Note that the order 7 is used to reject both vibrations at
$13\,\mathrm{}$ and $22\,\mathrm{Hz}$ and atmospheric turbulence. These RTFs
are determined for the case magnitude 8.5. All the other parameters may be
found in Tab.1
All the results of the simulations are summarized in Tab.4 in term of SR for
all the considered observing conditions. Fig.5 shows the SR at different
wavelengths at magnitude 8.5, Fig.6 shows the corresponding results at
magnitude 12.5.
Figure 5: Summary of simulation results for turbulence, pupil shifts and
vibrations at magnitude 8.5. Comparison between Integrator, IIR filter based
control, and autogain. Seeing 0.8′′.
Figure 6: Summary of simulation results for turbulence, pupil shifts and vibrations at magnitude 12.5. Comparison between Integrator, IIR filter based control, and autogain. Seeing 0.8′′. band | V | R | H | K | V | R | H | K | V | R | H | K
---|---|---|---|---|---|---|---|---|---|---|---|---
Controller | integrator | IIR filter | OMGI
magnitude | Strehl Ratio $\%$
Turbulence
8.5 | 54.8 | 73.4 | 94.5 | 96.8 | 59.7 | 76.7 | 95.2 | 97.2 | 56.2 | 74.4 | 94.7 | 97.0
12.5 | 14.9 | 37.1 | 83.3 | 90.1 | 34.1 | 57.5 | 90.4 | 94.4 | 27.2 | 51.1 | 88.4 | 93.3
Turbulence + Pupil shifts
8.5 | 53.3 | 72.4 | 94.2 | 96.7 | 56.3 | 74.5 | 94.7 | 97.0 | 52.1 | 71.5 | 94.0 | 96.5
12.5 | 9.3 | 28.6 | 79.1 | 87.5 | 32.5 | 56.1 | 90.0 | 94.2 | 24.5 | 48.3 | 87.5 | 92.7
Turbulence + Vibrations
8.5 | 23.0 | 40.8 | 81.6 | 88.9 | 56.8 | 74.7 | 94.8 | 97.0 | 32.2 | 52.9 | 88.0 | 93.0
12.5 | 2.2 | 8.4 | 45.3 | 60.5 | 38.5 | 61.2 | 91.4 | 95.0 | 12.9 | 31.0 | 78.5 | 87.0
Turbulence + Pupil shifts + Vibrations
8.5 | 22.1 | 39.7 | 81.1 | 88.6 | 52.1 | 71.5 | 94.0 | 96.5 | 29.2 | 50.1 | 87.1 | 92.4
12.5 | 1.9 | 7.6 | 43.1 | 58.3 | 27.1 | 50.9 | 88.3 | 93.2 | 11.1 | 28.4 | 76.9 | 86.0
Table 4: Summary of simulation results. Comparison between Integrator, IIR
filter based control, and OMGI. Seeing 0.8′′.
The integrator and the OMGI show similar results in case of turbulence and
pupil shifts, while the vibrations affect more the integrator. These results
is expected, because the OMGI has more degrees of freedoms, one for each mode,
with respect to the single degree of the integrator. In fact it can be shown
in Tab.3 that the OMGI maximum gain value is greater in case of vibrations, so
it has a greater bandwidth and can better reject the Tip/Tilt disturbances.
Note that in case of pupil shifts the gains are lower to guarantee greater
robustness. At the faintest magnitude the differences between the controller
are greater, because, as we expected, the IIR filters reject better the noise
without decreasing the bandwidth (see Sec.2.2 and Fig.2).
The IIR filter based control produces similar performance in all the tested
conditions: it can efficiently reject both calibration errors and
disturbances. In the worst case – turbulence, pupil shifts and vibrations –
the IIR filter at V band gives the same SR of the OMGI and about two times the
one of the integrator both in the condition without pupil shifts nor
vibrations.
## 5 Processing power estimate
In this section we compare the processing power required for the IIR filter
based control described above, compared to the integrator case. Referencing
the scheme in Fig.1, we can split the required processing power into three
major steps:
1. 1.
Reconstruction matrix
2. 2.
Controller
3. 3.
Modes to command projection matrix
Steps 1 and 3 are common to the different control strategies, while step 2 is
where the techniques differ.
For the estimate, we suppose the following system parameters (partially
following Tab.1):
* •
Number of controlled modes: 1127 (including tip-tilt)
* •
Number of DM actuators: 1127
* •
Number of slopes measured by the WFS: 2480
* •
Loop frequency: 1000 Hz
* •
System delay: 2 frames
Step 1 is a straightforward vector/matrix multiplication. The required
processing power is $N_{\mathrm{modes}}\times N_{\mathrm{slopes}}$ multiply-
accumulate (MAC, a basic operation employed by most Digital Signal Processors
- DSPs where two operands are multiplied and summed to an accumulator). In our
example it is about 2.8 Mega-MAC (MMAC). Step 3 is also a vector/matrix
multiplication, this time of dimension $N_{\mathrm{modes}}\times
N_{\mathrm{commands}}$. In our example this amounts to about 1.3 MMAC.
Assuming a total delay of 2 frames at 1 KHz, and further assuming that this
computation cannot take more than 1/10th of the available time in order to be
considered negligible, the processing power requirements for steps 1 and 3
combined is 20 GMAC/sec. This is comparable with existing systems, like the
LBT one[19] where the total processing power, distributed over 168 DSPs, is
about 25 GMAC/sec (only a fraction of this processing power is used on the LBT
system, since the reconstrucion matrices have smaller dimensions and the DSPs
also have more tasks to attend to). Both steps 1 and 3 can be efficiently
parallelized over such a large number of DSPs, since each row of the matrix
multiplication can be processed independently of the others.
For step 2, the integrator case needs to sum the measurement vector from the
previous frame to the current one. Additionally, a gain value must be
multiplied to each measurement value. This requires about 1 Kilo-MAC (KMAC),
which is completely negligible (about 1/4000th) when compared to the previous
numbers. The only effect (and a rather large one) is to prevent full
pipelining of the two matrix multiplications.
For the IIR filters, we take into account a filter whith order 7 for the tip-
tilt modes, and 3 for the other modes (as in Sec.2.1). With $N_{a}=7$ and
$N_{b}=7$, this requires only a few dozens MACs for the tip-tilt IIR filters,
and about 6 KMAC for the other modes. While this number is 6 times bigger than
the integrator, it is again negligible in the context of the processing power
required by steps 1 and 3. Furthermore, no additional requirements are posed
on the other processing steps, except for the full separation of the two
matrix multiplications, which was already required for the integrator with
modal gain.
We can therefore conclude that the IIR filtering strategy outlined in this
article can readily be implemented on existing or future AO systems with
little to no penalty in terms of loop delay.
## 6 Conclusions
In this paper we have shown that IIR filter based control is a control
solution that can deliver good performance even in difficult conditions. It is
a possible choice for AO systems requiring high performance and robustness in
case of calibration errors and telescope vibrations.
Moreover, thanks to the data driven approach, the presented method need only a
basic knowledge of the system, and the computational burden is focused in the
parameters optimization that can be made off-line. In fact, the computational
power requested to the Real Time Computer of the AO system will be negligible
in comparison to the integrator as we proved in Sec.5.
As further work, more investigation on the robustness, on the coefficient
$\eta$ and on the impact of IIR filter order $N_{a}$, $N_{b}$ on the
performance must be done.
###### Acknowledgements.
This study was supported by the TECNO INAF 2009 grant from the italian
Istituto Nazionale di Astrofisica (INAF).
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* [14] R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” Journal of Modern Optics 43, pp. 289–293, 1996.
* [15] P. Salinari, D. V. C., and B. V., “A study of an adaptive secondary mirror,” in Active and adaptive optics: ESO Conference and Workshop Proceedings, p. 247, 1994.
* [16] “iXon X3 897 data sheet,” 2011.
* [17] O. Daigle, J. Gach, C. Guillaume, C. Carignan, P. Balard, and O. Boissin, “L3CCD results in pure photon counting mode,” Proceedings of SPIE 5499, pp. 219–227, 2004.
* [18] F. Quirós-Pacheco, L. Busoni, G. Agapito, S. Esposito, E. Pinna, A. Puglisi, and A. Riccardi, “First light AO (FLAO) system for LBT: performance analysis and optimization,” in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 7736, July 2010.
* [19] A. Riccardi, G. Brusa, P. Salinari, S. Busoni, O. Lardiere, P. Ranfagni, D. Gallieni, R. Biasi, M. Andrighettoni, S. Miller, and P. Mantegazza, “Adaptive secondary mirrors for the Large binocular telescope,” in Astronomical Adaptive Optics Systems and Applications, R. K. Tyson & M. Lloyd-Hart, ed., Proc. SPIE 5169, pp. 159–168, 2003.
|
arxiv-papers
| 2012-07-16T11:41:44 |
2024-09-04T02:49:33.099489
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "G. Agapito, C. Arcidiacono, F. Quir\\'os-Pacheco, A. Puglisi, S.\n Esposito",
"submitter": "Guido Agapito",
"url": "https://arxiv.org/abs/1207.3634"
}
|
1207.3646
|
jcis@epacis.org
OGCOSMO: An auxiliary tool for the study of the Universe within hierarchical
scenario of structure formation
Eduardo S. Pereira111duducosmo@das.inpe.br, Oswaldo D. Miranda
222oswaldo@das.inpe.br
National Institute for Space Research, São José dos Campos, SP, Brazil
Received on September , 2010 / accepted on *****, 2010
###### Abstract
In this work is presented the software OGCOSMO. This program was written using
high level design methodology (HLDM), that is based on the use of very high
level (VHL) programing language as main, and the use of the intermediate level
(IL) language only for the critical processing time. The languages used are
PYTHON (VHL) and FORTRAN (IL). The core of OGCOSMO is a package called
OGC_lib. This package contains a group of modules for the study of
cosmological and astrophysical processes, such as: comoving distance, relation
between redshift and time, cosmic star formation rate, number density of dark
matter haloes and mass function of supermassive black holes (SMBHs). The
software is under development and some new features will be implemented for
the research of stochastic background of gravitational waves (GWs) generated
by: stellar collapse to form black holes, binary systems of SMBHs. Even more,
we show that the use of HLDM with PYTHON and FORTRAN is a powerful tool for
producing astrophysical softwares.
Keywords:Computational Physics, Cosmology, Gravitational Waves, Black Hole,
Structure Formation.
1\. INTRODUCTION
Around 300 thousand years after the Big Bang, matter and radiation decoupled
and the photons could freely travel through the space. Henceforth, today we
can obtain information about this period from the Cosmic Microwave Background
Radiation (CMBR). Moreover, about 840 million years after the Big Bang, the
Universe was reionized by the first stars. However, we have no direct
information from the period between 300 thousand years and 840 million years.
This period is so-called dark age (DA). On the other hand, the theory of
general relativity predicts the existence of gravitational waves (GWs) as
perturbations of space-time which propagate at the speed of light. The
detection of GWs will open a new astronomical window for observing the
Universe. In particular, they will allow us a deeper understand about the DA.
In this context, we are interested in to describe, by analytical and semi-
analytical models, some astrophysical and cosmological processes as, for
example, the cosmic star formation rate, and its connection with the
stochastic background of GWs [4], this kind of study will shed light on the
knowledge of the process that took place at the end of the DA.
As such, this modeling framework will provide us with an important auxiliary
tool for the reconstruction of the cosmic history. Thus, we decided to
organize the computational development as a software, it called OGCOSMO, whose
main characteristics are discussed in this work.
In section 2 it will be showed a short description of the astrophysical and
cosmological model adopted. In section 3 it will be presented the main
characteristics of the software and the type of programing used. In section 4
we present some results. Finally, in section 5 are the final considerations.
2\. The Model
In this work we are considering the general theory of relativity with cold
dark matter and cosmological constant ($\Lambda$CDM model) for background
cosmology.
Here is assumed the hierarchical scenario of structure formation, having as
base the Press-Schechter (PS) like formalism [6]. The basic idea behind this
scenario is that the formation of objects like galaxies and galaxy clusters
occur in the following way: First, when the mean density of dark matter
perturbation, within a given volume, is larger than a threshold level,
$\delta_{c}$, the matter leaves the linear regime and collapses to form small-
halo objects. These halos become gravitational wells that attract the baryonic
matter, that is the ordinary matter that form stars and planets. Durant all
this process, greater halos are formed by fusion of the minors and more
baryons fall into structures. The star formation starts and black holes grow
up by accretion of matter. The complete details, about the all considerations
and the main results obtained with this model, can be seen in [4, 5].
3\. The Software
The OGCOSMO software was written using object-oriented programing paradigm,
that permits to construct codes that are really reusable and clean. The main
programming language used was PYTHON [7], that is a very-high level language
programing, and FORTRAN only for the critical time parts of the code. This
form of writing codes is called of high level design [3, 2]. The external
modules used were basically: Tkinter [8] for construction of the Graphical
Using Interface (GUI), scipy and numpy for numerical methods [9].
The core of the OGCOSMO is the OGC_lib, that is a package containing the
following principal modules, up to now:
* •
OGC_cosmo: This module contains the class “Cosmo()” that is composed by the
cosmological background methods, and callbacks, that are, comoving distance
and comoving volume, relation between time and redshift ($z$), matter density
evolution (dark and baryonic matters), grow function of matter perturbation,
variance of matter linear density field, linear extrapolation of the critical
density for collapse of structures in a given $z$ , comoving volume of dark
matter halo.
* •
OGC_PS: This module contains the class “PressSchechter(Cosmo)”, that inherit
the methods of Cosmo class. This class contains the base of Press-Schechter
like formalism, that is, the functions used for study of structure formation
as, mass function and numerical density of dark haloes, fraction and accretion
rate of baryonic matter within structures and the cosmic star formation rate -
CSFR.
* •
OGC_SMBH: In this module is the “SuperBH(PressSchechter)” class, that is a
class under development and it will be used for the study of the evolution of
supermassive black holes through of its mass function and by the coalescence
of these objects [5].
4\. Results
4.1 The OGCOSMO
In the figure 1 we show the GUI of OGCOSMO. In A are the spaces for entry
parameters such as the cosmological and those associated with the star
formation rate. In B are the buttons that start the calculus of the CSFR,
supermassive black holes mass function and some buttons for creation of
specific plots as the CSFR (see figure 2).
In the figure 2 is represented the CSFR behavior, whose result can be seen by
pressing the button “Grafico CSFR”. The Salpeter exponent of initial mass
function is $x=1.35$, the characteristic scale for star formation is
$\tau=2.5\times 10^{9}$ years and the exponent of the star formation rate
$n=1$. For more details see [4].
Figure 1: GUI of OGCOSMO. In A are showed the entry parameters. In B are
showed the buttons for running the models and generate some graphics.
Figure 2: Plot of the cosmic star formation rate behavior. In this case was
considered $\Omega_{m}=0.24$, $\Omega_{b}=0.04$, $\Omega_{\Lambda}=0.76$ and
$h=0.73$, for the cosmological parameters, $x=1.35$ and $\tau=2.5\times
10^{9}$ years.
4.2 OGC_lib Usage
In Python a package is basically a hierarchical file directory structure which
defines a environment that consists of modules. This directory contains a file
named __ini__.py that identifies the directory as a package [1]. In the
context of this work, this means that if a user does not want to use the GUI
of OGCOSMO, he (or she) can use only determined methods by importing its
specific class from a module within OGC_lib. For example, an user want to call
the method age from the class Cosmo of the module OGC_cosmo:
$[1]>>>$ from OGC_lib.OGC_cosmo import Cosmo
$[2]>>>$ MyUniverse = Cosmo(0.04,0.24,0.73,0.76,6.0,20.0,’./trabalho’)
$[3]>>>$ Age = MyUniverse.age(5)
$[4]>>>$ print‘ Age = %3.9e’ %Age
$[5]$ Age = 1.189273236e+09
In line $[1]$ was imported, from OGC_cosmo module within OGC_lib package, the
class Cosmo. In line $[2]$ was created a new object (MyUniverse) and in line
$[3]$ was used the method age. The example above show that for
$\Omega_{m}=0.24$, $\Omega_{b}=0.04$, $\Omega_{\Lambda}=0.76$ and $h=0.73$,
the age of the Universe at $z=5$ is $1.2\times 10^{9}$ years. The last
argument of the class Cosmo (’./trabalho’) is the directory where the data
will be saved.
This result shows that OGC_lib can be used as a framework for fast an easy
development of astrophysical and cosmological applications for the study of
the cosmic history, within hierarchical scenario for structure formation.
5\. Final Considerations
In this work are presented both the initial stage of the OGCOSMO software and
the OGC_lib package, that have been developed to be an auxiliary tool for
reconstruction of the cosmic history. Moreover, it is possible for an user to
have access for a specific method of the OGC_lib. This is an important feature
to show that this package can be used as a framework for construction of
others softwares. An another aspect is that the program can be used as
didactic tools in lectures of theory of general relativity and cosmology.
The next steps will be to write classes to obtain the mass function of SMBHs;
research of stochastic background of gravitational waves (GW) generated by:
collapse of stars to black holes, binary systems of SMBH; calculate the
signal/noise rate for GW detectors (LISA, LIGO, Decigo, BBO).
Even more, it was showed that the use of PYTHON with FORTRAN for high-level
design program is a powerful tool for development of astrophysical softwares.
ACKNOWLEDGMENTS: E. S. Pereira would like to thank the Brazilian Agency CAPES
for support. O. D. Miranda would like to thank the Brazilian Agency CNPq for
partial support (grant 300713/2009-6)
## References
* [1] GUPTA, R, 2002. Making Use of Python. Wiley Publishing.
* [2] HINSEN, K, 2007. Parallel scripting wiht Python. Computing in science & engineering. Nov./Dec. 82-89.
* [3] HINSEN, K, LANGTANGEN, HP, SKAVHAUG, O, ØDEGÅRD, Å, 2006. Using BSP and Python to simplify parallel programming. Future Generation Computer Systems. 22: 123-157.
* [4] PEREIRA, ES, MIRANDA, OD, 2010. Stochastic background of gravitational waves generated by pre-galactic black holes. Mon. Not. R. Astron. Soc. 401: 1924-1932.
* [5] PEREIRA, ES, MIRANDA, OD, 2010. Massive Black Hole Binary Systems in Hierarchical Scenario of Structure Formation. International Journal of Modern Physics D. 19: 1271-1274
* [6] PRESS, WH, SCHECHTER, P, 1974. Formation of galaxies and clusters of galaxies by self-similar gravitational condesation. Apj, 193, 425-438.
* [7] http://www.python.org/ $<$accessed in 13 august 2010$>$
* [8] http://docs.python.org/library/tkinter.html $<$accessed in 13 august 2010$>$
* [9] http://www.scipy.org/ $<$accessed in 13 august 2010$>$
* [10] http://dirac.cnrs-orleans.fr/plone/software/scientificpython/ $<$accessed in 13 august 2010$>$
|
arxiv-papers
| 2012-07-16T12:20:16 |
2024-09-04T02:49:33.106939
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eduardo dos Santos Pereira and Oswaldo D. Miranda",
"submitter": "Eduardo Pereira",
"url": "https://arxiv.org/abs/1207.3646"
}
|
1207.3658
|
jcis@epacis.org
Programing Using High Level Design With Python and FORTRAN: A Study Case in
Astrophysics
Eduardo S. Pereira111duducosmo@das.inpe.br, Oswaldo D. Miranda
222oswaldo@das.inpe.br
Instituto Nacional de Pesquisas Espaciais - Divisão de Astrofísica,
Av. dos Astronautas 1758, São José dos Campos, 12227-010 SP, Brazil
Received on September , 2011 / accepted on *****, 2010
###### Abstract
In this work, we present a short review about the high level design
methodology (HLDM), that is based on the use of very high level (VHL)
programing language as main, and the use of the intermediate level (IL)
language only for the critical processing time. The languages used are Python
(VHL) and FORTRAN (IL). Moreover, this methodology, making use of the oriented
object programing (OOP), permits to produce a readable, portable and reusable
code. Also is presented the concept of computational framework, that naturally
appears from the OOP paradigm. As an example, we present the framework called
PYGRAWC (Python framework for Gravitational Waves from Cosmological origin).
Even more, we show that the use of HLDM with Python and FORTRAN produces a
powerful tool for solving astrophysical problems.
Keywords: Computational Physics, Cosmology, Programming Methodology, HLDM.
1\. INTRODUCTION
Python [1, 2] is a very-high level dynamic language programming. The Python
interpreter is available for different operational systems (OS). This means
that is possible to write a code which can run in different OS without
requiring any modification. However, in order to have the best performance,
the critical computational part of the code should be written in a compiled
language like c/c++ or FORTRAN. This way of writing codes is called of high
level design (HLD) [3, 4]. Another important fact is that, in general, $80$%
of the runtime is spent in $20$% of the code (Pareto Principle) [5]. Thus, the
use of a VHL for the principal part of the code permits a more agile
processing. Although the HLD goes beyond of a mixing of different languages,
an important feature of HLD is related to the use of oriented object
programming paradigm, OOP, working together an Unified Modeling Language (UML)
class diagram.
2\. The High Level Design
The first step to write an efficient code consists in dividing the problem in
classes. In this section we present programs that make this task easier, as
for example, dia [6] and dia2code [7]. The first one permits us to write class
diagrams, and with the second program we can generate a frame code. That is,
the class in the diagram image which is converted to a class in code
structure. Through this section these concepts will become clearer.
2.1 Planning Before Programing
We start this section showing an example of class diagram333All documentation
about how to install and use the dia software can be found in [6].. Through
this paper, we will use an example derived from cosmology. In particular, the
main characteristics of a cosmological model are: the age of the Universe, the
scale factor which describes how the radius of the Universe evolves with time,
and the density of matter/energy.
The class diagram that represents this cosmological model is showed in the
figure 1. The attributes of this class are the cosmological parameters, at the
present, for total matter (self.omegam - $\Omega_{m}$), barionic matter
(self.omegab -$\Omega_{b}$), dark energy (self.omegal - $\Omega_{\Lambda}$)
and Hubble parameter (self.h- $h$ )444The Hubble constant at the present time
is written in terms of $h$ by $H_{0}=100\,h\,{\rm km}\,{\rm s}^{-1}\,{\rm
Mpc}^{-1}$ (where $1\,{\rm Mpc}=3.086\times 10^{24}\,{\rm cm}$)..
The file is saved with the name cosmo.dia. Now, it is possible to generate a
structured code with dia2code (see [7] for details), using the command
dia2code cosmo.dia -t Python. The code generated by this example, and all
examples used in this article, can be downloaded from [8]. It is possible to
generate c++ and java structured code choosing the name of the equivalent
language from the dia2code command.
However, this way to structure a code class is only a start point. It is
necessary to do a better organization and fill the methods with the equivalent
operations.
Figure 1: cosmo.dia, an example of class diagram for the basic characteristics
of a cosmological model.
2.2 Mixing Python and FORTRAN
There is a very useful tool, called f2py [9], which permits to do a wrapper of
a FORTRAN 77 code to Python. That is, it compiles the FORTRAN subroutine in a
format which can be used by Python module. The f2py is contained in the
package numpy [10]. Below, we present a simple example:
C FILE hiword.f
subroutine hiword(a,b)
real*8 a,b
cf2py intent(in) a
cf2py intent(out) b
b = a*a
write(6,*) ’b = ’,b,’, a = ’,a
return
end
The comment cf2py allows the f2py wrapper can be identified with both the
input and output variables in the function hiword. Giving the name hiword.f to
the file contained in the above code, we can compile it from the following
command:
f2py -c hiword.f -m hiword
In this case, the -c means compile, and -m generate a Python module with name
hiword. Below, we present an example how to call the function hiword in a
Python code.
$[1]>>>$import hiword
$[2]>>>$print hiword.__doc__
$[3]>>>$ This module ’hiword’ is auto-generated with f2py (version:2).
$[4]>>>$Functions:
$[5]>>>$ b = hiword(a)
$[6]>>>$hiword.hiword(5)
$[7]>>>$ b = 25.000000000000000 , a = 5.0000000000000000
$[8]>>>$ 25.0
The text in front of $>>$ represents what is printed in the display. For more
details and examples see [9].
2.2 Optimizing the Code for Multi-Core Machines
Another interesting fact about Python is that it has a lot of modules. One of
this is the multiprocessing that permits to write a parallel code in an easy
way. As an example of using this module in scientific computing, consider the
following equation:
$f(x)=\int_{a}^{b}{g(x,k)}dk,$ (1)
where $a\leq k\leq b$.
In many cases $g(x,k)$ can not be written in a separated form. In this case,
the integral equation must be evaluated for each $x$ in a given range
$[x_{0},x_{f}]$. However, we can divide the range $[x_{0},x_{f}]$ by the
number of central processor units (CPU) of a cluster compute (or multi-core
machine), and so we can calculate $f(x)$ in parallel mode. In the figure 2, it
is showed the class diagram of ppvector, that is a class we developed to do
this type of operation in parallel model, for multi-core machine, based on the
module multiprocessing. The source code can be downloaded from [11].
Figure 2: ppvector, a Python module for construction of parallel scientific
code in a multi-core machine.
The code below shows the use of ppvector:
import multiprocessing as mpg
from ppvector import ppvector
from scipy.integrate import romberg
np=10000; zmax=20.0; deltaz=zmax/np
g= mpg.Array(’d’,[0 for i in range(np)]) # The d indicate duble precision
z= mpg.Array(’d’,[zmax-i*deltaz for i in range(np)])
#Define a function that will be calculate the integral in parallel
#k is the starter point of the sub-range
#E is the lenght of the range
#n is the number of CPU’s of machine
def f(x):
def f2(k):
return (x+k)**(-2.0)
return romberg(f2,5.0,20.0)
def fun(k,E,n):
k2=k+E
for i in range(k,k2+1):
zloc=z[k]
g[k]=f(zloc)
C1= ppvector(np,fun) # Star the ppvector class
C1.runProcess() # Executing the parallel calculus.
The function Array, of multiprocessing module, allocates a matrix in a global
memory which can be accessed by all CPU’s. In line $25$ is passed on the
length of the vector and the function that divides the job in sub-ranges. In
line $26$, the parallel code is called and executed.
3\. Python Framework for Cosmological Gravitational Waves - PYGRAWC
A framework is a set of classes, interfaces and patterns to solve a group of
problems. It is like a little application with statical and dynamical
structures to solve a set of restrict problems. So, a framework is more than a
simple library (we refer the reader to [12, 13, 14]).
In figure 3 is presented the class diagram of the core of PyGraWC. It is a
framework that we are developing to study gravitational waves from
cosmological origin. Here, it is only showed the class name and the relation
among their several components.
Figure 3: The PYGRAWC framework class diagram.
The class cosmo describes the background cosmology. The class PressSchechter
is based on a Press-Schechter-like formalism [15] and it describes both the
evolution of dark matter halos and the infall of barionic matter in these
halos. The class csfr describes the evolution of the cosmic star formation
rate. The class smbh describes the evolution of supermassive black holes in
the centers of galaxies. The classes bhestelar and bhmassivo calculate the
stochastic background of gravitational waves generated by: the collapse of
stars to form black holes [15] and the growth of supermassive black holes (in
progress). All details about the astrophysical model and the results obtained
from this framework can be seen in [15, 16, 17].
4\. Final Considerations
In this work is presented a High Level Design methodology (HLD), that consists
in the mixing of a very-hight level interpreted language (VHL) with an
intermediated compiled language (IL). Using tools of software engineering,
like UML, and also framework concept, we can write efficient scientific codes
without spending a lot of time in the development phase. Here, it was used
Python (VHL) and Fortran (IL) and it was showed that this combination can be
easily done giving excellent results, as can be seen by the presentation of
Python framework for Gravitational Waves from Cosmological origin (PyGraWC).
Acknowledgments: E.S. Pereira would like to thank the Brazilian Agency CAPES
for support. O.D. Miranda would like to thank the Brazilian Agency CNPq for
partial support (grant 300713/2009-6)
## References
* [1] http://www.Python.org/ $<$accessed in 09 September 2011$>$
* [2] GUPTA, R, 2002. Making Use of Python. Wiley Publishing.
* [3] HINSEN, K, LANGTANGEN, HP, SKAVHAUG, O, ØDEGÅRD, Å, 2006. Using BSP and Python to simplify parallel programming. Future Generation Computer Systems. 22: 123-157.
* [4] HINSEN, K, 2007. Parallel scripting wiht Python. Computing in science & engineering. Nov./Dec. 82-89.
* [5] BEHNEL, S, BRADSHAW, R, CITRO, C, DALCIN, L, SELJEBOTN, D, SMITH, K,.Cython: The Best of Both Worlds. CISE, 13, 2, 31-39.
* [6] http://projects.gnome.org/dia/ $<$accessed in 09 September 2011$>$
* [7] http://dia2code.sourceforge.net/ $<$accessed in 09 September 2011$>$
* [8] https://duducosmos@github.com/duducosmos/pereira_miranda_jcis2012.git $<$accessed in 10 September 2011$>$
* [9] PETERSON,P, 2009. F2PY: a tool for connecting Fortran and Python programs IJCSE, 4, 4, 296-305.
* [10] http://www.scipy.org/ $<$accessed in 09 September 2011$>$
* [11] https://duducosmos@github.com/duducosmos/ppvector.git $<$accessed in 09 September 2011$>$
* [12] FAYAD, M. E. 2000. Introduction to the computing surveys electronic symposium on object-oriented application frameworks. ACM Comput. Surv., 32,1,1-9.
* [13] Fayad, M. E., SCHIMIDT, D. C. 1997. Object-oriented application frameworks. Commun.ACM, 40, 10, 32-38.
* [14] GOVONI, D. 1999. Java Application Frameworks. John Wiley & Sons.
* [15] PEREIRA, ES, MIRANDA, OD, 2011. Supermassive Black Holes: Connecting the Growth to the Cosmic Star Formation Rate. accept in Mon. Not. R. Astron. Soc. Letter.
* [16] PEREIRA, ES, MIRANDA, OD, 2010. Stochastic background of gravitational waves generated by pre-galactic black holes. Mon. Not. R. Astron. Soc. 401: 1924-1932.
* [17] PEREIRA, ES, MIRANDA, OD, 2010. Massive Black Hole Binary Systems in Hierarchical Scenario of Structure Formation. International Journal of Modern Physics D. 19: 1271-1274
|
arxiv-papers
| 2012-07-16T12:50:30 |
2024-09-04T02:49:33.113214
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Eduardo dos Santos Pereira and Oswaldo D. Miranda",
"submitter": "Eduardo Pereira",
"url": "https://arxiv.org/abs/1207.3658"
}
|
1207.3673
|
# A concrete anti-de Sitter black hole with dynamical horizon having toroidal
cross-sections and its characteristics
Pouria Dadras Department of Physics, Sharif University of Technology, Tehran,
Iran J. T. Firouzjaee School of Physics and School of Astronomy, Institute
for Research in Fundamental Sciences (IPM), Tehran, Iran
j.taghizadeh.f@ipm.ir Reza Mansouri Department of Physics, Sharif University
of Technology, Tehran, Iran and
School of Astronomy, Institute for Research in Fundamental Sciences (IPM),
Tehran, Iran mansouri@ipm.ir
###### Abstract
We propose a special solution of Einstein equations in the general Vaidya form
representing a dynamical black hole having horizon cross sections with
toroidal topology. The concrete model enables us to study for the first time
dynamical horizons with toroidal topology, its area law, and the question of
matter flux inside the horizon, without using a cut-and-paste technology to
construct the solution.
###### pacs:
95.30.Sf,98.80.-k, 98.62.Js, 98.65.-r
## I Introduction
The topology of black hole horizons has been a matter of wide discussions in
the past. Starting with the Hawking’s theorem, stating that each connected
component of the event horizon of a stationary black hole in four dimensional
space time has the topology of a 2-sphere haw-book , most of the authors have
been interested in non-dynamical asymptotically flat space times, excluding
any kind of cosmological black holes man . Gannon was the first who opened the
possibility of a torus topology for a black hole horizon Gannon , generalizing
Hawking’s theorem by replacing stationarity by some weaker assumptions.
On the other hand, Chru´sciel and Wald wald showed that each connected
component of a cross-section of the event horizon of a stationary
asymptotically flat black hole must have spherical topology. Jacobson and
Venkataramani Jacobson proved that, under certain conditions, the topology of
the event horizon of a four dimensional asymptotically flat black-hole space
time must be 2-sphere. All theses studies have been supported by the
topological censorship theorem of Friedmann, Schleich and Witt, another
statement indicating the impossibility of non spherical horizons Friedman .
The theorem states that in a globally hyperbolic, asymptotically flat
spacetime, any two causal curves extending from past to future null infinity
are homotopic, meaning that a black hole with toroidal surface topology is a
possible violation of topological censorship theorem, as pointed out in
Jacobson . In fact, as was shown by Shapiro, Teutolsky and Winicour Teutolsky
, a temporarily toroidal horizon can be formed in a gravitational collapse, in
a way consistent with the theorem.
Now, a concrete model of a black hole with toroidal horizon has been
constructed by vanzo for which the thermodynamics and area law is also
considered. Vanzo’s model represents an isolated genus-one black hole in an
asymptotically anti-de Sitter space time; its extension to $d$ dimensions with
a negative cosmological constant is given in Birmingham . Recently, toroidal
and higher genus asymptotically AdS black holes have been put forward torus-
collapse through gluing of some special metrics such as Lemaitre-Tolman-
Bondi-like and McVittie solution to toroidal or higher genus asymptotically
AdS black holes. These pasted-manifolds, however, lack the dynamical features
of the black hole we are interested in.
Our purpose is to construct a dynamical topological black hole in an
asymptotically anti-de Sitter space time as a first step towards a better
understanding of cosmological black holesman and their thermodynamics. For a
more concrete definition of dynamical horizons and their difference to the
isolated ones we refer to ashtekar-03 and the references therein. A general
formalism for the area law in the case of dynamical black holes is also
formulated there for the first time, including the possibility of the toroidal
cross section of the horizon and the corresponding black hole area law. Using
a 3+1 decomposition and the Gauss-Bonnet theorem, they also found a formalism
which identifies the rate at which the radius of the cross-sections increases
precisely according to the matter flux and gravitational wave on the horizon
with cross-sections having a spherical topology and strictly positive
cosmological constants. However, in the case of zero and negative cosmological
constant, where the topology of horizon cross-sections may be toroidal, their
formalism is not conclusive.
In this paper we construct a non-stationary space-time which is asymptotically
anti-de Sitter and represents a dynamical horizon with toroidal cross-
sections. An area law is then written for this dynamical black hole showing an
increase of the horizon in accordance with the second law of black hole
thermodynamics. The matter flux, however, is non-vanishing, although the total
matter flux including the contribution from the cosmological constant
vanishes. Our dynamical horizon model with toroidal cross-sections reduce to
that of Vanzo vanzo with an isolated horizon if the space time is stationary.
We will follow in this paper definitions and notations used in ashtekar-03 .
They introduced a local definition of horizon as a three dimensional manifold
$H$ in space time which can be foliated by closed 2-dimensional surfaces $S$,
assuming special characteristic of the expansion on each leaf. The space-time
metric $g_{ab}$ has signature $(-,+,+,+)$ and its covariant derivative
operator will be denoted by $\nabla$. The Riemann tensor is defined by
$R_{abc}{}^{d}W_{d}:=2\nabla_{[a}\nabla_{b]}W_{c}$, the Ricci tensor by
$R_{ab}:=R_{acb}{}^{c}$, and the scalar curvature by $R:=g^{ab}R_{ab}$. The
unit normal to $H$ will be denoted by ${\widehat{\tau}}^{a}$;
$g_{ab}{\widehat{\tau}}^{a}{\widehat{\tau}}^{b}=-1$. The intrinsic metric and
the extrinsic curvature of $H$ are denoted by
$q_{ab}:=g_{ab}+{\widehat{\tau}}_{a}{\widehat{\tau}}_{b}$ and
$K_{ab}:={q_{a}}^{c}{q_{b}}^{d}\nabla\\!_{c}{\widehat{\tau}}_{d}$
respectively. $D$ is the covariant derivative operator on $H$ compatible with
$q_{ab}$, ${\mathcal{R}}_{ab}$ its Ricci tensor and ${\mathcal{R}}$ its scalar
curvature. The unit space-like vector orthogonal to $S$ and tangent to $H$ is
denoted by ${\widehat{r}}^{\,a}$. Quantities intrinsic to $S$ will be
generally written with a tilde. Thus, the two-metric on $S$ is
${\widetilde{q}}_{ab}$ and the extrinsic curvature of $S\subset H$ is
${\widetilde{K}}_{ab}:=\widetilde{q}_{a}^{\,\,\,\,c}\widetilde{q}_{b}^{\,\,\,\,d}D_{c}{\widehat{r}}_{d}$;
the derivative operator on $(S,{\widetilde{q}}_{ab})$ is $\widetilde{D}$ and
its Ricci tensor is ${\widetilde{{\mathcal{R}}}}_{ab}$. Finally, we can fix
the rescaling freedom in the choice of null normals
${\ell}^{a}:={\widehat{\tau}}^{\,a}+{\widehat{r}}^{\,a}$ and
$n^{a}:={\widehat{\tau}}^{\,a}-{\widehat{r}}^{\,a}$ such that
$\ell^{a}n_{a}=-2$. The shear tensor $\sigma_{ab}$ of the null vector $l^{a}$
and its trace $\sigma$ are defined as usual. A dynamical horizon is then
defined as a three dimensional space-like sub-manifold of space time such that
it can be foliated with closed orientable two dimensional surfaces on which
the expansion $\Theta_{(\ell)}$ vanishes, and the expansion $\Theta_{(n)}$ of
the other null normal is negative. Section 2 is devoted to a short review of
the area law formalism for dynamical horizons in different cases of the
cosmological constant. The area law for the dynamical horizon in a concrete
metric with AdS back ground ($\Lambda<0$) as an exact solution of the Einstein
equations is discussed in section III. We then conclude in section IV.
## II Area law for dynamical black holes
Now, having the necessary definitions and notations, we may write the first
consequence of the above definition of a dynamical horizon, using the
relations $\Theta_{(\ell)}=0$ and $\Theta_{(n)}<0$, in the form
${\widetilde{K}}=\tilde{q}^{ab}D_{a}{\widehat{r}}_{b}=\frac{1}{2}\,\tilde{q}^{ab}\nabla_{a}(\ell_{b}-n_{b})=-\frac{1}{2}\Theta_{(n)}>0.$
(1)
Hence, the area $a_{S}$ of $S$ will increase monotonically along
${\widehat{r}}^{\,a}$ by the change of the cross-sections, which is equivalent
to the second law of black hole mechanics on $H$. To obtain an explicit
expression for the area change the authors in ashtekar-03 take first two
fixed cross-sections $S_{1}$ and $S_{2}$ of $H$, and then integrate the result
on a portion $\Delta H\subset H$ bounded by $S_{1}$ and $S_{2}$ with the
corresponding radii $R_{1}$ and $R_{2}$. Note that $R$ is the area radius of
$S$ defined by $a_{S}=4\pi R^{2}$, $a_{S}$ being the surface area of $S$
independent of its topology. Following result is then obtained using Gauss-
Bonnet theoremashtekar-03 :
$\mathcal{I}\,(R_{2}-R_{1})=16\pi G\int_{\Delta H}(T_{ab}-\frac{\Lambda}{8\pi
G}g_{ab}){\widehat{\tau}}^{\,a}\xi_{(R)}^{b}\,d^{3}V+\int_{\Delta
H}N_{R}\left\\{|\sigma|^{2}+2|\zeta|^{2}\right\\}\,d^{3}V,$ (2)
where $\mathcal{I}$ is the Euler characteristic of $S$; The scalar $|\zeta|$
is the length of the vector
$\zeta^{a}=\tilde{q}^{ab}\hat{r}^{c}\nabla_{b}\ell_{c}$ and
$\xi_{(R)}^{a}=N_{R}l^{a}$ where the lapse function is given by
$N_{R}=|\partial R|$. The first term on the right hand side is usually called
the matter flux, $\mathcal{F}^{(R)}_{matter}$, and the second term is the
gravitational wave flux, $\mathcal{F}^{(R)}_{\rm grav}$. The discussion on the
topology of $S$ is now divided in three cases depending on the cosmological
constant.
_Case 1_ :
$\Lambda>0$. Since the stress energy tensor $T_{ab}$ is assumed to satisfy the
dominant energy condition, the right hand side is manifestly positive
definite. Due to the fact that the area increases along ${\widehat{r}}^{a}$,
we must have $R_{2}-R_{1}>0$. It then follows that $\mathcal{I}$ has to be
positive. Therefore, the closed, orientable 2-manifolds $S$ are necessarily
topologically 2-spheres and $\mathcal{I}=8\pi$. Eq.(2) now gives a measure for
the increase of the horizon cross-section areaashtekar-03 :
$\displaystyle\frac{R_{2}-R_{1}}{2G}$ $\displaystyle=$
$\displaystyle\int_{\Delta H}(T_{ab}-\frac{\Lambda}{8\pi
G}\,g_{ab}){\widehat{\tau}}^{\,a}\xi_{(R)}^{b}\,d^{3}V$ (3) $\displaystyle+$
$\displaystyle\frac{1}{16\pi G}\,\int_{\Delta
H}N_{R}\left\\{|\sigma|^{2}+2|\zeta|^{2}\right\\}\,d^{3}V\,.$
_Case 2_
: $\Lambda=0$. The right-hand side of eq.(2) is necessarily non-negative.
Hence, the topology of $S$ is either that of a 2-sphere (if the right hand
side is positive) or that of a 2-torus (if the right hand side vanishes). The
torus topology can occur if and only if $T_{ab}\ell^{b}$, $\sigma_{ab}$ and
$\zeta^{a}$ all vanish everywhere on $H$. Therefore, it may be concluded that
the scalar curvature ${\widetilde{{\mathcal{R}}}}$ of $S$ must also vanish on
every cross-section. The 2-manifold $S$ then has to be a flat torus. Using the
fact that $H$ is space-like, we conclude that in this case ${\cal
L}_{n}\,\Theta_{(\ell)}=0$ everywhere on $H$. Thus, in this case the dynamical
horizon cannot be a FOTH ashtekar-03 . Furthermore, since
$\Theta_{(\ell)},\sigma_{ab}$, and $R_{ab}\ell^{b}$ all vanish on $H$, the
Raychaudhuri equation now implies that ${\cal L}_{\ell}\,\Theta_{(\ell)}$ also
vanishes.
Note the following cases:
* •
In the case of torus topology the transition to the stationary case and
isolated horizon is not trivialashtekar-03 . Given that in the stationary case
the topology can not be a torus haw-book , a topology change is then
unavoidable horowitz-91 . The procedure used in ashtekar-03 to understand the
transition to the isolated horizon is based on the relation
$f\,{\cal L}\,_{n}\,\Theta_{(\ell)}=-\sigma^{2}-R_{ab}\ell^{a}\ell^{b},$ (4)
where $f$ is the length of a space-like vector orthogonal to $S$ and tangent
to $H$. Now, due to the fact that in the case of torus topology we have ${\cal
L}\,_{n}\,\Theta_{(\ell)}=0$, it can not be concluded that $f=0$ which is a
necessary condition for the isolated horizon with the sphere topology.
* •
The familiar matter flux as defined in (2) does vanish in the case of the
torus topology. However, by changing the definition of $\xi_{(R)}$ in (2) to
$\xi_{(R)}^{b}=cn^{b}$, where $c$ is an appropriate coefficient related to
$f$, we may arrive at a non-vanishing matter flux. That this is not always the
case, we will see on hand a concrete example in the following section. We may
also note that the positivity of the extrinsic curvature of $S$ along $r^{a}$
in the case of torus topology does not necessarily means an area increase.
This is due to the vanishing of the left hand side of (2).
_Case 3_ :
$\Lambda<0$. In this case there is no control on the sign of the right-hand
side of eq.(2). Hence, any topology is permissible. Stationary solutions with
quite general topologies are known for black holes which are locally
asymptotically anti-de Sittervanzo . Event horizons of these solutions are
potential asymptotic states of these dynamical horizons in the distant future.
## III Area law for a dynamical black hole in AdS back ground
We are interested in a solution of Einstein equations with negative
cosmological constant representing a black hole with a dynamical horizon
having torodial cross-sections. So far we have not found any exact solution of
Einstein equations having these features and not constructed through a cut and
paste technology. Solutions produced by cut and paste technology do not
represent a genuine dynamical black hole due to the build-in freezing of the
matching hypersurface. We propose then a solution in the general Vaidya form:
$ds^{2}=-f(v,r)dv^{2}+2dvdr+r^{2}(d\theta^{2}+d\phi^{2}),$ (5)
with the arbitrary function $f$ of coordinates $v$ and $r$, where $v$ is the
advanced time coordinate with $-\infty<v<\infty$, $r$ is the radial coordinate
with $0<r<\infty$, and $\theta,\phi$ are coordinates describing the two-
dimensional zero-curvature space generated by the two-dimensional commutative
Lie group $G_{2}$ of isometries vanzo .
The black-hole apparent horizon is space-like and located at
$f(v,r):=0\,.$ (6)
The expansions of the corresponding null normals are
$\Theta_{\ell}=\frac{f}{r},\qquad\textrm{and}\qquad\Theta_{n}=-\frac{4}{r}\,.$
(7)
Note that $\Theta_{n}$ is always negative and $\Theta_{\ell}$ vanishes
precisely at the horizon, as required by a dynamical horizon. The unit normal
to the horizon is given by
$\hat{\tau}_{a}=\frac{1}{\sqrt{|2\dot{f}f^{\prime}|}}[\dot{f},f^{\prime}]\qquad\textrm{and}\qquad\hat{\tau}^{a}=\frac{1}{\sqrt{|2\dot{f}f^{\prime}|}}[f^{\prime},\dot{f}]\,.$
(8)
The constant $r$ surfaces are the preferred cross-sections of the horizon and
the unit space-like normal $\hat{r}^{a}$ to these cross sections is given by
$\hat{r}_{a}=\frac{1}{\sqrt{|2\dot{f}f^{\prime}|}}[-\dot{f},f^{\prime}]\qquad\textrm{and}\qquad\hat{r}^{a}=\frac{1}{\sqrt{|2\dot{f}f^{\prime}|}}[f^{\prime},-\dot{f}].$
(9)
The properly rescaled null normals are then given by
$\ell^{a}=\frac{2|f^{\prime}|}{\sqrt{|2\dot{f}f^{\prime}|}}(1,0,0,0)\qquad\textrm{and}\qquad
n^{a}=\frac{2\dot{f}}{\sqrt{|2\dot{f}f^{\prime}|}}(0,1,0,0)\,.$ (10)
The lapse function corresponding to the radial coordinate $r$, which in this
case is identical to the area radius, is given by
$N_{r}=\left|\frac{\dot{f}}{2f^{\prime}}\right|^{1/2}.$ (11)
Thus the properly rescaled vector field corresponding to the radial coordinate
$r$ is $\xi_{(R=r)}^{a}=N_{r}\ell^{a}=(\partial/\partial v)^{a}$.
Now, take the following special solution with toroidal horizon configurations
suggested in lemos97
$ds^{2}=-\left(\alpha^{2}r^{2}-\frac{\beta
m(v)}{r}\right)dv^{2}+2dvdr+r^{2}(d\theta^{2}+d\phi^{2}),$ (12)
where $0\leq\theta<2\pi$, $0\leq\phi<2\pi$. The corresponding energy-momentum
tensor is then given by
$\displaystyle T_{ab}=\frac{\beta}{8\pi r^{2}}\frac{dm(v)}{dv}k_{a}k_{b},$
$\displaystyle\quad k_{a}=-\delta^{v}_{a},\quad k_{a}k^{a}=0,$ (13)
where $\alpha\equiv\sqrt{\frac{-\Lambda}{3}}$, $\beta=q/\alpha$, and $m(v)$ is
the Misner-Sharp mass. In these coordinates, lines with $v=$constant represent
incoming radial null vectors having tangent vectors in the form
$k^{a}=(0,-1,0,0)$, or $k_{a}=(-1,0,0,0)$. The energy momentum tensor depends,
in general, on $\Lambda$ and diverges as $\Lambda\rightarrow 0$. To assume a
toroidal cross-section and apply the eq.(2), we therefore ask for $\beta$ to
be independent of $\Lambda$, leading to $q=\frac{2\alpha}{\pi}$.
Now, noting that the energy-density of the radiation is
$\epsilon=\frac{1}{4\pi^{2}r^{2}}\frac{dm}{dv}$, one sees that the weak energy
condition for the radiation is satisfied whenever $\frac{dm}{dv}\geq 0$, i.e.,
the radiation is imploding.
The apparent horizon surface is now defined by
$qm|_{AH}=\alpha^{3}r|_{AH}^{3}.$ (14)
Having specified $f$, and noting that $\dot{f}=-\frac{q\dot{m}}{\alpha r}$ and
$f^{\prime}=\frac{2\alpha^{3}r^{3}+qm}{\alpha r^{2}}$, we are able to
calculate the gravitational flux leading to
$\mathcal{F}^{(R)}_{\rm grav}=\frac{1}{16\pi G}\,\int_{\Delta
H}N_{R}\left\\{|\sigma|^{2}+2|\zeta|^{2}\right\\}\,d^{3}V=0.$ (15)
Therefore, from the equation (2) we conclude that in our model of dynamical
black hole with the toroidal topology the total matter flux term including the
$\Lambda$-fluid across the horizon has to be zero, i.e.
$\mathcal{F}^{(R)}_{total}=0$. This means that the ordinary matter flux
$\mathcal{F}^{(R)}_{ordi}=\int_{\Delta
H}T_{ab}{\widehat{\tau}}^{\,a}\xi_{(R)}^{b}\,d^{3}V$, equals to the
corresponding $\Lambda$-fluid term, i.e.
$\mathcal{F}^{(R)}_{\Lambda}=\int_{\Delta H}\frac{\Lambda}{8\pi
G}g_{ab}){\widehat{\tau}}^{\,a}\xi_{(R)}^{b}\,d^{3}V$.
Therefore, in the general case of non-stationary metric where $m(v)$ is
everywhere time dependent, the horizon is dynamical, the total matter flux is
vanishing while the ordinary matter flux is non-zero, and the horizon is
space-like. Now, let us differentiate two special cases:
* •
$m$ is constant everywhere : In this case, the horizon is isolated. The
following coordinate transformations will then give us the stationary metric
suggested by Vanzo vanzo :
$t=v-\int\frac{dr}{\left(\alpha^{2}r^{2}-\frac{qm}{\alpha r}\right)}.$ (16)
Therefore, all results stated in vanzo for this metric including the formula
for the black hole area law are applicable here.
* •
$m$ is constant on the horizon ($m|_{AH}=constant$) :
In this limiting case we have $\dot{m}|_{AH}=0$, i.e. the matter flux in
addition to the total flux is zero, and ${\widehat{\tau}}^{\,a}$ will be null.
Thus, the horizon, being null now, has a constant surface. It is therefore an
isolated horizon although the metric is non-stationary.
## IV Conclusion
We have constructed a dynamical black hole having toroidal topology in its
cross-sections within an asymptotically anti-de Sitter space time as an exact
solution of Einstein equations not produced by any cut-and-paste technology.
The area law is written out and it has been shown that the total matter flux,
including the $\Lambda$-matter, is zero while the ordinary matter flux is non-
vanishing. This model for the first time, exemplifies the existence of
dynamical horizons with toroidal topology. The vanishing of the total matter
flux may be just a feature of the concrete model we have proposed. The model
leads in a limiting case to an isolated horizon. Assuming our general metric
to be stationary it reduces to the Vanzo metric vanzo .
## V ACKNOWLEDGMENT
We would like to thank Abhay Ashtekar for suggesting the problem.
## References
* (1) Hawking, S. W., Comm. Math. Phys 25, 152 (1972); Hawking, S. W. and Ellis, G. F. R., The Large Scale Sructure of Spacetime, Cambridge University Press (page 335).
* (2) Gannon, D., Gen. Rel. Grav. 7, 219 (1976).
* (3) P. T. Chrusciel and R. M. Wald, Class. Quantum Grav. 11, L147 (1994).
* (4) T. Jacobson and S. Venkataramani. Class. and Quantum Grav. 12, 1055 (1995).
* (5) J.L. Friedmann, K. Schleich and D.M. Witt. Phys. Rev. Lett. 71, 1486 (1993).
* (6) S.L. Shapiro, S.A. Teutolsky and J. Winicour. Phys. Rev. D52, 6982 (1995).
* (7) L. Vanzo, Phys. Rev. D 56, 6475 (1997).
* (8) Danny Birmingham, Class. Quantum Grav. 16 1197 (1999).
* (9) Filipe C. Mena, Jose Natario, Paul Tod, Adv.Theor.Math.Phys. 12 (2008) 1163-1181; Filipe C. Mena, Jose Natario, Paul Tod, Annales. Inst. Henri Poincar$\acute{e}$ 10:1359-1376, 2010.
* (10) J.T. Firouzjaee, Reza Mansouri, Gen. Relativity Gravitation. 42, 2431 (2010), [arXiv:0812.5108]; J.T. Firouzjaee, Int. J. Mod. Phys. D, 21, 1250039 (2012) [arXiv:1102.1062]; J. T. Firouzjaee, M. Parsi 1 Mood and Reza Mansouri, Gen. Relativ. Gravit. 44, 639 (2012) [arXiv:1010.3971]; J. T. Firouzjaee and Reza Mansouri, Europhys. Lett. 97, 29002 (2012) [arXiv:1104.0530].
* (11) A. Ashtekar and B. Krishnan, Phys. Rev. D 68, 104030 (2003). [arXiv:gr-qc/0308033].
* (12) Gray T Horowitz, Class. Quantum Grav. 8 587 (1991).
* (13) J. P. S. Lemos, Phys. Rev. D 57, 4600 (1998).
|
arxiv-papers
| 2012-07-16T13:27:08 |
2024-09-04T02:49:33.119322
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pouria Dadras, J. T. Firouzjaee and Reza Mansouri",
"submitter": "Javad Taghizadeh firouzjaee",
"url": "https://arxiv.org/abs/1207.3673"
}
|
1207.3695
|
# On some fundamental peculiarities of the traveling wave reactor
V.D. Rusov V.A. Tarasov I.V. Sharph V.N. Vaschenko E.P. Linnik T.N.
Zelentsova R. Beglaryan S. Chernegenko S.I. Kosenko V.P. Smolyar
Department of Theoretical and Experimental Nuclear Physics,
Odessa National Polytechnic University, Odessa, Ukraine State Ecological
Academy for Postgraduate Education, Kiev, Ukraine
###### Abstract
On the basis of the condition for nuclear burning wave existence in the
neutron-multiplicating media (U-Pu and Th-U cycles) we show the possibility of
surmounting the so-called dpa-parameter problem, and suggest an algorithm of
the optimal nuclear burning wave mode adjustment, which is supposed to yield
the wave parameters (fluence/neutron flux, width and speed of nuclear burning
wave) that satisfy the dpa-condition associated with the tolerable level of
the reactor materials radioactive stability, in particular that of the
cladding materials.
It is shown for the first time that the capture and fission cross-sections of
238U and 239Pu increase with temperature within 1000-3000 K range, which under
certain conditions may lead to a global loss of the nuclear burning wave
stability. Some variants of the possible stability loss due to the so-called
blow-up modes (anomalous nuclear fuel temperature and neutron flow evolution)
are discussed and are found to possibly become a reason for a trivial
violation of the traveling wave reactor internal safety.
###### keywords:
traveling wave reactor , nuclear burning wave , temperature blow-up regimes ,
Fukushima Plutonium effect
††journal: Annals of Nuclear Energy
## 1 Introduction
Despite the obvious and unique effectiveness of nuclear energy of the new
generation, there are difficulties of its understanding related to the
nontrivial properties of an ideal nuclear reactor of the future.
First, nuclear fuel should be natural, i.e. non-enriched uranium or thorium.
Second, traditional control rods should be absolutely absent in reactor active
zone control system. Third, despite the absence of the control rods, the
reactor must exhibit the so-called internal safety. This means that under any
circumstances the reactor active zone must stay at a critical state, i.e.
sustain a normal operation mode automatically, with no operator actions,
through physical causes and laws, that naturally prevent the explosion-type
chain reaction. Figuratively speaking, the reactors with internal safety are
”the nuclear devices that never explode” [1].
Surprisingly, reactors that meet such unusual requirements are really
possible. The idea of such self-regulating fast reactor was expressed for the
first time in a general form (the so-called breed-and-burn mode) by Russian
physicists Feynberg and Kunegin during the II Geneva conference in 1958 [2]
and was relatively recently ”reanimated” in a form of the self-regulating fast
reactor in traveling nuclear burning wave mode by Russian physicist Feoktistov
[3] and independently by American physicists Teller, Ishikawa and Wood [4].
The main idea of the reactor with internal safety is that the fuel components
are chosen in such a way that, first, the characteristic time $\tau_{\beta}$
of the active fuel component (the fissile component) nuclear burning is
significantly larger than the time of the delayed neutrons appearance; and
second, all the self-regulation conditions are sustained in the operation
mode. Particularly, the equilibrium concentration $\tilde{n}_{fis}$ of the
active fuel component, according to Feoktistov’s condition of the wave mode
existence, is greater than its critical concentration111Concentrations of the
active element (${}^{239}Pu$ and ${}^{233}U$ in cycles (1) and (2)), are
called equilibrium or critical when an equal number of the active element
nuclei or neutrons, respectively, is born and destroyed at the same time
during the nuclear cycle. $n_{crit}$ [3]. These conditions are very important,
though they are almost always practically implementable in case when the
nuclear transformations chain of Feoktistov’s uranium-plutonium cycle type [3]
is significant among other reactions in the reactor:
${{}^{238}U(n,\gamma)}\rightarrow{{}^{239}U}\xrightarrow[]{\beta^{-}}{{}^{239}Np}\xrightarrow[]{\beta^{-}}{{}^{239}Pu(n,fission)}$
(1)
The same is also true for the Teller-Ishikawa-Wood thorium-uranium cycle type
[4]
${{}^{232}Th(n,\gamma)}\rightarrow{{}^{233}Th}\xrightarrow[]{\beta^{-}}{{}^{233}Pa}\xrightarrow[]{\beta^{-}}{{}^{233}U(n,fission)},$
(2)
In these cases the fissionable isotopes form (${}^{239}Pu$ in (1) or
${}^{233}U$ in (2)) which are the active components of the nuclear fuel. The
characteristic time of such reaction depends on the time of the corresponding
$\beta$-decays, and approximately equals to $\tau_{\beta}=2.3/ln{2}\approx
3.3$ days in case (1) and $\tau_{\beta}\approx 39.5$ days in case (2) which is
many orders of magnitude higher than the corresponding time for the delayed
neutrons.
The effect of the nuclear burning process self-regulation is provided by the
fact that the system, being left by itself, cannot surpass the critical state
and enter the uncontrolled reactor runaway mode, because the critical
concentration is limited from above by a finite value of the active fuel
component equilibrium concentration (plutonium in (1) or uranium in (2)):
$\tilde{n}_{fis}>n_{crit}$ (the Feoktistov’s wave existence condition [3]).
Phenomenologically the process of the nuclear burning self-regulation is as
follows. Any increase in neutron flow leads to a quick burn-out of the active
fuel component (plutonium in (1) or uranium in (2)), i.e. to a reduction of
their concentration and neutron flow; meanwhile the formation of the new
nuclei by the corresponding active fuel component proceeds with the prior rate
during the time $\tau_{\beta}$. On the other hand, if the neutron flow drops
due to some external impact, the burn-out speed reduces and the active
component nuclei generation rate increases, followed by the increase of a
number of neutrons generated in the reactor during approximately the same time
$\tau_{\beta}$.
The system of kinetic equations for nuclei (the components of nuclear fuel)
and neutrons (in diffuse approximation) in such chains are rather simple. They
differ only by the depth of description [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] of all the
possible active fuel components and non-burnable poison222Here by poison we
mean the oxygen nuclei or other elements, chemically bound to heavy nuclides,
construction materials, coolant and the poison itself, i.e. the nuclei added
to the initial reactor composition in order to control the neutron balance..
Fig. 1 shows the characteristic solutions for such problem (equations (3)-(9)
in [30]) in a form of the soliton-like waves of the nuclear fuel components
and neutrons concentrations for uranium-plutonium cycle in a cylindrical
geometry case. Within the theory of the soliton-like fast reactors it is easy
to show that in general case the phase speed $u$ of soliton-like neutron wave
of nuclear burning is defined by the following approximate equality [30]:
$\frac{u\tau_{\beta}}{2L}\simeq\left(\frac{8}{3\sqrt{\pi}}\right)^{6}\exp{\left(-\frac{64}{9\pi}a^{2}\right)},~{}~{}a^{2}=\frac{\pi^{2}}{4}\cdot\frac{n_{crit}}{\tilde{n}_{fis}-n_{crit}},$
(3)
where $\tilde{n}_{fis}$ and $n_{crit}$ are the equilibrium and critical
concentration of the active (fissile) isotope, $L$ is the mean neutron
diffusion length, $\tau_{\beta}$ is the delay time, associated with the birth
of the active (fissile) isotope and equal to an effective $\beta$-decay period
of the intermediate nuclei in Feoktistov’s uranium-plutonium cycle (1) or in
Teller-Ishikawa-Wood thorium-uranium cycle (2).
Figure 1: Kinetics of the neutrons, 238U, 239U and 239Pu concentrations in the
core of a cylindrical reactor with radius of 125 cm and 1000 cm long at the
time of 240 days. Here $r$ is the transverse spatial coordinate axis (cylinder
radius), $z$ is the longitudinal spatial coordinate axis (cylinder length).
Temporal step of the numerical calculations is 0.1 s. Adopted from [30]
Let us note that expression (3) automatically incorporates a condition of
nuclear burning process self-regulation, since the fact of a wave existence is
obviously predetermined by the inequality $\tilde{n}_{fis}>n_{crit}$. In other
words, the expression (3) is a necessary physical condition of the soliton-
like neutron wave existence. Let us note for comparison that the maximal value
of the nuclear burning wave phase velocity, as follows from (3), is
characterized both for uranium and thorium by the equal average diffusion
length ($L\sim 5cm$) of the fast neutrons ($1~{}MeV$) and is equal to
$3.70~{}cm/day$ for uranium-plutonium cycle (4) and $0.31~{}cm/day$ for
thorium-uranium cycle (2).
Generalizing the results of a wide range of numerical experiments[5, 6, 7, 8,
9, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30], we can positively
affirm that the principal possibility of the main stationary wave parameters
control was reliably established within the theory of a self-regulating fast
reactor in traveling wave mode, or in other words, the traveling wave reactor
(TWR). It is possible both to increase the speed, the thermal power and the
final fluence as well as decrease them. Obviously, according to (3), it is
achieved by varying the equilibrium and critical concentrations of the active
fuel component, i.e. by the purposeful change of the initial nuclear fuel
composition.
The technological problems of TWR are actively discussed in science nowadays.
The essence of these problems usually comes to a principal impossibility of
such project realization, and is defined by the following insurmountable
flaws:
* 1.
High degree of nuclear fuel burn-up (over 20% in average) leading to the
following adverse consequences:
* (a)
High damaging dose of fast neutrons acting at at the constructional materials
($\sim$500 $dpa$)333For comparison – the claimed parameters for the Russian
FN-800 reactor are $93$ $dpa$. At the same time it is known that one of the
main tasks of the construction materials radioactive stability investigations
conducted at the Bochvar Hi-tech Institute for non-organic materials (Moscow)
is to achieve $133$ to $164$ $dpa$ by 2020!;
* (b)
High gas release, which requires an increased gas cavity length on top of a
long fuel rod as it is.
* 2.
Long active zone requiring the correspondingly long fuel rods, which makes
their parameters unacceptable from the technological use point of view. For
instance, this has to do with the parameters characterized by a significant
increase in:
* (a)
the value of a positive void coefficient of reactivity;
* (b)
hydraulic resistance;
* (c)
energy consumption for the coolant circulation through the reactor.
* 3.
The problem of nuclear waste associated with the unburned plutonium
reprocessing and nuclear waste utilization.
The main goal of the present paper is to solve the specified technological
problems of the TWR on the basis of a technical concept which makes it
impossible for the damaging dose of the fast neutrons in the reactor (fuel
rods jackets, reflection shield and reactor pit) to exceed the $\sim
200~{}dpa$ level. The essence of this technical concept is to provide the
given neutron flux on in-reactor devices by defining the speed of the fuel
movement relative to the nuclear burning wave speed. The neutron flux, wave
speed and fuel movement speed are in their turn predetermined by the chosen
parameters (equilibrium and critical concentrations of the active component in
the initial nuclear fuel composition).
Section 1 of this paper is dedicated to a brief analysis of the state-of-the-
art idea of a self-regulating fast reactor in traveling wave mode. Based on
this analysis we formulate the problem statement and chalk out the possible
ways to solve it. Chapter 2 considers the analytical solution for a non-
stationary 1D reactor equation in one-group approximation with negative
reactivity feedback (1D Van Dam [7] model). It yields the expressions for the
amplitude $\varphi_{m}$, phase $\alpha$ and phase speed $u$, as well as the
dispersion (FWHM) of the soliton-like burning wave. Knowing the FWHM we may
further estimate the spatial distribution of the neutron flux and thus a final
neutron fluence. Chapter 3 is dedicated to a description of the nontrivial
neutron fluence dependence on phase velocity of the solitary burn-up waves in
case of the fissible and non-fissible absorbents. It reveals a possibility of
the purposeful (in terms of the required neutron fluence and nuclear burning
wave speed values) variation of the initial nuclear fuel composition. Chapter
4 analyses the dependence of the damaging dose on neutron fluence, phase
velocity and dispersion of the solitary burn-up waves. Chapter 5 considers the
possible causes of the TWR internal safety violation caused by “Fukushima
plutonium” effect, or in other words, the temperature blow-up modes driven
either by temperature or neutron flux. Chapter 6 is dedicated to analysis of
the practical examples of the temperature blow-up modes in neutron-multiplying
media. The idea of an impulse thermo-nuclear TWR is also proposed. The
conclusion of the paper is presented in Chapter 7.
## 2 On entropy and dispersion of solitary burn-up waves.
Let us discuss the physical causes, defining the main characteristics of the
soliton-like propagation of “criticality” wave in the initially undercritical
environment, characterized by the infinite multiplication factor $k_{\infty}$
less than unit. Obviously, the supercritical area ($k_{\infty}>1$) must be
created by some external neutron source (e.g. by an accelerator or another
super-critical area). In the general case, the supercritical area is a result
of the breeding effects in fast nuclear systems or the burning of the fissible
absorbents (fuel components) in thermal nuclear systems444W. Seifritz (1995)
was the first to find theoretically a nuclear solitary burn-up wave in opaque
neutron absorbers [31]. The supercriticality waves in thermal nuclear reactors
are searched for and analysed in the papers by Akhiezer A.I. et al.[32, 33,
34, 35], where they show the possibility of both fast [32, 33, 34] and slow
[35] modes of nuclear burning distribution (i.e. the super-criticality waves)
in the framework of diffuse approximation.. Due to the gradual burn-out of the
neutron-multiplying medium in the supercritical area, this area looses its
supercritical properties, since $k_{\infty}$ becomes less than unit. The wave
would have to stop and diminish at this point in an ordinary case. However,
because of the neutrons, appearing during breeding and diffusely ”infecting”
the nearby areas, this ”virgin” area before the wavefront is forced to obtain
the properties of super-criticality, and the wave moves forward in this
direction. Apparently, the stable movement of such soliton-like wave requires
some kind of stabilizing mechanism. For example, the negative self-catalysis
or any other negative feedback. This is called the negative reactivity
feedback in traditional nuclear reactors555According to Van Dam [7], the
procedure of the reactivity introduction into the 1D non-stationary equation
of the reactor in one-group approximation, though implicitly, takes the
kinetic equations of the burn-out into account. Particularly, the production
of plutonium in U-Pu cycle or uranium in Th-U cycle.. Let us therefore
consider such an example qualitatively below.
For this purpose let us write down a 1D non-stationary equation of the reactor
in one-group approximation [36, 37, 38] with negative reactivity feedback:
$D\frac{\partial^{2}\varphi}{\partial
x^{2}}+\left(k_{\infty}-1+\gamma\varphi\right)\Sigma_{a}\varphi=\frac{1}{v}\frac{\partial\varphi}{\partial
t}$ (4)
where $\varphi$ is the neutron flux $[cm^{-2}s^{-1}]$; $D$ is the diffusion
coefficient, $[cm]$; $\gamma\varphi$ is the reactivity, dimensionless value;
$\Sigma_{a}$ is the total macroscopic absorption cross-section, $[cm^{-1}]$;
$v$ is the neutron speed, $[cm\cdot s^{-1}]$. In this case the negative
feedback $\gamma$ is defined mainly by the fact that the infinite
multiplication factor is used in (4), and therefore the flux density must be
corrected.
Let us search for the solution in an autowave form:
$\varphi(x,t)=\varphi(x-ut)\equiv\varphi(\xi),$ (5)
where $u$ is the wave phase velocity, $\xi$ is the coordinate in a coordinate
system, which moves with phase speed. In such case:
$\frac{1}{v}\frac{\partial\varphi}{\partial
t}=-\frac{u}{v}\frac{\partial\varphi}{\partial t}$ (6)
As is known [7], the relation $u/v$ by order of magnitude equals to $10^{-13}$
and $10^{-11}$ for fast and thermal nuclear systems respectively. Therefore
the partial time derivative in (4) may be neglected without loss of
generality. Further taking into account (5), the equation (4) may be presented
in the following form:
$L^{2}\frac{\partial^{2}\varphi}{\partial\xi^{2}}+\left[k_{\infty}(\psi)-1+\gamma\varphi\right]\varphi=0,$
(7)
where $L=(D/\Sigma_{a})^{1/2}$ is the neutron diffusion length, and $\psi$ is
the so-called neutron fluence function:
$\psi(x,t)=\int\limits_{0}^{t}\varphi(x,t^{\prime})dt^{\prime}.$ (8)
In order to find a physically sensible analytic solution of (7) by
substituting (8), we need to define some realistic form of the function
$k_{\infty}(\psi)$ (usually referred to as the burn-up function). Since the
real burn-up function $k_{\infty}(\psi)$ has a form of some asymmetric bell-
shaped dependence on fluence, normalized to its maximal value $\psi_{max}$
(fig. 1), following [7], let us define it in a form of parabolic dependency
without lose of generality:
$k_{\infty}=k_{max}+\left(k_{0}-k_{max}\right)\left(\frac{\psi}{\psi_{max}}-1\right)^{2},$
(9)
where $k_{max}$ and $k_{0}$ are the maximal and initial neutron multiplication
factors. Substituting (9) into (7) we obtain:
$L^{2}\varphi_{\xi\xi}+\rho_{max}\varphi+\gamma_{0}\varphi^{2}-\delta\left[\frac{\int\limits_{\xi}^{\infty}\varphi
d\xi}{u\psi_{m}}-1\right]^{2}\varphi=0,$ (10)
where $\rho_{max}=k_{max}-1$, $\delta=k_{max}-k_{0}$,
$\gamma_{0}\equiv\gamma$.
Figure 2: Asymmetric burn-up function as characteristic for realistic burn-up
function. Adapted from [7].
Suppose we are searching a partial solution of (10). Let us rewrite it in the
following form:
$L^{2}\frac{d^{2}\varphi}{d\xi^{2}}+\left(\rho_{max}+\gamma_{0}\varphi-\delta\left(\frac{\int\limits_{\xi}^{+\infty}\varphi
d\xi}{u\psi_{m}}-1\right)^{2}\right)\varphi=0.$ (11)
Let us introduce a new unknown function:
$\chi(\xi)=\int\limits_{\xi}^{+\infty}\varphi
d\xi\Rightarrow\varphi(\xi)=-\frac{d\chi(\xi)}{d\xi},$ (12)
that due to its non-negativity on the interval $\xi\in[0,\infty]$, must
satisfy the following boundary conditions: $\varphi=0$ for $\xi=0,\infty$.
The equation will take the form:
$L^{2}\frac{d^{3}\chi(\xi)}{d\xi^{3}}+\left(\rho_{max}-\gamma_{0}\frac{d\chi(\xi)}{d\xi}-\delta\left(\frac{\chi(\xi)}{u\psi_{m}}-1\right)^{2}\right)\frac{d\chi(\xi)}{d\xi}=0.$
(13)
In order to find a partial solution of (13), we require the following
additional condition to hold:
$\rho_{max}-\gamma_{0}\frac{d\chi(\xi)}{d\xi}-\delta\left(\frac{\chi(\xi)}{u\psi_{m}}-1\right)^{2}=f\left(\chi(\xi)\right),$
(14)
where $f(\xi)$ is an arbitrary function, the exact form of which will be
defined later. The condition (14) is chosen because it makes it possible to
integrate the equation (13). Really, if (14) is true, the equation (5) takes
the following form:
$L^{2}\frac{d^{3}\chi(\xi)}{d\xi^{3}}+f\left(\chi(\xi)\right)\frac{d\chi(\xi)}{d\xi}=0,$
(15)
That allows us reduce the order of the equation:
$L^{2}\frac{d^{2}\chi(\xi)}{d\xi^{2}}+F_{1}\left(\chi(\xi)\right)=C.$ (16)
Here $F_{1}(\chi)$ denotes a primitive of $f(\chi)$, and $C$ is an arbitrary
integration constant. The order of (16) may be further reduced by multiplying
both sides of the equation by $d\chi(\xi)/d\xi$:
$\frac{L^{2}}{2}\left(\frac{d\chi(\xi)}{d\xi}\right)^{2}+F_{2}\left(\chi(\xi)\right)-C\chi(\xi)=B,$
(17)
$F_{2}(\chi)$ here denotes a primitive of $F_{1}(\chi)$, i.e. “the second
primitive” of the function $f(\chi)$ introduced in (5), and $B$ is a new
integration constant.
The obtained equation (17) is a separable equation and may be rewritten in the
following form:
$d\xi=\pm\frac{d\chi}{\sqrt{\frac{2}{L^{2}}\left(B-F_{2}\left(\chi(\xi)\right)+C\chi(\xi)\right)}}.$
(18)
On the other hand, (14) may also be considered a separable equation relative
to $\chi(\xi)$. Then, separating variables in (14) we obtain:
$d\xi=\frac{d\chi}{\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}\left(\frac{\chi(\xi)}{u\psi_{m}}-1\right)^{2}-\frac{1}{\gamma_{0}}f\left(\chi(\xi)\right)}.$
(19)
Since the equations (18) and (19) are for the same function $\chi(\xi)$, by
comparing them, we derive that the following relation must hold:
$\pm\sqrt{\frac{2}{L^{2}}\left(B-F_{2}\left(\chi(\xi)\right)+C\chi(\xi)\right)}=\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}\left(\frac{\chi(\xi)}{u\psi_{m}}-1\right)^{2}-\frac{1}{\gamma_{0}}f\left(\chi(\xi)\right).$
(20)
In order to simplify (18) and (19), let us choose $f(\chi)$ in a polynomial
form of $\chi$. The order of this polynomial is $n$. Then $F_{2}(\xi)$,
obtained by double integration of $f(\chi)$, is a polynomial of order $(n+2)$.
Taking the square root, according to (20), should also lead to a polynomial of
the order $n$. Therefore $n+2=2n\Rightarrow n=2$.
Consequently, the function $f(\chi)$ may only be a second-order polynomial
under the assumptions made above.
$f(\chi)=s_{2}\chi^{2}+s_{1}\chi+s_{0},$ (21)
where $s_{0}$, $s_{1}$, $s_{2}$ are the polynomial coefficients.
Double integration of (21) leads to:
$F_{2}(\chi)=\frac{s_{2}}{12}\chi^{4}+\frac{s_{1}}{6}\chi^{3}+\frac{s_{0}}{2}\chi^{2}+c_{1}\chi+c2,$
(22)
where $c_{1}$ and $c_{2}$ are the integration constants.
Substituting (21) and (22) into (20) we get:
$\displaystyle\left(\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}\left(\frac{\chi}{u\psi_{m}}-1\right)^{2}-\frac{1}{\gamma_{0}}\left(s_{2}\chi^{2}+s_{1}\chi+s_{0}\right)\right)^{2}=$
$\displaystyle=\frac{2}{L^{2}}\left(B-\left(\frac{s_{2}}{12}\chi^{4}+\frac{s_{1}}{6}\chi^{3}+\frac{s_{0}}{2}\chi^{2}+c_{1}\chi+c_{2}\right)+C\chi\right)$
(23)
Further in (23) we set the coefficients at the same orders of $\chi$ equal:
$\left(\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}\right)^{2}=-\frac{s_{2}}{6L^{2}},$
(24)
$-2\left(\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}\right)\left(\frac{2\delta}{\gamma_{0}u\psi_{m}}-\frac{s_{1}}{\gamma_{0}}\right)=-\frac{s_{1}}{3L^{2}},$
(25)
$\left(\frac{2\delta}{\gamma_{0}u\psi_{m}}-\frac{s_{1}}{\gamma_{0}}\right)^{2}-2\left(\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}\right)\left(\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}-\frac{s_{0}}{\gamma_{0}}\right)=-\frac{s_{0}}{L^{2}};$
(26)
$2\left(\frac{2\delta}{\gamma_{0}u\psi_{m}}-\frac{s_{1}}{\gamma_{0}}\right)\left(\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}-\frac{s_{0}}{\gamma_{0}}\right)=-\left(\frac{2c_{1}}{L^{2}}+\frac{2C}{L^{2}}\right),$
(27)
$\left(\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}-\frac{s_{0}}{\gamma_{0}}\right)^{2}=\frac{2}{L^{2}}B-\frac{2c_{2}}{L^{2}}.$
(28)
Note that the first three equations are enough to find the coefficients
$s_{0}$, $s_{1}$ and $s_{2}$, and the remaining two equations may be satisfied
with the appropriate constants $B$, $C$, $c_{1}$, $c_{2}$.
$s_{0}=\rho_{max}-\delta;$ (29)
$s_{1}=\frac{2\delta}{u\psi_{m}}-\frac{\gamma_{0}}{L}\sqrt{\delta-\rho_{max}};$
(30)
$s_{2}=\frac{\delta\gamma_{0}}{3Lu\psi_{m}\sqrt{\delta-\rho_{max}}}-\frac{\gamma_{0}^{2}}{6L^{2}}-\frac{\delta}{u^{2}\psi_{m}^{2}}.$
(31)
After finding $s_{0}$, $s_{1}$, $s_{2}$ from this system, we may consider the
equation (19) in more detail, which takes the form:
$d\xi=\frac{d\chi}{-\left(\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}\right)\chi^{2}+\left(\frac{2\delta}{\gamma_{0}u\psi_{m}}+\frac{s_{1}}{\gamma_{0}}\right)\chi+\left(\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}-\frac{s_{0}}{\gamma_{0}}\right)}.$
(32)
Solving this equation yields:
$\frac{d\chi}{\left(\chi-K\right)^{2}-M^{2}}=-Nd\xi,$ (33)
where
$K=\frac{\frac{\delta}{\gamma_{0}u\psi_{m}}-\frac{s_{1}}{2\gamma_{0}}}{\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}},$
(34)
$M^{2}=\left(\frac{\frac{\delta}{\gamma_{0}u\psi_{m}}-\frac{s_{1}}{2\gamma_{0}}}{\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}}\right)+\frac{\left(\frac{\rho_{max}}{\gamma_{0}}-\frac{\delta}{\gamma_{0}}-\frac{s_{0}}{\gamma_{0}}\right)}{\left(\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}\right)},$
(35)
$N=\frac{\delta}{\gamma_{0}u^{2}\psi_{m}^{2}}+\frac{s_{2}}{\gamma_{0}}.$ (36)
Let us introduce a new variable $\chi_{1}$ into (33) by substituting:
$\chi-K=M\chi_{1},~{}~{}d\chi=Md\chi_{1}.$ (37)
Then the equation (33) will take the following form:
$\frac{d\chi_{1}}{(\chi_{1})^{2}-1}=-MNd\xi.$ (38)
Hence
$\chi_{1}=-\tanh{(MN\xi-D)},$ (39)
where $D$ is the integration constant. Taking into account that
$\chi-K=M\chi_{1}$:
$\chi=K-M\tanh{(MN\xi-D)}.$ (40)
Considering (12), we obtain the soliton-like solution in the form:
$\varphi(\xi)=M^{2}N\sec{h^{2}(MN\xi-D)}.$ (41)
Let us remind that together with introducing a new unknown function
$\chi(\xi)$ (see (12)) we obtained an obvious condition for this function:
$\lim\limits_{\xi\rightarrow\infty}\chi(\xi)=0.$ (42)
Let us show that this condition eventually leads to an autowave existence
condition. Obviously the condition (42) along with (40)
$\lim\limits_{\xi\rightarrow\infty}\chi(\xi)=\lim\limits_{\xi\rightarrow\infty}\left[K-M\tanh{(MN\xi-D)}\right]=0$
(43)
leads to
$K=M.$ (44)
This relation lets us define the amplitude $\varphi_{m}$, phase $\alpha$ and
phase velocity $u$ of the soliton-like wave:
$\alpha=MN=\frac{\sqrt{\delta-\rho_{max}}}{2L};$ (45)
$\varphi_{m}=M^{2}N=\frac{\delta-3\rho_{max}}{2\gamma_{0}}=\frac{3\rho_{max}-\delta}{2|\gamma_{0}|};$
(46)
$u=\frac{\varphi_{m}}{\alpha\psi_{m}}.$ (47)
From the condition of non-negative width (45) and amplitude (46) of the
nuclear burning wave, follows the condition of 1D autowave existence, or the
so-called “ignition condition” by van Dam [7]:
$3\rho_{max}-\delta=2k_{max}+k_{0}-3\geqslant 0,~{}~{}~{}where~{}1-k_{0}>0.$
(48)
It is noteworthy that the analogous results for a nonlinear one-group
diffusion 1D-model (4) with explicit feedback and burn-up effects were first
obtained by Van Dam[7]. The same results (see (45)-(47)) were obtained by Chen
and Maschek [12] while investigating the 3D-model by Van Dam using the
perturbation method. The only difference is that the value of neutron fluence
associated with the maximum of burn-up parameter $k_{\infty}$ was adapted to
the transverse buckling. In other words, they considered the transverse
geometric buckling mode as a basis for perturbation. Hence they introduced a
geometric multiplication factor $k_{GB}$ due to transverse buckling into two-
dimensional equation (4), which led to a change in some initial parameters
($\rho_{max}=k_{max}-k_{GB},~{}\delta=k_{max}-k_{0}$) and consequently – to a
change in the conditions of the autowave existence in 3D case:
$3\rho_{max}-\delta=2k_{max}+k_{0}-3k_{GB}\geqslant
0,~{}~{}~{}where~{}k_{GB}-k_{0}>0,$ (49)
that in the case of $k_{GB}=1$ is exactly the same as the so-called ignition
condition by Van Dam [7].
From the point of view of the more detailed Feoktistov model [3] analysis,
thoroughly considered in [30] and related to a concept of the nuclear systems
internal safety, the condition (49) is necessary, but not sufficient. On the
other hand, it is an implicit form of the necessary condition of wave
existence according to Feoktistov, where the equilibrium concentration
$\tilde{n}_{fis}$ of the active fuel component must be greater than its
critical concentration $n_{crit}$ ($\tilde{n}_{fis}>n_{crit}$) [3, 30]. The
physics of such hidden but simple relation will be explained below (see
Chapter 3).
Returning to a 1D reactor equation solution (10) in one-group approximation
with negative reactivity feedback, let us write it in a more convenient form
for analysis
$\varphi=\varphi_{m}\sec{h^{2}(\alpha\xi)}=\varphi_{m}\sec{h^{2}\left[\alpha(x-ut)\right]}.$
(50)
where $\varphi_{m}$ is the amplitude of the neutron flux; $1/\alpha$ is the
characteristic length proportional to the soliton wave width, which is a full
width at half-maximum (FWHM) by definition, and equals to
$\Delta_{1/2}=FWHM=2\ln{\left(1+\sqrt{2}\right)}\alpha^{-1},~{}~{}[cm].$ (51)
Apparently, integrating (49) yields the area under such soliton:
$A_{area}=2\varphi_{m}/\alpha,~{}~{}[cm^{2}].$ (52)
In order to estimate the extent of the found parameters influence on the
dynamics of the soliton-like nuclear burning wave stability, let us invoke an
information-probability approach, developed by Seifritz [39]. For this purpose
let us write down the expression for the mean value of information or more
precisely – the entropy of the studied process:
$S=-k_{B}\int\limits_{-\infty}^{\infty}p(x)\ln{p(x)}dx,$ (53)
where $p(x)$ is the function of probability density relative to a
dimensionless variable $x$; $\ln{1/p(x)}$ is the mean information value;
$k_{B}$ is the Boltzmann constant.
Substituting the soliton-like solution (49) into (53) we obtain
$S=-2k_{B}\int\limits_{0}^{\infty}\sec{h^{2}(\alpha\xi)\ln{\left[\sec{h^{2}(\alpha\xi)}\right]d\left(\frac{x}{u\tau_{\beta}}\right)}}=4k_{B}\int\limits_{0}^{\infty}\frac{\ln\cosh{(\beta
y)}}{\cosh^{2}{(\beta y)}}dy,$ (54)
where $\beta=\alpha(u\tau_{\beta})$ is another (dimensionless) scaling factor,
$\tau_{\beta}$ is the proper $\beta$-decay time of the active component of the
nuclear fuel. Let us point out the procedure of making the $x$ argument
dimensionless in (54), which takes into account the fact that the neutron flux
amplitude is proportional to the phase velocity of the nuclear burning wave
(see (47)), i.e. $\varphi_{m}\sim u$. Calculating the integral (54) leads to
the following quite simple expression for the entropy
$S=\frac{4(1-\ln{2})}{\beta}=4(1-\ln{2})\frac{k_{B}}{\alpha
u\tau_{\beta}}=\frac{4k_{B}(1-\ln{2})}{\sqrt{k_{GB}-k_{0}}}\frac{2L}{u\tau_{\beta}},$
(55)
that in the case of
$S\sim k_{B}\frac{2L}{u\tau_{\beta}}=const,$ (56)
points to an isentropic transport of the nuclear burning wave.
It is interesting to note here that if the width $\Delta_{1/2}\rightarrow 0$,
then due to isoentropicity of the process (56), the form of the soliton
becomes similar to the so-called Dirac $\delta$-function. Introducing two
characteristic sizes or two length scales ($l_{1}=1/\alpha$ and
$l_{2}=\varphi_{m}$), it is possible to see, that when the first of them is
small, the second one increases and vice versa. It happens because the area
$A_{area}$ (45) under the soliton must remain constant, since $A_{area}\propto
l_{1}l_{2}$. In this case the soliton entropy tends to zero because the
entropy is proportional to the ratio of these values ($S\propto
l_{1}/l_{2}\rightarrow 0$). These features are the consequence of the fact
that the scale $l_{1}$ is a characteristic of a dispersion of the process,
while another scale $l_{2}$ is a characteristic of the soliton non-linearity.
If $l_{1}\ll l_{2}$, then the process is weakly dispersed (fig. 3a). If
$l_{1}\gg l_{2}$, then the process is strongly dispersed (fig. 3b). In the
latter case the soliton amplitude becomes relatively large (a case of
$\delta$-function). And finally, if $l_{1}=l_{2}$, then the soliton speed
$u\propto(l_{1}^{2}l_{2}^{2})^{1/2}$ (see eq. (48)) is proportional to the
geometrical mean of the dispersion and non-linearity parameters.
Figure 3: (a) Weakly dispersive wave pattern, obtained by Chen and Maschek
[12] investigating a 3D-model by van Dam using perturbation method. Example
with the following parameters: $L=0.02~{}m$, $k_{GB}=1.04$, $k_{0}=1.02$ and
$k_{max}=1.06$; $\phi_{m}=10^{17}~{}m^{-2}s^{-1}$, $u=0.244~{}cm/day$; (b)
Strongly dispersive wave pattern, obtained Chen et. al [19] within a 3D-model
of traveling wave reactor. Example with the following parameters:
$L=0.017~{}m$, $k_{GB}=1.00030$, $k_{0}=0.99955$; $\phi_{m}=3\cdot
10^{15}~{}m^{-2}s^{-1}$, $u=0.05~{}cm/day$.
On the other hand, it is clear that according to (45) and (47), the burning
wave width $\Delta_{1/2}$ is a parameter that participates in formation of the
time for neutron fluence accumulation $\tau_{\beta}$ on the internal surface
of the TWR long fuel rod cladding material
$\psi_{m}\sim\frac{\Delta_{1/2}}{u}\varphi_{m}=\tau_{\varphi}\cdot\varphi_{m}.$
(57)
And finally, one more important conclusion. From the analysis of (45) and (46)
it is clear that the initial parameter $k_{0}$ for burning zone (fig. 4) is
predefined solely by the nuclear burning wave burn-up conditions, i.e. by the
parameters of an external neutron source and burn-up area composition (fig.
4). In other words, it means that by tuning the corresponding burn-up
conditions for a given nuclear fuel composition, we can set the certain value
of the nuclear burning front width. Moreover, by selecting the corresponding
equilibrium $\tilde{n}_{fis}$ and critical $n_{crit}$ concentrations of the
active nuclear fuel component, we can define the required value of the nuclear
burning wave speed $u$. Hence an obvious way for us to control the
corresponding neutron fluence $\tau_{\varphi}$ accumulation time in the
cladding material of the TWR fuel rod.
Figure 4: Schematic sketch of the two-zone cylindrical TWR.
Hence we can make an important conclusion that the realization of the TWR with
inherent safety requires the knowledge about the physics of nuclear burning
wave burn-up and the interrelation between the speed of nuclear burning wave
and the fuel composition. As is shown in [30], the properties of the fuel are
completely defined by the equilibrium $\tilde{n}_{fis}$ and critical
$n_{crit}$ (see (3)) concentrations of the active nuclear fuel component. We
examine this in more detail below.
## 3 Control parameter and condition of existence of stationary wave of
nuclear burning.
The above stated rises a natural question: ”What does the nuclear burning wave
speed in uranium-plutonium (1) and thorium-uranium (2) cycles mainly depend
on?” The answer is rather simple and obvious. The nuclear burning wave speed
in both cycles (far away from the burn-up source) is completely characterized
by its equilibrium $\tilde{n}_{fis}$ and critical $n_{crit}$ concentrations of
the active fuel component.
First of all, this is determined by a significant fact that the equilibrium
$\tilde{n}_{fis}$ and critical $n_{crit}$ concentrations of the active fuel
component completely identify the neutron-multiplying properties of the fuel
environment. They are the conjugate pair of the integral parameters, which due
to their physical content, fully and adequately characterize all the physics
of the nuclear transformations predefined by the initial fuel composition.
This is also easy to see from a simple analysis of the kinetics equations
solutions for the neutrons and nuclei, used in different models [5, 6, 7, 8,
9, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 27, 28, 29, 30]. It mean that
regardless of the nuclear cycle type and initial fuel composition, the nuclear
burning wave speed is defined by the equilibrium $\tilde{n}_{fis}$ and
critical $n_{crit}$ concentrations of the active fuel component through the
so-called para-parameter $a$ (see (3)). Consequently, as the numerical
simulation results show, (fig. 5 [30]), it follows the Wigner statistics.
Figure 5: The theoretical (solid line) and calculated (points) dependence of
$\Lambda(a_{*})=u\tau_{\beta}/2L$ on the parameter $a$ [30].
This is corroborated by the following fact. The papers [21, 22] study a
boundary-value problem for the stationarity of the nuclear burning wave,
formulated within the diffusion equation for the neutron fluence and the
kinetics equations for the nuclear density of the mother and daughter
nuclides:
$\frac{\partial\psi}{\partial t}=\nu\cdot
D\cdot\Delta\psi+G\left(\vec{N},\psi\right),$ (58)
$\frac{\partial\vec{N}}{\partial
t}=\widehat{\sigma}\cdot\vec{N}\cdot\frac{\partial\psi}{\partial
t}-\hat{\lambda}\vec{N},$ (59)
where
$G\left(\vec{N},\psi\right)=\int\limits_{0}^{\psi_{end}}g\left(\vec{N}\right)d\psi$
(60)
is a function666The expression (59) should be understood as an integral along
the system path in configuration space of the variables ($N$,$\psi$) for the
given spatial point. of neutron fluence generation, $g(\vec{N})$ is the
neutron generation function, which is a linear function of the nuclei
concentration $N_{i}$, $\vec{N}$ is the column of the mother and daughter
nuclides $N_{i}$, $\psi$ is the neutron fluence, $\psi_{end}$ is some final
neutron fluence, $\nu$ is the average neutron speed, $D$ is the diffusion
coefficient, $\hat{\sigma}$ is the matrix of the microscopic neutron
absorption and capture cross-sections, and $\hat{\lambda}$ is the matrix of
the radioactive decay constants for the $\beta$-active nuclei.
It is shown [21], that this boundary-value problem for the equations (58)-(59)
is a spectral non-linear differential problem. A dimensionless speed $W$ of
the nuclear burning wave was chosen as a spectral (free) parameter. Such
representation of the problem makes it possible to investigate the final
fluence behavior caused by the change of the absorbent properties, whose
concentration, according to [21, 22, 27, 28], controls (in a zero
approximation of perturbation theory) the value of the dimensionless wave
speed:
$W=\frac{u\tau_{2}}{L}=\frac{1}{b}\left(p_{0}-p\right),~{}~{}at~{}W\ll 1,$
(61)
where $u$ is the nuclear burning wave speed, $\tau_{2}$ is the internal time
scale, equal to the characteristic time of the intermediate nuclide
$\beta$-decay ($\tau_{2}=3.47$ days for ${}^{239}Np$ in U-Pu cycle and
$\tau_{2}=36.6$ days for ${}^{233}Pa$ in Th-U cycle), $L$ is the neutron
diffusion length, $1/b$ is the linear dependency slope (61), $p$ is the
dimensionless effective concentration of the absorbent, $p_{0}$ is the upper
limit of the absorbent concentration the nuclear wave can exist for.
Here we obtain an important result showing that the final fluence and the
absorbent concentration in the limiting case $W\rightarrow 0$ (zero order of
perturbation theory) are the solutions of the two equilibrium conditions for a
stationary nuclear burning wave:
$\int\limits_{0}^{\psi_{end}}gd\psi=0,$ (62)
$M\left(\psi_{end}\right)=\int\limits_{0}^{\psi_{end}}\left(\psi_{end}-\psi\right)gd\psi=0.$
(63)
For the sake of the more clear understanding of the physical sense of these
conditions, the authors of [21] suggest a simple, yet elegant and deep
analogy. The obtained conditions exactly coincide by form with the conditions
for a lever subject to a distributed perpendicular force $g(\psi)$ applied on
a segment $0\leqslant\psi\leqslant\psi_{end}$ along the lever. In other words,
according to this analogy, the conditions (62) and (63) represent the
conditions of the zero total force and total momentum respectively. Therefore,
if the first condition is an integral condition of the neutrons generation and
absorption equality, then by analogy the second condition may be called the
condition of the neutrons generation and absorption ”momenta” equality.
If the expression for $g\left(\vec{N}\right)$ is presented in the form of a
sum of the fuel $g^{F}$ and absorbent $g^{A}=p$ contributions in the neutron
generation:
$g=g^{F}-g^{A},$ (64)
then according to the conditions (62) and (63), it is easy to obtain a
modified condition (63) for the mean ”momenta” of the neutrons generation
$\left\langle M^{F}\right\rangle$ and absorption $\left\langle
M^{A}\right\rangle$ in the form [27]
$\left\langle
M^{F}\left(\psi_{end}\right)\right\rangle=\frac{\int\limits_{0}^{\psi_{end}}g^{F}\psi
d\psi}{\int\limits_{0}^{\psi_{end}}g^{F}d\psi}=\frac{\int\limits_{0}^{\psi_{end}}g^{A}\psi
d\psi}{\int\limits_{0}^{\psi_{end}}g^{A}d\psi}=\left\langle
M^{A}\left(\psi_{end}\right)\right\rangle,$ (65)
which by definition (62) has a trivial root $\psi=\psi_{end}$.
The main advantage of such presentation of the equation (65) is that its left
part, i.e. $\left\langle M^{F}\right\rangle$, is defined solely by the
properties of the fuel, while its right part, i.e. $\left\langle
M^{A}\right\rangle$, is defined solely by the absorbent. Moreover, according
to [27], the initial absorbent concentration is absent in the equation (65).
The plots for left and right parts of the equation (65) are presented at fig.
6 [27] for different cases.
Figure 6: Graphical solution of the equation (65) for the final fluence
$\psi_{end}$ with (a) different speeds of the absorbent burn-out and (b)
different enrichment of the ${}^{239}Pu$ for the non-burnable and burnable
adsorbents. At fig.6a the curve 1 is a graph of the right part of the
equation, and the curves 2-5 represent the left part for the absorbent cross-
sections of 0, 1, 2 and 3 barn respectively. At fig.6b the curves 1-4 are the
plots of the right part for the initial ${}^{239}Pu$ concentrations of 0, 3,
5, 7% respectively, and the curves 5 and 6 represent the left part of the
equation for the absorbent cross-sections of 0 and 2 barn respectively.
Adopted from [27].
A simple analysis of the fig. 6 shows that the increase of the non-burnable
absorbent concentration (curves 2-5 at fig. 6a) leads to the final
$\psi_{end}$ fluence increase, but according to (61) – to the nuclear burning
wave speed decrease at the same time. And vice versa, with the non-burnable
absorbent concentration increase (curves 1-4 at fig. 6b) the value of the
final fluence $\psi_{end}$ decreases, while the velocity of the nuclear
burning wave at $p=0$ also decreases in a complex manner (see fig. 7),
depending on the slope behavior in (61). The increase in the absorbent
concentration (curves 5-6 at fig. 6b) does not change the effect
qualitatively, but leads to a strong quantitative change. The final fluence
turns out to be very sensible to the burnable absorbent concentration: curve 6
at fig. 6b for the burnable absorbent crosses the curves 1-4 at much greater
values of final fluence.
Figure 7: Dependence of the nuclear burning wave speed on the non-burnable
absorbent concentration $p$, (a) taking into account the burn-out of the
${}^{239}Np$ (curve 1) and not taking into account the burn-out of the
${}^{239}Np$ (curve 2), (b) for the different initial fuel compositions: 1 -
100% ${}^{238}U$; 2 - 93% ${}^{238}U$ \+ 7% ${}^{235}U$; 3 - 93% ${}^{238}U$
\+ 7% ${}^{239}Pu$. Adopted from [27].
Thus, basing on the numerical solution of the boundary-value problem for the
stationary nuclear burning wave and the developed earlier analytical theory of
the nuclear burning wave [21, 22], it was shown in the papers [27, 28] (in a
zero order of perturbation theory for $W$) that the control parameter, which
makes it possible to change the speed of nuclear burning and hence the reactor
power, is the effective concentration of the absorbent. Moreover, this control
parameter allows both to increase and decrease the mentioned parameters.
According to [21, 22, 27, 28], ”…this is achieved by a purposeful change of
the initial reactor composition”.
The conclusion made by the authors of [21, 22, 27, 28] requires the following
comment. According to our earnest conviction, the results of these works are
indeed very interesting and informative, but the conclusion are not
satisfactory however strange it may seem. And here is why.
As a matter of fact, as it was shown in [30], the control parameter is not the
effective absorbent concentration ($p$) (or the nuclear burning wave speed
($W$), or the maximal neutron fluence ($\psi_{end}$)), but the so-called para-
parameter of the nuclear TWR burning. As it is shown above (see (3)), this
para-parameter is formed by a conjugated pair of the integral parameters, i.e.
by the equilibrium and critical concentrations of the active nuclear fuel
component. It is important to note that each of this concentrations varies
during the nuclear burning, but their ratio
$a=\frac{\pi}{2}\sqrt{\frac{n_{crit}}{\tilde{n}_{fis}-n_{crit}}}$ (66)
is a characteristic constant value for the given nuclear burning process [30].
In addition to that this para-parameter also determines (and it is extremely
important!) the conditions for the nuclear burning wave existence (3), the
neutron nuclear burning wave speed (see (3)) and the dimensionless width (66)
of the super-critical area in the burning wave of the active nuclear fuel
component.
Therefore, when the authors [21, 22, 27, 28] state that they control the
values of the parameters with purposeful variation of the initial reactor
composition, it actually means that changing the effective concentration of
the absorbent, they purposefully and definitely change (by definition (see [3,
30])) the equilibrium and critical concentrations of the active component,
i.e. the para-parameter (66) of the nuclear TWR-burning process.
It is necessary to note that the adequate understanding of the control
parameter determination problem is not a simple or even a scholastic task, but
is extremely important for the effective solution of another problem (the
major one, in fact!), related to investigation of the nuclear burning wave
stability conditions. In our opinion, the specified conditions for the
stationary nuclear burning wave existence (3) and (62)-(63) obtained in the
papers [30] and [21, 22, 27] respectively, reveal the path to a sensible
application of the so-called direct Lyapunov method [40] (the base theory for
the movement stability), and thus the path to a reliable justification of the
Lyapunov functional minimum existence (if it does exist) [40, 41, 42, 43].
Some variants of a possible solution stability loss due to anomalous evolution
of the nuclear fuel temperature are considered in Chapter 5.
At the same time one may conclude that the ”differential” [30] and ”integral”
[21, 22, 27] conditions for the stationary nuclear burning wave existence
provide a complete description of the wave reactor physics and can become a
basis for the future engineering calculations of a contemporary TWR project
with the optimal or preset wave properties.
## 4 On the dependence of the damaging dose on neutron fluence, phase
velocity and dispersion of the solitary burn-up waves
As follows from the expression for the soliton-like solution (50), it is
defined by three parameters – the maximal neutron flux $\psi_{m}$, the phase
$\alpha$ and the speed $u$ of nuclear burning wave. And even if we can control
them, it is still unclear, which condition determines the optimal values of
these parameters. Let us try to answer this question shortly.
It is known that a high cost effectiveness and competitiveness of the fast
reactors, including TWR, may be achieved only in case of a high nuclear fuel
burn-up777Since the maximum burn-up of the FN-600 reactor is currently $\sim
10\%$ [44], the burn-up degree of $\sim 20\%$ for the TWR may be considered
more than acceptable.. As the experience of the fast reactors operation shows,
the main hindrance in achieving the high nuclear fuel burn-up is the
insufficient radiation resistance of the fuel rod shells. Therefore the main
task of the radiation material science (along with the study of the physics
behind the process) is to create a material (or select among the existing
materials), which would keep the required level of performance characteristics
being exposed to the neutron irradiation. One of the most significant
phenomena leading to a premature fuel rods destruction is the vacansion
swelling of the shell material [45, 46, 47, 48]. Moreover, the absence of the
swelling saturation at an acceptable level and its acceleration with the
damaging dose increase leads to a significant swelling (volume change up to
30% and more) and subsequently to a significant increase of the active zone
elements size. The consequences of such effect are amplified by the fact that
the high sensitivity of the swelling to temperature and irradiation damaging
dose leads to distortions of the active zone components form because of the
temperature and irradiation gradients. And finally, one more aggravating
consequence of the high swelling is almost complete embrittlement of the
construction materials at certain level of swelling888It is known that the
fuel rod shell diameter increase due to swelling is accompanied by an
anomalously high corrosion damage of the shell by the fuel [44]. Consequently,
in order to estimate the possible amount of swelling, a damaging dose
(measured in dpa) initiated by the fast neutrons, e.g. in the fuel rod shells,
must be calculated (fig. 8).
Figure 8: Swelling of austenitic Phenix cladding compare to ferritic-
martensitic materials, ODS included. Adopted from [49].
Usually in order to evaluate the displacements per atom (DPA) created by the
spallation residues, the so-called modified NRT method is applied [50, 51],
which takes into account the known Lindhard correction [52]. Within such
modified NRT method, the total number of displacements produced by the
residues created by spallation reactions in the energetic window can be
calculated as the addition of the displacements produced by each of these
residues ($Z$,$A$), that in its turn leads to the following expression for the
displacements per atom:
$n_{dpa}=t\cdot\sum\limits_{i}^{N}\left\langle\sigma_{dpa}^{i}\right\rangle\int\limits_{E_{i}-1}^{E_{i}}\Phi\left(E_{i}\right)dE_{i}=\sum\limits_{i=1}^{N}\left\langle\sigma_{dpa}^{i}\right\rangle\varphi_{i}t,$
(67)
where
$\left\langle\sigma_{dpa}^{i}\right\rangle\left(E_{i}\right)=\left\langle\sigma_{d}\left(E_{i},Z,A\right)\right\rangle\cdot
d\left(E_{d}\left(Z,A\right)\right),$ (68)
while $\sigma_{d}\left(E_{i},Z,A\right)$ is the displacement cross-section of
recoil atom ($Z$,$A$), produced at incident particle energy $E_{i}$, $\Phi(E)$
is the energy-dependent flux of incident particles during time $t$ and
$d\left(E_{d}(Z,A)\right)$ is the number of displacements created at threshold
displacement energy $E_{d}$ of recoil atom ($Z$,$A$) or its so-called damage
function.
Actually, all the recoil energy of the residue $E_{r}$ is not going to be
useful to produce displacements because a part of it is lost inelastic
scattering with electrons in the medium. An estimation of the damage energy of
the residue can be calculated using the Lindhard factor $\xi$ [52]
$E_{dam}=E_{r}\xi.$ (69)
The number of displacements created by a residue ($Z$,$A$) are calculated
using this damage energy (69) and the NRT formula:
$d\left(E_{d}(Z,A)\right)=\eta\frac{E_{dam}}{2\left\langle
E_{d}\right\rangle},$ (70)
where $\left\langle E_{d}\right\rangle$ is the average threshold displacement
energy of an atom to its lattice site, $\eta=0.8$ [50, 51].
Figure 9: Operating condition for core structural materials in different power
reactors [53]. The upper yellow inset represents the data of Pukari M. &
Wallenius [49].
Consequently the condition of the maximal damaging dose for the cladding
materials of the fast neutron reactors, taking into account the metrological
data of IAEA [53] (see fig. 9) and contemporary estimates by Pukari M. &
Wallenius (see yellow inset at fig. 8) takes the form:
$n_{dpa}\simeq\left\langle\sigma_{dpa}\right\rangle\cdot\varphi\cdot\frac{2\Delta_{1/2}}{u}\leqslant
200~{}~{}[dpa].$ (71)
In this case the selection strategy for the required wave parameters and
allowed values of the neutron fluence for the future TWR project must take
into account the condition for the maximal damaging dose (70) for cladding
materials in fast neutron reactors, and therefore, must comply with the
following dpa-relation:
$\left\langle\sigma_{dpa}\right\rangle\cdot\varphi\cdot\frac{\Delta_{1/2}}{u}\leqslant
100~{}~{}[dpa].$ (72)
The question here is whether or not the parameters of the wave and neutron
fluence which can provide the burn-up level of the active nuclear fuel
component in TWR-type fast reactor of at least 20% are possible. Since we are
interested in the cladding materials resistible to the fast neutron damaging
dose, we shall assume that the displacement cross-section for the stainless
steel, according to Mascitti et al. [54] for neutrons with average energy 2
MeV equals $\left\langle\sigma_{dpa}\right\rangle\approx 1000~{}dpa$ (fig.
10).
Figure 10: Discretized displacement cross-section for stainless steel based on
the Lindhard model and ENDF/B scattering cross-section. Adopted from [54].
The analysis of the nuclear burning wave parameters in some authors’ models of
TWR [4, 5, 13, 14, 15, 19, 26, 27, 29, 30] presented in Table 1, shows that in
the case of U-Pu cycle none of the considered models satisfy the dpa-
parameter, while two Th-U cycle models by Teller [4] and Seifritz [5] groups
correspond well to the major requirements to wave reactors.
| $\Delta_{1/2}$ | $u$ | $\varphi$ | $\psi$ | $\left\langle\sigma_{dpa}\right\rangle$ | $\dfrac{n_{dpa}}{200}$ | Fuel | Solution
---|---|---|---|---|---|---|---|---
| [cm] | [cm/day] | [cm-2s-1] | [cm-2] | [barn] | burn-up
U-Pu cycle
Sekimoto [14] | 90 | 0.008 | 3.25 $\cdot$ 1015 | 3.2 $\cdot$ 10 23 | 1000 | 3.2 | $\sim$43% | No
Rusov [30] | 200 | 2.77 | 1018 | 6.2$\cdot$1024 | 1000 | 62 | $\sim$60% | No
Pavlovich [27] | – | 0.003 | – | 1.7$\cdot$1024 | 1000 | 17 | $\sim$30% | No
Fomin [15] | 100 | 0.07 | 2 $\cdot$ 1016 | 2.5$\cdot$1024 | 1000 | 25 | $\sim$30% | No
Fomin [13] | 125 | 1.7 | 5 $\cdot$ 1017 | 3.2$\cdot$1024 | 1000 | 32 | $\sim$40% | No
Chen [18] | 216 | 0.046 | 3 $\cdot$ 1015 | 1.2$\cdot$1024 | 1000 | 12 | $\sim$30% | No
T. Power [29] | – | – | – | – | – | 1.75 | $\sim$20% | No
Th-U cycle
Teller [4] | 70 | 0.14 | $\sim$2 $\cdot$ 1015 | 8.6$\cdot$1022 | 1000 | 0.96 | $\sim$50% | Yes
Seifritz [5] | 100 | 0.096 | 1015 | 9.0$\cdot$1022 | 1000 | 0.90 | $\sim$30% | Yes
Melnik [26] | 100 | 0.0055 | 0.5$\cdot$1016 | 7.9$\cdot$1024 | 1000 | $\sim$80 | $\sim$50% | No
U-Pu (+ moderator)
Example | 100 | 0.234 | 2.5 $\cdot$ 1015 | 9.2$\cdot$1023 | 100 | 0.92 | $\sim$20% | Yes
Ideal TWR | – | – | – | 1024 | 100 | 1.0 | $\geqslant$20% | Yes
Table 1: Results of the numerical experiments of the wave mode parameters
based on U-Pu and Th-U cycles
On the other hand, the authors of [13, 14, 15, 19, 26, 27, 29], obviously did
not take the problem of dpa-parameter in cladding materials into account,
since they were mainly interested in the fact of the wave mode of nuclear
burning existence in U-Pu and Th-U cycles at the time.
However, as the analysis of Table 1 shows, the procedure of account for the
dpa-parameter is not problematic, but it leads to unsatisfactory results
relative to the burn-out of the main fissle material.
In other words, from the analysis of Table 1 it follows that when
$\left\langle\sigma_{dpa}\right\rangle\approx 1000~{}dpa$, the considered
above dpa-condition for the maximum possible damaging dose for cladding
materials of the fast neutron reactors
$\psi_{1000}=\dfrac{\Delta_{1/2}}{u}\varphi\simeq 10^{23}~{}~{}[cm^{-2}],$
(73)
is not met by any example in Table 1. Here $\psi_{1000}$ is the neutron
fluence in case $\left\langle\sigma_{dpa}\right\rangle\approx 1000~{}dpa$,
$\varphi$ is the neutron flux, $\Delta_{1/2}$ and $u$ are the width and speed
of the soliton-like nuclear burning wave.
So on the one hand, the neturon fluence must be increased by an order of
magnitude to increase the burn-up level significantly, and on the other hand,
the maximum damaging dose for the cladding materials must also be reduced by
an order of magnitude. Such a controversial condition may be fulfilled
considered that the reduction of the fuel rod shell radioactive damage for a
given amount may be achieved by reducing the neutron flux density and energy
(see fig. 10). The latter is achieved by placing a specially selected
substance between fissile medium and fuel rod shell, which has the suitable
characteristics of neutron moderator and absorbent.
At the same time it is known from the reactor neutron physics [36, 55], that
the moderator layer width estimate $R_{mod}$ is:
$R_{mod}\simeq\dfrac{1}{\Sigma_{S}+\Sigma_{a}}\cdot\dfrac{1}{\xi}\ln{\dfrac{E_{fuel}}{E_{mod}}},$
(74)
where $\Sigma_{S}\approx\left\langle\sigma_{S}\right\rangle N_{mod}$ and
$\Sigma_{a}\approx\left\langle\sigma_{a}\right\rangle N_{mod}$ are the
macroscopic neturon scattering and absorption cross-sections respectively,
$\left\langle\sigma_{S}\right\rangle$ and
$\left\langle\sigma_{a}\right\rangle$ are the microscopic neutron scattering
and absorption cross-sections respectively averaged by energy interval of the
moderating neutrons from $E_{fuel}=2~{}MeV$ to $E_{mod}=0.1~{}MeV$, $N_{mod}$
is the moderator nuclei density,
$\xi=1+(A+1)^{2}\ln{\left[(A-1)/A+1\right]/2A}$ is the neutron energy
decrement of its moderation in the moderator-absorbent medium with atomic
number $A$.
It is clearly seen that the process of neutron moderation from 2.0 MeV to 0.1
MeV energy in moderator-absorbent of a given width (see Table 2) creates a
new, but satisfactory level of maximum possible damaging dose for the cladding
materials, corresponding to $\left\langle\sigma_{dpa}\right\rangle\approx
100~{}dpa$ (fig. 10). Therefore if we are satisfied with the main fissile
material burn-out level around $\sim$20%, then analyzing Table 1 and Table 2,
the conditions accounting for the dpa-parameter problem and contemporary level
of the radioactive material science will have the following form:
$\psi_{100}=\dfrac{\Delta_{1/2}}{u}\varphi\simeq 10^{24}~{}~{}[cm^{-2}],$ (75)
where $\psi_{100}$ is the neutron fluence (with 0.1 MeV energy) on cladding
materials surface in case $\left\langle\sigma_{dpa}\right\rangle\approx
100~{}dpa$ (see fig.10 and ”ideal” case in Table 1).
Moderator | Mass number, $A$ | Mean logarithmic energy, $\xi$ | density, $\rho$, g/cm3 | Impacts number required for moderating, $n$ | Neutron mean free path, $\lambda$ | Moderator layer width, $R_{mod}$, cm
---|---|---|---|---|---|---
Be | 9 | 0.21 | 1.85 | 11 | 1.39 | 15.3
C | 12 | 0.158 | 1.60 | 15 | 3.56 | 53.4
H2O | 18 | 0.924 | 1.0 | 2.5 | 16.7 | 41.6
H2O + B | | | | 2.5 | 10.0 | 25.0
He | 4 | 0.425 | 0.18 | 5.41 | 11.2 | 60.7
Table 2: Moderating and absorbing properties of some substances, moderator
layer width estimate for moderating neutron from $E_{fuel}=1.0~{}MeV$ to
$E_{mod}=0.1~{}MeV$.
And finally one can make the following intermediate conclusion. As shown above
in Chapter 3, the algorithm for determining the parameters (73) is mainly
defined by para-parameter that plays a role of a “response function” to all
the physics of nuclear transformations, predefined by initial fuel
composition. It is also very important that this parameter unequivocally
determines the conditions of the nuclear burning wave existence (3), the
neutron nuclear burning wave speed (see (3)) and the dimensionless width (66)
of the supercritical area in the wave of the active component burning.
Based on the para-parameter ideology [30] and Pavlovych group results [21, 22,
27], we managed to pick up a mode for the nuclear burning wave in U-Pu cycle,
having the parameters shown in Table 1 satisfying (73). The latter means that
the problem of dpa-parameter in cladding materials in the TWR-project is
currently not an insurmountable technical problem and can be successfully
solved.
In our opinion, the major problem of TWR are the so-called temperature blow-up
modes that take place due to coolant loss as observed during Fukushima nuclear
accident. Therefore below we shall consider the possible causes of the TWR
inherent safety breach due to temperature blow-up mode.
## 5 Possible causes of the TWR inherent safety failure: Fukushima plutonium
effect and the temperature blow-up mode
It is known that with loss of coolant at three nuclear reactors during the
Fukushima nuclear accident its nuclear fuel melted. It means that the
temperature in the active zone reached the melting point of uranium-oxide fuel
at some moments999Note that the third block partially used MOX-fuel enriched
with plutonium, i.e. $\sim$3000∘C.
Surprisingly enough, in scientific literature today there are absolutely no
either experimental or even theoretically calculated data on behavior of the
${}^{238}U$ and ${}^{239}Pu$ capture cross-sections depending on temperature
at least in 1000-3000∘C range. At the same time there are serious reasons to
suppose that the cross-section values of the specified elements increase with
temperature. We may at least point to qualitative estimates by Ukraintsev
[56], Obninsk Institute of Atomic Energetics (Russia), that confirm the
possibility of the cross-sections growth for ${}^{239}Pu$ in 300-1500∘C range.
Obviously, such anomalous temperature dependency of capture and fission cross-
sections of ${}^{238}U$ and ${}^{239}Pu$ may change the neutron and thermal
kinetics of a nuclear reactor drastically, including the perspective fast
uranium-plutonium new generation reactors (reactors of Feoktistov (1) and
Teller (2) type), which we classify as fast TWR reactors. Hence it is very
important to know the anomalous temperature behavior of ${}^{238}U$ and
${}^{239}Pu$ capture and fission cross-sections, as well as their influence on
the heat transfer kinetics, because it may turn into a reason of the positive
feedback101010Positive Feedback is a type of feedback when a change in the
output signal leads to such a change in the input signal, which leads to even
greater deviation of the output signal from its original value. In other
words, PF leads to the instability and appearance of qualitatively new (often
self-oscilation) systems. (PF) with the neutron kinetics leading to an
undesirable solution stability loss (the nuclear burning wave), and
consequently to a trivial reactor runaway with a subsequent nontrivial
catastrophe.
A special case of the PF is a non-linear PF, which leads to the system
evolution in the so-called blow-up mode [57, 58, 59, 60, 61, 62], or in other
words, in such a dynamic mode when one or several modeled values (e.g.
temperature and neutron flux) grows to infinity at a finite time. In reality,
instead of the infinite values, a phase transition is observed in this case,
which can become a first stage or a precursor of the future technogenic
disaster.
Investigation of the temperature dependency of ${}^{238}U$ and ${}^{239}Pu$
capture and fission cross-sections in 300-3000∘C range and the corresponding
kinetics of heat transfer and its influence on neutron kinetics in TWR is the
main goal of the chapter.
Heat transfer equation for uranium-plutonium fissile medium is:
$\displaystyle\rho\left(\vec{r},T,t\right)\cdot$ $\displaystyle
c\left(\vec{r},T,t\right)\cdot\dfrac{\partial
T\left(\vec{r},t\right)}{\partial t}=$
$\displaystyle=\aleph\left(\vec{r},T,t\right)\cdot\Delta
T\left(\vec{r},t\right)+\nabla\aleph\left(\vec{r},T,t\right)\cdot\nabla
T\left(r,t\right)+q_{T}^{f}\left(\vec{r},T,t\right),$ (76)
where the effective substance density is
$\rho\left(\vec{r},T,t\right)=\sum\limits_{i}N_{i}\left(\vec{r},T,t\right)\cdot\rho_{i},$
(77)
$\rho_{i}$ are tabulated values, $N_{i}\left(\vec{r},T,t\right)$ are the
components concentrations in the medium, while the effective specific heat
capacity (accounting for the medium components heat capacity values $c_{i}$)
and fissile material heat conductivity coefficient (accounting for the medium
components heat conductivity coefficients $\aleph_{i}(T)$) respectively are:
$c\left(\vec{r},T,t\right)=\sum\limits_{i}c_{i}(T)N_{i}\left(\vec{r},T,t\right),$
(78)
$\aleph\left(\vec{r},T,t\right)=\sum\limits_{i}\aleph_{i}(T)N_{i}\left(\vec{r},T,t\right).$
(79)
Here $q_{T}^{f}\left(\vec{r},T,t\right)$ is the heat source density generated
by the nuclear fissions $N_{i}$ of fissile metal components that vary in time.
Theoretical temperature dependency of heat capacity $c(T)$ for metals is
known: at low temperatures $c(t)\sim T^{3}$, and at high temperatures
$c(T)\rightarrow const$, and the constant value ($const\approx
6~{}Cal/(mol\cdot deg)$) is determined by Dulong-Petit law. At the same time
it is known that the thermal expansion coefficient is small for metals,
therefore the specific heat capacity at constant volume $c_{v}$ almost equals
to the specific heat capacity at constant pressure $c_{p}$. On the other hand,
the theoretical dependency of heat conductivity $\aleph_{i}(T)$ at high
temperature of ”fissile” metals is not known, while it is experimentally
determined that the heat conductivity coefficient $\aleph(T)$ of fissile
medium is a non-linear function of temperature (e.g. see [63], where the heat
conductivity coefficient is given for $\alpha$-uranium 238 and for metallic
plutonium 239, and also [64]).
While solving the heat conduction equations we used the following initial and
boundary conditions:
$T(r,t=0)=300~{}K~{}~{}~{}and~{}~{}j_{n}=\aleph\left[T(r\in\Re,t)-T_{0}\right],$
(80)
where $j_{n}$ is the normal (to the fissile medium boundary) heat flux density
component, $\aleph(T$) is the thermal conductivity coefficient, $\Re$ is the
fissile medium boundary, $T_{0}$ is the temperature of the medium adjacent to
the active zone.
Obviously, if the cross-sections of some fissile nuclides increase, then due
to the nuclei fission reaction exothermicity, the direct consequence of the
significantly non-linear kinetics of the parental and child nuclides in the
nuclear reactor is an autocatalyst increase of generated heat, similar to
autocatalyst processes of the exothermic chemical reactions. In this case the
heat flux density $q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ that characterizes
the generated heat amount will be:
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)=\Phi\left(\vec{r},T,t\right)\sum\limits_{i}Q_{i}^{f}\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)N_{i}\left(\vec{r},T,t\right),~{}~{}[W/cm^{3}],$
(81)
where
$\Phi\left(\vec{r},T,t\right)=\int\limits_{0}^{E^{max}_{n}}\Phi\left(\vec{r},E,T,t\right)dE$
is the full neutron flux density; $\Phi\left(\vec{r},E,T,t\right)$ is the
neutron flux density with energy $E$; $Q_{i}^{f}$ is the mean generated heat
emitted due to fission of one nucleus of the $i$-th nuclide;
$\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)=\int\limits_{0}^{E_{n}^{max}}\sigma_{f}^{i}(E,T)\rho\left(\vec{r},E,T,t\right)dE$
is the fission cross-section of the $i$-th nuclide averaged over the neutron
spectrum;
$\rho\left(\vec{r},E,T,t\right)=\Phi\left(\vec{r},E,T,t\right)/\Phi\left(\vec{r},T,t\right)$
is the neutron energy distribution probability density function;
$\sigma_{f}^{i}(E,T)$ is the microscopic fission cross-section of the i-th
nuclide that, as known, depends on the neutron energy and fissile medium
temperature (Doppler effect [36]); $N_{i}\left(\vec{r},T,t\right)$ is the
density of the $i$-th nuclide nuclei.
As follows from (81), in order to build the thermal source density function it
is necessary to derive the theoretical dependency of the cross-sections
$\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)$, averaged over the neutron
spectrum, on the reactor fuel temperature. As is known, the influence of the
nuclei thermal motion on the medium comes to a broadening and height reduction
of the resonances. By optical analogy, this phenomenon is referred to as
Doppler effect [36]. Since the resonance levels are observed only for heavy
nuclei in the low energy area, then Doppler effect is notable only during the
interaction of neutrons with such nuclei. And the higher environment
temperature the stronger is the effect.
Therefore a program was developed using Microsoft Fortran Power Station 4.0
(MFPS 4.0) that allows at the first stage to calculate the cross-sections of
the resonance neutron reactions depending on neutron energy taking into
account the Doppler effect. The cross-sections dependency on neutron energy
for reactor nuclides from ENDF/B-VII database [65], corresponding to 300K
environment temperature, were taken as the input data for the calculations.
For example, the results for radioactive neutron capture cross-sections
dependency on neutron energy for ${}^{235}U$ are given in fig. 11 for
different temperatures of the fissile medium in 300K-3000K temperature range.
Using this program, the dependency of scattering, fission and radioactive
neutron capture cross-sections for the major reactor fuel nuclides
${}_{92}^{235}U$, ${}_{92}^{238}U$, ${}_{92}^{239}U$ and ${}_{94}^{2}39Pu$ for
different temperatures in range 300K to 3000K were obtained.
Figure 11: Calculated dependency of radioactive neutron capture cross-section
on the energy for ${}^{235}_{92}U$ at different temperatures within 300K to
3000K.
At the second stage a program was developed to obtain the calculated
dependency of the cross-sections
$\overline{\sigma}_{f}^{i}\left(\vec{r},T,t\right)$ averaged over the neutron
spectrum for main reactor nuclides and for main neutron reactions for the
specified temperatures. The averaging of the neutron cross-sections for the
Maxwell distribution was performed using the following expression:
$\left\langle\sigma\left(E_{lim},T\right)\right\rangle=\dfrac{\int\limits_{0}^{E_{lim}}E^{1/2}e^{-E/kT}\sigma(E,T)dE}{\int\limits_{0}^{E_{lim}}E^{1/2}e^{-E/kT}dE},$
where $E_{lim}$ is the upper limit of the neutrons thermalization, while for
the procedure of neutron cross-sections averaging over the Fermi spectrum the
following expression was used:
$\left\langle\sigma\left(E_{lim},T\right)\right\rangle=\dfrac{\int\limits_{E_{lim}}^{\infty}\sigma(E,T)E^{-1}dE}{\int\limits_{E_{lim}}^{\infty}E^{-1}dE},$
During further calculations in our programs we used the results obtained at
the first stage i.e. the dependency of reaction cross-sections on neutron
energy and environment temperature (Doppler effect). The neutron spectrum was
specified in a combined way – by Maxwell spectrum $\Phi_{M}\left(E_{n}\right)$
below the limit of thermalization $E_{lim}$; by Fermi spectrum $\Phi_{F}(E)$
for a moderating medium with absorption above $E_{lim}$ but below $E_{F}$
(upper limit for Fermi neutron energy spectrum); by ${}^{239}Pu$ fission
spectrum [22, 23] above $E_{F}$, but below the maximal neutron energy
$E_{n}^{max}$. Here the neutron gas temperature for Maxwell distribution was
given by (82), described in [36]. According to this approach [36], the
drawbacks of the standard slowing-down theory for thermalization area may be
formally reduced if a variable $\xi(x)=\xi(1-2/z)$ is introduced instead of
the average logarithmic energy loss $\xi$, which is almost independent of the
neutron energy (as is known, the statement $\xi\approx 2/A$ is true for the
environment consisting of nuclei with $A>10$). Here $z=E_{n}/kT$, $E_{n}$ is
the neutron energy, $T$ is the environment temperature. Then the following
expression may be used for the neutron gas temperature in Maxwell spectrum of
thermal neutrons111111A very interesting expression revealing hidden
connection between the temperature of a neutron gas and the environment (fuel)
temperature.:
$T_{n}=T_{0}\left[1+\eta\cdot\dfrac{\Sigma_{a}(kT_{0})}{\langle\xi\rangle\Sigma_{S}}\right],$
(82)
where $T_{0}$ is the fuel environment temperature, $\Sigma_{a}(kT_{0})$ is an
absorption cross-section for energy $kT_{0}$, $\eta=1.8$ is the dimensionless
constant, $\langle\xi\rangle$ is averaged over the whole energy interval of
Maxwell spectrum $\xi(z)$ at $kT=1~{}eV$.
Fermi neutron spectrum for a moderating medium with absorption (we considered
carbon as a moderator and ${}^{238}U$, ${}^{239}U$ and ${}^{239}Pu$ as the
absorbers) was set in the form [36, 55]:
$\Phi_{Fermi}\left(E,E_{F}\right)=\dfrac{S}{\langle\xi\rangle\Sigma_{t}E}\exp{\left[-\int\limits_{E_{lim}}^{E_{f}}\dfrac{\Sigma_{a}\left(E^{\prime}\right)dE^{\prime}}{\langle\xi\rangle\Sigma_{t}\left(E^{\prime}\right)E^{\prime}}\right]},$
(83)
where $S$ is the total volume neutron generation rate,
$\langle\xi\rangle=\sum\limits_{i}\left(\xi_{i}\Sigma_{S}^{i}\right)/\Sigma_{S}$,
$\xi_{i}$ is the average logarithmic decrement of energy loss,
$\Sigma_{S}^{i}$ is the macroscopic scattering cross-section of the $i$-th
nuclide, $\Sigma_{t}=\sum\limits_{i}\Sigma_{S}^{i}+\Sigma_{a}^{i}$ is the
total macroscopic cross-section of the fissile material,
$\Sigma_{S}=\sum\limits_{i}\Sigma_{S}^{i}$ is the total macroscopic scattering
cross-section of the fissile material, $\Sigma_{a}$ is the macroscopic
absorption cross-section, $E_{F}$ is the upper neutron energy for Fermi
spectrum.
The upper limit of neutron thermalization $E_{lim}$ in our calculation was
considered a free parameter, setting the neutron fluxes of Maxwell and Fermi
spectra at a common energy limit $E_{lim}$ equal:
$\Phi_{Maxwell}\left(E_{lim}\right)=\Phi_{Fermi}\left(E_{lim}\right).$ (84)
The high energy neutron spectrum part ($E>E_{F}$) was defined by the fission
spectrum [55, 66, 67] in our calculations. Therefore the following expression
may be written for the total volume neutron generation rate $S$ in the Fermi
spectrum (83):
$S\left(\vec{r},T,t\right)=\int\limits_{E_{F}}^{E_{n}^{max}}\tilde{P}\left(\vec{r},E,T,t\right)\left[\sum\limits_{i}\nu_{i}(E)\cdot\Phi\left(\vec{r},E,T,t\right)\cdot\sigma_{f}^{i}(E,T)\cdot
N_{i}\left(\vec{r},T,t\right)\right]dE,$ (85)
where $E_{n}^{max}$ is the maximum energy of the neutron fission spectrum
(usually taken as $E_{n}^{max}\approx 10~{}MeV$), $E_{F}$ is the neutron
energy, below which the moderating neutrons spectrum is described as Fermi
spectrum (usually taken as $E_{F}\approx 0.2~{}MeV$);
$\tilde{P}\left(\vec{r},E,T,t\right)$ is the probability of neutron not
leaving the boundaries of the fissile medium which depends on the fissile
material geometry and the conditions at its border (e.g. presence of a
reflector).
The obtained calculation results show that the cross-sections averaged over
the spectrum may increase (fig. 12 for ${}^{239}Pu$ and fig. 14 for
${}^{238}U$) as well as decrease (fig. 13 for ${}^{235}U$) with fissile medium
temperature. As follows from the obtained results, the arbitrariness in
selection of the limit energy for joining the Maxwell and Fermi spectra does
not significantly alter the character of these dependencies evolution.
a)
b)
Figure 12: Temperature dependencies for the fission cross-section (a) and
radioactive capture cross-section (b) for ${}^{239}Pu$, averaged over the
Maxwell spectrum, on the Maxwell and Fermi spectra joining energy and
$\eta=1.8$ (see (82)).
a)
b)
Figure 13: Temperature dependencies for the fission cross-section (a) and
radioactive capture cross-section (b) for ${}^{235}U$, averaged over the
Maxwell spectrum, on the Maxwell and Fermi spectra joining energy and
$\eta=1.8$ (see (82)).
a)
b)
Figure 14: Temperature dependencies for the fission cross-section (a) and
radioactive capture cross-section (b) for ${}^{238}_{92}U$, averaged over the
combined Maxwell and Fermi spectra depending on the Maxwell and Fermi spectra
joining energy and $\eta=1.8$ (see (82)).
This can be justified by the fact that ${}^{239}Pu$ resonance area starts from
significantly lower energies than that of ${}^{235}U$, and with fuel
temperature increase the neutron gas temperature also increases producing the
Maxwell’s neutron distribution maximum shift to the higher neutron energies.
I.e. the neutron gas spectrum hardening, when more neutrons fit into resonance
area of ${}^{239}Pu$, is the cause of the averaged cross-sections growth.
This process in not as significant for ${}^{235}U$ because its resonance area
is located at higher energies. As a result, the ${}^{235}U$ neutron gas
spectrum hardening related to the fuel temperature increase (in the considered
interval) does not result in a significant increase of a number of neutrons
fitting into the resonance area. Therefore according to the known expressions
for ${}^{235}U$ determining the neutron reactions cross-sections behaviour
depending on their energy $E_{n}$ for non-resonance areas, we observe
dependency for the averaged cross-sections $\sigma_{nb}\sim 1/\sqrt{E_{n}}$.
The data on the averaged fission and capture cross-sections of ${}^{238}U$
presented at fig. 14 show that the averaged fission cross-section for
${}^{238}U$ is almost insensitive to the neutron spectrum hardening caused by
the fuel temperature increase – due to a high fission threshold $\sim$1 MeV
(see fig. 14a). At the same time they confirm the capture cross-section
dependence on temperature, since its resonance area is located as low as for
${}^{239}Pu$. Obviously, in this case the fuel enrichment with ${}^{235}U$
makes no difference because the averaged cross-sections for ${}^{235}U$, as
described above, behave in a standard way.
And finally we performed a computer estimate of the heat source density
dependence $q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ (81) on temperature for
the different compositions of the uranium-plutonium fissile medium with a
constant neutron flux density, presented at fig. 15. We used the dependencies
presented above at fig.12-14 for these calculations. Let us note that our
preliminary calculations were made not taking into account the change in the
composition and density of the fissile uranium-plutonium medium, which is a
direct consequence of the constant neutron flux assumption.
The necessity of such assumption is caused by the following. The reasonable
description of the heat source density
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ (81) temperature dependence requires
the solution of a system of three equations – two of them correspond to the
neutron kinetics equation (flux and fluence) and to the system of equations
for kinetics of the parental and child nuclides nuclear density (e.g. see [30,
27]), while the third one corresponds to a heat transfer equation of (76)
type. However, some serious difficulties arise here associated with the
computational capabilities available. And here is why.
One of the principal physical peculiarities of the TWR is the fact [21] that
fluctuation residuals of plutonium (or ${}^{233}U$ in Th-U cycle) over its
critical concentration burn out for the time comparable with reactor lifetime
of a neutron $\tau_{n}(x,t)$ (not considering the delayed neutrons), or at
least comparable with the reactor period121212The reactor period by definition
equals to $T(x,t)=\tau_{n}(x,t)/\rho(x,t)$, i.e. is a ratio of the reactor
neutron lifetime to reactivity. $T(x,t)$ (considering the delayed neutrons).
Meanwhile the new plutonium (or ${}^{233}U$ in Th-U cycle) is formed in a few
days (or a month) and not immediately. This means [21] that the numerical
calculation must be made with a temporal step about 10-6-10-7 in case of not
taking into account the delayed neutrons and $\sim$10-1-100 otherwise. At
first glance, taking into account the delayed neutrons, according to [21],
really “saves the day”, however it is not always true. If the heat transfer
equation contains a significantly non-linear source, then in the case of a
blow-up mode, the temperature may grow extremely fast under some conditions
and in 10-20 steps (with time step 10-6-10-7 s) reaches the critical amplitude
that may lead to (at least) a solution stability loss, or (as maximum) to a
blow-up bifurcation of the phase state, almost unnoticeable with a rough time
step.
According to these remarks, and considering the goal and format of this paper,
we did not aim at finding the exact solution of some specific system of three
joint equations described above. Instead, we found it important to illustrate
– at the qualitative level – the consequences of the possible blow-up modes in
case of a non-linear heat source presence in the heat transfer equation. As
said above, we made some estimate computer calculations of the heat source
density $q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ (81) temperature dependence
in 300-1400K range for some compositions of uranium-plutonium fissile medium
at a constant neutron flux (fig. 15).
Figure 15: Dependence of the heat source density
$q_{T}^{f}\left(r,\Phi,T,N_{i},t\right)~{}[eV]$ on the fissile medium
temperature (300-1400K) for several compositions of uranium-plutonium medium
(1 - 10% Pu; 2 - 5% Pu; 3 - 1% Pu) at the constant neutron flux density
$\Phi=10^{13}~{}n/(cm^{2}\cdot s)$.
The obtained dependencies for the heat source density
$q_{T}^{f}\left(\vec{r},\Phi,T,t\right)$ were successfully approximated by a
power function of temperature with an exponent of 4 (fig. 15). In other words,
we obtained a heat transfer equation with a significantly nonlinear heat
source in the following form:
$q_{T}(T)=const\cdot T^{(1+\delta)},$ (86)
where $\delta>1$ in case of non-linear thermal conductivity dependence on
temperature [57, 58, 59, 60, 61]. The latter means that the solutions of the
heat transfer equation (76) describe the so-called Kurdyumov blow-up modes
[57, 58, 59, 60, 61, 62], i.e. such dynamic modes when one of the modeled
values (e.g. temperature) turns into infinity for a finite time. As noted
before, in reality instead of reaching the infinite values, a phase transition
is observed (a final phase of the parabolic temperature growth), which
requires a separate model and is a basis for an entirely new problem.
Mathematical modeling of the blow-up modes was performed mainly using
Mathematica 5.2-6.0, Maple 10, Matlab 7.0, utilizing multiprocessor
calculations for effective application. A Runge–Kutta method of 8-9th order
and the numerical methods of lines [68] were applied for the calculations. The
numerical error estimate did not exceed 0.01%. The coordinate and temporal
steps were variable and chosen by the program in order to fit the given error
at every step of the calculation.
Below we give the solutions for the heat transfer equation (76) with nonlinear
exponential heat source (86) in uranium-plutonium fissile medium for boundary
and initial parameters corresponding to the industrial reactors. The
calculations were done for a cube of the fissile material with different
sizes, boundary and initial temperature values. Since the temperature
dependencies of the heat source density were obtained without account for the
changing composition and density of the uranium-plutonium fissile medium,
different blow-up modes can take place (HS-mode, S-mode, LS-mode) depending on
the ratio between the exponents of the heat conductivity and heat source
temperature dependences, according to [57, 58, 59, 60, 61, 62]. Therefore we
considered the cases for 1${}^{\text{st}}$, 2${}^{\text{nd}}$ and
4${}^{\text{th}}$ temperature order sources. Here the power of the source also
varied by varying the proportionality factor in (86)
($const=1.00J/(cm^{3}\cdot s\cdot K$) for the 1${}^{\text{st}}$ temperature
order source; $0.10J/(cm^{3}\cdot s\cdot K^{2})$, $0.15~{}J/(cm^{3}\cdot
s\cdot K^{2})$ and $1.00~{}J/(cm^{3}\cdot s\cdot K^{2})$ for the
2${}^{\text{nd}}$ temperature order source; $1.00~{}J/(cm^{3}\cdot s\cdot
K^{4})$ for the 4${}^{\text{th}}$ temperature order source).
During the calculations of the heat capacity $c_{p}$ (fig. 16a) and heat
conductivity $\aleph$ (fig. 16b) of a fissile medium dependence on temperature
in 300-1400K range the specified parameters were given by analytic
expressions, obtained by approximation of experimental data for${}^{238}U$
based on polynomial progression:
$c_{p}(T)\approx-7.206+0.64T-0.0047T^{2}+0.0000126T^{3}+2.004\cdot
10^{-8}T^{4}-1.60\cdot 10^{-10}T^{5}-2.15\cdot 10^{-13}T^{6},$ (87)
$\aleph(T)\approx 21.575+0.0152661T.$ (88)
Figure 16: Temperature dependence of the heat capacity $c_{P}$ and heat
conductivity $\chi$ of the fissile material. Points represent the experimental
values for the heat capacity and heat conductivity of ${}^{238}U$.
And finally the heat transfer equation (76) solution was obtained for the
constant heat conductivity ($27.5~{}W/(m\cdot K)$) and heat capacity
($11.5~{}J/(K\cdot mol)$), presented in fig. 17a, and also the solutions of
the heat transfer equation considering their temperature dependencies (fig.
17b-d).
Figure 17: Heat transfer equation (76) solution for 3D case (crystal sizes
0.001$\times$0.001$\times$0.001 mm; initial and boundary temperatures equal to
100K): a) The source is proportional to the 4${}^{\text{th}}$ order of
temperature; const = 1.00 $J/(cm^{3}\cdot s\cdot K^{4})$, heat capacity and
heat conductivity are constant and equal to 11.5 $J/(K\cdot mol)$ and 27.5
$W/(m\cdot K)$ respectively; b) the source is proportional to the
4${}^{\text{th}}$ order of temperature; const = 1.00 $J/(cm^{3}\cdot s\cdot
K^{4})$; c) The source is proportional to the 2${}^{\text{nd}}$ order of
temperature; const = 1.00 $J/(cm^{3}\cdot s\cdot K^{2})$; d) the source is
proportional to the 2${}^{\text{nd}}$ order of temperature; const = 0.10
$J/(cm^{3}\cdot s\cdot K^{2})$. Note: in the cases b) - d) the heat capacity
and heat conductivity were determined by (87) and (88) respectively.
These results point directly to a possibility of the local uranium-plutonium
fissile medium melting, with the melting temperature almost identical to that
of ${}^{238}U$, which is 1400K (fig. 16a-d). Moreover, these regions of the
local melting are not the areas of the so-called thermal peaks [69], and
probably are the anomalous areas of uranium surface melting observed by Laptev
and Ershler [70] that were also mentioned in [71]. More detailed analysis of
the probable temperature scenarios associated with the blow-up modes are
discussed below.
## 6 The blow-up modes in neutron-multiplying media and the pulse
thermonuclear TWR.
Earlier we noted the fact that due to a coolant loss at the nuclear reactors
during the Fukushima nuclear accident the fuel was melted, which means that
the temperature inside the active zone reached the melting temperature of
uranium-oxide fuel at some moment, i.e. $\sim$3000∘C.
On the other hand, we already know that the coolant loss may become a cause of
the nonlinear heat source formation inside the nuclear fuel, and therefore
become a cause of the temperature and neutron flux blow-up mode onset. A
natural question arises of whether it is possible to use such blow-up mode
(temperature and neutron flux) for the initiation of certain controlled
physical conditions under which the nuclear burning wave would regularly
“experience” the so-called “controlled blow-up” mode. It is quite difficult to
answer this question definitely, because such fast process has a number of
important physical vaguenesses, any of which can become experimentally
insurmountable for such process control.
Nevertheless such process is very elegant and beautiful from the physics point
of view, and therefore requires a more detailed phenomenological description.
Let us try to make it in short.
As we can see from the plots of the capture and fission cross-sections
evolution for ${}^{239}Pu$ (fig. 12), the blow-up mode may develop rapidly at
$\sim$1000-2000K (depending on the real value of the Fermi and Maxwell spectra
joining boundary), but at the temperatures over 2500-3000K the cross-sections
return almost to the initial values. If some effective heat sink is turned on
at that point, the fuel may return to its initial temperature. However, while
the blow-up mode develops, the fast neutrons already penetrate to the adjacent
fuel areas, where the new fissile material starts accumulating and so on (see
cycles (1) and (2)). After some time the similar blow-up mode starts
developing in this adjacent area and everything starts over again. In other
words, such hysteresis blow-up mode, closely time-conjugated to a heat takeoff
procedure, will appear on the background of a stationary nuclear burning wave
in a form of the periodic impulse bursts.
In order to demonstrate the marvelous power of such process, we investigated
the heat transfer equation with non-linear exponential heat source in uranium-
plutonium fissile medium with boundary and initial parameters emulating the
heat takeoff process. In other words, we investigated the blow-up modes in the
Feoktistov-type uranium-plutonium reactor (1), where the temperature inside
and at the boundary was deliberately fixed at 6000K, which corresponds to the
model of the georeactor131313Let us note that our model georeactor is not a
fast reactor. The possibility of the nuclear wave burning for a reactor
another than the fast one is examined in our next paper [72] [17]. Expression
(82) for the neutron gas temperature, used for the calculation of the cross-
sections averaged over the neutron spectrum, transforms in this case to the
following:
$T_{n}\approx\left[1+1.8\frac{8.0\cdot K_{2}}{<\xi>\cdot 4.5}\right]$ (89)
This equation is obtained for the supposed fissile medium composition of the
Uranium and Plutonium dicarbides [73, 74, 75, 17, 76], where the 238U was the
major absorber (its microscopic absorption cross-section for the
thermalization temperatures was set at $\sigma_{a}^{8}=8.0$ barn) and the 12C
was the major moderator (its microscopic scattering cross-section was set at
$\sigma_{s}^{12}=4.5$ barn). The 238U and 12C nuclei concentrations ratio was
set to the characteristic level for the dicarbides:
$K_{2}=\frac{N^{238}}{N^{12}}=0.5$
The Fermi spectrum for the neutrons in moderating and absorbing medium of the
georeactor (carbon played a role of the moderator, and the 238U), 239U and
239Pu played the role of the absorbers) was taken in the same form (83).
As en example Fig. 18 shows the calculated temperature dependences of the 235U
and 239Pu fission cross-sections averaged over the neutron spectrum.
Figure 18: The temperature dependences of the 239Pu fission cross-section
averaged over the neutron spectrum for the limit energy for the Fermi and
Maxwell spectra joining equal to 3kT. The analogous dependency for the 235U is
also shown.
The temperature choice is conditioned by the following important
consideration: “Is it possible to obtain a solution (i.e. a spatio-temporal
temperature distribution) in a form of the stationary solitary wave with a
limited amplitude instead of a $\delta$-function at some local spatial area,
under such conditions (6000K) emulating the time-conjugated heat takeoff (see
fig. 12)?” As shown below, such approach really works.
Below we present some calculation characteristics and parameters. During these
calculations we used the following expression for dependence of the heat
conductivity coefficient:
$\aleph=0.18\cdot 10^{-4}\cdot T,$
which was obtained using the Wiedemann–Franz law and the data on electric
conductivity of metals at temperature 6000K [77]. Specific heat capacity at
constant pressure was determined by value $c_{p}\approx 6~{}cal/(mol\cdot
deg)$ according to Dulong and Petit law.
The fissile uranium-plutonium medium was modeled as a cube with dimensions
10.0$\times$10.0$\times$10.0 m (fig.19). Here for heat source we used the 2nd
order temperature dependence (see (86)).
And finally fig. 19a-d present a set of solutions of heat transfer equation
(76) with nonlinear exponential heat source (86) in uranium-plutonium fissile
medium with boundary and initial conditions emulating such process of heat
takeoff that initial and boundary temperatures remain constant and equal to
6000K.
Figure 19: Heat transfer equation solution for a model georeactor (source
$\sim$ 2nd order temperature dependence, $const$ = 4.19 $J/(cm^{3}\cdot s\cdot
K^{2})$; initial and boundary temperatures equal to 6000 K; fissile medium is
a cube 10$\times$10$\times$10 m. The presented results correspond to the
following times of temperature field evolution: (a) (1-10)$\cdot 10^{-7}$ s,
(b) $10^{-6}$ s, (c) 0.5 s, (d) 50 s.
It is important to note here, that the solution set presented at fig. 19,
demonstrates the solution tendency towards its “stationary” state quite
clearly. This is achieved using the so-called “magnifying glass” approach,
when the solutions of the same problem are deliberately investigated at
different timescales. E.g. fig. 19a shows the solution at the time scale
$t\in[0,10^{-6}~{}s]$, while fig. 19b describes the spatial solution of the
problem (temperature field) for $t=10^{-6}~{}s$. The fig. 19c-d presents the
solution (spatial temperature distribution) at $t=0.5~{}s$ and $t=50~{}s$.
As one can see, the solution (fig. 19d) is completely identical to the
previous (fig.19c), i.e. to the distribution established in the medium in 0.5
seconds, which allowed us to make a conclusion on the temperature field
stability, starting from some moment. It is interesting that the established
temperature field creates the conditions suitable for the thermonuclear
synthesis reaction, i.e. reaching 108K, and such temperature field lifetime is
not less than 50 s. These conditions are highly favorable for a stable
thermonuclear burning, according to a known Lawson criterion, provided the
necessary nuclei concentration entering the thermonuclear synthesis reaction.
One should keep in mind though, that the results of this chapter are for the
purpose of demonstration only, since their accuracy is rather uncertain and
requires a careful investigation with application of the necessary
computational resources. Nevertheless, the qualitative peculiarities of these
solutions should attract the researchers’ attention to the nontrivial
properties of the blow-up modes – at least, with respect to the obvious
problem of the inherent TWR safety violation.
## 7 Conclusions
Let us give some short conclusions stimulated by the following significant
problems.
1. 1.
TWR and the problem of dpa-parameter in cladding materials. A possibility to
surmount the so-called problem of dpa-parameter based on the conditions of
nuclear burning wave existence in U-Pu and Th-U cycles is shown. In other
words it is possible to find a nuclear burning wave mode, whose parameters
(fluence/neutron flux, width and speed of the wave) satisfy the dpa-condition
(73) of the reactor materials radiation resistance, particularly, that of the
cladding materials. It can be done using the joined application of the
“differential”[30] and “integral”[21, 22, 27] conditions for nuclear burning
wave existence. The latter means that at the present time the problem of dpa-
parameter in cladding materials in the TWR-project is not an insurmountable
technical problem and can be satisfactorily solved.
Here we may add that this algorithm of an optimal nuclear burning wave mode
selection predetermines a satisfactory solution of another technical problems
mentioned in introduction. For example, the fuel rod length in the proposed
TWR variant (see the “ideal” case in Table1) is predetermined by the nuclear
burning wave speed, which in a given case equals to 0.254 cm/day$\equiv$ 85
cm/year, i.e. 20 years of TWR operation requires the fuel rod length $\sim$ 17
m. On the other hand, it is known [78] that for a twisted fuel rod form with
two- or four-bladed symmetry, the tension emerging from the fuel rod surface
cooling is 30% lower than that of a round rod with the same diameter, other
conditions being equal. The same reduction effect applies to the hydraulic
resistance in comparison to a round rod of the same diameter.
Another problem associated with the reactor materials swelling is also solved
rather simply. It is pertinent to note that if a ferritic-martensitic material
is chosen as a cladding material (fig. 8 [49]), then the swelling effect at
the end of operation will be only $\sim$0.5% [49]. We could discuss other
drawbacks mentioned in the introduction as well, but in our opinion, the rest
of the problems are not the super-obstacles for the contemporary level of
nuclear engineering, as compared to the main problem of dpa-condition, and can
be solved in a traditional way.
2. 2.
The consequences of the anomalous ${}^{238}U$ and ${}^{239}Pu$ cross-sections
behavior with temperature. It is shown that the capture and fission cross-
sections of ${}^{238}U$ and ${}^{239}Pu$ manifest a monotonous growth in
1000-3000K range. Obviously, such anomalous temperature dependence of
${}^{238}U$ and ${}^{239}Pu$ cross-sections changes the neutron and heat
kinetics of the nuclear reactors drastically. It becomes essential to know
their influence on kinetics of heat transfer because it may become the cause
of a positive feedback with neutron kinetics, which may lead not only to
undesirable loss of the nuclear burning wave stability, but also to a reactor
runaway with a subsequent disaster.
3. 3.
Blow-up modes and the problem of the nuclear burning wave stability. One of
the causes of possible fuel temperature growth is a deliberate or spontaneous
coolant loss similar to Fukushima nuclear accident. As shown above, the
coolant loss may become a cause of the nonlinear heat source formation in the
nuclear fuel and the corresponding mode with temperature and neutron flux
blow-up. In our opinion, the preliminary results of heat transfer equation
with nonlinear heat source investigations point to an extremely important
phenomenon of the anomalous behaviour of the heat and neutron flux blow-up
modes. This result poses a natural nontrivial problem of the fundamental
nuclear burning wave stability, and correspondingly, of a physically
reasonable application of the Lyapunov method to this problem.
It is shown that some variants of the solution stability loss are caused by
anomalous nuclear fuel temperature evolution. They can lead not only to the
TWR inherent safety loss, but – through a bifurcation of states (and this is
very important!) – to a new stable mode when the nuclear burning wave
periodically “experiences” the so-called “controlled blow-up” mode. At the
same time, it is noted that such fast (blow-up regime) process has a number of
physical uncertainties, which may happen to be experimentally insurmountable
for the purposes of such process control.
4. 4.
On-line remote neutrino diagnostics of the intra-reactor processes. The high-
power TWR or a nuclear fuel transmutation reactor are the projects with the
single-load, fuel burn-up and the subsequent burial of the reactor apparatus.
Thus an obvious necessity for the system of remote neutrino monitoring of the
nuclear burning wave in the normal operation mode and the neutron kinetics in
emergency situation. The details and peculiarities of the isotope composition
spatio-temporal distribution calculation in the active zone of the TWR are
presented in [79, 17, 76] in detail within the inverse problem of the intra-
reactor processes neutrino diagnostics.
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|
arxiv-papers
| 2012-07-16T14:50:32 |
2024-09-04T02:49:33.127770
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. D. Rusov, V. A. Tarasov, I. V. Sharf, V. M. Vaschenko, E. P.\n Linnik, T. N. Zelentsova, M. E. Beglaryan, S. A. Chernegenko, S. I. Kosenko,\n V. P. Smolyar",
"submitter": "Vladimir Smolyar",
"url": "https://arxiv.org/abs/1207.3695"
}
|
1207.3726
|
# Energy Level Alignment in Organic-Organic Heterojunctions: The TTF-TCNQ
Interface
Juan I. Beltrán juan.beltran@uam.es [ Fernando Flores
fernando.flores@uam.es [ José I. Martínez [ José Ortega [
###### Abstract
The energy level alignment of the two organic materials forming the TTF-TCNQ
interface is analyzed by means of a local orbital DFT calculation, including
an appropriate correction for the transport energy gaps associated with both
materials. These energy gaps are determined by a combination of some
experimental data and the results of our calculations for the difference
between the TTFHOMO and the TCNQLUMO levels. We find that the interface is
metallic, as predicted by recent experiments, due to the overlap (and charge
transfer) between the Density of States corresponding to these two levels,
indicating that the main mechanism controlling the TTF-TCNQ energy level
alignment is the charge transfer between the two materials. We find an induced
interface dipole of 0.7 eV in good agreement with the experimental evidence.
We have also analyzed the electronic properties of the TTF-TCNQ interface as a
function of an external bias voltage $\Delta$ between the TCNQ and TTF
crystals, finding a transition between metallic and insulator behavior for
$\Delta\sim 0.5$ eV.
Universidad Autónoma de Madrid]Departamento de Física Teórica de la Materia
Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain Universidad Autónoma de
Madrid]Departamento de Física Teórica de la Materia Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain Universidad Autónoma de
Madrid]Departamento de Física Teórica de la Materia Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain Universidad Autónoma de
Madrid]Departamento de Física Teórica de la Materia Condensada,
Universidad Autónoma de Madrid, E-28049 Madrid, Spain
## 1 Introduction
Figure 1: (color online) TTF-TCNQ bulk structure with its view-plane along the
b and a lattice vectors, which corresponds to $\hat{y}$ and $\hat{x}$ axis,
for the left and right pictures, respectively.
Organic electronics is an area of important research effort where organic
materials provide an immense variety of properties and atomic structures. In
these materials, interfaces play a key role in fundamental phenomena (e.g.
charge separation and charge recombination) for electronic applications such
as solar cells and light emitting diodes, to name a few 1, 2, 3.
TTF/TCNQ (tetrathiofulvalene-7,7,8,8,-tetracyanoquinodimethane) is a very
interesting organic bulk material that was already studied in the early 70’s
showing novel phenomena, like large e- conductivity and superconductivity 4.
TTF/TCNQ shows three phase transitions which have been related to condensed
charge-density waves, due to Peierls distortions, with transition temperatures
of 38, 49 and 54 K 4. Also, the experimental band widths of TTF and TCNQ
crystal-bulks are 0.95 and 2.5 eV respectively, while density functional
theory (DFT) calculations based on local and semi-local exchange-correlation
(XC) functional provide systematically, for TTF and TCNQ, band widths of
around 0.65 and 0.70 eV respectively 5, 6, 7, 8. These intriguing properties
have prompted a lot of experimental and theoretical studies on the properties
of bulk TTF/TCNQ, see e.g. 5, 6, 7, 8, 9, 10; thin films of mixed TTF-TCNQ
have also been studied on different substrates like graphene and Au(111) 11,
12.
However, the large amount of work on bulk and thin film properties of TTF/TCNQ
contrasts with the much lesser attention dedicated to the interface of the two
organic materials. One of the few studies 13 has found that the interface
retains its metallic character due to the charge transfer between the two
molecules. The existence of an interface dipole layer ($IDL$) 14, 15,
generated by charge transfer or/and by polarization effects, and its
corresponding vacuum level shift ($VLS$) 1, shows that the Schottky-Mott limit
is not satisfied at this interface. Careful spectroscopic characterization of
TTF-TCNQ interfaces over different substrates shows a $VLS$ of 0.6 eV and a
$\Delta_{HOMO}$ = HOMO${}_{TTF}-$HOMOTCNQ of either 2.1 or 2.4 eV 16, 17.
In this communication we consider the interface between TTF and TCNQ crystals,
and analyze the interface energy level alignment and barrier height formation,
using a local orbital DFT-approach in which an appropriate correction is
included to properly describe the transport gaps of both materials. In section
II we present the interface geometries, and in section III we discuss how we
determine the transport energy gaps for TTF and TCNQ and give some details
about our DFT-calculations. In section IV, we present our results. In
particular, our analysis shows that the TTF-TCNQ interface is metallic and
that the energy difference between the TCNQLUMO and TTFHOMO is pinned to
around 0.8 eV. We have also analyzed how the interface properties depend on
the application of an external voltage between the TCNQ and TTF crystals,
finding a transition from a metallic to an insulating interface for a voltage
of $\sim$ 0.5 eV. Finally, the interfacial geometry has also been analyzed by
means of a DFT calculation combined with a semi-empirical parametrization of
the van der Waals interaction.
Figure 2: (color online) Similar to 1 but for the bilayer interface.
## 2 Interface Geometry
At room temperature, TTF/TCNQ bulk presents a monoclinic structure with
lattice parameters a = 12.298 Å, b = 3.819 Å and c = 18.468 Å, and an angle
$\hat{\beta}$ = 104.46° 18 (see 1). However, detailed experimental information
of both the geometry and the energetics at the interface is still missing due
to inherent difficulties in its structural characterization. For the TTF-TCNQ
surface, though, it has been measured by scanning tunneling microscopy 19 and
angle-dependent near-edge X-ray adsorption fine structure 20 that the cleaved
surfaces are highly ordered and retain the periodicity of the bulk. Thus, in
this work we have considered the interfacial structure arising from selecting
the interface geometry between TTF and TCNQ layers from the TTF/TCNQ bulk
structure along the (100) direction (see 2). In the DFT calculations we have
used bulk periodic boundary conditions along b and c vectors, while for the a
direction we have considered two cases: a monolayer interface, with only one
layer of each material, and a bilayer interface, with 2 TTF and TCNQ layers.
This amounts to 2 TTF + 2 TCNQ or 4 TTF + 4 TCNQ molecules in the unit cell,
respectively. In the bilayer interface, the second layer is generated from the
interfacial plane by adding parallel molecules which are shifted in a similar
way as in their independent crystals for both the TTF and TCNQ bulk structures
– see Cambridge Structural Database – and satisfying the periodic boundary
conditions of keeping b and c vectors from the bulk structure (see 2).
Figure 3: (color online) Energy level diagram for the isolated TTF and TCNQ
molecules obtained from $\Delta$-SCF calculations (explained in the main
text).
## 3 TTF-TCNQ Interface Energy Level Alignment
### 3.1 $\Delta$-SCF calculations
As a first step in order to determine the energy level alignment at the TTF-
TCNQ interface, we have performed $\Delta$-SCF calculations for the isolated
TTF and TCNQ molecules. In these calculations, the HOMO and LUMO levels are
determined from total energy DFT calculations 21, 22 for molecules with $N$,
$N+1$ and $N-1$ electrons ($N$ corresponding to the neutral molecule):
$\displaystyle\varepsilon_{HOMO}$ $\displaystyle=$ $\displaystyle
E[N]-E[N-1],$ $\displaystyle\varepsilon_{LUMO}$ $\displaystyle=$
$\displaystyle E[N+1]-E[N],$ (1)
where $E[N_{\alpha}]$ is the total energy for a molecule with $N_{\alpha}$
electrons. The results of these calculations are summarized in 3. We obtain a
transport gap (difference between $\varepsilon_{LUMO}$ and
$\varepsilon_{HOMO}$) of 5.3 eV and 6.3 eV for TCNQ and TTF, respectively,
while the mid-gap positions are found to be located at 5.5 eV (TCNQ) or 2.5 eV
(TTF) below the Vacuum Level ($VL$).
Figure 4: (color online) Left panel: energy level diagram for the TCNQ-TTF
interface obtained from a combination of experimental and theoretical
information. Right panel: energy levels in the DFT calculation for the
periodic interface shown in 2. In this calculation the initial transport gaps
and mid-gap positions have been adjusted to reproduce those given in the left
panel of this Figure (see main text).
### 3.2 Energy levels at the interface
The energy level values for the isolated molecules have to be corrected due to
the interaction with the other molecules in the crystal and at the interface.
The effect of this ‘environment’ is three-fold: (a) broadening of the energy
levels into bands; (b) relative shift of the TTF and TCNQ molecular levels due
to electrostatic and Pauli exclusion effects; (c) Polarization or screening
effects, shifting in different directions occupied and empty states. Point (b)
gives rise to a Vacuum Level Shift ($VLS$), or interface potential, between
both crystals, while the effect of point (c) is a reduction of the transport
gaps at the interface (or in the crystals) as compared with the values for
isolated TTF or TCNQ molecules, $E_{g}^{T}=(E_{g}^{T})^{0}-\delta U$. For
example, for a molecule interacting with a metal, image potential effects
reduce the transport gap as follows: $E_{g}^{T}=(E_{g}^{T})^{0}-eV_{IM}$,
where $V_{IM}$ is the image potential 23. In the case of an organic crystal
(or interface) a similar effect is present, due to the screening/polarization
of the other molecules.
In order to obtain the energy level alignment for the TTF-TCNQ interface, we
have used a combination of experimental and theoretical information:
* (1)
The $VLS$ at the interface is 0.6 eV, as determined experimentally 17. In
these experiments it is also found that
$\Delta\varepsilon_{HOMO}=\varepsilon_{HOMO}(TTF)-\varepsilon_{HOMO}(TCNQ)=$
2.1 eV 17.
* (2)
The mid-gap energies obtained in the $\Delta$-SCF calculations mentioned above
are 5.5 eV (TCNQ) and 2.5 eV (TTF) below their corresponding Vacuum Level
($VL$) positions.
* (3)
As discussed below, we find that
$\varepsilon_{LUMO}(TCNQ)-\varepsilon_{HOMO}(TTF)\approx$ 0.8 eV, a result
that is related to the experimental observation that the interface is metallic
14, see below.
Using this information, we can deduce that at the interface the transport gap
is 2.9 eV for TCNQ, and 3.5 eV for TTF, i.e. screening/polarization effects
reduce the transport gaps by 2.4 eV (TCNQ) and 2.8 eV (TTF). 4 (left panel)
summarizes the resulting energy level alignment for the TCNQ-TTF interface.
Figure 5: (color online) A) VIDIS vs. $\Delta$ (initial shift of the TCNQ
levels w.r.t. the TTF levels, see text) for the monolayer and bilayer
interfaces; $\Delta$ = 0 corresponds to the realistic interface configuration.
The experimental $VLS$ (or VIDIS) value, 0.6 eV, is included as a dashed
horizontal line; the screening parameter for each of the two regimes is also
shown; B) Transfer of charge vs. $\Delta$ for the monolayer interface; and C)
Value of [$\varepsilon_{HOMO}$(TTF)$-\varepsilon_{LUMO}$(TCNQ)] vs. $\Delta$
for the monolayer interface.
### 3.3 DFT calculations
We have performed DFT calculations for the monolayer and bilayer periodic
interface structures presented in section II (see 2). In these calculations we
have used a local-orbital DFT code 24, and the initial transport gaps for TCNQ
and TTF have been adjusted to the values given in 4 (left panel) using a
scissor operator; besides, the initial TTF and TCNQ molecular levels are
shifted in such a way that their initial $VL$ positions have the same value,
and the initial mid-gap positions correspond to the values given in 4 (left
panel) (see e.g. Refs. 25, 26, 27 and references therein for details). Other
technical details are the use of the local density approximation (LDA) and an
optimized basis set of $s$ (H) and $sp^{3}d^{5}$ (C, N and S) Numerical
Atomic-like Orbitals 28, with the following cut-off radii (in a.u.): $s$=4.1
for H, and $s$=4.0, 3.6, 4.2, $p$=4.5, 4.1, 4.7, and $d$=5.4, 5.2, 5.5 for C,
N and S, respectively.
Figure 6: (color online) Density of states projected on either the TCNQ or TTF
of the monolayer interface for $\Delta$ = 0.
## 4 Results and Discussion
The right panel of 4 shows the TCNQ-TTF interface energy level diagram
obtained in our DFT calculations (explained in third subsection of previous
section) with the initial conditions as explained above. This self-consistent
result is quite similar to the energy level diagram obtained in second
subsection of previous section, shown in the left panel of 4. The band-gaps
and interface dipole are quite similar in both panels of 4, however the mid-
gap positions move slightly upwards by around 0.3-0.4 eV. This shift is due to
the Pauli repulsion (related to the Pauli exclusion principle).
In order to analyze the mechanism responsible for the energy level alignment
at the TTF-TCNQ interface, we have performed DFT calculations as described in
third subsection of previous section, in which we shift by $\Delta$ the
initial (before self-consistency) position of the molecular levels of TCNQ
with respect to the molecular levels of TTF; $\Delta$ = 0 corresponds to equal
initial vacuum level positions, and $\Delta>$ 0 corresponds to a
destabilization of the TCNQ levels, w.r.t. the TTF levels. This shift
simulates the application of a bias potential of value $\Delta$ between the
two crystals. 4 (right panel) represents the final levels (after self-
consistency) for the $\Delta$ = 0 case. A change $\Delta$ in the initial
relative position of the TCNQ and TTF energy levels will influence the induced
dipole and the rest of the electronic properties at the interface. The final
$VLS$ will depend on the initial misalignment setup $\Delta$ (e.g. the
external bias between the two crystals) and on the induced potential after
electronic self-consistency, $V_{IDIS}$, in such a way:
$VLS=\Delta+V_{IDIS}.$ (2)
In 5A we depict $V_{IDIS}$ as a function of $\Delta$ for the mono and bilayer-
interfaces. In both graphs two different regimes are clearly observed, split
by a crossing point located at $\Delta$ = 0.2 (0.45) eV for the monolayer
(bilayer) case. For both regimes, we see that:
$\delta V_{IDIS}=-(1-S)\delta\Delta,$ (3)
$S$ being a screening parameter 23, taking the value 0.03 for $\Delta<$ 0.2
(0.45) eV and 0.93 for $\Delta>$ 0.2 (0.45) eV. The case of $\Delta<$
0.2(0.45) eV corresponds to a metallic phase in which the
$\varepsilon_{LUMO}(TCNQ)$ and $\varepsilon_{HOMO}(TTF)$ Densities of States
(DOS) are overlapping, as shown in 6 and 7; in this case, the difference
between $\varepsilon_{LUMO}(TCNQ)$ and $\varepsilon_{HOMO}(TTF)$ is almost
constant ($\approx$ 0.8 eV), due to the transfer of charge between these two
levels. For the other phase, $\Delta>$ 0.2(0.45) eV, we find a typical
heterojunction with small screening and a small induced potential, $V_{IDIS}$.
Figure 7: (color online) DOS of the monolayer and bilayer interfaces projected
on each material for $\Delta$ = 0. The PDOS intensity of the monolayer
interface has been scaled according with the number of atoms included in the
bilayer case.
This is illustrated in 5B-C, where we depict, for the monolayer interface, the
charge transfer (positive if electron charge flows from TTF to TCNQ) and the
value of $[\varepsilon_{HOMO}(TTF)-\varepsilon_{LUMO}(TCNQ)]$ vs. $\Delta$.
For both plots the crossing point between the two different regimes is located
at the same position than for $V_{IDIS}$, close to $\Delta=$ 0.2 eV, dividing
the charge transfer plot into a regime with a large increase of charge
transfer as $\Delta$ is reduced, for $\Delta<$ 0.2 eV, and another in which
the transfer of charge is small and changes rather slowly as $\Delta$ is
increased, for $\Delta>$ 0.2 eV. In 5C we see that for $\Delta>$ 0.2 eV the
value of $[\varepsilon_{HOMO}(TTF)-\varepsilon_{LUMO}(TCNQ)]$ vs. $\Delta$
presents a slope close to -1, while for $\Delta<$ 0.2 eV
$[\varepsilon_{HOMO}(TTF)-\varepsilon_{LUMO}(TCNQ)]$ is approximately
constant, with a value of -(0.7-0.8) eV. This value
($\varepsilon_{HOMO}(TTF)-\varepsilon_{LUMO}(TCNQ)\approx$ 0.8 eV) has been
used in second subsection of previous section (see left panel of 4) to deduce
the TCNQ and TTF energy level positions at the interface. It is worth
mentioning that zero charge transfer corresponds to the case of $\Delta=$ 3.12
eV; for this alignment $V_{IDIS}$ is not, however, zero, but takes the values
0.10 and 0.11 eV for the mono and the bilayer cases, respectively. These
induced potentials are created by the polarization effects of the molecule.
In 5 we observe that from the monolayer to the bilayer case the crossing point
moves around 0.25 eV to positive values of $\Delta$. This is explained in 7,
that shows, for the monolayer and bilayer interfaces, the DOS projected on
each material for a value of $\Delta$ close to the crossing point. We notice
that there is a broadening for the bilayer case of the $LUMO(TCNQ)$ DOS, which
shifts by 0.25 eV the position where this DOS starts to overlap with the DOS
corresponding to the $HOMO(TTF)$, as compared with the monolayer case.
Figure 8: (color online) Energy of the monolayer-interface as a function of
the distance between the two planes.
Finally, as a check of the interface geometry used (see 2), we have calculated
the total energy as a function of the distance between the TTF and TCNQ layers
at the interface. 8 shows the total energy curves for the LDA calculation and
the van der Waals (vdW) interactions energy; this energy is included in a
semiempirical fashion as an atractive $-f_{D}(R)C_{6}/R^{6}$ atom-atom term.
$f_{D}(R)$ is the Grimme damping expresion 29 and $C_{6}$=0.7$\times
C_{6}^{\prime}$, where $C_{6}^{\prime}$ is the interatomic parameter
calculated using the London dispersion relation as a function of the atomic
polarizability and the first ionization potential 30; 0.7 is a constant
correction calculated from a detailed analysis of the C–C interaction 31,
which is applied to every $C_{6}^{\prime}$ parameter. In order to prevent for
the over-counting that would appear including both, the exchange-correlation
energy provided by a conventional LDA and the correlation energy associated
with the long-range vdW interaction, we have also performed a “corrected LDA”
calculation in which the exchange-correlation energy for the complete system
is calculated as the sum of the exchange-correlation energies for each
subsystem, taken each one independently, neglecting in this way the effect of
the overlapping densities in the exchange-correlation energy 32. In 8 this
interaction energy is labeled weak chemical interaction (WCI). The minimum of
the WCI+vdW total energy curve is around 2.5 Å, which is close to the
experimental value for the distance between TTF and TCNQ layers in bulk
TTF/TCNQ; notice, however, the very flat energy profile (within 0.02 eV) from
2.2 to 2.7 Å. This result suggests that starting from the bulk structure to
generate the interface geometry is a good approximation.
## 5 Conclusions
In conclusion, we have presented a DFT analysis of the energy level alignment
in the TTF/TCNQ interface, introducing in a self-consistent fashion
appropriate transport gaps for both materials as determined from a combination
of experimental and theoretical information. In these calculations we have
also analyzed the metal-insulator phase-transition occurring in this interface
when applying a bias voltage between the two materials. We find that at zero
bias the interface is metallic while for bias larger than $\sim$ 0.5 eV
(shifting TTF towards higher binding energies w.r.t. TCNQ) the system becomes
an insulator. Our results have been favorably compared with the experimental
data, indicating that the main mechanism controlling energy level alignment in
the TTF/TCNQ interface is the charge transfer between both organic
semiconductors.
## 6 Acknowledgements
This work is supported by Spanish MICIIN under contract FIS2010-16046, the CAM
under contract S2009/MAT-1467 and the European Project MINOTOR (Grant
FP7-NMP-228424). JIB gratefully acknowledges financial support by the European
Project MINOTOR. JIM acknowledges funding from Spanish MICIIN through Juan de
la Cierva Program. Special thanks to Dr. Daniel González for the development
of the Xeo software package, which has been extensively used for the data
analysis.
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|
arxiv-papers
| 2012-07-16T17:01:24 |
2024-09-04T02:49:33.144417
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Juan I. Beltr\\'an, Fernando Flores, Jos\\'e I. Mart\\'inez, and Jos\\'e\n Ortega",
"submitter": "Jose Ignacio Martinez",
"url": "https://arxiv.org/abs/1207.3726"
}
|
1207.3815
|
# The Geography of Recent Genetic Ancestry across Europe.
Peter Ralph1 and Graham Coop2
1 Department of Evolution and Ecology & Center for Population Biology,
University of California, Davis.
2 Department of Molecular and Computational Biology,
University of Southern California.
To whom correspondence should be addressed:
plralph@ucdavis.edu,gmcoop@ucdavis.edu
## Abstract
The recent genealogical history of human populations is a complex mosaic
formed by individual migration, large-scale population movements, and other
demographic events. Population genomics datasets can provide a window into
this recent history, as rare traces of recent shared genetic ancestry are
detectable due to long segments of shared genomic material. We make use of
genomic data for 2257 Europeans (in the POPRES dataset) to conduct one of the
first surveys of recent genealogical ancestry over the past three thousand
years at a continental scale.
We detected 1.9 million shared long genomic segments, and used the lengths of
these to infer the distribution of shared ancestors across time and geography.
We find that a pair of modern Europeans living in neighboring populations
share around 2–12 genetic common ancestors from the last 1500 years, and
upwards of 100 genetic ancestors from the previous 1000 years. These numbers
drop off exponentially with geographic distance, but since these genetic
ancestors are a tiny fraction of common genealogical ancestors, individuals
from opposite ends of Europe are still expected to share millions of common
genealogical ancestors over the last 1000 years. There is also substantial
regional variation in the number of shared genetic ancestors. For example,
there are especially high numbers of common ancestors shared between many
eastern populations which date roughly to the migration period (which includes
the Slavic and Hunnic expansions into that region). Some of the lowest levels
of common ancestry are seen in the Italian and Iberian peninsulas, which may
indicate different effects of historical population expansions in these areas
and/or more stably structured populations.
Population genomic datasets have considerable power to uncover recent
demographic history, and will allow a much fuller picture of the close
genealogical kinship of individuals across the world.
## Author Summary
Few of us know our family histories more than a few generations back.
Therefore, it is easy to overlook the fact that we are all distant cousins,
related to one another via a vast network of relationships. Here we use
genome-wide data from European individuals to investigate these relationships
over the past three thousand years, by looking for long stretches of shared
genome between pairs of individuals inherited from common genetic ancestors.
We quantify this ubiquitous recent common ancestry, showing that for instance
even pairs of individuals on opposite ends of Europe share hundreds of genetic
common ancestors over this time period. Despite this degree of commonality,
there are also striking regional differences. For instance, southeastern
Europeans share large numbers of common ancestors which date to the era of the
Slavic and Hunnic expansions around 1500 years ago, while most common
ancestors that Italians share with other populations lived longer ago than
2500 years. The study of long stretches of shared genetic material holds the
promise of rich information about many aspects of recent population history.
## 1 Introduction
Even seemingly unrelated humans are distant cousins to each other, as all
members of a species are related to each other through a vastly ramified
family tree (their pedigree). We can see traces of these relationships in
genetic data when individuals inherit shared genetic material from a common
ancestor. Traditionally, population genetics has studied the distant bulk of
these genetic relationships, which in humans typically date from hundreds of
thousands of years ago (e.g. Cann et al., 1987; Takahata, 1993). Such studies
have provided deep insights into the origins of modern humans (e.g. Li and
Durbin, 2011), and into recent admixture between diverged populations (e.g.
Moorjani et al., 2011; Henn et al., 2012a).
Although most such genetic relationships among individuals are very old, some
individuals are related on far shorter time scales. Indeed, given that each
individual has $2^{n}$ ancestors from $n$ generations ago, theoretical
considerations suggest that all humans are related genealogically to each
other over surprisingly short time scales (Chang, 1999; Rohde et al., 2004).
We are usually unaware of these close genealogical ties, as few of us have
knowledge of family histories more than a few generations back, and these
ancestors often do not contribute any genetic material to us (Donnelly, 1983).
However, in large samples we can hope to identify genetic evidence of more
recent relatedness, and so obtain insight into the population history of the
past tens of generations. Here we investigate such patterns of recent
relatedness in a large European dataset.
The past several thousand years are replete with events that may have had
significant impact on modern European relatedness, such as the Neolithic
expansion of farming, the Roman empire, or the more recent expansions of the
Slavs and the Vikings.
Our current understanding of these events is deduced from archaeological,
linguistic, cultural, historical, and genetic evidence, with widely varying
degrees of certainty. However, the demographic and genealogical impact of
these events is still uncertain (e.g. Gillett, 2006). Genetic data describing
the breadth of genealogical relationships can therefore add another dimension
to our understanding of these historical events.
Work from uniparentally inherited markers (mtDNA and Y chromosomes) has
improved our understanding of human demographic history (e.g. Soares et al.,
2010). However, interpretation of these markers is difficult since they only
record a single lineage of each individual (the maternal and paternal
lineages, respectively), rather than the entire distribution of ancestors.
Genome-wide genotyping and sequencing datasets have the potential to provide a
much richer picture of human history, as we can learn simultaneously about the
diversity of ancestors that contributed to each individual’s genome.
A number of genome-wide studies have begun to reveal quantitative insights
into recent human history (Novembre and Ramachandran, 2011). Within Europe,
the first two principal axes of variation of the matrix of genotypes are
closely related to a rotation of latitude and longitude (Menozzi et al., 1978;
Novembre et al., 2008; Lao et al., 2008), as would be expected if patterns of
ancestry are mostly shaped by local migration (Novembre and Stephens, 2008).
Other work has revealed a slight decrease in diversity running from south-to-
north in Europe, with the highest haplotype and allelic diversity in the
Iberian peninsula (e.g. Lao et al., 2008; Auton et al., 2009; Nelson et al.,
2012), and the lowest haplotype diversity in England and Ireland (O’Dushlaine
et al., 2010). Recently, progress has also been made using genotypes of
ancient individuals to understand the prehistory of Europe (Patterson et al.,
2012; Skoglund et al., 2012; Keller et al., 2012). However, we currently have
little sense of the time scale of the historical events underlying modern
geographic patterns of relatedness, nor the degrees of genealogical
relatedness they imply.
In this paper, we analyze those rare long chunks of genome that are shared
between pairs of individuals due to inheritance from recent common ancestors,
to obtain a detailed view of the geographic structure of recent relatedness.
To determine the time scale of these relationships, we develop methodology
that uses the lengths of shared genomic segments to infer the distribution of
the ages of these recent common ancestors. We find that even geographically
distant Europeans share ubiquitous common ancestry even within the past 1000
years, and show that common ancestry from the past 3000 years is a result of
both local migration and large-scale historical events. We find considerable
structure below the country level in sharing of recent ancestry, lending
further support to the idea that looking at runs of shared ancestry can
identify very subtle population structure (e.g. Lawson et al., 2012).
Our method for inferring ages of common ancestors is conceptually similar to
the work of Palamara et al. (2012), who use total amount of long runs of
shared genome to fit simple parametric models of recent history, as well as to
Li and Durbin (2011) and Harris and Nielsen (2012), who use information from
short runs of shared genome to infer demographic history over much longer time
scales. Other conceptually similar work includes Pool and Nielsen (2009) and
Gravel (2012), who used the length distribution of admixture tracts to fit
parametric models of historical admixture. We rely less on discrete, idealized
populations or parametric demographic models than these other works, and
describe continuous geographic structure by obtaining average numbers of
common ancestors shared by many populations across time in a relatively non-
parametric fashion.
### 1.1 Definitions: Genetic ancestry and identity by descent
Figure 1: (A) A hypothetical portion of the pedigree relating two sampled
individuals, which shows six of their genealogical common ancestors, with the
portions of ancestral chromosomes from which the sampled individuals have
inherited shaded grey. The IBD blocks they have inherited from the two genetic
common ancestors are colored red, and the blue arrow denotes the path through
the pedigree along which one of these IBD blocks was inherited. (B) Cartoon of
the spatial locations of ancestors of two individuals – circle size is
proportional to likelihood of genetic contribution, and shared ancestors are
marked in grey. Note that common ancestors are likely located between the two,
and their distribution becomes more diffuse further back in time.
We can only hope to learn from genetic data about those common ancestors from
whom two individuals have both inherited the same genomic region. If a pair of
individuals have both inherited some genomic region from a common ancestor,
that ancestor is called a “genetic common ancestor”, and the genomic region is
shared “identical by descent” (IBD) by the two. Here we define an “IBD block”
to be a contiguous segment of genome inherited (on at least one chromosome)
from a shared common ancestor without intervening recombination (see figure
1A). A more usual definition of IBD restricts to those segments inherited from
some prespecified set of “founder” individuals (e.g. Fisher, 1954; Donnelly,
1983; Chapman and Thompson, 2002), but we allow ancestors to be arbitrarily
far back in time. Under our definition, everyone is IBD everywhere, but mostly
on very short, old segments (Powell et al., 2010). We measure lengths of IBD
segments in units of Morgans (M) or centiMorgans (cM), where 1 Morgan is
defined to be the distance over which an average of one recombination (i.e. a
crossover) occurs per meiosis. Segments of IBD are broken up over time by
recombination, which implies that older shared ancestry tends to result in
shorter shared IBD blocks.
Sufficiently long segments of IBD can be identified as long, contiguous
regions over which the two individuals are identical (or nearly identical) at
a set of Single Nucleotide Polymorphisms (SNPs) which segregate in the
population. Formal, model-based methods to infer IBD are only computationally
feasible for very recent ancestry (e.g. Brown et al., 2012), but recently,
fast heuristic algorithms have been developed that can be applied to thousands
of samples typed on genotyping chips (e.g. Browning and Browning, 2011; Gusev
et al., 2009).
The relationship between numbers of long, shared segments of genome, numbers
of genetic common ancestors, and numbers of genealogical common ancestors can
be difficult to envision. Since everyone has exactly two biological parents,
every individual has exactly $2^{n}$ paths of length $n$ meioses leading back
through their pedigree, each such path ending in a grandn-1parent. However,
due to Mendelian segregation and limited recombination, genetic material will
only be passed down along a small subset of these paths (Donnelly, 1983). As
$n$ grows, these paths proliferate rapidly and so the genealogical paths of
two individuals soon overlap significantly. (These points are illustrated in
in figure 1.) By observing the number of shared genomic blocks, we learn about
the degree to which their genealogies overlap, or the number of common
ancestors from which both individuals have inherited genetic material.
At least one parent of each genetic common ancestor of two individuals is also
a genetic common ancestor, so the number of genetic common ancestors at each
point back in time is strictly increasing. A more relevant quantity is the
rate of appearance of most recent common genetic ancestors. This quantity can
be much more intuitive, and is closely related to the coalescent rate (Hudson,
1990), as we demonstrate later. For this reason, when we say “genetic common
ancestor” or “rate of genetic common ancestry”, we are referring to only the
most recent genetic common ancestors from which the individuals in question
inherited their shared segments of genome.
## 2 Results
We applied the fastIBD method, implemented in BEAGLE v3.3 (Browning and
Browning, 2011), to the European subset of the POPRES dataset (dbgap accession
phs000145.v1.p1, Nelson et al., 2008), which includes language and country-of-
origin data for several thousand Europeans genotyped at 500000 SNPs. Our
simulations showed that we have good power to detect long IBD blocks
(probability of detection 50% for blocks longer than 2cM, rising to 98% for
blocks longer than 4cM), and a low false positive rate (discussed further in
section 4.2 of the methods). We excluded from our analyses individuals who
reported grandparents originating from non-European countries or more than one
distinct country (and refer to the remainder as “Europeans”). After removing
obvious outlier individuals and close relatives, we were left with 2257
individuals who we grouped using reported country of origin and language into
40 populations, listed with sample sizes and average IBD levels in table 1.
For geographic analyses, we located each population at the largest population
city in the appropriate region. Pairs of individuals in this dataset were
found to share a total of 1.9 million segments of IBD, an average of 0.74 per
pair of individuals, or 831 per individual. The mean length of these blocks
was 2.5cM, the median was 2.1cM and the 25th and 75th quantiles are 1.5cM and
2.9cM respectively. The majority of pairs sharing some IBD shared only a
single block of IBD (94%). The total length of IBD blocks an individual shares
with all others ranged between 30% and 250% (average 128%) of the length of
the genome (greater than 100% is possible as individuals may share IBD blocks
with more than one other at the same genomic location).
E group | $n$ | self | other | N group | $n$ | self | other
---|---|---|---|---|---|---|---
Albania | AL | 9 | 14.5 | 1.7 | Denmark | DK | 1 | – | 0.9
Austria | AT | 14 | 1.3 | 0.9 | Finland | FI | 1 | – | 1.2
Bosnia | BO | 9 | 4.1 | 1.6 | Latvia | LV | 1 | – | 1.6
Bulgaria | BG | 1 | – | 1.3 | Norway | NO | 2 | 2.0 | 0.8
Croatia | HR | 9 | 2.8 | 1.6 | Sweden | SE | 10 | 3.4 | 1.0
Czech Republic | CZ | 9 | 2.1 | 1.3 | | | | |
Greece | EL | 5 | 1.8 | 0.9 | W group | $n$ | self | other
Hungary | HU | 19 | 1.9 | 1.2 | Belgium | BE | 37 | 1.1 | 0.6
Kosovo | KO | 15 | 9.9 | 1.7 | England | EN | 22 | 1.3 | 0.7
Montenegro | ME | 1 | – | 1.8 | France | FR | 86 | 0.7 | 0.5
Macedonia | MA | 4 | 2.5 | 1.4 | Germany | DE | 71 | 1.1 | 0.9
Poland | PL | 22 | 3.8 | 1.5 | Ireland | IE | 60 | 2.6 | 0.6
Romania | RO | 14 | 2.1 | 1.2 | Netherlands | NL | 17 | 1.9 | 0.7
Russia | RU | 6 | 4.3 | 1.4 | Scotland | SC | 5 | 2.2 | 0.7
Slovenia | SI | 2 | 5.0 | 1.3 | Swiss French | CHf | 839 | 1.3 | 0.6
Serbia | RS | 11 | 2.7 | 1.5 | Swiss German | CHd | 103 | 1.6 | 0.6
Slovakia | SK | 1 | – | 0.7 | Switzerland | CH | 17 | 1.1 | 0.5
Ukraine | UA | 1 | – | 1.5 | United Kingdom | UK | 358 | 1.2 | 0.7
Yugoslavia | YU | 10 | 3.4 | 1.5 | | | | |
| | | | | I group | $n$ | self | other
TC group | $n$ | self | other | Italy | IT | 213 | 0.6 | 0.5
Cyprus | CY | 3 | 2.7 | 0.4 | Portugal | PT | 115 | 1.9 | 0.5
Turkey | TR | 4 | 2.2 | 0.5 | Spain | ES | 130 | 1.5 | 0.4
Table 1: Populations, abbreviations, sample sizes ($n$), mean number of IBD
blocks shared by a pair of individuals from that population (“self”), and mean
IBD rate averaged across all other populations (“other”); sorted by regional
groupings described in the text.
The observed genomic density of long IBD blocks (per cM) can be affected by
recent selection (Albrechtsen et al., 2010) and by cis-acting recombination
modifiers. We find that the local density of IBD blocks of all lengths is
relatively constant across the genome, but in certain regions the length
distribution is systematically perturbed (see supplemental figure S1),
including around certain centromeres and the large inversion on chromosome 8
(Giglio et al., 2001), also seen by Albrechtsen et al. (2010). Somewhat
surprisingly, the MHC does not show an unusual pattern of IBD, despite having
shown up in other genomic scans for IBD (Albrechtsen et al., 2010; Gusev et
al., 2012). However, there are a few other regions where differences in IBD
rate are not predicted by differences in SNP density. Notably, there are two
regions, on chromosomes 15 and 16, which are nearly as extreme in their
deviations in IBD as the inversion on chromosome 8, and may also correspond to
large inversions segregating in the sample. These only make up a small portion
of the genome, and do not significantly affect our other analyses (and so are
not removed); we leave further analysis for future work.
### 2.1 Substructure and recent migrants
We should expect significant within-population variability, as modern
countries are relatively recent constructions of diverse assemblages of
languages and heritages. To assess the uniformity of ancestry within
populations, we used a permutation test to measure, for each pair of
populations $x$ and $y$, the uniformity with which relationships with $x$ are
distributed across individuals from $y$. Most comparisons show statistically
significant heterogeneity (supplemental figure S2), which is probably due to
population substructure (as well as correlations introduced by the pedigree).
A notable exception is that nearly all populations showed no significant
heterogeneity of numbers of common ancestors with Italian samples, suggesting
that most common ancestors shared with Italy lived longer ago than the time
that structure within modern-day countries formed.
Figure 2: Substructure, in (A) Italian, and (B) UK samples. The leftmost
plots of (A) show histograms of the numbers of IBD blocks that each Italian
sample shares with any French-speaking Swiss (top) and anyone from the UK
(bottom), overlaid with the expected distribution (Poisson) if there was no
dependence between blocks. Next is shown a scatterplot of numbers of blocks
shared with French-speaking Swiss and UK samples, for all samples from France,
Italy, Greece, Turkey, and Cyprus. We see that the numbers of recent ancestors
each Italian shares with the French-speaking Swiss and with the United Kingdom
are both bimodal, and that these two are positively correlated, ranging
continuously between values typical for Turkey/Cyprus and for France. Figure
(B) is similar, showing that the substructure within the UK is part of a
continuous trend ranging from Germany to Ireland. The outliers visible in the
scatterplot of figure 2B are easily explained as individuals with immigrant
recent ancestors – the three outlying UK individuals in the lower left share
many more blocks with Italians than all other UK samples, and the individual
labeled “SK” is a clear outlier for the number of blocks shared with the
Slovakian sample.
Two of the more striking examples of substructure are illustrated in figure 2.
Here, we see that variation within countries can be reflective of continuous
variation in ancestry that spans a broader geographic region, crossing
geographic, political, and linguistic boundaries. Figure 2A shows the
distinctly bimodal distribution of numbers of IBD blocks that each Italian
shares with both French-speaking Swiss and the UK, and that these numbers are
strongly correlated. Furthermore, the amount that Italians share with these
two populations varies continuously from values typical for Turkey and Cyprus,
to values typical for France and Switzerland. Interestingly, the Greek samples
(EL) place near the middle of the Italian gradient. It is natural to guess
that there is a north-south gradient of recency of common ancestry along the
length of Italy, and that southern Italy has been historically more closely
connected to the eastern Mediterranean.
In contrast, within samples from the UK and nearby regions we see negative
correlation between numbers of blocks shared with Irish and numbers of blocks
shared with Germans. From our data, we do not know if this substructure is
also geographically arranged within the UK (our sample of which which may
include individuals from Northern Ireland). However, an obvious explanation of
this pattern is that individuals within the UK differ in the number of recent
ancestors shared with Irish, and that individuals with less Irish ancestry
have a larger portion of their recent ancestry shared with Germans. This
suggests that there is variation across the UK – perhaps a geographic gradient
– in terms of the amount of Celtic versus Germanic ancestry.
The first two principal components of the matrix of genotypes, after suitable
manipulations, can reproduce the geographic positions of European populations
(e.g. Menozzi et al., 1978; Novembre et al., 2008; Lao et al., 2008).
Therefore, it is natural to compare the structure we see within populations in
terms of IBD sharing to the positions on the principal components map. (A PCA
map of these populations (produced by EIGENSTRAT, Price et al., 2006) is shown
in supplemental figure S4.) It is not known what the geographic resolution of
the principal components map is, but if relative positions within populations
is meaningful, then comparison of IBD to PCA can stand in for comparison to
geography. Indeed, as seen in supplemental figures S6 and S5, the substructure
of figure 2 correlates well with the position on certain principal components,
further suggesting that the structure is geographically meaningful.
Conversely, since the substructure we see is highly statistically significant,
this demonstrates that the scatter of positions within populations on the
European PCA map is at least in part signal, rather than noise.
### 2.2 Europe-wide patterns of relatedness
Individuals usually share the highest number of IBD blocks with others from
the same population, with some exceptions. For example, individuals in the UK
share more IBD blocks on average, and hence more close genetic ancestors, with
individuals from Ireland than with other individuals from the UK (1.26 versus
1.09 blocks at least 1cM per pair, Mann-Whitney $p<10^{-10}$), and Germans
share similarly more with Polish than with other Germans (1.24 versus 1.05,
$p=5.7\times 10^{-6}$), a pattern which could be due to recent asymmetric
migration from a smaller population into a larger population. In figure 3 we
depict the geography of rates of IBD sharing between populations, i.e. the
average number of IBD blocks shared by a randomly chosen pair of individuals.
Above, maps show the IBD rate relative to certain chosen populations, and
below, all pairwise sharing rates are plotted against the geographic distance
separating the populations. It is evident that geographic proximity is a major
determinant of IBD sharing (and hence recent relatedness), with the rate of
pairwise IBD decreasing relatively smoothly as the geographic separation of
the pair of populations increases. Note that even populations represented by
only a single sample are included, as these showed surprisingly consistent
signal despite the small sample size.
Superimposed on this geographic decay there is striking regional variation in
rates of IBD. To further explore this variation, we divided the populations
into the four groups listed in table 1, using geographic location and
correlations in the pattern of IBD sharing with other populations (shown in
supplemental figure S7). These groupings are defined as: Europe “E”, lying to
the east of Germany and Austria; Europe “N”, lying to the north of Germany and
Poland; Europe “W”, to the west of Germany and Austria and including these;
the Iberian and Italian peninsulas “I”; and Turkey/Cyprus “TC”. Although the
general pattern of regional IBD variation is strong, none of these groups have
sharp boundaries – for instance, Germany, Austria, and Slovakia are
intermediate between E and W. Furthermore, we suspect that the Italian and
Iberian peninsulas likely do not group together because of higher shared
ancestry with each other, but rather because of similarly low rates of IBD
with other European populations. The overall mean IBD rates between these
regions are shown in table 2, and comparisons between different groupings are
colored differently in figure 3G–I, showing that rates of IBD sharing between
E populations and between N populations average a factor of about three higher
than other comparisons at similar distances. Such a large difference in the
rates of IBD sharing between regions, cannot be explained by plausible
differences in false positive rates or power between populations, since this
pattern holds even at the longest length scales, where block identification is
nearly perfect.
To better understand IBD within these groupings, we show in figures 3G–I how
average numbers of IBD blocks shared, in three different length categories,
depend on the geographic distance separating the two populations. Even without
taking into account regional variation, mean numbers of shared IBD blocks
decay exponentially with distance, and further structure is revealed by
breaking out populations by the regional groupings described above. The
exponential decays shown for each pair of groupings emphasize how the decay of
IBD with distance becomes more rapid for longer blocks. This is expected under
models where migration is mostly local, since as one looks further back in
time, the distribution of each individual’s ancestors is less concentrated
around the individual’s location (recall figure 1B). Therefore, the expected
number of ancestors shared by a pair of individuals decreases as the
geographic distance between the pair increases; and this decrease is faster
for more recent ancestry.
This wider spread of older blocks can also explain why the decay of IBD with
distance varies significantly by region even if dispersal rates have been
relatively constant. For instance, the gradual decay of sharing with the
Iberian and Italian peninsulas could occur because these blocks are inherited
from much longer ago than blocks of similar lengths shared by individuals in
other populations.
Conversely, there is a high level of sharing for “E–E” relationships over a
broad range of distances. This is especially true for our shortest (oldest)
blocks: individuals in our E grouping share on average more short blocks with
individuals in distant E populations than do pairs of individuals in the same
W population. We argue below that this is because modern individuals in these
locations have a larger proportion of their ancestors in a relatively small
population that subsequently expanded.
Figure 3: In all figures, colors give categories based on the regional groupings of table 1. (A–F) The area of the circle located on a particular population is proportional to the mean number of IBD blocks of length at least 1cM shared between random individuals chosen from that population and the population named in the label (also marked with a star). Both regional variation of overall IBD rates and gradual geographic decay are apparent. (G–I) Mean number of IBD blocks of lengths 1–3cM (oldest), 3–5cM, and $>$5cM (youngest), respectively, shared by a pair of individuals across all pairs of populations; the area of the point is proportional to sample size (number of distinct pairs), capped at a reasonable value; and lines show an exponential decay fit to each category (using a Poisson GLM weighted by sample size). Comparisons with no shared IBD are used in the fit but not shown in the figure (due to the log scale). “E–E”, “N–N”, and “W–W” denote any two populations both in the E, N, or W grouping, respectively; “TC-any” denotes any population paired with Turkey or Cyprus; “I-(I,E,N,W)” denotes Italy, Spain, or Portugal paired with any population except Turkey or Cyprus; and “between E,N,W” denotes the remaining pairs (when both populations are in E, N, or W, but the two are in different groups). The exponential fit for the N–N points is not shown due to the very small sample size. See supplemental figure S8 for an SVG version of these plots where it is possible to identify individual points. IBD rate | E | I | N | TC | W
---|---|---|---|---|---
E | 2.57 | 0.44 | 0.99 | 0.62 | 0.53
I | 0.44 | 0.80 | 0.43 | 0.41 | 0.45
N | 0.99 | 0.43 | 2.62 | 0.33 | 0.86
TC | 0.62 | 0.41 | 0.33 | 1.43 | 0.25
W | 0.53 | 0.45 | 0.86 | 0.25 | 0.93
Table 2: Rates of IBD within and between each geographic grouping given in
table 1.
Having seen the continent-wide patterns of IBD in figure 3, it is natural to
wonder if similar information is contained in single-site summaries of
relatedness, such as mean Identity by State (IBS) values across European
populations. The mean IBS between populations $x$ and $y$ is defined as the
probability that two randomly chosen alleles from $x$ and $y$ are identical
(“By State”), averaging over SNPs and individuals. In the analogous plot of
IBS against geographic distance (supplemental figure S9), some of the patterns
seen in figure 3 are present, and some are not. For instance, there is a
continuous decay with geographic distance (linear, not exponential), and
comparisons to the southern “I” group and to Cyprus/Turkey are even more well-
separated below the others. On the other hand, the “E-E” comparisons do not
show higher IBS than the bulk of the remaining comparisons.
### 2.3 Ages and numbers of common ancestors
Each block of genome shared IBD by a pair of individuals represents genetic
material inherited from one of their genetic common ancestors. Since the
distribution of lengths of IBD blocks differs depending on the age of the
ancestors – e.g. older blocks tend to be shorter – it is possible to use the
distribution of lengths of IBD blocks to infer numbers of most recent pairwise
genetic common ancestors back through time averaged across pairs of
individuals. For this inference, we restricted to blocks longer than 2cM,
where we had good power to detect true IBD blocks. We obtain dates in units of
generations in the past, and for ease of discussion convert these to years ago
(ya) by taking the mean human generation time to be 30 years (Fenner, 2005).
#### Nature of the results on age inference
There are two major difficulties to overcome, however. First, detection is
noisy: we do not detect all IBD segments (especially shorter ones), and some
of our IBD segments are false positives. This problem can be overcome by
careful estimation and modeling of error, described in section 4.2. The second
problem is more serious and unavoidable: as described in section 4.7, the
inference problem is extremely “ill conditioned” (in the sense of Petrov and
Sizikov, 2005), meaning in this case that there are many possible histories of
shared ancestry that fit the data nearly equally well. For this reason, there
is a fairly large, unavoidable limit to the temporal resolution, but we still
obtain a good deal of useful information.
We deal with this uncertainty by describing the set of histories (i.e.
historical numbers of common genetic ancestors) that are consistent with the
data, summarized in two ways. First, it is useful to look at individual
consistent histories, which gives a sense of recurrent patterns and possible
historical signals. Figure 4 shows for several populations both the best-
fitting history (in black) and the smoothest history that still fits the data
(in red). We can make general statements if they hold across all (or most)
consistent histories. Second, we can summarize the entire set of consistent
histories by finding confidence intervals (bounds) for the total number of
common ancestors aggregated in certain time periods. These are shown in figure
5, giving estimates (colored bands) and bounds (vertical lines) for the total
numbers of genetic common ancestors in each of three time periods, roughly
0–500ya, 500–1500ya, and 1500–2500ya (“ya” denotes “years ago”). Supplemental
figures S12 (and S13) is a version of figure 5 with more populations (in
coalescent units, respectively), and plots analogous to figure 4 for all these
histories are shown in supplemental figure S16. For a precise description of
the problem and our methods, see section 4.7. We validated the method through
simulation (details in supplemental document), and found that it performed
well to the temporal resolution discussed here. We note that in simulations
where the population size changes smoothly, the maximum likelihood solution is
often overly peaky, whereas the smoothed solution can smear out the signal of
rapid change in population size. In light of that we encourage the reader to
view truth as lying somewhere between these two solutions, and to not
overinterpret specific peaks in the maximum likelihood, which may occur due to
numerical properties of the inference. That said, there are a number of sharp
peaks in common ancestry shared across many population comparisons older than
2000 years ago, which may potentially indicate demographic events in a shared
ancestral population. A more thorough investigation of these older shared
signals would potentially need a more model-based approach, so we restrict
ourselves here to talking about the broad differences between the distribution
of common shared ancestors between regions.
Figure 4: Estimated average number of most recent genetic common ancestors
per generation back through time, shared by (A) pairs of individuals from “the
Balkans” (former Yugoslavia, Bulgaria, Romania, Croatia, Bosnia, Montenegro,
Macedonia, Serbia, and Slovenia, excluding Albanian speakers); and, shared by
one individual from the Balkans with one individual from (B) Albanian speaking
populations; (C) Italy; or (D) France. The black distribution is the maximum
likelihood fit; shown in red is smoothest solution that still fits the data,
as described in section 4.7. (E) shows the observed IBD length distribution
for pairs of individuals from the Balkans (red curve), along with the
distribution predicted by the smooth (red) distribution in (A), as a stacked
area plot partitioned by time period in which the common ancestor lived. The
partitions with significant contribution are labeled on the left vertical axis
(in generations ago), and the legend in (J) gives the same partitions, in
years ago; the vertical scale is given on the right vertical axis. The second
column of figures (F–J) is similar, except that comparisons are relative to
samples from the UK.
The time periods we use for these bounds are quite large, but this is
unavoidable, because of a trade off between temporal resolution and
uncertainty in numbers of common ancestors. Also note that the lower bounds on
numbers of common ancestors during each time interval are often close to zero.
This is because one can (roughly speaking) obtain a history with equally good
fit by moving ancestors from that time interval into the neighboring ones,
resulting in peaks on either side of the selected time interval (see figure
S14), even though these do not generally reflect realistic histories. The
reader should also bear in mind that we do not depict the dependence of
uncertainty between intervals.
Figure 5: Estimated average total numbers of genetic common ancestors shared
per pair of individuals in various pairs of populations, in roughly the time
periods 0–500ya, 500–1500ya, 1500–2500ya, and 2500–4300ya. We have combined
some populations to obtain larger sample sizes: “S-C” denotes Serbo-Croatian
speakers in former Yugoslavia, “PL” denotes Poland, “R-B” denotes Romania and
Bulgaria, “DE” denotes Germany, “UK” denotes the United Kingdom, “IT” denotes
Italy, and “Iber” denotes Spain and Portugal. For instance, the green bars in
the leftmost panels tell us that Serbo-Croatian speakers and Germans most
likely share 0–0.25 most recent genetic common ancestor from the last 500
years, 3–12 from the period 500–1500 years ago, 120–150 from 1500–2500 ya, and
170–250 from 2500–4400 ya. Although the lower bounds appear to extend to zero,
they are significantly above zero in nearly all cases except for the most
recent period 0–540ya.
#### Results of age inference
In figure 4 we show how the age and number of shared pairwise genetic common
ancestors changes as we move away from the Balkans (left column) and the UK
(right column), along with two examples of how the observed block length
distribution is composed of ancestry from different depths. (The average
number of shared pairwise genetic common ancestors from generation $n$ is the
probability that the most recent common ancestor of a pair at a single site
lived in generation $n$ (i.e. the coalescent rate) multiplied by the expected
number of segments that recombination has broken a pair of individuals’
genomes into that many generations back, as shown in section 4.8.) More plots
of this form are shown in supplemental figure S16, and coalescent rates
between pairs of populations are shown in the (equivalent) supplemental figure
S15.
Most detectable recent common ancestors lived between 1500 and 2500 years ago,
and only a small proportion of blocks longer than 2cM are inherited from
longer ago than 4000 years. Obviously, there are a vast number of genetic
common ancestors older than this, but the blocks inherited from such common
ancestors are sufficiently unlikely to be longer than 2cM that we do not
detect many. For the most part, blocks longer than 4cM come from 500–1500
years ago, and blocks longer than 10cM from the last 500 years.
In most cases, only pairs within the same population are likely to share
genetic common ancestors within the last 500 years. Exceptions are generally
neighboring populations (e.g. UK and Ireland). During the period 500–1500ya,
individuals typically share tens to hundreds of genetic common ancestors with
others in the same or nearby populations, although some distant populations
have very low rates. Longer ago than 1500ya, pairs of individuals from any
part of Europe share hundreds of genetic ancestors in common, and some share
significantly more.
#### Regional variation: interesting cases
We now examine some of the more striking patterns we see in more detail.
There is relatively little common ancestry shared between the Italian
peninsula and other locations, and what there is seems to derive mostly from
longer ago than 2500ya. An exception is that Italy and the neighboring Balkan
populations share small but significant numbers of common ancestors in the
last 1500 years, as seen in supplemental figures S16 or S17. The rate of
genetic common ancestry between pairs of Italian individuals seems to have
been fairly constant for the past 2500 years, which combined with significant
structure within Italy suggests a constant exchange of migrants between
coherent subpopulations.
Patterns for the Iberian peninsula are similar, with both Spain and Portugal
showing very few common ancestors with other populations over the last 2500
years. However, the rate of IBD sharing within the peninsula is much higher
than within Italy – during the last 1500 years the Iberian peninsula shares
fewer than 2 genetic common ancestors with other populations, compared to
roughly 30 per pair within the peninsula; Italians share on average only about
8 with each other during this period.
The higher rates of IBD between populations in the “E” grouping shown in
figure 3 seem to derive mostly from ancestors living 1500–2500ya, but also
show increased numbers from 500–1500ya, as shown in figure 5 and supplemental
figures S17. For comparison, the IBD rate is high enough that even
geographically distant individuals in these eastern populations share about as
many common ancestors as do two Irish or two French-speaking Swiss.
By far the highest rates of IBD within any populations is found between
Albanian speakers – around 90 ancestors from 0–500ya, and around 600 ancestors
from 500–1500ya (so high that we left them out of figure 5; see supplemental
figure S12). Beyond 1500ya, the rates of IBD drop to levels typical for other
populations in the eastern grouping.
There are clear differences in the number and timing of genetic common
ancestors shared by individuals from different parts of Europe, These
differences reflect the impact of major historical and demographic events,
superimposed against a background of local migration and generally high
genealogical relatedness across Europe. We now turn to discuss possible causes
and implications of these results.
## 3 Discussion
Genetic common ancestry within the last 2500 years across Europe has been
shaped by diverse demographic and historical events. There are both
continental trends, such as a decrease of shared ancestry with distance,
regional patterns, such as higher IBD in eastern and northern populations, and
diverse outlying signals. We have furthermore quantified numbers of genetic
common ancestors that populations share with each other back through time,
albeit with a (unavoidably) coarse temporal resolution. These numbers are
intriguing not only because of the differences between populations, which
reflect historical events, but the high degree of implied genealogical
commonality between even geographically distant populations.
#### Ubiquity of common ancestry
We have shown that typical pairs of individuals drawn from across Europe have
a good chance of sharing long stretches of identity by descent, even when they
are separated by thousands of kilometers. We can furthermore conclude that
pairs of individuals across Europe are reasonably likely to share common
genetic ancestors within the last 1000 years, and are certain to share many
within the last 2500 years. From our numerical results, the average number of
genetic common ancestors from the last 1000 years shared by individuals living
at least 2000km apart is about 1/32 (and at least 1/80); between 1000–2000ya
they share about one; and between 2000–3000ya they share above ten. Since the
chance is small that any genetic material has been transmitted along a
particular genealogical path from ancestor to descendent more than 8
generations deep (Donnelly, 1983) – about .008 at 240ya, and $2.5\times
10^{-7}$ at 480ya – this implies, conservatively, thousands of shared
genealogical ancestors in only the last 1000 years even between pairs of
individuals separated by large geographic distances. At first sight this
result seems counterintuitive. However, as 1000 years is about 33 generations,
and $2^{33}\approx 10^{10}$ is far larger than the size of the European
population, so long as populations have mixed sufficiently, by 1000 years ago
everyone (who left descendants) would be an ancestor of every present day
European. Our results are therefore one of the first genomic demonstrations of
the counter-intuitive but necessary fact that all Europeans are genealogically
related over very short time periods, and lends substantial support to models
predicting close and ubiquitous common ancestry of all modern humans (Rohde et
al., 2004).
The fact that most people alive today in Europe share nearly the same set of
(European, and possibly world-wide) ancestors from only 1000 years ago seems
to contradict the signals of long term, albeit subtle, population genetic
structure within Europe (e.g. Novembre et al., 2008; Lao et al., 2008). These
two facts can be reconciled by the fact that even though the distribution of
ancestors (as cartooned in figure 1B) has spread to cover the continent, there
remain differences in degree of relatedness of modern individuals to these
ancestral individuals. For example, someone in Spain may be related to an
ancestor in the Iberian peninsula through perhaps 1000 different routes back
through the pedigree, but to an ancestor in the Baltic region by only 10
different routes, so that the probability that this Spanish individual
inherited genetic material from the Iberian ancestor is roughly 100 times
higher. This allows the amount of genetic material shared by pairs of extant
individuals to vary even if the set of ancestors is constant.
#### Relation to single-site summaries
Other work has studied fine-scale differentiation between populations within
Europe based on statistics such as $F_{ST}$, IBS (e.g. Lao et al., 2008;
O’Dushlaine et al., 2010), or PCA (Novembre et al., 2008), that summarize in
various ways single-marker correlations, averaged across loci. Like rates of
IBD, these measures of differentiation can be thought of as weighted averages
of past coalescent rates (Malécot, 1969; Slatkin, 1991; Rousset, 2002; McVean,
2009), but take much of their information from much more distant times (tens
thousands of generations). As expected, we have seen both strong consistency
between these measures and IBD (e.g. the decay with geographic distance), as
well as distinct patterns (e.g. higher sharing in eastern Europe). These
results highlight the fact that long segments of IBD contain information about
much more recent events than do single-site summaries, information that can be
leveraged to learn about the timing of these events.
#### Limitations of Sampling
A concern about our results is that the European individuals in the POPRES
dataset were all sampled in either Lausanne or London. This might bias our
results, for instance, if an immigrant community originated mostly from a
particular small portion of their home population, thereby sharing a
particularly high number of recent common ancestors with each other. We see
remarkably little evidence that this is the case: there is a high degree of
consistency in numbers of IBD blocks shared across samples from each
population, and between neighboring populations. For instance, we could argue
that the high degree of shared common ancestry among Albanian speakers was
because most of these sampled originated from a small area rather than
uniformly across Albania and Kosovo. However, this would not explain the high
rate of IBD between Albanian speakers and neighboring populations. Even
populations from which we only have one or two samples, which we at first
assumed would be unusably noisy, provide generally reliable, consistent
patterns, as evidenced by e.g. supplemental figure S3.
Conversely, it might be a concern that individuals sampled in Lausanne or
London are more likely to have recent ancestors more widely dispersed than is
typical for their population of origin. This is a possibility we cannot
discard, and if true, would mean there is more structure within Europe than
what we detect. However, by the incredibly rapid spread of ancestry, this is
unlikely to have an effect over more than a few generations and so does not
pose a serious concern about our results about the ubiquitous levels of common
ancestry. Fine-scale geographic sampling of Europe as a whole is needed to
address these issues, and these efforts are underway in a number of
populations (e.g. Price et al., 2009; Jakkula et al., 2008; Tyler-Smith and
Xue, 2012; Winney et al., 2011).
Finally, we have necessarily have taken a narrow view of European ancestry as
we have restricted our sample to individuals who are not outliers with respect
to genetic ancestry, and when possible to those having all four grandparents
drawn from the same county. Clearly the ancestry of Europeans is far more
diverse than those represented here, but such steps seemed necessary to make
best initial use of this dataset.
#### Ages of particular common ancestors
We have shown that the problem of inferring the average distribution of
genetic common ancestors back through time has a large degree of fundamental
uncertainty. The data effectively leave a large number of degrees of freedom
unspecified, so one must either describe the set of possible histories, as we
do, and/or use prior information to restrict these degrees of freedom.
A related but far more intractable problem is to make a good guess of how long
ago a certain shared genetic common ancestor lived, as personal genome
services would like to do, for instance: if you and I share a 10cM block of
genome IBD, when did our most recent common ancestor likely live? Since the
mean length of an IBD block inherited from 5 generations ago is 10cM, we might
expect the average age of the ancestor of a 10cM block to be from around 5
generations. However, a direct calculation from our results says that the
typical age of a 10cM block shared by two individuals from the UK is between
32 and 52 generations (depending on the inferred distribution used). This
discrepancy results from the fact that you are a priori much more likely to
share a common genetic ancestor further in the past, and this acts to skew our
answers away from the naive expectation – even though it is unlikely that a
10cM block is inherited from a particular shared ancestor from 40 generations
ago, there are a great number of such older shared ancestors. This also means
that estimated ages must depend drastically on the populations’ shared
histories: for instance, the age of such a block shared by someone from the UK
with someone from Italy is even older, usually from around 60 generations ago.
This may not apply to ancestors from the past very few (perhaps less than
eight) generations, from whom we expect to inherit multiple long blocks – in
this case we can hope to infer a specific genealogical relationship with
reasonable certainty (e.g. Huff et al., 2011; Henn et al., 2012b), although
even then care must be taken to exclude the possibility that these multiple
blocks have not been inherited from distinct common ancestors.
Although the sharing of a long genomic segment can be an intriguing sign of
some recent shared ancestry, the ubiquity of shared genealogical ancestry only
tens of generations ago across Europe (and likely the world, Rohde et al.,
2004) makes such sharing unsurprising, and assignment to particular
genealogical relationships impossible. What is informative about these chance
sharing events from distant ancestors is that they provide a fine-scale view
of an individual’s distribution of ancestors (e.g. figure 3), and that in
aggregate they can provide an unprecedented view into even small-scale human
demographic history.
#### Where do your $n^{\mathrm{th}}$ cousins live?
Our results also offer a way to understand the geographic location of
individuals of a given degree of relatedness. The values of figure 5 (and S12)
can be interpreted as the distribution of distant cousins for any reference
population – for instance, the set of bars for Poland (“PL”) in the top row
shows that a randomly chosen distant cousin of a Polish individual with the
common ancestor living in the past 500 years most likely lives in Poland but
has reasonable chance of living in the Balkan peninsula or Germany. Here
“randomly chosen” means chosen randomly proportional to the paths through the
pedigree – concretely, take a random walk back through the pedigree to an
ancestor in the appropriate time period, and then take a random walk back
down. If one starts in Poland, then the chance of arriving in, say, Romania is
proportional to the average number of (genetic) common ancestors shared by a
pair from Poland and Germany, which is exactly the number estimated in figure
5.
### The signal of history
As we have shown, patterns of IBD provide ample but noisy geographic and
temporal signals, which can then be connected to historical events. Rigorously
making such connections is difficult, due to the complex recent history of
Europe, controversy about the demographic significance of many events, and
uncertainties in inferring the ages of common ancestors. Nonetheless, our
results can be plausibly connected to several historical and demographic
events.
#### The migration period
One of the striking patterns we see is the relatively high level of sharing of
IBD between pairs of individuals across eastern Europe, as high or higher than
that observed within other, much smaller populations. This is consistent with
these individuals having a comparatively large proportion of ancestry drawn
from a relatively small population that expanded over a large geographic area.
The “smooth” estimates of figure 4 (and more generally figures 5, and S17),
suggest that this increase in ancestry stems from around 1000–2000 years ago,
since during this time pairs of eastern individuals are expected to share a
substantial number of common ancestors, while this is only true of pairs of
non-eastern individuals if they are from the same population. For example,
even individuals from widely separated eastern populations share about the
same amount of IBD as do two Irish individuals (see supplemental figure S3),
suggesting that this ancestral population may have been relatively small.
This evidence is consistent with the idea that these populations derive a
substantial proportion of their ancestry from various groups that expanded
during the “migration period” from the fourth through ninth centuries (Davies,
2010). This period begins with the Huns moving into eastern Europe towards the
end of the fourth century, establishing an empire including modern-day Hungary
and Romania; and continues in the fifth century as various Germanic groups
moved into and ruled much of the western Roman empire. This was followed by
the expansion of the Slavic populations into regions of low population density
beginning in the sixth century, reaching their maximum by the 10th century
(Barford, 2001). The eastern populations with high rates of IBD is highly
coincident with the modern distribution of Slavic languages, so it is natural
to speculate that much of the higher rates were due to this expansion. The
inclusion of (non-Slavic speaking) Hungary and Romania in the group of eastern
populations sharing high IBD could indicate the effect of other groups (e.g.
the Huns) on ancestry in these regions, or because some of the same group of
people who elsewhere are known as Slavs adopted different local cultures in
those regions. Greece and Albania are also part of this putative signal of
expansion, which could be because the Slavs settled in part of these areas
(with unknown demographic effect), or because of subsequent population
exchange. However, additional work and methods would be needed to verify this
hypothesis.
The highest levels of IBD sharing are found in the Albanian-speaking
individuals (from Albania and Kosovo), an increase in common ancestry deriving
from the last 1500 years. This suggests that a reasonable proportion of the
ancestors of modern-day Albanian speakers (at least those represented in
POPRES) are drawn from a relatively small, cohesive population that has
persisted for at least the last 1500 years. These individuals share similar
but slightly higher numbers of common ancestors with nearby populations than
do individuals in other parts of Europe (see figure S3), implying that these
Albanian speakers have not been a particularly isolated population so much as
a small one. Furthermore, our Greek and Macedonian samples share much higher
numbers of common ancestors with Albanian speakers than with other neighbors,
possibly a result of historical migrations, or else perhaps smaller effects of
the Slavic expansion in these populations. It is also interesting to note that
the sampled Italians share nearly as much IBD with Albanian speakers as with
each other. The Albanian language is a Indo-European language without other
close relatives (Hamp, 1966) that persisted through periods when neighboring
languages were strongly influenced by Latin or Greek, suggesting an intriguing
link between linguistic and genealogical history in this case.
#### Italy, Iberia, and France
On the other hand, we find that France and the Italian and Iberian peninsulas
have the lowest rates of genetic common ancestry in the last 1500 years (other
than Turkey and Cyprus), and are the regions of continental Europe thought to
have been least affected by the Slavic and Hunnic migrations. These regions
were, however, moved into by Germanic tribes (e.g. the Goths, Ostrogoths, and
Vandals), which suggests that perhaps the Germanic migrations/invasions of
these regions entailed a smaller degree of population replacement, than the
Slavic and/or Hunnic, or perhaps that the Germanic groups were less
genealogically cohesive. This is consistent with the argument that the Slavs
moved into relatively depopulated areas, while Gothic “migrations” may have
been takeovers by small groups of extant populations (Halsall, 2005;
Kobyliński, 2005).
In addition to the very few genetic common ancestors that Italians share both
with each other and with other Europeans, we have seen significant modern
substructure within Italy (i.e. figure 2) that predates most of this common
ancestry, and estimate that most of the common ancestry shared between Italy
and other populations is older than about 2300 years (supplemental figure
S16). Also recall that most populations show no substructure with regards to
the number of blocks shared with Italians, implying that the common ancestors
other populations share with Italy predate divisions within these other
populations. This suggests significant old substructure and large population
sizes within Italy, strong enough that different groups within Italy share as
little recent common ancestry as other distinct, modern-day countries,
substructure that was not homogenized during the migration period. These
patterns could also reflect in part geographic isolation within Italy as well
as a long history of settlement of Italy from diverse sources.
In contrast to Italy, the rate of sharing of IBD within the Iberian peninsula
is similar to that within other populations in Europe. There is furthermore
much less evidence of substructure within our Iberian samples than within the
Italians, as shown in supplemental figure S2. This suggests that the reduced
rate of shared ancestry is due to geographic isolation (by distance and/or the
Pyrenees) rather than long-term stable substructure within the peninsula.
#### Future directions
Our results show that patterns of recent identity by descent both provide
evidence of ubiquitous shared common ancestry and hold the potential to shed
considerable light on the complex history of Europe. However, these inferences
also quickly run up against a fundamental limit to our ability to infer
pairwise rates of recent common genetic ancestry. In order to make a fuller
model of European history, we will need to make use of diverse sources of
genomic information from large samples, including IBD segments and rare
variants (Nelson et al., 2012; Tennessen et al., 2012), and develop methods
that can more fully utilize this information across more than pairs of
populations. Another profound difficulty is that Europe – and indeed any large
continental region – has such complex layers of history, through which
ancestry has mixed so greatly, that attempts to connect genetic signals in
extant individuals to particular historical events requires the corroboration
of other sources of information from many disciplines. For example, the
ability to isolate ancient autosomal DNA from individuals who lived during
these time periods (as do Skoglund et al., 2012; Keller et al., 2012) will
help to overcome some of these these profound difficulties. More generally,
the quickly falling cost of sequencing, along with the development of new
methods, will shed light on the recent demographic and genealogical history of
populations of recombining organisms, human and otherwise.
## 4 Materials and Methods
### Description of data and data cleaning
We used the two European subsets of the POPRES dataset – the CoLaus subset,
collected in Lausanne, Switzerland, and the LOLIPOP subset, collected in
London, England; the dataset is described in Nelson et al. (2008). Those
collected in Lausanne reported parental and grandparental country of origin;
those collected in London did not. We followed Novembre et al. (2008) in
assigning each sample to the common grandparental country of origin when
available, and discarding samples whose parents or grandparents were reported
as originating in different countries. We took further steps to restrict to
individuals whose grandparents came from the same geographic region, first
performing principal components analysis on the data using SMARTPCA (Patterson
et al., 2006), and excluding 41 individuals who clustered with populations
outside Europe (the majority of such were already excluded by self-reported
non-European grandparents). These individuals certainly represent an important
part of the recent genetic ancestry of Europe, but are excluded because we aim
to study events stemming from older patterns of gene flow, and because we do
not model the whole-genome dependencies in recently admixed genomes. We then
used PLINK’s inference of the fraction of single-marker IBD (Z0, Z1, and Z2,
Purcell et al., 2007) to identify very close relatives, finding 25 pairs that
are first cousins or closer (including duplicated samples), and excluded one
individual from each pair. We grouped samples into populations mostly by
reported country, but also used reported language in a few cases. Because of
the large Swiss samples, we split this group into three by language: French-
speaking (CHf), German-speaking (CHd), or other (CH). Many samples reported
grandparents from Yugoslavia; when possible we assigned these to a modern-day
country by language, and when this was ambiguous or missing we assigned these
to “Yugoslavia”. Most samples from the United Kingdom reported this as their
country of origin; however, the few that reported “England” or “Scotland” were
assigned this label. This left us with 2257 individuals from 40 populations;
for sample sizes see table 1. Supplemental table S2 further breaks this down,
and unambiguously gives the composition of each population. Physical distances
were converted to genetic distances using the hg36 map, and the average human
generation time was taken to be 30 years (Fenner, 2005).
All figures were produced in R R Development Core Team (2012), with color
palettes from packages colorspace Zeileis et al. (2009) and RColorBrewer Plate
(2011). Code implementing all methods described below is distributed along
with IBD block data sufficient to reproduce the historical analyses through
http://www.github.com/petrelharp/euroibd and in the Dryad digital repository
Ralph and Coop (2013).
### 4.1 Calling IBD blocks
To find blocks of IBD, we used fastIBD (implemented in BEAGLE, Browning and
Browning, 2011), which records putative genomic segments shared IBD by pairs
of individuals, along with a score indicating the strength of support. As
suggested by the authors, in all cases we ran the algorithm 10 times with
different random seeds, and postprocessed the results to obtain IBD blocks.
Based on our power simulations described below, we modified the postprocessing
procedure recommended by Browning and Browning (2011) to deal with spurious
gaps or breaks introduced into long blocks of IBD by low marker density or
switch error, as follows: We called IBD segments by first removing any
segments not overlapping a segment seen in at least one other run (as
suggested by Browning and Browning (2011), except with no score cutoff); then
merging any two segments separated by a gap shorter than at least one of the
segments and no more than 5cM long; and finally discarding any merged segments
that did not contain a subsegment with score below $10^{-9}$. As shown in
figure 6, this resulted in a false positive rate of between 8–15% across
length categories, and a power of at least 70% above 1cM, reaching 95% by 4cM.
After post-processing, we were left with 1.9 million IBD blocks, 1 million of
which were at least 2cM long (at which length we estimate 85% power and a 10%
false positive rate).
Figure 6: (A) Bias in inferred length with lines $x=y$ (dotted) and a loess
fit (solid). Each point is a segment of true IBD (copied between individuals),
showing its true length and inferred length after postprocessing. Color shows
the number of distinct, nonoverlapping segments found by BEAGLE, and the
length of the vertical line gives the total length of gaps between such
segments that BEAGLE falsely inferred was not IBD (these gaps are corrected by
our postprocessing). (B) Estimated false positive rate as a function of
length. Observed rates of IBD blocks, per pair and per cM, are also displayed
for the purpose of comparison. “Nearby” and “Distant” means IBD between pairs
of populations closer and farther away than 1000km, respectively. (C) Below,
the estimated power as a function of length (black line), together with the
parametric fit $c(x)$ of equation (1) (red dotted curve).
### 4.2 Power and false positive simulations
All methods to identify haplotypic IBD rely on identifying long regions of
near identical haplotypes between pairs of individuals (referred to as
identical by state, IBS). However, long IBS haplotypes could potentially also
result from the concatenation of multiple shorter blocks of true IBD. While
such runs can contain important information about deeper population history
(e.g. Li and Durbin, 2011; Harris and Nielsen, 2012), we view them as a false
positives as they do not represent single haplotypes shared without
intervening recombination. The chance of such a false positive IBD segment
decreases as the genetic length of shared haplotype increases. However, the
density of informative markers also plays a role, because such markers are
necessary to infer regions of IBS.
#### Power
If we are to have a reasonable false positive rate, we must accept imperfect
power. Power will also vary with the density and informativeness of markers
and length of segment considered. For example, it is intuitive that segments
of genome containing many rare alleles are easier identify as IBD. Conversely,
rare immigrant segments from a population with different allele frequencies
may, if they are shared by multiple individuals within the population, cause
higher false positive rates. For these reasons, when estimating statistical
power and false positive rate, it is important to use a dataset as similar to
the one under consideration as possible. Therefore, to determine appropriate
postprocessing criteria and to estimate our statistical power, we constructed
a dataset similar to the POPRES with known shared IBD segments as follows: we
copied haploid segments randomly between 60 trio-phased individuals of
European descent (using only one from each trio) from the HapMap dataset
(haplotypes from release #21, 17/07/06 International HapMap Consortium et al.,
2007), reoriented alleles to match the strand orientation of POPRES,
substituted these for 60 individuals from Switzerland in the POPRES data, and
ran BEAGLE on the result as before. These segments were copied between single
chromosomes of randomly chosen individuals, for random lengths 0.5–20cM, with
gaps of at least 2cM between adjacent segments and without copying between the
same two individuals twice in a row. When copying, we furthermore introduced
genotyping error by flipping alleles independently with probability .002 and
marking the allele missing with probability .023 (error rates were determined
from duplicated individuals in the sample as given by Nelson et al. (2008)).
An important feature of the inferred data was that BEAGLE often reported true
segments longer than about 5cM as two or more shorter segments separated by a
short gap, which led us to merge blocks as described above.
#### Length bias
We also need a reasonably accurate assessment of our bias and false positive
rates for our inference of numbers of genetic ancestors from the IBD length
spectrum. Although the estimated IBD lengths were approximately unbiased, we
fit a parametric model to the relationship between true and inferred lengths
after removing inferred blocks less than 1cM long. A true IBD block of length
$x$ is missed entirely with probability $1-c(x)$, and is otherwise inferred to
have length $x+\epsilon$; with probability $\gamma(x)$ the error $\epsilon$ is
positive; otherwise it is negative and conditioned to be less than $x$. In
either case, $\epsilon$ is exponentially distributed; if $\epsilon>0$ its mean
is $1/\lambda_{+}(x)$, while if $\epsilon<0$ its (unconditional) mean is
$1/\lambda_{-}(x)$. The parametric forms were chosen by examination of the
data; these are, with final parameter values:
$\displaystyle\begin{split}c(x)&=1-1/\left(1+.077x^{2}\exp(.54x)\right)\\\
\gamma(x)&=.34\left(1-(1+.51(x-1)^{+}\exp(.68(x-1)^{+}))^{-1}\right)\\\
\lambda_{+}(x)&=1.40\\\ \lambda_{-}(x)&=\min(.40+1/(.18x),12)\end{split}$ (1)
where $z^{+}=\max(z,0)$. The parameters were found by maximum likelihood,
using constrained optimization as implemented in the R package optim (R
Development Core Team, 2012) separately on three independent pieces: the
parameters in $c(x)$ and $\gamma(x)$; the parameters in $\lambda_{-}$; and
finally the parameters in $\lambda_{+}$; the fit is shown in supplemental
figure S10.
#### False positive rate
To estimate the false positive rate, we randomly shuffled segments of diploid
genome between individuals from the same population (only those 12 populations
with at least 19 samples) so that any run of IBD longer than about 0.5cM would
be broken up among many individuals. Specifically, as we read along the genome
we output diploid genotypes in random order; we shuffled this order by
exchanging the identity of each output individual with another at independent
increments chosen uniformly between 0.1 and 0.2cM. This ensured that no output
individual had a continuous run of length longer than 0.2cM copied from a
single input individual, while also preserving linkage on scales shorter than
0.1cM. The results are shown in figure 6B; from these we estimate that the
mean density of false positives $x$ cM long per pair and per cM is
approximately
$f(x)=\exp(-13-2x+4.3\sqrt{x}),$ (2)
a parametric form again chosen by examination of the data and fit by maximum
likelihood.
We found that overall, the false positive rate was around $1/10^{\mathrm{th}}$
of the observed rate, except for very long blocks (longer than 5cM or so,
where it was close to zero), and for very short blocks (less than 1cM, where
it approached 0.4). As fastIBD depends on estimating underlying haplotype
frequencies, it is expected to have a higher false positive rate in
populations that are more differentiated from the rest of the sample. There
was significant variation in false positive rate between different
populations, with Spain, Portugal, and Italy showing significantly higher
false positive rates than the other populations we examined – see supplemental
figure S11 – however, the variation was significant only for blocks shorter
than 2cM across all population pairs, with the exception of pairs of
Portuguese individuals, where the upwards bias may be significant as high as
4cM.
#### Differential sample sizes
Finally, one concern is that as fastIBD calls IBD based on a model of
haplotype frequencies in the sample it may be unduly affected by the large-
scale sample size variation across the POPRES sample. In particular, the
French-speaking Swiss sample is very large, which could lead to systematic
bias in calling IBD in populations closely to the Swiss samples. To
investigate this, we randomly discarded 745 French-speaking Swiss (all but 100
of these), and a random sampling of the remaining populations (removing 812 in
total, leaving 1445). We then ran BEAGLE on chromosome 1 of these individuals,
postprocessing in the same way as for the full sample. Reassuringly, there was
high concordance between the two – we found that 98% (95%) of the blocks
longer than 2cM found in the analysis with the full dataset (respectively,
with the subset) were found in both analyses. Overall, more blocks were found
by the analysis with the smaller dataset; however, by adjusting the score
cutoff by a fixed amount this difference could be removed, leaving nearly
identical length distributions between the two analyses. This is a known
attribute of the fastIBD algorithm, and can alternatively be avoided by
adjusting the model complexity .
We then tested the extent to which the effect of sample size varied by
population, for IBD blocks in several length categories (binning block lengths
at 1, 2, 4, and 10cM). Suppose that $F_{xy}$ is the number of IBD blocks found
between populations $x$ and $y$ in the analysis of the full dataset, and
$S_{xy}$ is the number found in the analysis of the smaller dataset (counted
between the same individuals each time). We then assume that $F_{xy}$ and
$S_{xy}$ are Poisson with mean $\lambda^{F}_{xy}$ and $\lambda^{S}_{xy}$,
respectively, so that conditioned on $N_{xy}=F_{xy}+S_{xy}$ (the total number
of blocks), $S_{xy}$ is binomial with parameters $N_{xy}$ and
$p_{xy}=\lambda^{S}_{xy}/(\lambda^{S}_{xy}+\lambda^{F}_{xy})$. We are looking
for deviations from the null model under which the effect of smaller sample
size affects all population pairs equally, so that
$\lambda^{S}_{xy}=C\;\lambda^{F}_{xy}$ for some constant $C$. We therefore fit
a binomial GLM (McCullagh and Nelder, 1989) with a logit link, with terms
corresponding to each population – in other words,
$\displaystyle
p_{xy}=\left(1+\exp\left(-\alpha_{0}-\alpha_{x}-\alpha_{y}\right)\right)^{-1}.$
We found statistically significant variation by population (i.e. several
nonzero $\alpha_{x}$), but all effect sizes were in the range of 0–4%;
estimated parameters are listed in supplemental table S1. Notably, the
coefficient corresponding to the French-speaking Swiss (the population with
the largest change in sample size) was fairly small. We also fit the model not
assuming additivity when $x=y$, i.e. adding coefficients $\alpha_{xx}$ to the
formula for $p_{xx}$, but these were not significant. These results suggest
that sample size variation across the POPRES data has only minor effects on
the distribution of IBD blocks shared across populations.
### 4.3 IBD rates along the genome
To look for regions of unusual levels of IBD and to examine our assumption of
uniformity, we compared the density of IBD tracts of different lengths along
the genome,
in supplemental figure S1. To do this, we first divided blocks up into
nonoverlapping bins based on length, with cutpoints at 1, 2.5, 4, 6, 8, and
10cM. We then computed at each SNP the number of IBD blocks in each length bin
that covered that site. To control for the effect of nearby SNP density on the
ability to detect IBD, we then computed the residuals of a linear regression
predicting number of overlapping IBD blocks using the density of SNPs within
3cM. To compare between bins, we then normalized these residuals, subtracting
the mean and dividing by the standard deviation; these “z-scores” for each SNP
are shown in figure S1.
### 4.4 Correlations in IBD rates across populations
We noted repeated patterns of IBD sharing across multiple populations (seen in
supplemental figure S3), in which certain sets of populations tended to show
similar patterns of sharing. To quantify this, we computed correlations
between mean numbers of IBD blocks; in supplemental figure S7 we show
correlations in numbers blocks of various lengths. Specifically, if $I(x,y)$
is the mean number of IBD blocks of the given length shared by an individual
from population $x$ with a (different) individual from population $y$, there
are $n$ populations, and $\bar{I}(x)=(1/(n-1))\sum_{y\neq x}I(x,y)$, then
figure S7 shows for each $x$ and $y$,
$\frac{1}{n-2}\sum_{z\notin\\{x,y\\}}(I(x,z)-\bar{I}(x))(I(y,z)-\bar{I}(y)),$
(3)
the (Pearson) correlation between $I(x,z)$ and $I(y,z)$ ranging across
$z\notin\\{x,y\\}$. Other choices of block lengths are similar, although
shorter blocks show higher overall correlations (due in part to false
positives) and longer blocks show lower overall correlations (as rates are
noisier, and sharing is more restricted to nearby populations). The geographic
groupings of table 1 were then chosen by visual inspection.
### 4.5 Substructure
We assessed the overall degree of substructure within each population, by
measuring, for each $x$ and $y$, the degree of inhomogeneity across
individuals of population $x$ for shared ancestry with population $y$. We
measured inhomogeneity by the standard deviation in number of blocks shared
with population $y$, across individuals of population $x$. We assessed the
significance by a permutation test, randomly reassigning each block shared
between $x$ and $y$ to a individual chosen uniformly from population $x$, and
recomputing the standard deviation, 1000 times. (If there are $k$ blocks
shared between $x$ and $y$ and $m$ individuals in population $x$, this is
equivalent to putting $k$ balls in $m$ boxes, tallying how many balls are in
each box, and computing the sample standard deviation of the resulting list of
numbers.) Note that some degree of inhomogeneity of shared ancestry is
expected even within randomly mating populations, due to randomness of the
relationship between individuals in the pedigree. These effects are likely to
be small if the relationships are suitably deep, but this is still an area of
active research (Henn et al., 2012b; Carmi et al., 2012). The resulting
$p$-values are shown in supplemental figure S2. We did not analyze these in
detail, particularly as we had limited power to detect substructure in
populations with few samples, but note that a large proportion (47%) of the
population pairs showed greater inhomogeneity than in all 1000 permuted
samples (i.e. $p<.001$). Some comparisons even with many samples in both
populations (where we have considerable power to detect even subtle
inhomogeneity) showed no structure whatsoever – in particular, the
distribution of numbers of Italian IBD blocks shared by Swiss individuals is
not distinguishable from Poisson, indicating a high degree of homogeneity of
Italian ancestry across Switzerland.
### 4.6 Single-site summaries
To assess the single marker measures of relatedness across the POPRES sample
we calculated pairwise identity by state, the probability that two alleles
sampled at random from a pair of individuals are identical, averaged across
SNPs. This was calculated for all pairs of individuals using the “\--genome”
option in PLINK v1.07 (Purcell et al., 2007,
http://pngu.mgh.harvard.edu/purcell/plink), and is shown in Supplementary
Figure S9 with points colored as in Figure 3.
We also calculated principal components of the POPRES genotype data using the
EIGENSOFT package v3.0 (Patterson et al., 2006), which were used in
identifying outlying individuals and in producing figures S4, S5, and S6.
### 4.7 Inferring ages of common ancestors
Here, our aim is to use the distribution of IBD block lengths to infer how
long ago the genetic common ancestors were alive from which these IBD blocks
were inherited. A pair of individuals who share a block of IBD of genetic
length at least $x$ have each inherited contiguous regions of genome from a
single common ancestor $n$ generations ago that overlap for length at least
$x$. If we start with the population pedigree, those ancestors from which the
two individuals might have inherited IBD blocks are those that can be
connected to both by paths through the pedigree. The distribution of possible
IBD blocks is determined by the number of links (i.e. the number of meioses)
occurring along the two paths.
Throughout the article we informally often refer to ancestors living a certain
“number of generations in the past” as if humans were semelparous with a fixed
lifetime. Keeping with this, it is natural to write the number of IBD blocks
shared by a pair of individuals as the sum over past generations of the number
of IBD blocks inherited from that generation. In other words, if $N(x)$ is the
number of IBD blocks of genetic length at least $x$ shared by two individual
chromosomes, and $N_{n}(x)$ is the number of such IBD blocks inherited by the
two along paths through the pedigree having a total of $n$ meioses, then
$N(x)=\sum_{n}N_{n}(x)$. Therefore, averaging over possible choices of pairs
of individuals, the mean number of shared IBD blocks can be similarly
partitioned as
$\mathbb{E}[N(x)]=\sum_{n\geq 1}\mathbb{E}[N_{n}(x)].$ (4)
In each successive generation in the past each chromosome is broken up into
successively more pieces, each of which has been inherited along a different
path through the pedigree, and any two such pieces of the two individual
chromosomes that overlap and are inherited from the same ancestral chromosome
contribute one block of IBD. Therefore, the mean number of IBD blocks coming
from $n/2$ generations ago is the mean number of possible blocks multiplied by
the probability that a particular block is actually inherited by both
individuals from the same genealogical ancestor in generation $n/2$. Allowing
for overlapping generations, the first part we denote by $K(n,x)$, the mean
number of pieces of length at least $x$ obtained by cutting the chromosome at
the recombination sites of $n$ meioses, and the second part we denote by
$\mu(n)$, the probability that the two chromosomes have inherited at a
particular site along a path of total length $n$ meioses (e.g. their common
ancestor at that site lived $n/2$ generations ago). Multiplying these and
summing over possible paths, we have that
$\mathbb{E}[N(x)]=\sum_{n\geq 1}\mu(n)K(n,x),$ (5)
i.e. the mean rate of IBD is a linear function of the distribution of the time
back to the most recent common ancestor averaged across sites. The
distribution $\mu(n)$ is more precisely known as the coalescent time
distribution (Kingman, 1982; Wakeley, 2005), in its obvious adaptation to
population pedigrees. As a first application, note that the distribution of
ages of IBD blocks above a given length $x$ depends strongly on the
demographic history – a fraction $\mu(n)K(n,x)/\sum_{m}\mu(m)K(m,x)$ of these
are from paths $n$ meioses long.
Furthermore, it is easy to calculate that for a chromosome of genetic length
$G$,
$K(n,x)=\left(n(G-x)+1\right)\exp(-xn),$ (6)
assuming homogeneous Poisson recombination on the genetic map (as well as
constancy of the map and ignoring the effect of interference, which is a
reasonable for the range of $n$ we consider). The mean number of IBD blocks of
length at least $x$ shared by a pair of individuals across the entire genome
is then obtained by summing (5) across all chromosomes, and multiplying by
four (for the four possible chromosome pairs).
Equations (5) and (6) give the relationship between lengths of shared IBD
blocks and how long ago the ancestor lived from whom these blocks are
inherited. Our goal is to invert this relationship to learn about $\mu(n)$,
and hence the ages of the common ancestors underlying our observed
distribution of IBD block lengths. To do this, we first need to account for
sampling noise and estimation error. Suppose we are looking at IBD blocks
shared between any of a set of $n_{p}$ pairs of individuals, and assume that
$N(y)$, the number of observed IBD blocks shared between any of those pairs of
length at least $y$, is Poisson distributed with mean $n_{p}M(y)$, where
$\displaystyle M(y)$ $\displaystyle=\int_{y}^{G}f(z)+\sum_{n\geq
1}\mu(n)\left(\int_{0}^{G}c(x)R(x,z)dK(n,x)\right)dz,\quad\mbox{with}$ (7)
$\displaystyle R(x,y)$
$\displaystyle=\begin{cases}\gamma(x)\lambda_{+}(x)\exp(-\lambda_{+}(x)(y-x))\quad&\mbox{for}\;y>x\\\
(1-\gamma(x))\lambda_{-}\exp(-\lambda_{-}(x)(x-y))/(1-\exp(-\lambda_{-}(x)x))\quad&\mbox{for}\;y<x.\end{cases}$
(8)
Here the false positive rate $f(z)$, power $c(x)$ and the components of the
error kernel $R(x,y)$ are estimated as above, with parametric forms given in
equations (2) and (1). The Poisson assumption has been examined elsewhere
(e.g. Fisher, 1954; Huff et al., 2011), and is reasonable because there is a
very small chance of having inherited a block from each pair of shared
genealogical ancestors; there a great number of these, and if these events are
sufficiently independent, the Poisson distribution will be a good
approximation (see e.g. Grimmett and Stirzaker, 2001). If this holds for each
pair of individuals, the total number of IBD blocks is also Poisson
distributed, with $M$ given by the mean of this number across all constituent
pairs. (Note that this does not assume that each pair of individuals has the
same mean number, so does not assume that our set of pairs are a homogeneous
population.)
We have therefore a likelihood model for the data, with demographic history
(parametrized by $\mu=\\{\mu(n):n\geq 1\\}$) as free parameters.
Unfortunately, the problem of inferring $\mu$ is ill-conditioned (unsuprising
due to its similarity of the kernel (6) to the Laplace transform, see Epstein
and Schotland, 2008), which in this context means that the likelihood surface
is flat in certain directions (“ridged”): for each IBD block distribution
$N(x)$ there is a large set of coalescent time distributions $\mu(n)$ that fit
the data equally well. A common problem in such problems is that the
unconstrained maximum likelihood solution is wildly oscillatory; in our case,
the unconstrained solution is not so obviously wrong, since we are helped
considerably by the knowledge that $\mu\geq 0$. For reviews of approaches to
such ill-conditioned inverse problems, see e.g. Petrov and Sizikov (2005) or
Stuart (2010); the problem is also known as “data unfolding” in particle
physics (Cowan, 1998). If one is concerned with finding a point estimate of
$\mu$, most approaches add an additional penalty to the likelihood, which is
known as “regularization” (Tikhonov and Arsenin, 1977) or “ridge regression”
(Hoerl and Kennard, 1970).
However, our goal is parametric inference, and so we must describe the limits
of the “ridge” in the likelihood surface in various directions, (which can be
seen as maximum a posteriori estimates under priors of various strengths).
To do this, we first discretize the data, so that $N_{i}$ is the number of IBD
blocks shared by any of a total of $n_{p}$ distinct pairs of individuals with
inferred genetic lengths falling between $x_{i-1}$ and $x_{i}$. We restrict to
blocks having a minimum length of 2cM long, so that $x_{0}=2$. To find a
discretization so that each $N_{i}$ has roughly equal variance, we choose
$x_{i}$ by first dividing the range of block lengths into 100 bins with equal
numbers of blocks falling in each; discard any bins longer than 1cM; and
divide the remainder of the range up into 1cM chunks. To further reduce
computational time, we also discretize time, effectively requiring $\mu_{n}$
to be constant on each interval $n_{j}\leq n<n_{j+1}$, with
$n_{j+1}-n_{j}=\lfloor{j/10}\rfloor$, for $1\leq j\leq 360$ – so the
resolution is finest for recent times, and the maximum time depth considered
is 6660 meioses, or 99900 years ago. (The discretization and upper bound on
time depth were found to not affect our results.) We then compute by numerical
integration (using the function integrate in R) the matrix $L$ discretizing
the kernel given in (7), so that
$L_{in}=\int_{x_{i-1}}^{x_{i}}\int_{0}^{G}c(x)R(x,z)dK(n,x)dz$ is the kernel
that applied to $\mu$ gives the mean number of true IBD blocks per pair
observed with lengths between $x_{i-1}$ and $x_{i}$, and
$f_{i}=\int_{x_{i-1}}^{x_{i}}f(z)dz$ is the mean number of false positives per
pair with lengths in the same interval. We then sum across chromosomes, as
before. The likelihood of the data is thus
$\exp\left(-n_{p}\sum_{i,n}L_{in}\mu_{n}+f_{i}\right)\prod_{i}\frac{1}{N_{i}!}\left(n_{p}\sum_{n}L_{in}\mu_{n}+f_{i}\right)^{N_{i}}.$
(9)
To the (negative) log likelihood we add a penalization $\gamma$, after
rescaling by the number of pairs $n_{p}$ (which does not affect the result but
makes penalization strengths comparable between pairs of populations), and use
numerical optimization (the L-BFGS-B method in optim, R Development Core Team,
2012) to minimize the resulting functional (which omits terms independent of
$\mu$)
$\mathcal{L}(\mu;\gamma,N)=\sum_{i,n}L_{in}\mu_{n}-\sum_{i}\frac{N_{i}}{n_{p}}\log\left(\sum_{n}L_{in}\mu_{n}+f_{i}\right)+\frac{\gamma(\mu)}{n_{p}}.$
(10)
Often we will fix the functional form of the penalization and vary its
strength, so that $\gamma(\mu)=\gamma_{0}z(\mu)$, in which case we will write
$\mathcal{L}(\mu;\gamma_{0},N)$ for $\mathcal{L}(\mu;\gamma_{0}\,z(\mu),N)$.
For instance, the leftmost panels in figure 4 show the minimizing solutions
$\mu$ for $\gamma(\mu)=0$ (no penalization) and for
$\gamma(\mu)=\gamma_{0}\sum_{n}(\mu_{n+1}-\mu_{n})^{2}$ (“roughness”
penalization). Because our aim is to describe extremal reasonable estimates
$\mu$, in this and in other cases, we have chosen the strength of penalization
$\gamma_{0}$ to be “as large as is reasonable”, choosing the largest
$\gamma_{0}$ such that the minimizing $\mu$ has log likelihood differing by no
more than 2 units from the unconstrained optimum. This choice of cutoff can be
justified as in Edwards (1984), gave quite similar answers to other methods,
and performed well on simulated population histories (see supplemental
document).
This can be thought of as taking the strongest prior that still gives us
“reasonable” maximum a posteriori answers. Note that the optimization is over
nonnegative distributions $\mu$ also satisfying $\sum_{n}\mu(n)\leq 1$
(although the latter condition does not enter in practice).
We would also like to determine bounds on total numbers of shared genetic
ancestors who lived during particular time intervals, by determining e.g. the
minimum and maximum numbers of such ancestors that are consistent with the
data. Such bounds are shown in figure 5. To obtain a lower bound for the time
period between $n_{1}$ and $n_{2}$ generations, we penalized the total amount
of shared ancestry during this interval, using the penalizations
$\gamma_{-}(\mu)=\gamma_{0}^{-}\left(\sum_{n=n_{1}}^{n_{2}}\mu(n)\right)^{2}$,
and choosing $\gamma_{0}^{-}$ to give a drop of 2 log likelihood units, as
described above. The lower bound is then the total amount of coalescence
$\sum_{n=n_{1}}^{n_{2}}\mu_{-}(n)$ for $\mu_{-}$ minimizing
$\mathcal{L}(\cdot;\gamma_{-},N)$. The upper bound is found by penalizing
total shared ancestry outside this interval, i.e. by applying the penalization
$\gamma_{+}(\mu)=\gamma_{0}^{+}\left(\sum_{n<n_{1}}\mu(n)+\sum_{n>n_{2}}\mu(n)\right)^{2}$.
It is almost always the case that lower bounds are zero, since there is
sufficient wiggle room in the likelihood surface to explain the observed block
length distribution using peaks just below $n_{1}$ and above $n_{2}$. Examples
are shown in supplemental figure S14. On the other hand, upper bounds seem
fairly reliable.
In the above we have assumed that the minimizer of $\mathcal{L}$ is unique,
thus glossing over e.g. finding appropriate starting points for the
optimization. In practice, we obtained good starting points by solving the
natural approximating least-squares problem, using quadprog (Turlach and
Weingessel, 2011) in R. We then evaluated uniqueness of the minimizer by using
different starting points, and found that if necessary, adding only a very
small penalization term was enough to ensure convergence to a unique solution.
#### Testing the method
To test this method, we implemented a whole-genome coalescent IBD simulator,
and applied our inference methods to the results under various demography
scenarios. We also used these simulations to evaluate the sensitivity of our
method to misestimation of power or false positive rates. The simulations, and
the results, are described in Supplemental Document 1; in all cases, the
simulations showed that the method performed well to the level of uncertainty
discussed throughout the text and confirmed our understanding of the method
described above. We also found that misestimation of false positive rate only
affects estimated numbers of common ancestors by a comparable amount, and that
misestimation of blocks less than 4cM long mostly affects estimates older than
about 2000 years. Therefore, if our false positive rates above 2cM are off by
10% (the range that seems reasonable), which would change our estimated
numbers of blocks by about 1%, this would only change our estimated numbers of
shared ancestors by a few percent.
#### Extending to shorter blocks
We only used blocks longer than 2cM to infer ages of common ancestors, in part
because the model we use does not seem to fit the data below this threshold.
Attempts to apply the methods to all blocks longer than 1cM reveals that there
is no history of rates of common ancestry that, under this model, produces a
block length distribution reasonably close to the one observed – small, but
significant deviations occur below about 2cM. This occurs probably in part
because our estimate of false positive rate is expected to be less accurate at
these short lengths. Furthermore, our model does not explicitly model the
overlap of multiple short IBD segments to create on long segment deriving from
different ancestors, which could start to have a significant effect at short
lengths. (The effect on long blocks we model as error in length estimation.)
This could be incorporated into a model (in a way analogous to Li and Durbin,
2011), but consideration of when several contiguous blocks of IBD might have
few enough differences to be detected as a long IBD block quickly runs into
the need for a model of IBD detection, which we here treat as a black box. Use
of these shorter blocks, which would allow inference of older ancestry, will
need different methods, and probably sequencing rather than genotyping data.
### 4.8 Numbers of common ancestors
Estimated numbers of genetic common ancestors can be found by simply solving
for $N(0)$ using an estimate of $\mu(n)$ in equations (5) and (6) (still
restricting to genetic ancestors on the autosomes). These tell us that given
the distribution $\mu(n)$, the mean number of genetic common ancestors coming
from generation $n/2$ – i.e. the mean number of IBD blocks of any length
inherited from such common ancestors – is
$N(0)=\mu(n)\sum_{k=1}^{22}\left(nG_{k}+1\right)$, where $G_{k}$ is the total
sex-averaged genetic length of the $k^{\mathrm{th}}$ human chromosome. Since
the total sex-averaged map length of the human autosomes is about 32 Morgans,
this is about $\mu(n)(32n+22)$. This procedure has been used in figures 4 and
5.
Converting shared IBD blocks to numbers of shared genealogical common
ancestors is more problematic. Suppose that modern-day individuals $a$ and $b$
both have $c$ as a grandn-1parent. Using equation (6) at $x=0$, we know that
the mean number of blocks that $a$ and $b$ both inherit from $c$ is $r(2n)$,
with $r(n):=2^{-n}(32n+22)$, since each block has chance $2^{-2n}$ of being
inherited across $2n$ meioses. First treat the endpoints of each distinct path
of length $n$ back through the pedigree as a grandn-1parent, so that everyone
has exactly $2^{n}$ grandn-1parents, and some ancestors will be
grandn-1parents many times over. Then if $a$ and $b$ share $m$ genetic
grandn-1parents, a moment estimator for the number of genealogical
grandn-1parents is $m/r(n)$. However, the geometric growth of $r(n)$ means
that small uncertainties in $n$ have large effects on the estimated numbers of
genealogical common ancestors – and we have large uncertainties in $n$.
Despite these difficulties, we can still get some order-of-magnitude
estimates. For instance, we estimate that someone from Hungary shares on
average about 5 genetic common ancestors with someone from the UK between 18
and 50 generations ago. Since $1/r(36)=5.8\times 10^{7}$, we would
conservatively estimate that for every genetic common ancestor there are tens
of millions of genealogical common ancestors. Most of these ancestors must be
genealogical common ancestors many times over, but these must still represent
at least thousands of distinct individuals.
## Acknowledgements
Thanks to Razib Khan, Sharon Browning, and Don Conrad for several useful
discussions, and to Jeremy Berg, Ewan Birney, Yaniv Brandvain, Joe Pickrell,
Jonathan Pritchard, Alisa Sedghifar, and Joel Smith for useful comments on
earlier drafts. We also thank the four anonymous reviewers, as well as Amy
Williams (at Haldane’s sieve), for their helpful suggestions.
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## Supplemental material
Figure S1: Normalized density of IBD blocks of different lengths, corrected
for SNP density, across all autosomes (see section 4.3 for details). Marked
with a grey bar and “c” are the centromeres; and marked with “8p” is a large,
segregating inversion (Giglio et al., 2001). The grey curve along the bottom
shows normalized SNP density. Figure S2: Two measures of overdispersal of
block numbers across individuals (i.e. substructure): Suppose we have $n$
individuals from population $x$, and $N_{iy}$ is the number of IBD blocks of
length at least 1cM that individual $i$ shares with anyone from population
$y$. Our statistic of substructure within $x$ with respect to $y$ is the
variance of these numbers,
$s_{xy}=\frac{1}{n-1}\left(\sum_{i}N_{iy}^{2}-\frac{1}{n}\left(\sum_{i}N_{iy}\right)^{2}\right)$.
We obtained a “null” distribution for this statistic by randomly reassigning
all blocks shared between $x$ and $y$ to an individual from $x$, and used this
to evaluate the strength and the statistical significance of this
substructure. (A) Histogram of the “$p$-value”, of the proportion of 1000
replicates that showed a variance greater than or equal to the observed
variance $s_{xy}$, for all pairs of populations $x$ and $y$ with at least 10
individuals in population $y$. (B) The “$z$ score”, which is observed value
$s_{xy}$ minus mean value divided by standard deviation, estimated using 1000
replicates. The population $x$ is shown on the vertical axis, with text labels
giving $y$, so for instance, Italians show much more substructure with most
other populations than do Irish. Note that sample size still has a large
effect – it is easier to see substructure with respect to the Swiss French
($x=$CHf) because the large number of Swiss French samples allows greater
resolution. A vertical line is shown at $z=5$. Only pairs of populations with
at least 3 samples in country $x$ and 10 samples in country $y$ are shown.
Because of the log scale, only pairs with a positive $z$ score are shown, but
no comparisons had $z<-2.5$, and only three had $z<-2$. Figure S3: (A) Mean
numbers of IBD blocks of length at least 1cM per pair of individuals, shown as
a modified Cleveland dotchart, with $\pm$2 standard deviations shown as
horizontal lines. For instance, on the bottom row we see that someone from the
UK shares on average about one IBD block with someone else from the UK and
slightly less than 0.2 blocks with someone from Turkey. Note that in most
cases, the distribution of block numbers is fairly concentrated, and that
nearby populations show quite similar patterns. Figure S4: The positions of
our sample of the first two principal components of the genotype matrix, as
produced by EIGENSTRAT (Price et al., 2006). Population centroids are marked
by text and a transparent circle. Note the correspondence to a map of Europe,
after a rotation and flip. Figure S5: Comparison of figure 2A in the main
text to figure S4 – the axes are self-explanatory; the colors and symbols are
the same as in figure 2A. Figure S6: Comparison of figure 2B in the main
text to figure S4 – the axes are self-explanatory; the colors and symbols are
the same as in figure 2B. The four outlying UK individuals are, as in figure
2B, three who share a very high number of IBD blocks with Italians, and one
who shares a very high number with the Slovakian sample. Figure S7:
Correlations in IBD rates, for six different length windows (omitted length
windows are similar). If there are $n$ populations, $I(x,y)$ is the mean
number of blocks in the given length range shared by a pair from populations
$x$ and $y$, and $\bar{I}(x)=(1/(n-1))\sum_{z\neq x}I(x,z)$, shown is
$(1/(n-2))\sum_{z\notin\\{x,y\\}}(I(x,z)-\bar{I}(x))(I(y,z)-\bar{I}(y))$.
http://www.eve.ucdavis.edu/plralph/ibd/sharing-rates.svg
Figure S8: The same plot as figure 3G–I, but rendered as an SVG figure with
tooltips that allow identification of individual points (using R Plate, 2011)
– open the file in a reasonably compliant browser (e.g. Firefox, Opera) or SVG
browser (e.g. squiggle) and hover the mouse over a point of interest to see
the label. Figure S9: Mean IBS (“Identity by State”) against geographic
distance, calculated using plink (Purcell et al., 2007) as described in the
main text, using the same groups and fitting the same curves as in figure 3 of
the main text. The lowest set of points, roughly following a line, are mean
IBS with Turkey; unlike with IBD, mean IBS with Cyprus was significantly
higher. In fact, the other rough line of points (between the comparisons to
Turkey and the orange points) is almost entirely mean IBS with Cyprus, as well
as mean IBS to Slovakia. Since Slovakia is only represented by a single
individual in the dataset, we cannot reach further conclusions. Figure S10:
Goodness-of-fit for our estimated error distribution – points show data from
simulations (described in the text), and lines show the parametric forms of
equations (1). Each simulated IBD block of length $x$ was either found by
BEAGLE (and passed our filters) or was not; and if it was found, it had
inferred length $y=x+\epsilon$, i.e. with length error $\epsilon$. The top
figure shows the probability that a segment of a given length is missed
entirely (and $1-c(x)$) in green, the probability that $\epsilon>0$ given the
segment was found (and $\gamma(x)$) in black, and the probability that
$\epsilon\leq 0$ given the segment was found (and $1-\gamma(x)$) in red. The
second figure shows the probability density of all positive $\epsilon$ (in
black, with $\lambda_{+}(x)$), and probability densities of positive
$\epsilon$ for various categories of true length $x$ (colors). The third
figure is similar to the second, except that it shows negative $\epsilon$.
Note that blocks with inferred length $y<1$ were omitted. Figure S11:
Estimated false positive rates per pair, compared to the observed rate, as a
function of block length. The black dotted curves show the mean number of IBD
blocks per pair observed in the false positive simulations (see section 4.2 of
the main text), per centiMorgan, binned at 0, 1.2, 1.5, 1.8, 2.1, 2.4, 2.7,
3.2, 4.5, and 7.5cM, and the parametric fit described in the text. The colored
curves show the same quantity, separately for each pair of country
comparisons, with the extreme values labeled. No comparisons other than
Portugal–Portugal show any significant deviations from the parametric fit
above 2cM. For comparison, the black solid curve shows the mean observed IBD
rate across the same set of individuals; note that e.g. the false positive
rate pairs of Portuguese individuals is higher than this at short lengths
because the observed IBD rate between Portuguese at short block lengths is
much higher than the overall mean. Figure S12: Estimated total numbers of
genetic common ancestors shared by various pairs of populations, in roughly
the time periods 0–500ya, 500–1500ya, 1500–2500ya, and 2500–4300ya. The
population groupings are: “AL”, Albanian speakers (Albania and Kosovo); “S-C”,
Serbo-Croatian speakers in Bosnia, Croatia, Serbia, Montenegro, and
Yugoslavia; “R-B”, Romania and Bulgaria; “UK”, United Kingdom, England,
Scotland, Wales; “Iber”, Spain and Portugal; “Bel”, Belgium and the
Netherlands; “Bal”, Latvia, Finland, Sweden, Norway, and Denmark; and denotes
a single population with the same abbreviations as in table 1 otherwise.
Figure S13: For those who are used to thinking in effective population sizes,
the equivalent figure to figure S12, except with coalescent rate on the
vertical axis, rather than numbers of most recent genetic common ancestors.
Figure S14: An example of the set of consistent histories (as coalescent
distributions $\mu(n)$) used to find upper and lower bounds in figures S12 and
5. The example shown is Poland–Germany; “MLE” is the maximum likelihood
history; “smooth” is the smoothest consistent history; and the remaining plots
show the histories giving lower and upper bounds for the referenced time
intervals (in numbers of generations). In each case, the segment of time on
which we are looking for a bound is shaded. Figure S15: For those who are
used to thinking in effective population sizes, the equivalent figure to
figure 4, except with coalescent rate on the vertical axis, rather than
numbers of most recent genetic common ancestors. Figure S16: The maximum
likelihood history (grey) and smoothest consistent history (red) for all pairs
of population groupings of figure S12 (including those of figure 5). Each
panel is analogous to a panel of figure 4; time scale is given by vertical
grey lines every 500 years. For these plots on a larger scale see supplemental
figure S17.
http://www.eve.ucdavis.edu/$\sim$plralph/ibd/boxplotted-inversions.pdf
Figure S17: All inversions shown in S16, one per page (225 pages total).
There is one page per pair of comparisons used in figure 5. On each page,
there is one large plot, showing 10 distinct consistent histories (numbers of
genetic ancestors back through time), and below are 10 histograms of IBD block
length, one for each consistent history, showing both the observed
distribution and the partitioning of blocks into age categories predicted by
that history. The names of the two groupings are shown in the upper right:
“pointy” is the unconstrained maximum likelihood solution; “smooth” is the
smoothest consistent history; “$a$–$b$ lower” is the history used to find the
lower bound for the time period $a$–$b$ generations ago in figure 5; and
“$a$–$b$ upper” is the history used to find the corresponding upper bound.
Each of these are described in more detail in the Methods.
| 0-1 cM | 1-2 cM | 2-4 cM | 4-10 cM
---|---|---|---|---
(Intercept) | 0.08313 | 0.1436 ** | 0.07034 | -0.0193
Albania | -0.00097 | 0.0063 | -0.04232 | -0.0323
Austria | 0.36424 | 0.0365 | 0.05874 | -0.1399
Belgium | -0.02009 | 0.0863 | 0.02874 | 0.0374
Bosnia | -0.13914 | -0.0327 | 0.02277 | 0.1032
Bulgaria | -0.14857 | 0.0467 | -0.21836 | 0.2427
Croatia | 0.08645 | -0.0223 | 0.02045 | -0.0371
Cyprus | -0.15556 | -0.0775 | 0.08306 | 0.2158
Czech Republic | -0.14301 | 0.1013 | 0.02656 | 0.1607
Denmark | 0.66101 | 0.1265 | 0.03675 | 0.3095
England | 0.12131 | 0.0854 | 0.03784 | 0.0199
Finland | 0.24617 | 0.1491 | -0.09775 | 0.9455
France | 0.05049 | 0.0533 | 0.09670 ** | 0.1655
Germany | 0.06837 | 0.0454 | 0.08617 ** | 0.1288
Greece | 0.06752 | -0.2266 | 0.00086 | 0.3340
Hungary | 0.04491 | -0.0070 | 0.11499 * | 0.1179
Ireland | 0.07873 | 0.0676 | 0.04292 | 0.0466
Italy | 0.02728 | 0.0218 | 0.06694 * | 0.1607
Kosovo | 0.17256 | 0.0384 | -0.02022 | 0.0109
Latvia | 0.76499 | 0.1805 | -0.07177 | -0.0787
Macedonia | -0.12898 | 0.1414 | -0.04692 | -0.0093
Montenegro | 0.50084 | -0.0845 | -0.01400 | 0.2746
Netherlands | 0.10448 | 0.0842 | 0.11875 * | 0.0423
Norway | -0.46112 | -0.0176 | 0.07972 | 0.4042
Poland | 0.21095 | 0.0021 | 0.05353 | 0.1314
Portugal | 0.03987 | -0.0019 | 0.06771 * | 0.0664
Romania | -0.24251 | 0.0545 | -0.05095 | -0.0845
Russia | 0.12047 | -0.0094 | 0.00484 | 0.1568
Scotland | 0.25426 | 0.0183 | 0.03205 | -0.1172
Serbia | 0.06965 | 0.0384 | -0.04688 | 0.1397
Slovakia | -0.00988 | -0.0458 | 0.21389 | -0.3070
Slovenia | -0.19993 | 0.1456 | -0.00855 | 0.4673
Spain | 0.02470 | 0.0268 | 0.08928 ** | 0.1084
Sweden | 0.02043 | -0.0076 | 0.20759 ** | 0.2666
Swiss French | 0.11099 | 0.1394 ** | 0.07782 * | 0.0525
Swiss German | 0.14555 | 0.0943 ** | 0.08075 ** | 0.0564
Switzerland | 0.13339 | 0.0102 | 0.04665 | 0.1135
Turkey | -0.01107 | 0.0762 | 0.13784 | -0.1003
Ukraine | -0.33535 | -0.0543 | 0.03820 | -1.2068
United Kingdom | 0.10379 | 0.0868 ** | 0.11605 ** | 0.1309
Yugoslavia | -0.10508 | 0.0436 | -0.00218 | 0.0018
Table S1: Estimated coefficients describing the effect of changing population sample size, as described in the text (section 4.2, “Differential sample sizes”). Stars denote statistical significance: “*” corresponds to $p<.05$ and “**” corresponds to $p<.01$. The coefficients are from a binomial GLM with a logit link function, applied to the number of IBD segments detected in the same set of individuals run with and without an additional 812 individuals. For instance, the top three entries in the lefthand column tell us that if $F$ is the number of segments greater than 1cM found between Albanian and Austrian individuals in analysis with the full dataset, and $S$ is the corresponding number in the analysis with only the subset, that the model predicts that $S/(S+F)\approx(1+\exp(-0.08313+0.00097-0.36424))^{-1}=0.61$ (plus binomial sampling noise). Note that coefficients producing effect sizes larger than 4% (e.g. Austria for 0–1cM) all correspond to populations with small sample sizes, and are not significant. COUNTRY_SELF | COUNTRY_GFOLX | PRIMARY_LANGUAGE | Population | $n$
---|---|---|---|---
Albania | Albania | Albanian | Albania | 3
Yugoslavia | Serbia | Albanian | Albania | 1
Yugoslavia | Yugoslavia | Albanian | Albania | 5
Austria | | German | Austria | 3
Austria | Austria | German | Austria | 10
Spain | Austria | German | Austria | 1
Belgium | Belgium | Dutch | Belgium | 4
Belgium | Belgium | Flemish | Belgium | 3
Belgium | Belgium | French | Belgium | 28
Germany | Belgium | French | Belgium | 1
Switzerland | Belgium | French | Belgium | 1
Bosnia | Bosnia | Bosnian | Bosnia | 4
Bosnia | Bosnia | Serbian | Bosnia | 1
Bosnia | Bosnia | Serbo-Croatian | Bosnia | 4
Bulgaria | Bulgaria | Bulgarian | Bulgaria | 1
Croatia | | Croatian | Croatia | 1
Croatia | Croatia | Croatian | Croatia | 6
Yugoslavia | Yugoslavia | Croatian | Croatia | 1
Croatia | Croatia | Serbo-Croatian | Croatia | 1
Cyprus | | English | Cyprus | 1
Cyprus | | Greek | Cyprus | 1
Cyprus | Cyprus | Greek | Cyprus | 1
Czech Republic | Czech Republic | Czech | Czech Republic | 9
Denmark | | Danish | Denmark | 1
England | England | English | England | 18
Turkey | England | English | England | 1
United Kingdom | England | English | England | 3
Finland | Finland | Finnish | Finland | 1
France | | French | France | 2
France | France | French | France | 82
Germany | France | French | France | 1
Switzerland | France | French | France | 1
Germany | | | Germany | 1
Germany | | English | Germany | 2
Germany | Germany | French | Germany | 1
Germany | | German | Germany | 1
Germany | Germany | German | Germany | 63
Switzerland | Germany | German | Germany | 1
Hungary | Germany | Hungarian | Germany | 1
Germany | Germany | Polish | Germany | 1
Switzerland | Greece | French | Greece | 1
Greece | Greece | Greek | Greece | 4
(continued on next page)
Table S2: The composition of our populations. “COUNTRY_SELF” is the reported country of origin; “COUNTRY_GFOLX” is the country of origin of all reported grandparents (individuals with reported grandparents from different countries were removed); “PRIMARY_LANGUAGE” is the reported primary language; “Population” is our population label; and $n$ gives the number of individuals falling in this category. COUNTRY_SELF | COUNTRY_GFOLX | PRIMARY_LANGUAGE | Population | $n$
---|---|---|---|---
Hungary | Hungary | French | Hungary | 1
Hungary | Hungary | Hungarian | Hungary | 17
Hungary | Hungary | Russian | Hungary | 1
Ireland | | | Ireland | 19
Ireland | | English | Ireland | 38
England | Ireland | English | Ireland | 1
Ireland | Ireland | English | Ireland | 1
Ireland | Ireland | French | Ireland | 1
Italy | | | Italy | 1
France | Italy | French | Italy | 1
Italy | Italy | French | Italy | 8
Switzerland | Italy | French | Italy | 9
Italy | Italy | German | Italy | 1
Italy | | Italian | Italy | 3
France | Italy | Italian | Italy | 1
Italy | Italy | Italian | Italy | 170
Romania | Italy | Italian | Italy | 1
Sweden | Italy | Italian | Italy | 1
Switzerland | Italy | Italian | Italy | 17
Kosovo | | | Kosovo | 1
Yugoslavia | Kosovo | Albanian | Kosovo | 10
Yugoslavia | Kosovo | Kosovan | Kosovo | 2
Yugoslavia | Kosovo | Serbo-Croatian | Kosovo | 2
Latvia | Latvia | Latvian | Latvia | 1
Macedonia | Macedonia | Macedonian | Macedonia | 4
Yugoslavia | Montenegro | Serbian | Montenegro | 1
Netherlands | Netherlands | Dutch | Netherlands | 15
Holland | | English | Netherlands | 1
Netherlands | Netherlands | French | Netherlands | 1
Norway | Norway | Norwegian | Norway | 2
France | Poland | French | Poland | 1
Poland | | Polish | Poland | 4
France | Poland | Polish | Poland | 2
Poland | Poland | Polish | Poland | 15
France | Portugal | Portuguese | Portugal | 1
Portugal | Portugal | Portuguese | Portugal | 114
Romania | Romania | Romanian | Romania | 14
Romania | Russia | Romanian | Russia | 1
Russia | Russia | Russian | Russia | 5
Scotland | | English | Scotland | 3
Scotland | Scotland | English | Scotland | 2
Yugoslavia | Serbia | Hungarian | Serbia | 1
Serbia | Serbia | Serbian | Serbia | 1
Yugoslavia | Serbia | Serbian | Serbia | 4
Yugoslavia | Yugoslavia | Serbian | Serbia | 2
Croatia | Serbia | Serbo-Croatian | Serbia | 1
Yugoslavia | Serbia | Serbo-Croatian | Serbia | 2
(continued on next page)
Table S3: Continuation of table S2. COUNTRY_SELF | COUNTRY_GFOLX | PRIMARY_LANGUAGE | Population | $n$
---|---|---|---|---
Slovakia | Slovakia | Slovakian | Slovakia | 1
Italy | Slovenia | Slovene | Slovenia | 1
Slovenia | Slovenia | Slovene | Slovenia | 1
Spain | Spain | Columbia | Spain | 2
Switzerland | Spain | Columbia | Spain | 2
Spain | Spain | French | Spain | 5
Switzerland | Spain | French | Spain | 2
Spain | Spain | Galician | Spain | 2
Spain | | Spanish | Spain | 4
Spain | Spain | Spanish | Spain | 106
Switzerland | Spain | Spanish | Spain | 7
Sweden | | | Sweden | 1
Sweden | Sweden | Swedish | Sweden | 9
Switzerland | | French | Swiss French | 1
Belgium | Switzerland | French | Swiss French | 1
Czech Republic | Switzerland | French | Swiss French | 1
France | Switzerland | French | Swiss French | 7
Poland | Switzerland | French | Swiss French | 1
Portugal | Switzerland | French | Swiss French | 1
Spain | Switzerland | French | Swiss French | 1
Switzerland | Switzerland | French | Swiss French | 826
Switzerland | Switzerland | German | Swiss German | 103
Italy | Switzerland | Italian | Switzerland | 2
Switzerland | Switzerland | Italian | Switzerland | 12
Switzerland | Switzerland | Patois | Switzerland | 1
Switzerland | Switzerland | Romansch | Switzerland | 1
Spain | Switzerland | Spanish | Switzerland | 1
Turkey | Turkey | Turkish | Turkey | 4
Ukraine | Ukraine | Ukranian | Ukraine | 1
United Kingdom | | | United Kingdom | 87
United Kingdom | | English | United Kingdom | 270
United Kingdom | United Kingdom | English | United Kingdom | 1
Yugoslavia | | | Yugoslavia | 1
Yugoslavia | Yugoslavia | French | Yugoslavia | 1
Yugoslavia | Yugoslavia | Romanian | Yugoslavia | 1
Yugoslavia | Yugoslavia | Serbo-Croatian | Yugoslavia | 3
Yugoslavia | Yugoslavia | Yugoslavian | Yugoslavia | 4
Table S4: Continuation of table S3.
## Supplemental material
## S1 Simulation methods
To test our inference procedure, we implemented a simple whole-genome pedigree
simulator, which we briefly describe here. We simulate in reverse time,
keeping track of those parts of the pedigree along which genomic material have
actually passed; effectively constructing the ancestral recombination graph
(Griffiths and Marjoram, 1997) in its entireity. This is computationally
taxing, but it is still feasible to generate all relationships between 100
sampled humans (with the actual chromosome numbers and lengths) going back 300
generations in an hour or so on a modern machine with only 8GB of RAM; or on
the same machine, 1000 sampled humans going back 150 generations, overnight.
(Since memory use and time to process a generation scale linearly with the
number of generations and the number of samples, a machine with more RAM could
produce longer or larger simulations.) The demographic scenarios are
restricted to (arbitrary) discrete population models with changing population
sizes and migration rates. The code (python and R) is freely available at
http://github.org/petrelharp.
The process we want to simulate is as follows: we have $n$ sampled diploid
individuals in the present day, and at some time $T$ in the past, wish to know
which of the $2n$ sampled haplotypes inherited which portions of genome from
the same ancestral haplotypes (i.e. are IBD by time $T$). We work with diploid
individuals, always resolved into maternal and paternal haplotypes, and only
work with the autosomes. We treat all chromosomes in a common coordinate
system by laying them down end-to-end, with the chromosomal endpoints
$g_{1}<\cdots<g_{c}$ playing a special role. The algorithm iterates through
previous generations, and works as follows to produce the state at $t+1$
generations ago from the state at $t$ generations ago. Each sampled haplotype
can be divided into segments inheriting from distinct ancestral haplotypes
from $t$ generations ago. For each of the sampled haplotypes, indexed by
$1\leq i\leq 2n$, we record the sequence of genomic locations separating these
segments as $b(i,t)=(b_{1}(i,t),\ldots,b_{B(i,t)}(i,t))$, and the identities
of the corresponding ancestors from $t$ generations ago as
$a(i,t)=(a_{1}(i,t),\ldots,a_{B(i,t)}(i,t))$, where $B(i,t)$ is the total
number of segments the $i^{\mathrm{th}}$ sampled haplotype is divided into $t$
generations ago. The first genomic location $b_{1}(i,t)$ is always 0, and for
notational convenience, let $b_{B(i,t)+1}(i,t)=g_{c}$ (the total genome
length). The meaning of $a(i,t)$ is that if two samples $i$ and $j$ match on
overlapping segments, i.e. for some $k$ and $\ell$, $a_{k}(i,t)=a_{\ell}(i,t)$
and $[x,y]=[b_{k}(i,t),b_{k+1}(i,t)]\cap[b_{\ell}{j,t},b_{\ell+1}{j,t}]$, then
both have inherited the genomic segment $[x,y]$ from the same ancestral
haplotype $a_{k}(i,t)$ alive at time $t$, and are thus IBD on that segment
from sometime in the past $t$ generations.
As parameters, we are given $N_{t}(u)$, the effective population size in
subpopulation $u$ at time $t$ in the past.
Figure S1: An illustration of the update procedure, moving to the previous
generation. At the bottom of (A) is the current state in generation $t$, with
haplotype segments colored by which generation-$t$ ancestral haplotype they
derive from (e.g. $a(i,t)$), only showing colors for segments inheriting from
the depicted two ancestors (all segments in fact have labels). In (B) these
have been updated to be colored corresponding to which of the
generation-$(t+1)$ haplotypes they derive from. The two depicted ancestors are
half sibs. The smaller symbols with partially-colored chromosomes depict the
location of recombination breakpoints, which are located on the sampled
haplotypes by arrows and vertical dotted lines.
To update from $t$ to $t+1$, we need to pick parents for each generation-$t$
ancestor, choose the recombination breakpoints for the meiosis leading to each
generation-$t$ haplotype – so, each unique value of $a(\cdot,t)$ has a
corresponding (diploid) parent and a set of recombination breakpoints. These
steps are performed iteratively along each haplotype, checking if parents and
recombination breakpoints have been chosen already for each ancestor, and
randomly generating these if not. Recombinations are generated as a Poisson
process of unit rate along the genome (so, lengths are in Morgans); to this
set each chromosomal endpoint is added independently with probability $1/2$
each. To choose a parent, stretches of genome between alternating
recombination breakpoints are assigned to the two haplotypes of the parent.
For instance, if the recombination breakpoints of a generation-$t$ haplotype
labeled $a$ are at $r_{1}<\cdots<r_{R}$, and the maternal and paternal
haplotypes of the parent of $a$ are labeled $h_{m}$ and $h_{p}$ respectively,
any $a_{k}(i,t)$ with $b_{k}(i,t)<r_{1}$ would be changed to $h_{m}$, while
those with $r_{1}\leq b_{k}(i,t)<r_{2}$ would be changed to $h_{p}$, and new
segments are added when breakpoints fall inside of an existing segment
($a_{k}(i,t)<r_{1}\leq a_{k+1}(i,t)$).
The algorithm is run for a given number of generations, after which an
algorithm iterates along all sampled haplotypes in parallel, writing out any
pairwise blocks of IBD longer than a given threshold. An IBD block here is a
contiguous piece of a single chromosome over which both sampled chromosomes
share the same state.
## S2 Test of inference methods
We simulated from three simple demographic scenarios, with parameters chosen
to roughly match the mean number of IBD blocks per pair longer than 2cM that
we see in the data. The scenarios are as follows:
* (A)
Constant effective population size $10^{5}$ – average 0.79 IBD blocks longer
than 2cM per pair.
* (B)
Exponential growth, starting from (constant) effective population size
$1.5\times 10^{4}$ prior to 100 generations ago, and approaching $3\times
10^{6}$ exponentially, as $N_{e}(t)=3\times 10^{6}-(3\times 10^{6}-1.5\times
10^{4})\exp(-0.077(100-t))$ – average 0.51 IBD blocks longer than 2cM per
pair.
* (C)
Exponential growth as in B, except expanding only 50 generations ago, and
beginning with an effective population size of $3\times 10^{4}$ – average 1.11
IBD blocks longer than 2cM per pair.
* (D)
A more complex scenario: constant size $4\times 10^{4}$ older than 60
generations ago; growing logistically to $8\times 10^{5}$ between 60 and 30
generations ago; decreasing logistically to $3\times 10^{4}$ between 30 and 15
generations ago; constant between 5 and 15 generations ago; and growing again
to $4\times 10^{6}$ until the present – average 1.09 IBD blocks longer than
2cM per pair. (Imagine a population that grows large, has a small group split
off gradually, which then grows in the present day; not motivated by any
specific history, but chosen to test the methods when the true history is more
”bumpy”.)
For computational convenience, we simulated only up to 300 generations ago,
and retained only blocks longer than 0.5cM (but often restricted analysis to
those longer than 2cM); as in the paper we merged any blocks separated by a
gap that was shorter than at least one adjacent block and shorter than 5cM.
Coalescent rates and block length distributions are shown in figure S2. Even
though we have not modeled gap removal, the results still closely match
theory, since very few blocks fell so close to each other.
Figure S2: Coalescent rate (left) and IBD length spectra (right) for the
three scenarios. For the length distributions, the value given is observed
blocks per pair and per centiMorgan; for each, the dotted line gives the
theoretical value predicted from the theoretical coalescent time distribution,
and the solid line is the observed distribution.
For each scenario, we applied the inversion procedure described in the text to
the full, error-free set of blocks as well as to various subsets and
modifications of it. The inversion procedure we followed for each was as
follows. We chose a discretization for block lengths as described in the text,
by starting with the percentiles of the distribution, and refining further so
that the largest bin length was 1cM. We then computed the matrix $L$ as
described in the text, except with no error distribution or false positive
rate, so that if the $i^{\mathrm{th}}$ length bin is $[x_{i},x_{i+1})$, then
$L_{in}=\sum_{g=1}^{22}K_{g}(n,\min(x_{i+1},G_{g}))-K(n,\min(x_{i},G_{g}))$,
with $G_{g}$ the length of the $g^{\mathrm{th}}$ chromosome and
$K_{g}(n,x)=(n(G_{g}-x)+1)\exp(-nx)$. For most simulations, we did not
discretize time any further, but allowed $n$ to range from 1 up to 300
generations. We then used constrained optimization as implemented in the R
package optim (L-BFGS-B method, R Development Core Team, 2012) to maximize the
penalized likelihoods described in the text, beginning at the solution to the
natural approximating least-squares problem (using quadprog, Turlach and
Weingessel, 2011). For each case, we show the “maximum likelihood solution”
(estimated by adding a small amount of smoothness penalty to ensure numerical
uniqueness, allowing the algorithm to converge), and the “smoothest consistent
solution” – the largest $\gamma$ so that the solution has decreased in log-
likelihood by no more than 2 units.
In each case, we also show the exact coalescent distribution, as well as the
block length spectrum predicted by theory from the true coalescent
distribution and each coalescent distribution found by penalized maximum
likelihood.
Note that we could have incorporated false positives, missed blocks, or length
misestimation into the simulations, and subsequently modified the kernel $L$
to incorporate these rates, but this would only add additional layers of
simulation code, and would not make the task of inference more difficult,
since we account for these effects analytically. The sensitivity of the
methods to misestimation of these rates is a concern, but this amounts to
misestimation of the kernel $L$, which we investigate below.
In figure S3 we show the results of the inference procedure applied to the
full set of blocks longer than 0.5cM; figure S4 is the same, except using only
blocks longer than 2cM, and figure S5 shows the results for scenario D
separately. Comparing these, we see that the short blocks 0.5–2cM does not
significantly improve the resolution in recent times, but does allow better
estimation of coalescent rates longer ago in time. Using blocks longer than
2cM gives us good resolution on the time scale we consider (the past 100
generations), and including those down to 0.5cM does not make the likelihood
much less ridged (as expected from theory).
One counter-intuitive result we obtained was that the coalescent history could
have a dramatic effect on the estimation of ages of blocks given their
lengths. In figure S6 we show the probability distribution of the ages of
blocks of various lengths under the four scenarios, i.e. how many generations
ago the ancestors lived from whom the samples inherited blocks of that length.
These are counter-intuitive because a block inherited from $n$ generations ago
has mean length $50/n$ cM, but the age distributions of blocks in practice
show that the converse is not true – blocks $x$ cM long are usually much older
than $50/n$ generations. This is computed simply as follows: the mean number
of IBD blocks of length $x$ per unit of coalescence from $n/2$ generations ago
(from paths of $n$ meioses) is
$K(n,x)=\sum_{i=1}^{2}2n(n(G_{i}-x)+1)\exp(-nx)$; so the probability that a
block of length $x$ came from $n/2$ generations ago is
$K(n,x)/\sum_{m}K(m,x)$.
Figure S3: Results of the inference procedure applied to all data (all blocks
at least 0.5cM) with 300 generations as the upper limit. Above are true
(green) and inferred (shaded) coalescent rates; below are block length
distributions (density per pair), observed (black) and predicted by the
inferred coalescent distributions in the respective upper panel. (A–B),
scenario A; (C–D), scenario B; and (E–F), scenario C. The dangling line at the
end of several plots is due to a few rare long blocks and is not a significant
deviation from the expectation. Figure S4: Longer blocks only – as in figure
S3, except in each case we have only used blocks above at least 2cM. Figure
S5: Scenario D: Here we show results for the more complex demographic
scenario. (A–C) use all blocks (down to 0.5cM), with (A) and (C) the same as
in figure S3, and (B) the same as (A) except zooming in on more recent times.
(D–F) is as (A–C), except using only blocks longer than 2cM. Figure S6: Age
distributions of blocks under each of the four scenarios – each curve shows
the probability distribution for the age of a 2cM, 5cM, and 8cM block under
each of the four scenarios. For the age distribution of blocks $x$ cM long,
the vertical dotted line is at $50/x$ generations, the naive expectation for
the typical age of such blocks.
## S3 Sensitivity analysis
We also used these simulations to evaluate our sensitivity to error. Of
particular concern is error arising due to misestimation of the false positive
rate – we have seen that false positive rate at short lengths can vary
somewhat by population – we estimate as much as 10% around 2cM. To evaluate
the effect on inference, we added to the numbers of blocks observed in each
category some number of “false positives”, but applied the inference methods
without accounting for these (so we still have $f=0$). The numbers of false
positives added to each length bin are Poisson with mean equal to the
theoretical mean predicted for that bin, multiplied by a factor that depends
on the length and decreases (so there is an artifical inflation of short
blocks). The results for three different false positive rates are shown in
figure S7. From these, we see that if IBD rate is only increased by a maximum
of 10% – even if the effect extends out to 6 or 8cM – the effect on the
inferred coalescent distribution is minor. It is also useful to add a
unrealistically high level of unaccounted-for false positives, as it is
natural to suspect that an excess of short blocks will only increase the
coalescent rate at relatively older time periods. This is indeed the case –
doubling the distribution at the short end (about 2–4cM) only affects inferred
coalescent rates beyond about 100 generations, because this is when the bulk
of the 2–4cM blocks have come from.
Figure S7: False positives – each row shows inference results from the three
scenarios with different amounts of spurious false positives added on. If the
predicted number of blocks in the bin with length midpoint $x$cM is $m(x)$, we
added a Poisson number of blocks with mean $h(x)m(x)$ to the number observed,
with $h$ varying. In the first row (A–C), $h(x)=0.1\exp(2-x)$, in the second
row (D–F), $h(x)=0.1\exp((2-x)/4)$, and in the third row (G–I),
$h(x)=\exp(2-x)$. The third scenario is not thought to be realistic, but
demonstrates that misestimation at short lengths only affects inference at
older times.
|
arxiv-papers
| 2012-07-16T20:24:15 |
2024-09-04T02:49:33.156010
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Peter Ralph, Graham Coop",
"submitter": "Peter Ralph",
"url": "https://arxiv.org/abs/1207.3815"
}
|
1207.3907
|
# Haplotype-based variant detection from short-read sequencing
Erik Garrison and Gabor Marth
###### Abstract
The direct detection of haplotypes from short-read DNA sequencing data
requires changes to existing small-variant detection methods. Here, we develop
a Bayesian statistical framework which is capable of modeling multiallelic
loci in sets of individuals with non-uniform copy number. We then describe our
implementation of this framework in a haplotype-based variant detector,
FreeBayes.
## 1 Motivation
While _statistical phasing_ approaches are necessary for the determination of
large-scale haplotype structure [Browning and Browning, 2007, Li et al., 2010,
Delaneau et al., 2012, Howie et al., 2011], sequencing traces provide short-
range phasing information that may be employed directly in primary variant
detection to establish phase between proximal alleles. Present read lengths
and error rates limit this _physical phasing_ approach to variants clustered
within tens of bases, but as the cost of obtaining long sequencing traces
decreases [Branton et al., 2008, Clarke et al., 2009], physical phasing
methods will enable the determination of larger haplotype structure directly
using only sequence information from a single sample.
Haplotype-based variant detection methods, in which short haplotypes are read
directly from sequencing traces, offer a number of benefits over methods which
operate on a single position at a time. Haplotype-based methods ensure
semantic consistency among described variants by simultaneously evaluating all
classes of alleles in the same context. The use of locally-phased genotype
data can lower the computational burden of genotype imputation by reducing the
possible space of haplotypes which must be considered. Locally phased
genotypes can be used to improve genotyping accuracy in the context of rare
variations that can be difficult to impute due to sparse linkage information.
Similarly, they can assist in the design of genotyping assays, which can fail
in the context of undescribed variation at the assayed locus. Provided
sequencing errors are independent, the use of longer haplotypes in variant
detection can improve detection by increasing the signal to noise ratio of the
genotype likelihood space that is used in analysis. This follows from the fact
that the space of possible erroneous haplotypes expands dramatically with
haplotype length, while the space of true variation remains constant, with the
number of true alleles less than or equal to the ploidy of the sample at a
given locus.
The direct detection of haplotypes from alignment data presents several
challenges to existing variant detection methods. As the length of a haplotype
increases, so does the number of possible alleles within the haplotype, and
thus methods designed to detect genetic variation over haplotypes in a unified
context must be able to model multiallelism. However, most variant detection
methods establish estimates of the likelihood of polymorphism at a given loci
using statistical models which assume biallelism [Li, 2011, Marth et al.,
1999] and uniform, typically diploid, copy number [DePristo et al., 2011].
Moreover, improper modeling of copy number impedes the accurate detection of
small variants on sex chromosomes, in polyploid organisms, or in locations
with known copy-number variations, where called alleles, genotypes, and
likelihoods should reflect local copy number and global ploidy.
To enable the application of population-level inference methods to the
detection of haplotypes, we generalize the Bayesian statistical method
described by Marth et al. [1999] to allow multiallelic loci and non-uniform
copy number across the samples under consideration. We have implemented this
model in FreeBayes [Garrison, 2012].
## 2 Generalizing variant detection to multiallelic loci and non-uniform copy
number
### 2.1 Definitions
At a given genetic locus we have $n$ samples drawn from a population, each of
which has a copy number or multiplicity of $m$ within the locus. We denote the
number of copies of the locus present within our set of samples as
$M=\sum_{i=1}^{n}m_{i}$. Among these $M$ copies we have $K$ distinct alleles,
$b_{1},\ldots,b_{K}$ with allele frequencies $f_{1},\ldots,f_{K}$. Each
individual has an unphased genotype $G_{i}$ comprised of $k_{i}$ distinct
alleles $b_{i_{1}},\ldots,b_{k_{i}}$ and corresponding genotype allele
frequencies $f_{i_{1}},\ldots,f_{i_{k_{i}}}$, which may be equivalently
expressed as a multiset of alleles $B_{i}:|B_{i}|=m_{i}$. For the purposes of
our analysis, we assume that we cannot accurately discern phasing information
outside of the haplotype detection window, so our $G_{i}$ are unordered and
all $G_{i}$ containing equivalent alleles and frequencies are regarded as
equivalent. Assume a set of $s_{i}$ sequencing observations
$r_{i_{1}},\ldots,r_{i_{s_{i}}}=R_{i}$ over each sample in our set of $n$
samples such that there are $\sum_{i=1}^{n}s_{i}$ reads at the genetic locus
under analysis. $Q_{i}$ denotes the mapping quality, or probability that the
read $r_{i}$ is mis-mapped against the reference.
### 2.2 A Bayesian approach
To genotype the samples at a specific locus, we could simply apply a Bayesian
statistic relating $P(G_{i}|R_{i})$ to the likelihood of sequencing errors in
our reads and the prior likelihood of specific genotypes. However, this
maximum-likelihood approach limits our ability to incorporate information from
other individuals in the population under analysis, which can improve
detection power.
Given a set of genotypes $G_{1},\ldots,G_{n}$ and observations observations
$R_{1},\ldots,R_{n}$ for all individuals at the current genetic locus, we can
use Bayes’ theorem to related the probability of a specific combination of
genotypes to both the quality of sequencing observations and _a priori_
expectations about the distribution of alleles within a set of individuals
sampled from the same population:
$P(G_{1},\ldots,G_{n}|R_{1},\ldots,R_{n})={P(G_{1},\ldots,G_{n})P(R_{1},\ldots,R_{n}|G_{1},\ldots,G_{n})\over
P(R_{1},\ldots,R_{n})}\\\ $ (1)
$P(G_{1},\ldots,G_{n}|R_{1},\ldots,R_{n})={P(G_{1},\ldots,G_{n})\prod_{i=1}^{n}P(R_{i}|G_{i})\over\sum_{\forall{G_{1},\ldots,G_{n}}}(P(G_{1},\ldots,G_{n})\prod_{i=1}^{n}P(R_{i}|G_{i}))}$
(2)
Under this decomposition,
$P(R_{1},\ldots,R_{n}|G_{1},\ldots,G_{n})=\prod_{i=1}^{n}P(R_{i}|G_{i})$
represents the likelihood that our observations match a given genotype
combination (our data likelihood), and $P(G_{1},\ldots,G_{n})$ represents the
prior likelihood of observing a specific genotype combination. We estimate the
data likelihood as the joint probability that the observations for a specific
individual support a given genotype. We use a neutral model of allele
diffusion conditioned on an estimated population mutation rate to estimate the
prior probability of sampling a given collection of genotypes.
Except for situations with small numbers of samples and alleles, we avoid the
explicit evaluation of the posterior distribution as implied by (2), instead
using a number of optimizations to make the algorithm tractable to apply to
very large datasets (see section 3.3).
### 2.3 Estimating the probability of a sample genotype given sequencing
observations, $P(R_{i}|G_{i})$
Given a set of reads $R_{i}=r_{i_{1}},\ldots,r_{i_{s_{i}}}$ of a sample at a
given locus, we can extract a set of $k_{i}$ observed alleles
$B^{\prime}_{i}=b^{\prime}_{1},\ldots,b^{\prime}_{k_{i}}$ which encapsulate
the potential set of represented variants at the locus in the given sample,
including erroneous observations. Each of these observed alleles
$b^{\prime}_{l}$ has a frequency $o^{\prime}_{f}$ within the observations of
the individual sample $:\sum_{j=1}^{k_{i}}o^{\prime}_{j}=s_{i}$ and
corresponds to a true allele $b_{l}$.
If we had perfect observations of a locus, $P(R_{i}|G_{i})$ for any individual
would approximate the probability of sampling observations $R_{i}$ out of
$G_{i}$ with replacement. This probability is given by the multinomial
distribution in $s_{i}$ over the probability $P(b_{l})$ of obtaining each
allele from the given genotype, which is ${f_{i_{j}}\over m_{i}}$ for each
allele frequency in the frequencies which define $G_{i}$,
$f_{i_{1}},\ldots,f_{i_{k_{i}}}$.
$P(R_{i}|G_{i})\approx
P(B^{\prime}_{i}|G_{i})={s_{i}!\over{\prod_{j=1}^{k_{i}}o^{\prime}_{j}!}}\prod_{j=1}^{k_{i}}{\left({f_{i_{j}}\over
m_{i}}\right)^{o^{\prime}_{j}}}$ (3)
Our observations are not perfect, and thus we must account for the probability
of errors in the reads. We can use the per-base quality scores provided by
sequencing systems to develop the probability that an observed allele is drawn
from an underlying true allele, $P(B^{\prime}_{i}|R_{i})$. We assume a mapping
between sequencing quality scores and allele qualities such that each observed
allele $b^{\prime}_{l}$ has a corresponding quality $q_{l}$ which approximates
the probability that the observed allele is incorrect.
Furthermore, we must sum $P(R_{i}|G_{i})$ for all possible $R_{i}$
combinations $\forall(R_{i}\in G_{i}:|R_{i}|=k_{i})$ drawn from our genotype
to obtain the joint probability of $R_{i}$ given $G_{i}$, as each
$\prod_{l=1}^{s_{i}}{P(b^{\prime}_{l}|b_{l})}$ only accounts for the marginal
probability of the a specific $R_{i}$ given $B^{\prime}_{i}$.
This extends $P(R_{i}|G_{i})$ as follows:
$P(R_{i}|G_{i})=\sum_{\forall(R_{i}\in
G_{i})}\left({s_{i}!\over{\prod_{j=1}^{k_{i}}o^{\prime}_{j}!}}\prod_{j=1}^{k_{i}}{\left({f_{i_{j}}\over
m_{i}}\right)^{o^{\prime}_{j}}}\prod_{l=1}^{s_{i}}{P(b^{\prime}_{l}|b_{l})}\right)$
(4)
In summary, the probability of obtaining a set of reads given an underlying
genotype is proportional to the probability of sampling the set of
observations from the underlying genotype, scaled by the probability that our
reads are correct. As $q_{l}$ approximates the probability that a specific
$b_{l}$ is incorrect, $P(b^{\prime}_{l}|b_{l})=1-q_{l}$ when $b_{l}\in G_{i}$
and $P(b^{\prime}_{l}|b_{l})=q_{l}$ when $b_{l}\notin G_{i}$.
### 2.4 Priors for unphased genotype combinations, $P(G_{1},\ldots,G_{n})$
#### 2.4.1 Decomposition of prior probability of genotype combination
Let $G_{1},\ldots,G_{n}$ denote the set of genotypes at the locus and
$f_{1},\ldots,f_{k}$ denote the set of allele frequencies which corresponds to
these genotypes. We estimate the prior likelihood of observing a specific
combination of genotypes within a given locus by decomposition into resolvable
terms:
$P(G_{1},\ldots,G_{n})=P(G_{1},\ldots,G_{n}\cap f_{1},\ldots,f_{k})$ (5)
The probability of a given genotype combination is equivalent to the
intersection of that probability and the probability of the corresponding set
of allele frequencies. This identity follows from the fact that the allele
frequencies are derived from the set of genotypes and we always will have the
same $f_{1},\ldots,f_{k}$ for any equivalent $G_{1},\ldots,G_{n}$.
Following Bayes’ Rule, this identity further decomposes to:
$P(G_{1},\ldots,G_{n}\cap
f_{1},\ldots,f_{k})=P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})P(f_{1},\ldots,f_{k})$
(6)
We now can estimate the prior probability of $G_{1},\ldots,G_{n}$ in terms of
the genotype combination sampling probability,
$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})$, and the probability of observing a
given allele frequency in our population, $P(f_{1},\ldots,f_{k})$.
#### 2.4.2 Genotype combination sampling probability
$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})$
The multinomial coefficient ${M\choose f_{1},\ldots,f_{k}}$ gives the number
of ways which a set of alleles with frequencies $f_{1},\ldots,f_{k}$ may be
distributed among $M$ copies of a locus. For phased genotypes $\hat{G_{i}}$
the probability of sampling a specific $\hat{G_{1}},\ldots,\hat{G_{n}}$ given
allele frequencies $f_{1},\ldots,f_{k}$ is thus provided by the inverse of
this term:
$P(\hat{G_{1}},\ldots,\hat{G_{n}}|f_{1},\ldots,f_{k})={M\choose
f_{1},\ldots,f_{k}}^{-1}$ (7)
However, our model is limited to unphased genotypes because our primary data
only allows phasing within a limited context. Consequently, we must adjust (7)
to reflect the number of phased genotypes which correspond to the unphased
genotyping $G_{1},\ldots,G_{n}$. Each unphased genotype corresponds to as many
phased genotypes as there are permutations of the alleles in $G_{i}$. Thus,
for a given unphased genotyping $G_{1},\ldots,G_{n}$, there are
$\prod_{i=1}^{n}{m_{i}\choose f_{i_{1}},\ldots,f_{i_{k_{i}}}}$ phased
genotypings.
In conjunction, these two terms provide the probability of sampling a
particular unphased genotype combination given a set of allele frequencies:
$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})={M\choose
f_{1},\ldots,f_{k}}^{-1}\prod_{i=1}^{n}{m_{i}\choose
f_{i_{1}},\ldots,f_{i_{k_{i}}}}=\frac{1}{M!}\prod_{l=1}^{k}f_{l}!\prod_{i=1}^{n}\frac{m_{i}!}{\prod_{j=1}^{k_{i}}f_{i_{j}}!}$
(8)
In the case of a fully diploid population, the product of all possible
multiset permutations of all genotypes reduces to $2^{h}$, where $h$ is the
number of heterozygous genotypes, simplifying (8) to:
$P(G_{1},\ldots,G_{n}|f_{1},\ldots,f_{k})=2^{h}{M\choose
f_{1},\ldots,f_{k}}^{-1}$ (9)
#### 2.4.3 Derivation of $P(f_{1},\ldots,f_{k})$ by Ewens’ sampling formula
Provided our sample size $n$ is small relative to the population which it
samples, and the population is in equilibrium under mutation and genetic
drift, the probability of observing a given set of allele frequencies at a
locus is given by Ewens’ sampling formula [Ewens, 1972]. Ewens’ sampling
formula is based on an infinite alleles coalescent model, and relates the
probability of observing a given set of allele frequencies to the number of
sampled chromosomes at the locus ($M$) and the population mutation rate
$\theta$.
The application of Ewens’ formula to our context is straightforward. Let
$a_{f}$ be the number of alleles among $b_{1},\ldots,b_{k}$ whose allele
frequency within our set of samples is $f$. We can thus transform our set of
frequencies $f_{1},\ldots,f_{k}$ into a set of non-negative frequency counts
$a_{1},\ldots,a_{M}:\sum_{f=1}^{M}{fa_{f}}=M$. As many $f_{1},\ldots,f_{k}$
can map to the same $a_{1},\ldots,a_{M}$, this transformation is not
invertible, but it is unique from $a_{1},\ldots,a_{M}$ to
$f_{1},\ldots,f_{k}$.
Having transformed a set of frequencies over alleles to a set of frequency
counts over frequencies, we can now use Ewens’ sampling formula to approximate
$P(f_{1},\ldots,f_{k})$ given $\theta$:
$P(f_{1},\ldots,f_{k})=P(a_{1},\ldots,a_{M})={M!\over\theta\prod_{z=1}^{M-1}(\theta+z)}\prod_{j=1}^{M}{\theta^{a_{j}}\over
j^{a_{j}}a_{j}!}$ (10)
In the bi-allelic case in which our set of samples has two alleles with
frequencies $f_{1}$ and $f_{2}$ such that $f_{1}+f_{2}=M$:
$P(a_{f_{1}}=1,a_{f_{2}}=1)={M!\over\prod_{z=1}^{M-1}(\theta+z)}{\theta\over
f_{1}f_{2}}$ (11)
While in the monomorphic case, where only a single allele is represented at
this locus in our population, this term reduces to:
$P(a_{M}=1)={(M-1)!\over\prod_{z=1}^{M-1}(\theta+z)}$ (12)
In this case, $P(f_{1},\ldots,f_{k})=1-\theta$ when $M=2$. This is sensible as
$\theta$ represents the population mutation rate, which can be estimated from
the pairwise heterozygosity rate of any two chromosomes in the population
[Watterson, 1975, Tajima, 1983].
## 3 Direct detection of phase from short-read sequencing
By modeling multiallelic loci, this Bayesian statistical framework provides
the foundation for the direct detection of longer, multi-base alleles from
sequence alignments. In this section we describe our implementation of a
haplotype-based variant detection method based on this model.
Our method assembles haplotype observations over minimal, dynamically-
determined, reference-relative windows which contain multiple segregating
alleles. To be used in the analysis, haplotype observations must be derived
from aligned reads which are anchored by reference-matching sequence at both
ends of the detection window. These haplotype observations have derived
quality estimations which allow their incorporation into the general
statistical model described in section 2. We then employ a gradient ascent
method to determine the maximum _a posteriori_ estimate of a mutual genotyping
over all samples under analysis and establish an estimate of the probability
that the loci is polymorphic.
### 3.1 Parsing haplotype observations from sequencing data
In order to establish a range of sequence in which multiple polymorphisms
segregate in the population under analysis, it is necessary to first determine
potentially polymorphic windows in order to bound the analysis. This
determination is complicated by the fact that a strict windowing can
inappropriately break clusters of alleles into multiple variant calls. We
employ a dynamic windowing approach that is driven by the observation of
multiple proximal reference-relative variations (SNPs and indels) in input
alignments.
Where reference-relative variations are separated by less than a configurable
number of non-polymorphic bases in an aligned sequence trace, our method
combines them into a single haplotype allele observation, $H_{i}$. The
observational quality of these haplotype alleles is given as
$\min(q_{l}\,\forall\,b^{\prime}_{i}\in H_{i},\,Q_{i})$, or the minimum of the
supporting read’s mapping quality and the minimum base quality of the
haplotype’s component variant allele observations.
### 3.2 Determining a window over which to assemble haplotype observations
At each position in the reference, we collect allele observations derived from
alignments as described in 3.1. To improve performance, we apply a set of
input filters to exclude alleles from the analysis which are highly unlikely
to be true. These filters require a minimum number of alternate observations
and a minimum sum of base qualities in a single sample in order to incorporate
a putative allele and its observations into the analysis.
We then determine a haplotype length over which to genotype samples by a
bounded iterative process. We first determine the allele passing the input
filters which is longest relative to the reference. For instance, a longer
allele could be a multi-base indel or a composite haplotype observation
flanked by SNPs. Then, we parse haplotype observations from all the alignments
which fully overlap this window, finding the rightmost end of the longest
haplotype allele which begins within the window. This rightmost position is
used to update the haplotype window length, and a new set of haplotype
observations are assembled from the reads fully overlapping the new window.
This process repeats until the rightmost end of the window is not partially
overlapped by any haplotype observations which pass the input filters. This
method will converge given reads have finite length and the only reads which
fully overlap the detection window are used in the analysis.
### 3.3 Detection and genotyping of local haplotypes
Once a window for analysis has been determined, we parse all fully-overlapping
reads into haplotype observations which are anchored at the boundaries of the
window. Given these sets of sequencing observations
$r_{i_{1}},\ldots,r_{i_{s_{i}}}=R_{i}$ and data likelihoods $P(R_{i}|G_{i})$
for each sample and possible genotype derived from the putative alleles, we
then determine the probability of polymorphism at the locus given the Bayesian
model described in section 2.
To establish a maximum _a posteriori_ estimate of the genotype for each
sample, we employ a convergent gradient ascent approach to the posterior
probability distribution over the mutual genotyping across all samples under
our Bayesian model. This process begins at the genotyping across all samples
$G_{1},\ldots,G_{n}$ where each sample’s genotype is the maximum-likelihood
genotype given the data likelihood $P(R_{i}|G_{i})$:
$G_{1},\ldots,G_{n}=\underset{G_{i}}{\operatorname{argmax}}\;P(R_{i}|G_{i})$
(13)
The posterior search then attempts to find a genotyping $G_{1},\ldots,G_{n}$
in the local space of genotypings which has higher posterior probability under
the model than this initial genotyping. In practice, this step is done by
searching through all genotypings in which a single sample has up to the $N$th
best genotype when ranked by $P(R_{i}|G_{i})$, and $N$ is a small number (e.g.
2). This search starts with some set of genotypes $G_{1},\ldots,G_{n}=\\{G\\}$
and attempts to find a genotyping $\\{G\\}^{\prime}$ such that:
$P(\\{G\\}^{\prime}|R_{1},\ldots,R_{n})>P(\\{G\\}|R_{1},\ldots,R_{n})$ (14)
$\\{G\\}^{\prime}$ is then used as a basis for the next update step. This
search iterates until convergence, but in practice must be bounded at a fixed
number of steps in order to ensure optimal performance. As the quality of
input data increases in coverage and confidence, this search will converge
more quickly because the maximum-likelihood estimate will lie closer to the
maximum _a posteriori_ estimate under the model.
This method incorporates a basic form of genotype imputation into the
detection method, which in practice improves the quality of raw genotypes
produced in primary allele detection and genotyping relative to methods which
only utilize a maximum-likelihood method to determine genotypes. Furthermore,
this method allows for the determination of marginal genotype likelihoods via
the marginalization of assigned genotypes for each sample over the posterior
probability distribution.
### 3.4 Probability of polymorphism
Provided a maximum _a posteriori_ estimate of the genotyping of all the
individuals in our sample, we might like establish an estimate of the quality
of the genotyping. For this, we can use the probability that the locus is
polymorphic, which means that the number of distinct alleles at the locus,
$K$, is greater than 1. While in practice the space of possible genotypings is
too great to integrate over, it is possible to derive the probability that the
loci is polymorphic in our samples by summing across the monomorphic cases:
$P(K>1|R_{1},\ldots,R_{n})=1-P(K=1|R_{1},\ldots,R_{n})$ (15)
Equation (15) thus provides the probability of polymorphism at the site, which
is provided as a quality estimate for each evaluated locus in the output of
FreeBayes.
### 3.5 Marginal likelihoods of individual genotypes
Similarly, we can establish a quality estimate for a single genotype by
summing over the marginal probability of that specific genotype and sample
combination under the model. The marginal probability of a given genotype is
thus:
$P(G_{j}|R_{i},\ldots,R_{n})=\sum_{\forall(\\{G\\}:G_{j}\in\\{G\\})}P(\\{G\\}|R_{i},\ldots,R_{n})$
(16)
In implementation, the development of this term is more complex, as we must
sample enough genotypings from the posterior in order to obtain well-
normalized marginal likelihoods. In practice, we marginalize from the local
space of genotypings in which each individual genotype is no more than a small
number of steps in one sample from the maximum _a posteriori_ estimate of
$G_{i},\ldots,G_{n}$. This space is similar to that used during the posterior
search described in section 3.3. We apply (16) to it to estimate marginal
genotype likelihoods for the most likely individual genotypes, which are
provided for each sample at each site in the output of our implementation.
## References
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* Browning and Browning [2007] S. R. Browning and B. L. Browning. Rapid and accurate haplotype phasing and missing-data inference for whole-genome association studies by use of localized haplotype clustering. _Am. J. Hum. Genet._ , 81(5):1084–1097, Nov 2007\.
* Clarke et al. [2009] J. Clarke, H. C. Wu, L. Jayasinghe, A. Patel, S. Reid, and H. Bayley. Continuous base identification for single-molecule nanopore DNA sequencing. _Nat Nanotechnol_ , 4(4):265–270, Apr 2009.
* Delaneau et al. [2012] O. Delaneau, J. Marchini, and J. F. Zagury. A linear complexity phasing method for thousands of genomes. _Nat. Methods_ , 9(2):179–181, Feb 2012.
* DePristo et al. [2011] M. A. DePristo, E. Banks, R. Poplin, K. V. Garimella, J. R. Maguire, C. Hartl, A. A. Philippakis, G. del Angel, M. A. Rivas, M. Hanna, A. McKenna, T. J. Fennell, A. M. Kernytsky, A. Y. Sivachenko, K. Cibulskis, S. B. Gabriel, D. Altshuler, and M. J. Daly. A framework for variation discovery and genotyping using next-generation DNA sequencing data. _Nat. Genet._ , 43(5):491–498, May 2011.
* Ewens [1972] W. J. Ewens. The sampling theory of selectively neutral alleles. _Theor Popul Biol_ , 3:87–112, Mar 1972.
* Garrison [2012] E. Garrison. FreeBayes source repository. https://github.com/ekg/freebayes, 2012.
* Howie et al. [2011] B. Howie, J. Marchini, and M. Stephens. Genotype imputation with thousands of genomes. _G3 (Bethesda)_ , 1(6):457–470, Nov 2011.
* Li [2011] H. Li. A statistical framework for SNP calling, mutation discovery, association mapping and population genetical parameter estimation from sequencing data. _Bioinformatics_ , 27(21):2987–2993, Nov 2011\.
* Li et al. [2010] Y. Li, C. J. Willer, J. Ding, P. Scheet, and G. R. Abecasis. MaCH: using sequence and genotype data to estimate haplotypes and unobserved genotypes. _Genet. Epidemiol._ , 34(8):816–834, Dec 2010\.
* Marth et al. [1999] G. T. Marth, I. Korf, M. D. Yandell, R. T. Yeh, Z. Gu, H. Zakeri, N. O. Stitziel, L. Hillier, P. Y. Kwok, and W. R. Gish. A general approach to single-nucleotide polymorphism discovery. _Nat. Genet._ , 23:452–456, Dec 1999.
* Tajima [1983] F. Tajima. Evolutionary relationship of DNA sequences in finite populations. _Genetics_ , 105(2):437–460, Oct 1983.
* Watterson [1975] G. A. Watterson. On the number of segregating sites in genetical models without recombination. _Theor Popul Biol_ , 7(2):256–276, Apr 1975.
|
arxiv-papers
| 2012-07-17T07:53:38 |
2024-09-04T02:49:33.180674
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Erik Garrison and Gabor Marth",
"submitter": "Erik Garrison",
"url": "https://arxiv.org/abs/1207.3907"
}
|
1207.3923
|
Copyright 2011–2012, University of Glasgow noleaders
# Managing Research Data in Big Science
Norman Gray Tobia Carozzi and Graham Woan
SUPA School of Physics and Astronomy University of Glasgow
(2011 July 14, v1.1)
Version | v1.1
---|---
URL | http://purl.org/nxg/projects/mrd-gw/report
Distribution | Public
This report was prepared as part of the RDMP strand of the JISC programme
Managing Research Data.
## Abstract
The project which led to this report was funded by JISC in 2010–2011 as part
of its ‘Managing Research Data’ programme, to examine the way in which Big
Science data is managed, and produce any recommendations which may be
appropriate.
Big science data is different: it comes in large volumes, and it is shared and
exploited in ways which may differ from other disciplines. This project has
explored these differences using as a case-study Gravitational Wave data
generated by the LSC, and has produced recommendations intended to be useful
variously to JISC, the funding council (STFC) and the LSC community.
In Sect. 1 we define what we mean by ‘big science’, describe the overall data
culture there, laying stress on how it necessarily or contingently differs
from other disciplines.
In Sect. 2 we discuss the benefits of a formal data-preservation strategy, and
the cases for open data and for well-preserved data that follow from that.
This leads to our recommendations that, in essence, funders should adopt
rather light-touch prescriptions regarding data preservation planning: normal
data management practice, in the areas under study, corresponds to notably
good practice in most other areas, so that the only change we suggest is to
make this planning more formal, which makes it more easily auditable, and more
amenable to constructive criticism.
In Sect. 3 we briefly discuss the LIGO data management plan, and pull together
whatever information is available on the estimation of digital preservation
costs.
The report is informed, throughout, by the OAIS reference model for an open
archive. Some of the report’s findings and conclusions were summarised in [1].
See the document history on page About this document.
This report was prepared for, and funded by, the RDMP strand of the JISC
programme Managing Research Data.
Release: 7b021525c2d7, 2012-07-17 09:55 +0100
. This work is licensed under the Creative Commons Attribution-Share Alike 2.5
UK: Scotland Licence. To view a copy of this licence, visit
http://creativecommons.org/licenses/by-sa/2.5/scotland/.
###### Contents
1. 0 Introduction
1. 0.1 Project Background
2. 0.2 How to read this document
3. 0.3 Working with communities – pragmatics
2. 1 Data management in Big Science
1. 1.1 LIGO in perspective: LIGO, big science, and astronomy
2. 1.2 Data volumes
3. 1.3 Data management styles in the physical sciences
4. 1.4 Astronomy data
1. 1.4.1 Strasbourg Data Centre (CDS) as a disciplinary repository
2. 1.4.2 Collaborations in astronomy
5. 1.5 High Energy Physics data
6. 1.6 Gravitational wave physics
1. 1.6.1 Gravitational wave consortia
2. 1.6.2 GW data
3. 1.6.3 Gravitational wave data releases
4. 1.6.4 Summary: big-science preservation challenges
7. 1.7 A contrast: social science data
8. 1.8 Babylonian data management (less contrast than you’d think)
9. 1.9 Bibliographic repositories
10. 1.10 Virtual Observatories
11. 1.11 Data products and proprietary periods: reifying data management and release
3. 2 The responsibilities for data preservation
1. 2.1 Visualising benefits
2. 2.2 The case for open data
3. 2.3 The case for data preservation
4. 2.4 Should raw data be preserved?
5. 2.5 OAIS: suitability and motivation
6. 2.6 What should big-science funders require, or provide?
4. 3 The practicalities of data preservation
1. 3.1 Modelling the archive
1. 3.1.1 The OAIS model
2. 3.1.2 The DCC Curation Lifecycle model
2. 3.2 Software preservation
3. 3.3 Data management planning
1. 3.3.1 DMP in space
2. 3.3.2 Current and future DMP in the LSC
4. 3.4 Data preservation costs
5. 3.5 The GW community and the AIDA toolkit
5. 4 Conclusions and recommendations
6. A Case study
7. B AIDA assessment
8. About this document
## 0 Introduction
Astronomy is as old as human culture. Early agricultural civilisations
required reliable predictions of the positions and motions of the Sun and
Moon, in order to predict in turn seasons, tides, and river risings. Even in
the absence of an extensive scientific model, these predictions relied on
careful observations, preserved in the form of almanacs or ephemerides.
Documents such as these associate astronomy with not only the first data
archives but, since these artifacts still exist, also the oldest data archives
in the world. Long-term digital preservation in astronomy is nothing new. We
cannot resist saying more about this, in Sect. 1.8.
Astronomical archiving does however evolve, and in the last few decades both
astronomy and particle physics have had to become leaders in large-scale data
management.
Although astronomical images (now all born digital) have always been
substantial in size, they have generally been reasonably manageable. Newer
astronomical techniques – and we are thinking of 21st century radio astronomy
and gravitational astronomy – are capable of generating truly challenging
quantities of data; and particle physics has been generating, and addressing,
intimidating data problems for decades. These problems cover both the
management and preservation of large data volumes, as technical problems, and
the preservation of the data’s information content, on substantially varying
timescales.
### 0.1 Project Background
The Managing Research Data/Gravitational Waves project (MRD-GW) is concerned
with the data management arrangements of the LIGO Scientific Collaboration
(LSC), and of the broader Gravitational Wave (GW) community. It is one of the
six projects in the RDMP strand of the JISC Managing Research Data (MRD)
programme [2].
The GW community was selected by the Science and Technology Facilities Council
(STFC), at JISC’s invitation, as a representative example of big-science data
management practice – as we elaborate below, it has features of both the
traditional astronomy and HEP communities, without being identifiable with
either of them. While many of the specifics, below, relate to this community,
we believe much of the discussion is relevant to the other disciplines. Here,
we are focusing on the big-science projects which receive strategic support
from STFC, rather than the smaller-scale projects funded by specific research
grants, since it is these large-scale projects that are distinctive about
STFC-funded research. We assume that the outputs of the smaller projects will
be managed through disciplinary repositories, in a manner which more closely
resembles that of other research councils.
The MRD-GW project exists to inform three sets of stakeholders:
* •
Although the Joint Information Systems Committee (JISC) and the Digital
Curation Centre (DCC) have extensive experience with digital libraries and
digital curation in general, there are problems specific to ‘big science’ data
which JISC would like to better understand.
* •
The Research Councils have recently started to require bidders to include a
‘data management plan’ within project proposals. However there is no consensus
on what such a plan should look like for science funded by the STFC. The US
National Science Foundation (NSF) has recently placed binding requirements on
projects to produce data management plans [3].
* •
The LSC community has considerable internal software and administration
experience, and has solved a large number of data management problems focused
on large-scale data storage and transport. However there is an awareness that
(partly because there have been no immediate imperatives to do so) there was
until recently no published plan for a long-term data archive.
The existence of these three groups is reflected in the overall structure of
the document.
This project’s context also includes the broad Virtual Observatory (VO)
movement, which aims to develop standards and areas of consensus which help
scientists have ready access to astronomical data across sub-disciplines and
wavelengths. All the stakeholder groups have interests in the success of the
VO movement.
The project aims to bring together two sets of practice, namely the long-term
digital preservation perspectives represented by the OAIS reference model in
the abstract and the DCC in particular, and the very considerable experience
of practical large-scale data management, embedded within the LSC
community.††margin: For OAIS, see [4] and Sect. 3.1.1; in this report the
‘DCC’ is the JISC Digital Curation Centre, not the LIGO Document Control
Center. ††margin: For OAIS, see [4] and Sect. 3.1.1; in this report the ‘DCC’
is the JISC Digital Curation Centre, not the LIGO Document Control Center.
### 0.2 How to read this document
This document is organised into three main sections, broadly corresponding to
the three audiences we are addressing.
Sect. 1 is about data management in big science. It is addressed to the JISC
and to the data preservation community in general, and is intended to
illuminate the ways in which scientists in these areas have distinctive data
management requirements, and a distinctive data culture, which contrasts
informatively with other disciplines.
Sect. 2 is primarily addressed to STFC and other similar funders of this type
of science. It is concerned with the responsibilities which are imposed on
funders by the wider society, and which are passed on to the funded through
requirements on the governance of projects and the availability of data. The
recommendations here are concerned with how best to express these
responsibilities.
Finally, Sect. 3 is primarily addressed to the LSC, as a proxy for similar
big-science projects. The explicit recommendations here are intended to be of
as much interest to projects, as actions they may wish to take, as to funders,
as behaviour it may be prudent or productive to require.
### 0.3 Working with communities – pragmatics
This report is the result of a fruitful collaboration with the GW community.
It may be useful to note some of the features of the project, and the
community, which contributed to this.
* •
The project team, as part of Glasgow University, has current involvement in
the community, and the project director (Woan) is a senior figure there.
* •
The LIGO community is already aware of the general need for data management,
and the specific need for preservation (see [5]).
* •
The project personnel have relevant scientific background, and are to some
extent in the position of being informatics-for-astronomy specialists (ie
we’re ‘insiders’).
* •
The community is large and (via studies such as [6]) has some experience of
being ‘studied’.
* •
The existing LVC workshop series meant that we could contact relevant people
easily in a context where newcomers were expected, and we didn’t have to add
our own data management workshop.
## 1 Data management in Big Science
### 1.1 LIGO in perspective: LIGO, big science, and astronomy
What is ‘big science’?
Big science projects tend to share many features which distinguish them from
the way that experimental science has worked in the past. Such projects share
(non-independent) features such as:
big discoveries
These projects are expected to be amongst the most important ones of their
generation. Although there is very high confidence that their headline science
goals (for example the Higgs and GW searches) will be successful, they are
also expected to produce long lists of unexpected results, and a broad range
of engineering spinoffs.
big money
These are decades-long projects, supported by country-scale funders and
billion-Euro budgets (the total budget for the LHC is around three billion
Euros††margin:
http://askanexpert.web.cern.ch/AskAnExpert/en/Accelerators/LHCgeneral-en.html
††margin:
http://askanexpert.web.cern.ch/AskAnExpert/en/Accelerators/LHCgeneral-en.html
, not including the detectors, nor the personnel and hardware costs directly
supported by country funders, which cost between one and two times that sum).
big author lists
The projects involve collaborations of hundreds of people (the LSC author list
runs at around 600 people (cf Sect. 1.6.1), and the LHC’s ATLAS detector
author list is around 3000).
big data
Enhanced- and Advanced-LIGO (for example) will produce of order
$1\text{\,}\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$, comparable to the ATLAS
detector’s $10\text{\,}\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$; the eventual
SKA data volumes will dwarf these.
big admin
MOUs, councils, workshop series.
big careers
Individuals may make the journey from PhD to chair on a single project.
There is a discussion of the features of ‘big science’, and LIGO’s progress
towards that style of working, in [7], with an extended history of the sub-
discipline in [6].
Because of the large costs involved and because there is usually little
immediate commercial value in this research (though of course there are
substantial long-term economic payoffs for the investing countries), these
large projects are funded at the national or international level, so that
taxpayers are the ultimate stakeholders. Even putting aside the scientific and
scholarly need for adequate data preservation, these national investments make
it necessary for funders both to demonstrate that projects are being
efficiently exploited to produce macro-economic value, and to make the data
products available for public use. We discuss open data in Sect. 2.2
### 1.2 Data volumes
The most immediate problem with data curation and sharing in these scientific
areas – though in the end not the most significant one – is the data volumes
involved. The current volume of LIGO data is of the order of hundreds of
terabytes, and the data rates is expected to grow, over the course of the
project, from its current $100\text{\,}\mathrm{TB}\text{\,}{\mathrm{yr}}^{-1}$
to around $1\text{\,}\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$ (see Table 1,
which shows the variation in data size for science runs three to six††margin:
In the context of larger-scale projects, a ‘science run’ is a period when the
equipment is run more-or-less continuously, gathering scientifically useful
data. Between science runs, the experiment will either be down for
maintenance, or on a planned ‘engineering run’; data from engineering runs is
generally stored, but is not expected to be useful to scientists. ††margin:
In the context of larger-scale projects, a ‘science run’ is a period when the
equipment is run more-or-less continuously, gathering scientifically useful
data. Between science runs, the experiment will either be down for
maintenance, or on a planned ‘engineering run’; data from engineering runs is
generally stored, but is not expected to be useful to scientists. ).
| S3 | S4 | S5 | S6
---|---|---|---|---
L0 | $57$ | $32$ | $816$ | $261$
L1 | $8.24$ | $4.04$ | $119$ | $76$
L2 | $1.55$ | | |
L3 | $0.97$ | $0.86$ | $9.70$ | $3$
(duration/day) | $70$ | $29$ | $695$ | $482$
Table 1: LIGO data set size estimates in TB, and run lengths in days, for
science runs three to six (‘S$n$’), and various data types (size data taken
from [8]; there were a total of six science runs in LIGO; L0 is the run’s raw
dataset, and L1 to L3 are progressively reduced).
LIGO is just one of several other existing or planned big physics projects,
including the LHC, the Square Kilometre Array (SKA), and various European
Space Agency (ESA)/NASA space missions. In comparison with these projects,
LIGO’s data handling requirements are relatively modest. The LHC will have
data volumes of tens of $\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$††margin:
ATLAS, one of the two larger LHC detectors, stores
$3\text{\,}\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$ by itself; see
http://atlas.ch/pdf/atlas_factsheet_4.pdf for some entertaining numbers.
††margin: ATLAS, one of the two larger LHC detectors, stores
$3\text{\,}\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$ by itself; see
http://atlas.ch/pdf/atlas_factsheet_4.pdf for some entertaining numbers.
Further in the future, the SKA (which is due to be commissioned around 2020)
has predicted requirements up to
$1\text{\,}\mathrm{Tbit}\text{\,}{\mathrm{s}}^{-1}$ locally and
$100\text{\,}\mathrm{Gbit}\text{\,}{\mathrm{s}}^{-1}$ intercontinentally; this
involves transporting, though not necessarily storing, around
$1\text{\,}\mathrm{TB}\text{\,}{\mathrm{min}}^{-1}$ or
$0.5\text{\,}\mathrm{EB}\text{\,}{\mathrm{yr}}^{-1}$ [9]. This is 0.05% of the
predicted $1\text{\,}\mathrm{ZB}\text{\,}{\mathrm{yr}}^{-1}$ _total_ worldwide
IP traffic for 2015 [10].††margin: $1\text{\,}\mathrm{PB}$ is
$1000\text{\,}\mathrm{TB}$; $1\text{\,}\mathrm{EB}$ is
$1000\text{\,}\mathrm{PB}$; $1\text{\,}\mathrm{ZB}$ is
$1000\text{\,}\mathrm{EB}$; note that the unit B refers to bytes, not bits
††margin: $1\text{\,}\mathrm{PB}$ is $1000\text{\,}\mathrm{TB}$;
$1\text{\,}\mathrm{EB}$ is $1000\text{\,}\mathrm{PB}$; $1\text{\,}\mathrm{ZB}$
is $1000\text{\,}\mathrm{EB}$; note that the unit B refers to bytes, not bits
Large-scale physical science experiments have long produced significant data
volumes, but in recent years datasets appear to be increasing in volume and in
complexity at an overwhelming rate, and this may present a qualitatively
different data management problem. This is sometimes described in rather
apocalyptic terms – as a ‘data deluge’ or the like – and some of the
challenges and opportunities are described in [11].
### 1.3 Data management styles in the physical sciences
It seems useful to discuss, here, some of the distinctive features of data
collection and management in the experimental physical sciences, since these
have an impact on both the expectations for, and the problems with, the data.
Big-science research projects have a number of relevant common features:
Large data sets
Such projects’ data sets are ‘large’ in the objective sense that the projects
are typically so greedy for data storage, that their holdings are near the
edge of what it is technically feasible to store and transport.
Innovative data management
As part of the response to their need for large data volumes, big-science
projects are often extremely innovative in their solutions to data management
problems, to the extent that they are willing to work with experimental
filesystem types, or adapt and extend operating system software or network
transport protocols (see http://lcg.web.cern.ch/ to get an impression of the
scope of development efforts here).
Specialised software
Because the instruments and their data sets are so complicated, these projects
typically generate large custom data analysis software suites. These may
require specialised and unwritten knowledge to use, and therefore appear to
represent a significant software preservation challenge.
Beyond the substantial software engineering challenges described above, the
physical sciences tend to have few ‘IT’ problems, since the communities
contain plenty of people with sufficient technological nous to address
essentially all day-to-day computing-related problems, and these communities
are therefore generally reasonably well-organised with regard to backups,
storage, and basic file sharing (see also the discussion of technological
readiness in Sect. 3.5). At the same time, however, the communities are rather
conservative from the point of view of a computer scientist, and sometimes
rather informal from the point of view of a software engineer. That is, the
attitude to custom computing solutions is very similar to the attitude to
custom lab hardware: it may need to be creative and experimental, but never
for its own sake; it must be stable, but is never frozen; it is accurately
made, but rarely polished. The analogy with lab hardware and software holds to
the extent that, in the LHC community, data management groups are regarded as
detector subsystem groups; that is, they have the same general status as the
magnet or accelerator engineers, and expected to produce agile and innovative
computing services very different from the more routine, lab-wide, provision
of CERN IT services.††margin: One of the DCC researchers, commenting on this
report, quoted a GridPP survey respondent commenting that the LHC computing
task: ‘‘Providing resilient services that maintain access to data for the
experiment users 24/7 – services are complex, bleeding edge, and are
constantly being updated. Controlling that process, whilst also maintaining
service up-time is very challenging’’. ††margin: One of the DCC researchers,
commenting on this report, quoted a GridPP survey respondent commenting that
the LHC computing task: ‘‘Providing resilient services that maintain access to
data for the experiment users 24/7 – services are complex, bleeding edge, and
are constantly being updated. Controlling that process, whilst also
maintaining service up-time is very challenging’’.
The result of this is that lab software represents functional solutions to
immediate-term problems, generally with flexibility enough to respond to
medium-term problems, but without much attention being given to the
imponderables of the long term, after the experiment has completed. It is
precisely these Long Term preservation questions, in the OAIS sense of more
than one technology generation, that are the concern of this report.
There is plenty of prior art in this area. See reference [12] for a review of
data management practice in a variety of scholarly areas, which additionally
covers several proposed life-cycle models, and analysis techniques. There is a
similar overview in the PARSE.Insight case-studies report [13], which examines
data management practice in HEP, earth observation, and social science and
humanities. These case-studies were conducted via interviews, and
participation in ongoing efforts within the communities. The same project
produced a gap analysis and roadmap, which make valuable reading.
This is a good place to stress that ‘big science’ generally handles its data
well, and can even be regarded as exemplary (compare Sect. 3.5). There are a
few features which naturally encourage good data management practice in the
large-scale physical sciences.
* •
These are often relatively well-resourced projects, with plenty of computing
experience and lots of engineering management. There is lots of obvious
infrastructure in the development of a large collaborative experiment, which
gives data management an obvious budgetary home, where it is not competing
with funding which directly supports researchers.
* •
Astronomy and HEP projects have always produced ‘large’ data volumes: this
makes _ad hoc_ data management manifestly unattractive, and encourages
explicit data management planning and discipline.
* •
The scale of these experiments means that they tend to be shared facilities
providing documented services to their users, so that documented interfaces
and SLAs are natural.
* •
These projects rarely if ever produce commercially sensitive data, so that the
confidentiality concerns are well circumscribed, concerning professional
priority rather than IPR or other financial worries.
Although these features are to a greater or lesser extent specific to this
type of science, they have given rise to the notions of _data products_ and
explicit _proprietary periods_ , which we believe would be useful in other
areas, and which we discuss in Sect. 1.11.
Although it is GW data which is our nominal focus in this report, it is
convenient to first describe general astronomy data, then distinguish that
from High Energy Physics (HEP) data, which has a somewhat different data
culture, and then describe how the GW community, which is in many ways
intermediate between the two, handles its data.
### 1.4 Astronomy data
Astronomy (excluding GW astronomy for the moment) has probably the most
straightforward data management practices in the physical sciences. When an
optical telescope takes an image (or a spectrum, which for our present
purposes is technically equivalent to an image), either as part of a
systematic survey of the whole sky or as a pointed observation requested by an
astronomer, the image is typically moved from the telescope’s detector
straight into its archive, from where it can be later retrieved by the
astronomer, accompanied by automatic or manually-added metadata.
Non-optical astronomers (covering the rest of the spectrum, from radio to
gamma rays), and most satellite missions, have a somewhat more complicated
route from observation to image, and a broader set of data products, but have
essentially the same model, and the same discipline and expectations around
archives. From the point of view of data management therefore, we can elide
the differences between the various branches of astronomy. Gravitational wave
and neutrino astronomy, in contrast, are not studying the electromagnetic
spectrum, and partly as a consequence their study more closely resembles
particle physics (see Sects. 1.5 and 1.6 below).
Most large telescopes, satellites and instruments††margin: An ‘instrument’ in
this context is the light-sensitive detector attached to the telescope or
satellite optics (the camera, in effect). It is replaceable or swappable, and
regarded as a separate piece of engineering from the telescope. The days when
observers would travel to the telescope carrying their own instrument are now
largely past. ††margin: An ‘instrument’ in this context is the light-
sensitive detector attached to the telescope or satellite optics (the camera,
in effect). It is replaceable or swappable, and regarded as a separate piece
of engineering from the telescope. The days when observers would travel to the
telescope carrying their own instrument are now largely past. operate partly
or exclusively according to a model in which astronomers are awarded
‘telescope time’, ranging from a few hours to a few nights, as the results of
competitive bids closely analogous to grant bids. The resulting data generally
has a proprietary period, extending for perhaps 12, 18 or 24 months after the
data is taken, during which only the observer who requested it can retrieve
it, but after which it automatically becomes retrievable by anyone (‘embargo’
would be a better term, though unconventional). Similarly, instruments built
by consortia generally have proprietary periods during which the data is only
available to consortium members. The proprietary periods are partly for the
benefit of the consortium individuals – it is their reward for the initiative
and possibly decadal effort of building the instrument – but they are also a
pragmatic reflection of the length of time it may take to calibrate and
validate acquired data, ready for deposit in an open archive. As a result, the
lengths and terms of proprietary periods are the subject of negotiations
between the instrument builders and their ultimate funders, though the
negotiations are always about the length of the delay before a general data
release, and never question the necessity for the release itself.
NASA missions now typically have 12-month proprietary periods, but this has
varied historically, and for example the 1990 COBE mission, which included
significant technological novelty, and whose performance was therefore rather
unpredictable, had a 36-month proprietary period.
Not all instruments have formal release plans, and the proprietary periods
that exist may be adjusted informally. Caltech is one of the few private
institutions which is rich enough to own, or have a significant share in,
world-class telescopes (Palomar and Keck). It has no declared policy on data
management or data sharing, beyond a broad tacit expectation that data will be
published as appropriate for normal scientific practice. As a second example,
during the ‘science demonstration phase’ of the commissioning of the Herschel
telescope (that is, the last commissioning phase, verifying that the science
goals were achievable), the instrument team invited††margin: See ‘Herschel
Observers’ Manual’ §1.1.4,
http://herschel.esac.esa.int/Docs/Herschel/html/ch01.html; thanks to Haley
Gomez for bringing this to our attention. ††margin: See ‘Herschel Observers’
Manual’ §1.1.4, http://herschel.esac.esa.int/Docs/Herschel/html/ch01.html;
thanks to Haley Gomez for bringing this to our attention. observers to
nominate part of their scheduled observations to be performed early, during
this still-experimental commissioning phase. When the observations proved
successful (as they generally were), the observing teams were given the choice
either of making the data immediately public, in time for the opening of the
Herschel archive and a journal special issue, and having the observation time
re-credited to them; or else retaining the 12 month proprietary period without
the re-credit.
Image data is the archetypal astronomy data, and is generally stored as files,
but another important category is the astronomical ‘catalogue’, of object
positions, spectra and other properties, usually stored in relational
databases. Astronomical archives range from quite small ones (at one extreme,
a small specialised instrument may have its ‘archive’ consisting of a file
server looked after by a graduate student) to very large professionally
managed archives which are both the primary sources of some data sets, and
mirrors of others.
Astronomy data is potentially very long-lived. Although astronomers are
naturally drawn to the newest instruments with the greatest sensitivity, it is
not unusual to draw on relatively old archive data. In most cases, this will
be still be born-digital data, but digitised versions of century-old
astronomical plates are used in precise astrometry, and to identify the
precursors of supernovae and other one-off events (see for example the
Edinburgh SSA††margin: http://surveys.roe.ac.uk/ssa/ ††margin:
http://surveys.roe.ac.uk/ssa/ , further discussed, with background, at [14];
and a more discursive account of plate scanning, including discussion of some
of the archival challenges, in [15, 16]). Even babylonian and ancient chinese
astronomical data has been used for contemporary science, helping measure the
rate at which the earth’s spin rate, and thus the length of day, is changing
[17]; similarly, 3 000-year-old egyptian data has been used to measure the
change in the orbital behaviour of the three stars in the Algol system [18].
The cosmos changes slowly on our timescales, so that the great majority of
astronomical observations are repeatable; the exceptions are those cases where
long time-bases are necessary (precise astrometry) or where the object of
study is a one-off, and therefore unrepeatable, event such as a recent or
historical supernova.
Astronomy data is also intelligible in the long term: although untranscribed
babylonian tablets can only be read by specialists, contemporary astronomers
can basically understand the data published in Kepler’s 1627 _Rudolphine
Tables_ , and with some assistance can understand the content of the 11th- to
12th-century Toledan Tables [19]. Although biologists might be able to make
similar claims with respect to, for example, Linnæus’s observations, it is
hard to find equally long-lived data in the physical sciences, or born-
analogue physics data where there is a similar contemporary pressure for
digitisation.
There is essentially no file-format problem in (electromagnetic) astronomy,
since the Flexible Image Transport System (FITS) format is universal [20, 21].
Though not perfect, this is a relatively simple and well-defined format,
combining binary or table data with keyword-value metadata.
Astronomical data also has a well-developed notion of _data products_. These
are datasets which contain, not raw data, but data which has been processed to
a greater or lesser extent. We can distinguish at least three levels of data
in this context; most large instruments will have more than one level of
derived products.
Raw data
This is the lowest-level data, consisting of the direct output of a detector
or other instrument, or the raw satellite telemetry. This data is made
meaningful only by processing with software which is to some extent specific
to the equipment in question. Though it will be preserved as a matter of
course, it is rarely published, nor used by, nor useful to, other researchers,
except in unusual circumstances. In the case of a particularly subtle effect –
or less commonly, a debate over a theoretical analysis or calibration – a
researcher might return to the raw data, but this will generally be done with
the collaboration of the instrument scientists, and may be otherwise
infeasible, to the extent that any results obtained without such insider
knowledge might not be believable by the broader community.
Data products
After it is gathered, (raw) data must be processed (‘reduced’) to turn it into
scientifically meaningful numbers (interpreting engineering or telemetry data
streams, and calibration) and to remove various instrumental and observational
artefacts. Data products are usually made available in standard formats (in
astronomy, generally FITS files), whereas raw data, if it is made available at
all, may well be in an instrument-specific form.
Publications
Sitting above the data products is a class of high-level outputs, including
scientific papers, and other peer-reviewed outputs such as published catalogue
s. Journal articles are curated at publisher sites and the Astrophysical Data
Service (ADS), and article preprints at the arXiv (cf Sect. 1.9). Modest
volumes of data can be published as digital appendices to journal articles in,
for example, Astronomy and Astrophysics Supplements; these are curated at the
journal and at VizieR.
It is the data products which are the outputs which are sufficiently free from
observational artefacts to be the starting point for scientific analysis
(high-level products are sometimes referred to, informally, as ‘science
data’), and which represent the class of data which is naturally archived,
most carefully documented, and which will eventually be made public. There may
be multiple levels of data products, with lower-level products carrying more
information, but using which requires more detailed knowledge of the
subtleties of the instrument and its processing pipeline. To a much greater
extent than is true for HEP data, for example, the highest level astronomy
data products are both useful and generally intelligible – everyone is, after
all, looking at the same sky – but researchers will often use intermediate-
level products, if they can invest the time to learn about them, or have
collaborators who have experience with them. Those researchers who are more
intimately involved with an instrument will be comfortable using lower-level
data products, because they will have the knowledge which enables them to run,
or experimentally re-run, the pipelines in a scientifically meaningful way.
That said, in OAIS terms, astronomical data can be characterised as having a
broad Designated Community and well understood Representation Information.
Publications are in the province of libraries and similar repositories, and
are not considered further in this report.
Optical astronomy (that is, with observations made using visible light) has
the most straightforward data, so that the distinction between raw data and
data products is slight to the point of being rather artificial: astronomers
reusing optical data would expect to recalibrate the raw or nearly raw data,
and would not anticipate having difficulty doing so.
We conclude with some examples: A typical telescope archive is the UKIRT
archive at http://archive.ast.cam.ac.uk/ukirt_arch/; there are several image
and spectrum archives at the Royal Observatory Edinburgh’s Wide Field Archive
Unit††margin: http://www.roe.ac.uk/ifa/wfau/ ††margin:
http://www.roe.ac.uk/ifa/wfau/ ; and there is a large collection of catalogue
s available at Strasbourg Data Centre (CDS) (see Sect. 1.4.1 below).
The ESA Hipparcos astrometry mission††margin:
http://www.esa.int/science/hipparcos ††margin:
http://www.esa.int/science/hipparcos flew between 1989 and 1993, and produced
a high-precision catalogue of 100 000 stars [22]. The catalogue is available
online as queriable databases at ESA††margin:
http://www.rssd.esa.int/index.php?project=HIPPARCOS ††margin:
http://www.rssd.esa.int/index.php?project=HIPPARCOS and CDS, as CDs, and as
PDFs which match the catalogue’s 17-volume printed version. The printed
version is an interesting case: as discussed in the catalogue (vol. 1,
§2.11.3), the printed pages are designed with a per-page checksum, to help
with re-scanning the catalogue from paper, in the presumed-likely future case
that the digital version becomes unreadable and only copies of the paper book
survive. The Tycho catalogue, from the same mission, comprises around 20 times
the number of stars, at lower precision, and is only available online.
There is some discussion of preservation costs in Sect. 3.4.
#### 1.4.1 Strasbourg Data Centre (CDS) as a disciplinary repository
CDS is a large disciplinary repository for astronomy [23]. It stores a broad
range of catalogues, of various sizes, in its VizieR service (see [24] and
http://vizier.u-strasbg.fr/) and provides a large librarian-curated collection
of data from, measurements of, and references to, individual astronomical
objects. It cooperates closely with ADS.
CDS was created, and is supported, by the french agency in charge of ground-
based astronomy – first CNRS/INAG then CNRS/INSU – as a joint venture with
Strasbourg University. The main support is through permanent positions from
the CNRS/INSU and the University (researchers, computer engineers, and
specialised librarians), with additional contracts supported by funding from
various sources.
CDS is administratively located within a research structure, Strasbourg
Observatory, providing an active research environment for CDS astronomers. The
preservation aspects have never been separated from the provision of services
and the maintenance of local expertise on data management and
preservation.††margin: We are most grateful to Françoise Genova, of CDS, for
this discussion of CDS’s history and support. ††margin: We are most grateful
to Françoise Genova, of CDS, for this discussion of CDS’s history and support.
This can be seen as an example of a very successful disciplinary repository.
There appear to be several key features of this success.
* •
CDS has established, and actively maintains, international leadership in the
curation of astronomical data, by virtue of collaborating widely and investing
effort in projects (such as the International Virtual Observatory Alliance
(IVOA)) which support and promote data sharing.
* •
As a result of the intimate relationship between the repository, the
observatory and the university (to the extent that the boundaries between the
three can seem rather vague to outsiders), CDS personnel have practical
knowledge of how their data is used, and what researchers need.
* •
The core funding for CDS comes from the french state, but it is conceived as
an internationally visible project.
#### 1.4.2 Collaborations in astronomy
The most visible collaborations in astronomy are large terrestrial and
satellite-borne telescopes and other instruments. At the risk of
oversimplifying, these are generally not the many-person collaborations usual
in GW or HEP physics, but are instead facilities created by space agencies or
consortia of national funders. Although they are highly innovative leading-
edge facilities, they are not seen as _experiments_ in the same way as LIGO or
the LHC are (massive) items of specialised hardware built to answer a
delimited set of scientific questions. They are instead _observatories_ : the
data management in these facilities is part of their general operating
infrastructure, and the research and research data they produce is ‘owned’ (at
least in an academic rather than a legal sense) by the scientist _users_ of
the facilities, rather than the facility itself.
Astronomy does however have a variety of data-analysis collaborations. These
are semi-formal collaborations concerned, mostly, with multi-wavelength
studies of multiple archives, and include for example UKIDSS-UDS, the Herschel
Atlas collaboration, HerMES and GAMA††margin:
http://www.nottingham.ac.uk/astronomy/UDS/, http://www.h-atlas.org/,
http://hermes.sussex.ac.uk/, http://www.gama-survey.org/ ††margin:
http://www.nottingham.ac.uk/astronomy/UDS/, http://www.h-atlas.org/,
http://hermes.sussex.ac.uk/, http://www.gama-survey.org/ . These have between
20 and 60 collaborating members scattered over perhaps a dozen institutions
but, crucially, no ‘corporate’ existence, and little or no direct funding.
Instead, they are funded indirectly via individual fellowships or rolling
grants: participation in the collaboration might be a strong feature of a
grant application, but it is not the collaboration as such that receives the
direct support. They do have governance structures, but these tend not to be
particularly formal, because they remain small enough that there is little
perceived need. These collaborations exist to derive high-value derived data
products from the lower-level data products of the archives they are analysing
(for example Herschel is an ESA observatory mission: this means that
individuals can bid for observations, but that ESA does not have it as part of
its remit to provide more than minimally reduced science data).
The collaborations distribute their results in papers, and associated
datasets; they typically build archives to support and distribute their work,
but there’s no expectation (beyond the usual cooperative academic norms) that
they will help others work on the data, or release it. It is hard to see how
there could be such an expectation, much less an obligation, since they
receive little direct funding, and their indirect funding comes from a
multinational set of entities with potentially very different Data Management
and Preservation (DMP) policies.
### 1.5 High Energy Physics data
Astronomy is essentially an observational science: telescopes, their optics,
and the detectors which hang off them, are constructed to create a path from
nature to data which is as nearly as possible unmediated. This means that it
is both reasonably obvious what things are to be archived, and that the nature
and processing of observational artefacts are well and commonly understood.
This means that astronomy, unusual in the physical sciences for needing to
preserve data long-term, is in the happy position of having its data readily
preservable.
HEP data is different. HEP is a participative science, where objects ranging
in size from electrons all the way up to nuclei are disassembled, and data
about the messy results of this disassembly is examined to retrieve
information about the interior structure of the original. This reconstruction
from collision data depends on a shifting engineering understanding of rivers
of data, out of instruments which are one-off works of art, designed and
assembled by a thousand-strong community, close-packed into a detector the
size of a small cathedral, attached to a machine with its own postcode.
The result of this is that HEP data analysis is rather tricky, with many steps
between data and science, each of which depends on software which encodes a
detailed understanding of the data’s provenance. In consequence, although HEP
data is typically distributed with multiple levels of reduction, almost none
of these levels (with the exception of formal publications) are
straightforwardly suitable for long-term preservation. This is because
interpretation of this data is heavily dependent on software, the use of which
requires detailed experimental knowledge which it may be infeasible to
preserve. In OAIS terms, the designated community is tiny because the
Representation Information is hugely complex.
In addition to this, HEP data has a considerably shorter shelf-life than
astronomy data, as discussed above. In contrast, old HEP data is typically
made redundant by new data, obtained from more powerful accelerators. Also in
contrast to astronomy data, HEP data is not expected to be generally
intelligible for very long: two- or three-decade old data might potentially be
useful or intelligible, but much beyond that would count as archaeology. At
the risk of being whimsical, we can compare the roughly millennial lifespan of
astronomical data with the roughly three-decade lifespan of HEP data, and
conclude that the latter goes ‘off’ about 30 times faster than the former.
Although facilities make very considerable efforts to manage data safely while
an experiment is running, there is little real pressure to preserve HEP data
into the long term.
Of course, things are not quite as straightforward as that in fact. (i) The
LHC gains interaction energy at the expense of a messier collision, so there
are potentially some features that will be detectable in one dataset (for
example the HERA p-e data) which would not be findable in the LHC. While
interaction energy is the most prominent metric of an accelerator’s
performance, it is not the only one, so that larger accelerators will not
render smaller ones obsolete as inevitably as we may have suggested above.
Similarly to this, (ii) data reduction errors may be dominated by theoretical
uncertainties rather than experimental ones, and these will only be improved,
and the data re-reduced, after the experiment is over. Finally (iii) there are
no accelerators bigger than the LHC currently scheduled, so that this dataset
may remain the highest-energy one for a relatively long time. The archaeology
is illustrated in [25] and the problem further explored in [26], which also
discusses the HEP community’s developing plans for data preservation.
Qualifications notwithstanding, the overall timescales in HEP are shorter than
in astronomy, and the solutions described in [26] are concerned with
prolonging a continuous low-level relationship with a dataset rather than
being able to return to a dataset cold.
Unlike astronomy, HEP has for the last few decades been organised into larger
and larger collaborations, and these collaborations have developed intricate,
and socially fascinating, cultures for managing this. The two larger
instruments at the LHC, ATLAS and Compact Muon Solenoid (CMS), each have
author lists of order 3 000 people, so that the various CERN collaborations
account for around 10 000 research-active individuals. There is extensive
discussion of the history and structure of the LHC collaborations in [27] and
in the outputs of the pegasus project,††margin:
http://www.pegasus.lse.ac.uk/research.htm and in particular [28] ††margin:
http://www.pegasus.lse.ac.uk/research.htm and in particular [28] but many of
the collaborations’ relevant organisational features are echoed in the GW
community: this is discussed in Sect. 1.6.1 and we do not discuss them here.
### 1.6 Gravitational wave physics
The gravitational wave community has astronomical goals, but in the scale of
the LIGO project, and in the amount of novel technology involved, as well as
in the fact that many of the personnel involved came originally from a HEP
background, the project’s culture more closely resembles that of a HEP
experiment than of an astronomical telescope. We discuss some specific
features of LIGO data in [29]; here we discuss where GW data, and the
discipline’s organisation structure, fits on the spectrum between astronomical
and HEP data.
#### 1.6.1 Gravitational wave consortia
There are three principal sources of recent GW data available to UK
researchers: LIGO, GEO600 and Virgo. There are other detectors which are
either smaller efforts (in terms of consortium sizes), which have stopped
taking data (TAMA-300), or which are still at the planning stage. See [30] for
an overview of current detectors, and of detector physics.
LIGO Lab is a collaboration between Caltech and MIT, which designs and runs
three interferometers in Hanford, WA, and Livingston, LA, in the US. GEO is a
German/British collaboration, which runs the GEO600 interferometer. The three
LIGO interferometers were shut down in October 2010 to refit for Advanced LIGO
(aLIGO); the GEO600 interferometer is still currently running. The LSC is the
result of a network of Memoranda of Understanding between LIGO Lab (or more
loosely the LSC) and multiple other institutions of various size.††margin:
The MOU which created the LVC is at [31], but MOUs are not routinely made
public. ††margin: The MOU which created the LVC is at [31], but MOUs are not
routinely made public. These relationships involve hardware, resources, and
data access of various types. Most typically, the resources in question are
personnel, and an institution such as a university physics department, which
wishes access to LIGO data, will contribute in return fractions of staff from
permanent staff, through post-docs, to PhD students, for a broad spectrum of
activities including data analysis, instrument fabrication and shift-work in
the detector control room. However in some cases, the MOUs are concerned with
data swaps, and set up limited data releases with other scientists: for
example, there are a few MOUs between the LSC, Virgo and other observatories,
which describe what data is to be shared, in what volumes, and the outline
authorship arrangements for any subsequent papers. GEO’s MOU describes a
particularly close relationship with LIGO Lab, but most of the MOUs are
broadly similar to each other, and the process of creating one is by now
streamlined. In total (as of June 2010),††margin: The definition of LSC
membership is included in [32] and the construction of the author list in
[33]. ††margin: The definition of LSC membership is included in [32] and the
construction of the author list in [33]. the LSC consists of a little over
1300 ‘members’; of these, 615 spend more than 50% of their time dedicated to
the project and so have a place on the LSC author list.
The term ‘LIGO’ has a number of not quite equivalent meanings: sometimes it
refers to LIGO Lab, sometimes to LIGO Lab plus the LSC, and the phrase ‘LIGO
detectors’ is generally understood to refer to the LIGO Lab and GEO detectors.
The Italian/French Virgo consortium has its own detector and analysis
pipeline, and has a data-sharing agreement with the LSC, represented by the
LVC.††margin: The term ‘LVC’ is not an initialism. It colloquially refers to
the data-sharing agreement [31] and joint meetings between the LSC and the
Virgo Collaboration. Though there are ‘LSC/Virgo collaboration groups’, there
is no formal big-C Collaboration. ††margin: The term ‘LVC’ is not an
initialism. It colloquially refers to the data-sharing agreement [31] and
joint meetings between the LSC and the Virgo Collaboration. Though there are
‘LSC/Virgo collaboration groups’, there is no formal big-C Collaboration. As
with LIGO, the Virgo detector will shut down between 2011 and roughly 2015.
Virgo has 246 members (with a slightly different definition from the LSC), and
GEO600 around 100.
There is an attempt to summarise these relationships in Fig. 1.
Figure 1: The relationships between various GW consortia.
These experiments have a common purpose: they exist to detect signatures of
gravitational waves, which are confidently predicted by the General Theory of
Relativity, but the actual observation of which would be a major scientific
event (there exists an LSC data processing flowchart which includes the not
entirely serious branch ‘‘Call Stockholm!’’).
Gravitational waves are sufficiently weak, however, that the existing
equipment will not become sensitive enough to have a good chance of detecting
them until after its refit, which began in late 2010 (when the project entered
the phase known as aLIGO), and which is scheduled to be completed when the new
detectors are commissioned in 2015.
#### 1.6.2 GW data
Although the consortia have (as expected) announced no detection so far, they
nonetheless produce a large volume of auxiliary data, representing background
and calibration signals of various types, and this, together with the core
data, means that the LSC collectively produces data at a rate of approximately
one $\mathrm{PB}\text{\,}{\mathrm{yr}}^{-1}$.
We can readily identify the levels of data which were discussed in Sect. 1.4:
Raw data
The lowest-level GW data consists of the signals from the core detectors. This
data is made meaningful only by processing with software which is completely
specific to the detectors in question. This is stored in ‘frame format’, which
is a very simple format intelligible to all the primary data analysis software
in the community, and which is multiply replicated across North America,
Europe and Australia. Although the disk format is common, the semantic content
of the raw data is specific to detectors and software, so that preserving it
long-term would represent a significant curation challenge.
Data products
The raw data is processed into calibrated ‘strain data’, which is the data
channel in which a GW signal will eventually be found (this is possibly, but
not necessarily, also held in frame format). This is the class of data
products which will eventually be made public. Unusually, it turns out that GW
raw data is in a semi-standard format, and the data products are specific to
the analysis pipeline which produced them.
Publications
Sitting above the data products is a class of high-level data products,
scientific papers, and other peer-reviewed outputs. The GW projects have
announced no detections of gravitational waves, but have nonetheless produced
a broad range of astrophysically significant negative results [30, §6.2].
As with the general astronomy data products discussed in Sect. 1.4, the
distinction between the ‘raw data’ and the ‘data products’ is that the latter
datasets, alongside their supporting documentation, will be available for use
and reuse by scientists who do not have an intimate connection with, and
knowledge of, the instrument.
Both the ‘data product’ and ‘publication’ groups are broad classes of objects.
The practical boundary between them is clear, however: what we are calling
‘publications’ are entities such as journal articles or derived catalogues
whose long-term curation is not the responsibility of the LSC data archive,
though they may be held in some separate LSC paper archive, which is as such
out of scope for this project.
#### 1.6.3 Gravitational wave data releases
Because the LSC has not announced the detection of any signal so far, and
because the data will remain proprietary to the consortium until well after
such an announcement, there are no distributed data products so far, and so
the issues surrounding formats and documentation have not yet been addressed.
However it is the eventual public data products which are the highest-value
outputs from the experiment, and which are the products which it will be most
important to archive indefinitely.
At present, LIGO data is available only to members of the LSC. This is an open
collaboration, and research groups which join the LSC have access to all of
the LIGO data††margin: http://www.ligo.org/about/join.php ††margin:
http://www.ligo.org/about/join.php . In return, they contribute personnel to
the project (including for example people to do shift-work manning the
detectors), and accept the collaboration’s publication policies, which require
that all publications based on LIGO data are reviewed by the entire
collaboration, and carry the complete 800-person author list. At present, and
in the future, data which is referred to by an LSC publication is made
publicly available. See Sect. 3.3.2 for further details on LIGO’s DMP plan.
#### 1.6.4 Summary: big-science preservation challenges
In the three sections above, we have tried to describe both differences and
commonalities between three large-scale scientific disciplines. Possibly the
biggest difference between the three areas is that high-level astronomical
data products are much more generally intelligible than even the highest-level
HEP products. In each case, however, we have a ladder of reasonably well-
defined data products, with each rung generated from the lower ones by
sophisticated data reduction pipelines.
The situation is not as rosy, from the point of view of long-term
preservation, as this account may suggest. Because the pipelines have
developed organically over a number of years, under the influence of
experience with earlier versions and increased understanding of the
instrument, the knowledge they represent is sometimes encoded within them in a
less structured way than would be desirable. Sometimes, metadata is encoded in
filenames, or in configuration files, or wikis, or even private emails. Of
course, one could simply argue that this information should be documented
better, but it would be hard to argue that the costs of this work would be
justifiable, to service a future theoretical need that few believe would even
become an actual one. In consequence, although the resulting data product will
be regarded as perfectly reliable, it may be infeasible to redo the analysis
other than by preserving and rerunning the pipeline software (even if it were
feasible, it would be prohibitively expensive, and rarely seen as valuable;
see also Sect. 2.4). For this reason, software preservation has some role in
the overall data preservation strategy. However it is not clear to us what
this role should be, and the thorny issue of software preservation is
addressed at greater length in Sect. 3.2.
### 1.7 A contrast: social science data
It is possibly instructive to contrast the data management practices discussed
here, with the very different problems faced by data managers in the social
sciences. In [34], the authors survey a number of social science projects,
with a particular focus on two large (for the social sciences) programmes
funded by the Economic and Social Research Council (ESRC) (the UK social
science research council) with substantial responsibilities for data
preservation and sharing.††margin: This work was part of the ‘Data Management
Planning for ESRC Research Data-Rich Investments’ project (DMP-ESRC)
(http://www.data-archive.ac.uk/create-manage/projects/JISC-DMP), funded by
JISC, like the present project, as part of the Managing Research Data
programme. ††margin: This work was part of the ‘Data Management Planning for
ESRC Research Data-Rich Investments’ project (DMP-ESRC) (http://www.data-
archive.ac.uk/create-manage/projects/JISC-DMP), funded by JISC, like the
present project, as part of the Managing Research Data programme.
For the ESRC projects, the artefacts being stored are simple things, at the
level of Content Information: they are conventional Word documents and audio
files, rather than the heavily structured and still somewhat experimental big
science data objects. The ESRC archive contents will remain broadly
intelligible to future researchers, without much archive-specific effort to
define Representation Information or a Designated Community. In contrast to
this simplicity, however, the ESRC archives have to cope with a broad range of
associated contextualising metadata, which is different for different
projects, and inconsistently or incompletely specified by the originating
researchers, perhaps as an afterthought. This makes archive ingest a
complicated problem, in contrast to the big science cases, where archive
ingest fundamentally involves little more than copying a self-contained set of
artefacts from working storage to some preservation store. In particular, the
ESRC projects have a complicated set of anxieties about copyright, IPR,
confidentiality, anonymization and consent; while LIGO cares intricately about
data access and security, it does so in the rather formal context of
professional ethics rather than family secrets.
This illustrates two further notable differences between physical science data
and that of social science or broader archival resources.
Firstly, the responsibility for ESRC data in practice lies with more junior
researchers, helped by part-funded archivists [34, §§5.2.1 & 5.4]. For big
science projects, it is funders and senior collaboration members who drive the
preservation efforts.
Secondly, essentially all physics data is born digital and complete, meaning
that all of the information to be archived is present at the time of deposit.
Of course, this is not complete from the point of view of reproducibility
(that requires journal articles and personal knowledge) and does not discount
the subsequent addition of subjective metadata as finding aids, but it is
completely specified from the point of view of conventional future analysis.
The distinction is that experimental data is a complete and objective account
of everything that was believed to be relevant in recording a physical event
which happened at a specific time. One can disagree with the experimenters’
beliefs about completeness (this shades into questions of reproducibility and
tacit knowledge), complain that some details might be recorded in notebooks
rather than digital records (more true of lab-scale than facility-scale
experiments), or in extreme cases argue about the nature of objectivity, but a
natural science experiment has a much clearer boundary, in space, time and
documentary extent, and so a more natural expectation of documentary
completeness, than will be usual for an experiment in the social or human
sciences. This is different from the traditional archive problem, where the
problems of interpretation are more visible and acknowledged, and the problem
of incompleteness more evident.††margin: For a vivid and illuminating
discussion of the complications and physicality of reproducing experiments,
see [35] and references therein (by coincidence, this describes observations
amongst gravitational wave experimenters in Glasgow); that discussion is
reprised in a larger context in [6, ch.35]. The question of tacit knowledge is
discussed at length in [36]. For a discussion of different types of reuse, see
[37, §3]. ††margin: For a vivid and illuminating discussion of the
complications and physicality of reproducing experiments, see [35] and
references therein (by coincidence, this describes observations amongst
gravitational wave experimenters in Glasgow); that discussion is reprised in a
larger context in [6, ch.35]. The question of tacit knowledge is discussed at
length in [36]. For a discussion of different types of reuse, see [37, §3].
The summary is not that the ESRC or the big science archives have an easier
job overall, but that the complications express themselves in different parts
of the mapping from OAIS abstractions to local fact. Big science archives must
preserve large complicated objects for a hard-to-describe Designated
Community, but because they are essentially always project-specific archives,
their implementation does not have to be generic, and many of the ingestion
issues can be baked into the original archive design.
### 1.8 Babylonian data management (less contrast than you’d think)
Contemporary astronomy began, in the west, in Mesopotamia in the fifth and
fourth centuries bce. Although earlier datasets exist – the _Venus tablet of
Ammisaduqa_ is a cluster of 7th C bce copies of 17th C or 16th C data
recording the rise times of Venus over a 21 year period – these earlier omen
texts seem to have been preserved for largely cultural reasons.††margin: See
[38] for background and further references, and [39, ch.4] for very detailed
discussion of the physical tablets. The precise date of the observations is of
considerable scholarly interest, since an agreed date would provide an
absolute fix for the otherwise relative chronology of the Late Bronze Age Near
East. ††margin: See [38] for background and further references, and [39,
ch.4] for very detailed discussion of the physical tablets. The precise date
of the observations is of considerable scholarly interest, since an agreed
date would provide an absolute fix for the otherwise relative chronology of
the Late Bronze Age Near East.
Figure 2: Calculated ephemeris for the period 104 bce March 23 to 101 bce
April 18, written on Seleucid year 209, month IX, day 18 (103 bce December
20?). Comparison with a JPL ephemeris shows that the text conjunction times
remain within a couple of hours of the correct values, with an offset
attributable to an error in the initial value. For detailed discussion, see
[40]. British Museum item Sp-II.52, ©Trustees of the British Museum.
Distinct from these, there is a large set of 4–500 other texts, ranging from
4th C bce to 75 ce with a smattering going back as far as mid-8th C bce, and
spanning the development of Babylonian theoretical astronomy during the 4th C
bce. These are a mixture of observations, calculated ephemerides (such as Fig.
2), and telegraphically obscure technical documentation. The observation texts
– ‘astronomical diaries’, forming the majority of the texts – describe in
sequence celestial and meterological observations, daily commodity prices,
river levels, and topical events. The observations of the Sun, Moon and
planets were of good enough quality, and preserved over a long enough time,
that when babylonian mathematical models were fitted to them they produced
values for the synodic and anomalistic months and (implicitly) the orbital
periods of the planets, which are very respectably close to their currently-
determined values (out by a factor of $3\times 10^{-7}$, in the case of the
synodic month). These were used to predict the first and last appearances of
planets, and the times of lunar (but not solar) eclipses.
The information in these texts is sometimes available on multiple tablets,
although it is not clear whether these duplicates were backups, mirrors, or
media refreshes. Many tablets have acquisition metadata, added in ink by the
archives, millennia apart, in Babylon and Bloomsbury.
It is clear that the tablets that have survived represent only a small
fraction of the total. but both the data, and the mathematical technology that
reduced the data and generated the ephemerides, were available and fully
intelligible to Hipparchus (_c_ 150 bce) and, either via him or directly, to
Ptolemy (_c_ 150 ce). The Babylon Data Centre was still active in the first
century ce, though funding cuts meant new acquisitions were by then minimal,
and it was operating in the collapsing ruins of the desert city.
The Content Information in the texts is sufficiently well preserved that if
the texts can be dated at all (in some cases through contemporary ingest
metadata), they can generally be dated to the very day; the technical
Representation Information, in contrast, is so terse as to make sense only
after the procedure being documented is reconstructed from the Content. The
cuneiform presents a challenge, but once this has been transliterated, the
datasets are fundamentally intelligible to current astronomers. The
preservation strategy is a daring one: by effectively founding western
astronomy, and arranging that the data was preserved just long enough that it
could be taken over by the (hellenic) successor civilisation,††margin: This
can be classed as a ‘high-risk’ data preservation strategy, and is not
included amongst this report’s Recommendations to STFC. ††margin: This can be
classed as a ‘high-risk’ data preservation strategy, and is not included
amongst this report’s Recommendations to STFC. the babylonians ensured that
their coordinate system (based on the zodiac) and number system (with angles
in degrees, subdivided into base-60 fractions) would still be in use by
astronomers 25 centuries later.
### 1.9 Bibliographic repositories
Though it is not strictly data, it seems useful to make parenthetical mention
of the big science communities’ literature repositories, since they seem to
illustrate the way in which the communities have learned to act collectively.
The preprint archive at arXiv.org started in 1991 as an electronic version of
the long-established practice of distributing preprints of accepted journal
articles around the high-energy physics community, by post. It currently
receives around 6000 submissions per month, predominantly in HEP, astronomy,
condensed matter physics and mathematics; it probably receives copies of
nearly 100% of the HEP community’s output.††margin:
http://arxiv.org/Stats/hcamonthly.html ††margin:
http://arxiv.org/Stats/hcamonthly.html Authors most typically submit papers at
the point when they have been accepted by the journal, but some submit earlier
versions, and a few are not further published at all. Although the journals
are still providing an _imprimatur_ , many papers are now principally read as
preprints, and many journals permit citations by arXiv reference. ArXiv is
supported by requesting contributions from its heaviest institutional users,
on a sliding scale rising to $4 000/year.††margin:
http://arxiv.org/help/support/whitepaper ††margin:
http://arxiv.org/help/support/whitepaper JISC Collections is one of these
‘tier 1’ supporters, on behalf of UK colleges and universities.
The NASA ADS at the Smithsonian Astrophysical Observatory preserves
bibliographic information for the astronomy literature, holds references to or
copies of journal article full texts, and curates digitised copies of older
articles sometimes unavailable from publishers. It also curates links between
these publications and the arXiv, and between publications and data. See [41]
for context, and some discussion of the arXiv numbers mentioned above.
The publication paradigm represented by arXiv (and similar smaller-scale
efforts) is underpinned by the peer review processes of journals. However as
journal subscription costs rise, journals are progressively cancelled, in a
process which may ultimately damage the reviewing process on which the
paradigm depends. The SCOAP3 consortium††margin: http://scoap3.org/about.html
††margin: http://scoap3.org/about.html aims to break out of this cycle by
directly supporting a small number of HEP journals, through a levy on the
funding agencies which support the field, in proportion to the share of HEP
publishing they support. In return for this the journals will remove both
subscription charges and page charges for these journals.
### 1.10 Virtual Observatories
A Virtual Observatory is an astronomical data-sharing system, composed of a
network of archives and data-access protocols. The goal is that the data
appears to be integrated and ideally appears to be local.
The earliest VOs were Astrogrid in the UK, the US-VO in the US (which became
NVO and then VAO), and the Astrophysical Virtual Observatory in Europe (which
became Euro-VO). They, along with a growing collection of smaller national or
regional VOs, formed the IVOA in 2002.††margin: http://www.astrogrid.org,
http://www.usvao.org/, and http://www.euro-vo.org; plus http://www.ivoa.net.
††margin: http://www.astrogrid.org, http://www.usvao.org/, and
http://www.euro-vo.org; plus http://www.ivoa.net. The IVOA exists to broker
portable network protocols for sharing data, on the part of cooperating
archives, and accessing it, on the part of client applications. The IVOA
focuses primarily on ‘traditional’ astronomy, and so has poor coverage of
solar physics and more broadly geophysics (and certainly provides no access to
GW data).
From this has grown the more general notion of the ‘VxO’, which is ‘‘[a]
service that ensures that all resources from sub-field $x$ are known,
discoverable, and easily accessible. It looks to the user like a uniform data
provider, but it is virtual.’’††margin: See further commentary in
http://lwsde.gsfc.nasa.gov/VxO_Report_Decadal_Survey_5_2011.pdf ††margin: See
further commentary in
http://lwsde.gsfc.nasa.gov/VxO_Report_Decadal_Survey_5_2011.pdf Examples
include the Virtual Solar-Terrestrial Observatory [42], HELIO, and NASA’s
Heliophysics Data Environment.††margin: http://www.helio-vo.eu and
http://lwsde.gsfc.nasa.gov/ ††margin: http://www.helio-vo.eu and
http://lwsde.gsfc.nasa.gov/
### 1.11 Data products and proprietary periods: reifying data management and
release
A common feature of the various data styles above is the notion of the _data
product_ , and it seems useful to recap and stress the salient features of
this here. Data products A data product is a designed and documented output of
an instrument, intended to be both archivable and immediately useful to other
researchers, by virtue of having observational artefacts removed as much as
possible. Depending on the discipline and the engineering complexity of the
instrument, data products may be anything from the raw data to a highly
processed derivative of the raw data; the ideal data product contains all the
scientifically relevant information with none of the experimental artefacts.
Researchers are not restricted to using only data products, but it will only
rarely be necessary for them to resort to reanalysing raw data (see the
discussion on p.1.4).
Data products correspond closely to the ‘Information Packages’ of the OAIS
model (see Sect. 3.1.1). In our experience, there tends to be little practical
difference between Submission Information Packages (SIPs) and Archival
Information Packages (AIPs), and where there are distinct Dissemination
Information Packages (DIPs), they tend to be available in addition to the
available SIPs and AIPs. An exception to this is archives such as the Wide-
Field Astronomy Unit at Edinburgh,††margin: http://www.roe.ac.uk/ifa/wfau/
††margin: http://www.roe.ac.uk/ifa/wfau/ which specialises in astronomical
survey science, and develops enhanced archives (which is to say, value-added
AIPs) as part of its participation in collaborative astronomy projects.
When the various Packages differ, they tend to be regarded as successively
higher-level, as opposed to alternative, data products.
The notion of data products has a number of concrete advantages.
* •
Most immediately, the existence of a stable and documented output makes it
easier for researchers to use and repurpose experimental and observational
results.
* •
Because the products are so central to an instrument’s output, they, and the
pipelines that produce them, are designed and costed at early stages of an
instrument’s production.
* •
Researchers can produce and share software which processes well-defined
products, possibly from more than one instrument.
* •
Because they are so explicit, they form well-defined start and end points of
discussions about interoperability between instruments. Indeed, the VO
programme could be characterised as an extended effort to negotiate new common
products which archives and software developers agree can be successfully
generated (by archives) from existing AIPs.
There is of course a cost associated with the design and development of data
products, but we believe that this will in most cases be much smaller than the
costs associated with the retrospective documentation and distribution of _ad
hoc_ datasets.
Another notion that is well-known in the physical sciences, but which as far
as we are aware is rare outside, is that of explicit _proprietary periods_ for
data. Proprietary period A ‘proprietary period’ is a period after data is
acquired, and therefore archived, by a shared instrument, during which it is
private to the observer or observers who requested it, and after which the
data (usually automatically) becomes public. The term ‘embargo period’ would
possibly be more generally intelligible, but ‘proprietary’ is conventional.
The notion is discussed elsewhere in this document (see for example Sect.
1.4), but we stress it here because it usefully concretizes a number of
otherwise vague questions about data release.
Instead of rather broad questions of the how, when, why and whether of data
management and release, we instead have questions such as ‘what are the data
products?’, ‘whom are they documented for, and how expensively?’, ‘how long is
the proprietary period?’ or ‘what is the quid pro quo for this
period?’††margin: Compare the comments about Herschel data in Sect. 1.4.
††margin: Compare the comments about Herschel data in Sect. 1.4. These
questions don’t magically become easy to answer, but they become a lot easier
to ask, and invite concrete answers and negotiation rather than _ad hoc_
argument.
There is nothing in the notions of data products and proprietary periods which
is obviously specific to the physical sciences. The notions have become well-
established in this area probably because it has long experience, of
necessity, of using large shared instruments which are operated to a greater
or lesser extent as services. This is less often the case in disciplines with
more bench-scale experimental norms, but even some areas of biology are now
more often using shared facilities, and in other disciplines, data products
and proprietary periods would become more natural, the more that preservation-
aware storage is used [43].
We commend the notions of data products and proprietary periods, and the data
culture they engender, to the broader research community. Indeed, we recommend
that ††margin: Recommendation 1 ††margin: Recommendation 1 data managers
should consider adopting the language of data products and explicit
proprietary periods when designing and documenting their holdings.
## 2 The responsibilities for data preservation
### 2.1 Visualising benefits
Why do funders wish to preserve data? Because they perceive _benefits_ to that
preservation.
Building on this truism, it seems useful to explicitly articulate these
benefits. The JISC-funded project Keeping Research Data Safe (KRDS) (see
http://www.beagrie.com/krds.php and [44]) described a collection of studies
and tools supporting data preservation. Amongst the KRDS innovations was a
typology of _benefits_ , describing three dimensions: direct to indirect,
near- to long-term and public to private. In a slight extension to the work in
KRDS, we can take the notion of ‘dimensions’ perfectly literally, assign any
particular benefit to a position along each of the three axes, and plot the
result in a three-dimensional space; see Fig. 3.
Figure 3: Visualizing benefits
In this figure we identify four benefits which might be associated with a big-
science project – namely the existence of data-reduction software, good
metadata, the provision of open data and the existence of system
administrators – and we sketch the approximate volumes they might occupy along
the three axes (in blue). On the same diagram, we can indicate (in red) the
approximate areas of interest of four sample stakeholders.
In the example here, ‘sysadmin support’ can be seen as an indirect benefit to
researchers, typically private to an institution, but creating value in the
near- and long term; it is therefore spread along the ‘near-long term’ axis,
but at one extreme of the other two dimensions. We can put on the same diagram
the approximate areas of interest of various research stakeholders. For
simplicity, we are here conceiving of individual researchers as selfish and
short-termist, though the same researchers will have long-term interest when
they have a collaboration or institutional hat on, and indirect public
interests in the long-term health of their discipline when they are serving on
a funding council grants panel; below we will take the term ‘funders’ to refer
both to the officials of funding bodies, acting as proxies for the wider
interests of society, and to members of the research community discharging
service roles.
We should not take this diagram too literally – it is not clear that the axes
are independent, and the extent and even the gross positions of the various
interests and benefits are debatable. The diagram is nonetheless thought-
provoking. For example, it visually predicts that much of the research
community is not particularly interested in ‘open data’††margin: (unless it’s
_other people’s_ open data, of course) ††margin: (unless it’s _other
people’s_ open data, of course) and only incompletely interested in ‘good
metadata’ (in-collaboration researchers care when a dataset was acquired,
because they need that information to perform their analyses, but they have
little interest in dissemination and licensing metadata, for example, because
that is the long-term concern of funders and their proxies). We can therefore
naturally conceive of the funders taking the role of the conscience of a
discipline, worrying about long-term imponderables so that individual
researchers don’t have to. It follows from this, that the open data case made
to funders, for example, will be an institutionally self-interested one, but
that the case made to researchers must be qualitatively different, and be
either pragmatic (‘you must care because your funders care’) or high-minded
(‘your socio-cultural duty is…’). Neither of these is a poor argument, nor
indeed a cynical one, but we are acknowledging here that, to a busy and
distracted researcher, the self-interest argument in isolation may have little
purchase.
### 2.2 The case for open data
Internationally, there is a push towards such data sharing in the more general
context of scholarly research (see for example [45] or [46]). The most
explicit statement here is in the NSF’s GC-1 document [47], which in section
41 states that ‘‘[NSF] expects investigators to share with other researchers,
at no more than incremental cost and within a reasonable time, the data,
samples, physical collections and other supporting materials created or
gathered in the course of the work. It also encourages grantees to share
software and inventions or otherwise act to make the innovations they embody
widely useful and usable.’’ This is reiterated in almost the same words in
their 2010 data sharing policy [3]. They additionally require a brief
statement, attached to proposals, of how the proposal would conform to NSF’s
data-sharing policy.
STFC, in common with the other UK research councils, requires that ‘‘the full
text of any articles resulting from the grant that are published in journals
or conference proceedings […] must be deposited, at the earliest opportunity,
in an appropriate e-print repository’’††margin:
http://www.scitech.ac.uk/rgh/rghDisplay2.aspx?m=s&s=64 ††margin:
http://www.scitech.ac.uk/rgh/rghDisplay2.aspx?m=s&s=64 ; it has not yet made
any corresponding statement on data releases.
The year 2009 saw some excitement (relating to the incident inevitably
labelled ‘climategate’, and to some other data-release disputes††margin:
http://www.guardian.co.uk/environment/2010/apr/20/climate-sceptic-wins-data-
victory ††margin: http://www.guardian.co.uk/environment/2010/apr/20/climate-
sceptic-wins-data-victory ) related to the management and release of climate
data. This illustrated the political and social significance of some science
data sets; the contrast between what scientists know, and the public believes,
to be normal scientific practice; and some of the issues involved in the
generation, ownership, use and publication of data.††margin: UEA’s Climate
Research Unit is a partner in the ACRID project, also funded by the JISC MRD
programme: http://www.cru.uea.ac.uk/cru/projects/acrid/ ††margin: UEA’s
Climate Research Unit is a partner in the ACRID project, also funded by the
JISC MRD programme: http://www.cru.uea.ac.uk/cru/projects/acrid/ The cases
during that year illustrate a number of complications involved in data
releases.
1. 1.
Data is often passed from researchers or groups directly to others, across
borders, with no general permission to distribute it further.
2. 2.
Data collection may be onerous, and the result of significant professional and
personal investments.
3. 3.
Raw data is generally useless without the more or less significant processing
which cleans it of artefacts and makes it useful for further analysis.
4. 4.
However not all disciplines have the clear notion of published data products
which is found in astronomy and which is implicit in the OAIS notion of
archival deposit.
5. 5.
Science is a complicated social process.††margin: The last point is
simultaneously obvious and deeply intricate. Unpacking it would distract us
here, but there is further discussion, in a very apposite historical context,
in [6], elaborated in [36]. ††margin: The last point is simultaneously
obvious and deeply intricate. Unpacking it would distract us here, but there
is further discussion, in a very apposite historical context, in [6],
elaborated in [36].
In science, we preserve data so that we can make it available later. This is
on the grounds that scientific data should generally be universally available,
partly because it is usually publicly paid for, but also because the public
display of corroborating evidence has been part of science ever since the
modern notion of science began to emerge in the 17th century (ce) – witness
the Royal Society’s motto, ‘nullius in verba’, which the Society glosses as
‘take nobody’s word for it’. Of course, the practice is not quite as simple as
the principle, and a host of issues, ranging across the technical, political,
social and personal, complicate the social, evidential and moral arguments for
general data release.
The arguments _against_ general data releases are practical ones: data
releases are not free, and may have significant financial and effort costs (cf
Sect. 3.4). Many of these costs come from (preparation for) data preservation,
since it is formally archived data products that are the most naturally
releasable objects: releasing raw or low-level data _may_ be cheap, but may
also have little value, since raw underdocumented datasets are likely to be
useless; or more pessimistically they may have a negative value, if they end
up fostering misunderstandings which are time-consuming to counter (this point
obviously has particular relevance to politicised areas such as climate
science). In consequence of this, the ‘open data question’ overlaps with the
question of data preservation – if the various costs and sensitivities of data
preservation are satisfactorily handled, then a significant subset of the
practical problems with open data release will promptly disappear. We discuss
the data preservation question below, in Sect. 2.3.
It seems worth noting, in passing, that the physical sciences broadly perform
better here than other disciplines, both in the technical maturity of the
existing archives and in the community’s willingness to allocate the time and
money to see this done effectively.
What all this indicates is that there is a need for an explicit framework for
discussing the pragmatics of open data (cf point 4 above). We can go further
and suggest (it is almost a Recommendation) that the OAIS model’s notion of an
AIP, and its reflection in the notion of a _data product_ should be central to
this discussion.
### 2.3 The case for data preservation
The case for data preservation in astronomy was implicitly made in Sect. 1.4:
as an observational science, much astronomy data is repeatable, but there are
important cases where what is being observed is a slow secular change, or some
unpredictable (usually ultimately explosive) event; sometimes data can be
opportunistically reanalysed to extract information distinct from the
information the observation was designed for. Astronomical data is potentially
useful _and_ usable almost indefinitely. Thus there is a reasonable
expectation that the data can be and will be exploited by unknown astronomers,
far into the future.
HEP data is somewhat different (as noted in Sect. 1.5). As an experimental
science, it is generally very much in control of what it observes, and is able
to design experiments of considerable ingenuity, in order to make measurements
of exquisite discriminatory power. A consequence of this is firstly that HEP
experiments have a much stronger tendency to become obsolete with each
technological generation, and secondly that the complication of the apparatus
makes it hard to communicate into the future a level of understanding
sufficient to make plausible use of the data. Experimental apparatus will
generally be understood better and better as time goes on (this is also true
of satellite-borne detectors in astronomy), so that data gathered early in an
experiment will be periodically reanalysed with increased accuracy. However
this understanding is generally not preserved formally, but is pragmatically
communicated through wikis, workshops, word of mouth, configuration and
calibration files, and internal and external reports. Even if all of the
tangible records were magically preserved with complete fidelity, and
supposing that the more formal records do contain all the information required
to analyse the raw data, an archive would still be missing the word-of-mouth
information which a new postgrad student (for example) has to acquire before
they can understand the more complete documentation. We can think of this as a
‘bootstrap problem’. In OAIS terms, the Representation Network for HEP data is
particularly intricate, and while the Representation Information nearest to
the Data Object may be complete, it may be infeasible to gather the
Representation Information necessary to let a naive researcher make sense of
it. The Designated Community for HEP data may therefore be null in the long
term.
This sounds pessimistic, but [26] describes a number of scenarios in which HEP
data can and should be reanalysed some decades after an experiment has
finished, and describes ongoing work on the development of consensus models
for preserving data for long enough to enable such post-experiment
exploitation. This provides a strong case for a style of preservation somewhat
different from the astronomical one. What these models have in common is a
commitment of staff to actively conserve and continuously exploit the data.
This post-experiment staff can therefore be conceived as a form of walking
Representation Information so that, while they are still involved, the data
might have a Designated Community which corresponds to those individuals in a
position to undertake an extended apprenticeship in the data analysis (this
model is further discussed on p.3.4).
GW data is, as usual, somewhere between these two extremes. As astronomy, the
GW data consists of unrepeatable measurements which will potentially be of
value to astronomers well into the future; as a HEP-style experiment it makes
those measurements using two or three generations of highly sophisticated
apparatus, each generation of which will improve on the sensitivity of its
predecessors by orders of magnitude. An additional feature, however, is that
no-one has ever convincingly detected a gravitational wave, though there have
been repeated claims of detection in the past, so that the first claims by
LIGO or aLIGO will be scrutinized particularly closely.
Finally, and as noted in Sect. 2.2, if data is well archived, then most of the
pragmatic objections to opening that data do not apply. Thus, to the extent
that general data release is a good in itself, it is a further argument in
favour of a well supported archive.
### 2.4 Should raw data be preserved?
In the data-preservation world, there is often an automatic expectation that
‘everything should be preserved’, so that an experiment can be redone, results
reanalysed, or an analysis repeated, later. Is this actually true? Or if it is
at least desirable, how much effort should be expended to make it true? This
question is implicit in, for example, the discussion of software preservation
in Sect. 3.2.
When a physical experiment is set up and working, it is usual to avoid
tinkering with it as much as possible, to avoid any unexpectedly significant
change. That is, even with a small-scale lab-bench experiment, it is accepted
that not everything can be effectively documented, and that an experiment
might not be immediately replicable purely from published information (cf [6,
ch.35] and Sect. 1.7). This expectation (or rather, lack of expectation) is
also true of larger-scale experiments, which might be financially,
professionally or, at the largest scales, politically infeasible to
replicate.††margin: It is because very large-scale experiments are impossible
to replicate, and even hard for an external reviewer of an article to
criticise meaningfully, that large collaborations submit their publications to
extremely scrupulous internal review. See
http://stuver.blogspot.com/2011/03/big-dog-in-envelope.html for a post-mortem
account of such a review. ††margin: It is because very large-scale
experiments are impossible to replicate, and even hard for an external
reviewer of an article to criticise meaningfully, that large collaborations
submit their publications to extremely scrupulous internal review. See
http://stuver.blogspot.com/2011/03/big-dog-in-envelope.html for a post-mortem
account of such a review. Perhaps this attitude should extend to other aspects
of the experimental process.
In many cases, the pipeline for reducing raw data seems to fall into this
category: it encodes hard-to-document information, but is itself hard to
document, hard to use, and unlikely ever to be reused in fact. If this
software is not preserved, then the raw data is effectively unreadable, which
means there is no case for preserving it. There is therefore a case that at
least some details of the experimental environment – digital as well as
physical – are not reasonably preservable, and that as a result little effort
should be expended on preserving them.
It is data products that make raw data less necessary. It is feasible to
document the scientific meaning of data products, and the community expects
that a project will provide this documentation as part of the publication of
the products (indeed, it the documentation that makes these _products_ rather
than just a casual data snapshot). The data products allow researchers to dig
beneath the conclusions of a particular article (or indeed the contents of a
higher-level data product), and to criticise and build on what they find
there. Higher-level products are the result of higher-level scientific
judgements, and it is normal for these to be regenerated by researchers other
than the originators, either using their own software or the originators’
pipelines. These later-stage pipelines are more formally supported by
projects, which involves making them reasonably portable, so that they are
both easier to preserve as well as being more valuable objects of
preservation.
We should stress that we are not advocating deliberately deleting raw data,
and its associated pipelines – it _might_ be useful, and it _might_ be usable
– but simply noting that one should not overstate its value.
### 2.5 OAIS: suitability and motivation
In Sect. 3.1.1, we provide an overview of the OAIS model, and describe how it
relates to astronomical data.
The OAIS standard is formally a product of the Consultative Committee for
Space Data Systems (CCSDS),††margin: http://www.ccsds.org ††margin:
http://www.ccsds.org and with this in its lineage it is quite naturally
matched to the data management problems of the physical sciences. Essentially
all the explicit and implicit assumptions of the OAIS standard are true in the
area we are studying: the data producer (a satellite or a detector) is usually
obvious, the various Information Package s (or data products) well understood,
and the Designated Community easily identified.††margin: There is no
contradiction here with the remarks in Sect. 1.7 about the difficulty of
describing the Designated Community of science archive users. It is easy to
name a science Designated Community, but it may be hard to describe ahead of
time what those community members can be expected to know. A social science
archive may have an unpredictably broad range of ultimate users, but using the
archive will need little specialist knowledge; in contrast a particle physics
dataset will probably be of interest only to particle physicists, but the
normal education of such a physicist three decades hence, and thus the content
and extent of the specialised Representation Information that Community will
need, might be very hard to guess at. ††margin: There is no contradiction
here with the remarks in Sect. 1.7 about the difficulty of describing the
Designated Community of science archive users. It is easy to name a science
Designated Community, but it may be hard to describe ahead of time what those
community members can be expected to know. A social science archive may have
an unpredictably broad range of ultimate users, but using the archive will
need little specialist knowledge; in contrast a particle physics dataset will
probably be of interest only to particle physicists, but the normal education
of such a physicist three decades hence, and thus the content and extent of
the specialised Representation Information that Community will need, might be
very hard to guess at.
The motivation for a digital preservation standard, as discussed in the OAIS
standard itself [4, §2], is that digital preservation represents a double
problem: (i) digital information is intrinsically harder to preserve than
traditional information, which is capable of sitting on a shelf in a well-
understood and intelligible format, and mouldering at a well-understood and
graceful rate; and (ii) more and more organisations are producing digital
information _and_ are implicitly expected to archive their own material. This
means that these non-specialist archives have a complicated task to perform,
which is potentially at odds with the daily urgencies of their main business.
This _appears_ to mean in turn (and in JISC contexts it is often taken to mean
in practice) that these organisations need as much detailed and prescriptive
help as possible, ideally devolving their archive responsibilities to a
central discipline- or funder-specific archive, to the extent possible while
respecting the low-level complications and friction alluded to in Sect. 1.7.
This is not the model which is appropriate for big-science datasets.
### 2.6 What should big-science funders require, or provide?
We have described several common features of big-science data management in
Sect. 1.3. and we have outlined some particular contrasts with other
communities in Sect. 1.7. As noted in Sect. 0.1, our focus here is on STFC’s
strategically funded projects, rather than the smaller projects funded by
individual research grants.
Big-science data sets are generally intimately coupled to solutions to
leading-edge technical challenges, and cannot usefully be regarded as
incremental changes to previous solutions. This, coupled with the general
availability of extensive technical expertise within such communities, means
that any generic solution is very unlikely to be appropriate, and that it is
both reasonable and feasible to require custom archiving solutions for such
projects. There is no _recipe_ for data preservation on this scale, and all
that can be hoped for is a structured approach to a custom solution. Having
said this, not even the most innovative science experiments are so completely
_sui generis_ that they warrant a data preservation approach which is
reimagined from scratch. It is therefore wasteful to ignore the considerable
intellectual investments in the OAIS model, the growing penumbra of
commentaries on and developments of it, and the minor industry of validation
and auditing efforts related to it.
We are therefore led to the conclusion that the most effective overall
strategy for effective data management in the large-scale experimental
physical sciences is that ††margin: Recommendation 2 ††margin:
Recommendation 2 funders should simply require that a project develop a high-
level DMP plan as a suitable profile of the OAIS specification [4]. This
profile should be detailed enough to require negotiation with the funder and
with the experiment’s community, but can leave many of the implementation
details to the good engineering judgement of the project’s management. We
believe the LIGO DMP plan [5] can be taken to be exemplary in this regard.
Big-science projects have the technical skills, the management structures, and
the budgets to take on such a task, and to deliver a custom archive which can
be shown to meet identified goals. We recommend that ††margin: Recommendation
3 ††margin: Recommendation 3 funders should support projects in creating per-
project OAIS profiles which are appropriate to the project and meet funders’
strategic priorities and responsibilities.
The discussion in Sect. 3.5 suggests that one result of the development of an
OAIS-based DMP plan is that the resulting plan is explicit enough to generate
useful deliverables, and to benefit from the growing interest in OAIS
‘validation’.
We suggest the following specific funder actions.
* •
Actively engage with projects to help them develop an OAIS profile. This will
include overview literature, including the OAIS specification, tutorial
reports such as [48], and commentary such as [49], or perhaps specialised
workshops if necessary. These are high-level introductions, rather than
procedure-based tick-lists.
* •
Develop or support expertise in criticising and validating such OAIS profiles.
For example, the CASPAR consortium (see for example [50]) has developed
strategies for detailed validation of projects’ claims about long-term data
migration. Similar work – for example validating a project’s assumptions about
its Designated Community – would reassure the wider community that the archive
design is likely to achieve its goals for the future.
The first of these is reasonably straightforward, consisting of little more
than gathering resources. The second is a longer-term project which may
require some expertise to be built up and supported at a funder-supported
facility (such as RAL, in the UK), or through liaison with the DCC.
A corollary of this more active engagement is that funders must financially
support the preservation work they require. See Sect. 3.4.
## 3 The practicalities of data preservation
### 3.1 Modelling the archive
#### 3.1.1 The OAIS model
We introduce here the main concepts of the OAIS model. Full details are in [4]
with a useful introductory guide in [48] and some discussion in the LSC
context in [5]; the OAIS motivation is further discussed in Sect. 2.5.
Figure 4: The highest-level structure of an OAIS archive, annotated with the
corresponding labels from conventional astronomical practice (redrawn from [4,
Fig. 2-4]). The dissemination data products will typically be the same as the
submitted ones, but archives can sometimes create value-added ones of their
own.
The term _OAIS_ stands for an _Open Archival Information System_. The word
‘open’ is not intended to imply that the archived data is freely available
(though it may be), but instead that the process of defining and developing
the system is an open one. The principal concern of an OAIS is to preserve the
usability of digital artefacts for a pragmatically defined long term. An OAIS
is not only concerned with storing the lowest-level _bits_ of a digital object
(though this part of its concern, and is not a trivial problem), but with
storing enough _information_ about the object, and defining an adequately
specified and documented _process_ for migrating those bits from system to
system over time, that the information or knowledge those bits represent can
be retrieved from them at some indeterminate future time. The OAIS model can
therefore be seen as addressing an administrative and managerial problem,
rather than an exclusively technical one.
The OAIS specification’s principal output is the _OAIS reference model_ ,
which is an explicit (but still rather abstract) set of concepts and
interdependencies which is believed to exhibit the properties that the
standard asserts are important (Fig. 4). The OAIS model can be criticised for
being so high-level that ‘‘almost any system capable of storing and retrieving
data can make a plausible case that it satisfies the OAIS conformance
requirements’’ [49], and there exist both efforts to define more detailed
requirements [49], and efforts to devise more stringent and more auditable
assessments of an OAIS’s actual ability to be appropriately responsive to
technology change [50].
An OAIS archive is conceived as an entity which preserves objects (digital or
physical) in the Long Term, where the ‘Long Term’ is defined as being long
enough to be subject to technological change. The archive accepts objects
along with enough Representation Information to describe how the digital
information in the object should be interpreted so as to extract the
information within it (for example, the FITS specification is Representation
Information for a FITS file). That Information may need further context – for
example, to say that a file is an ASCII file requires one to define what ASCII
means – and the collection of such explanations turns into a Representation
Network. This information is all submitted to the archive in the form of a SIP
agreed in some more or less formal contract between the archive and its data
producers.
Once the information is in the archive, the long-term responsibility for its
preservation is _transferred_ from the provider to the archive, which must
therefore have an explicit plan for how it intends to discharge this.
The archive distributes its wares to Consumers in one or more Designated
Communities, by transforming them, if necessary, into the DIP which
corresponds to a ‘data product’. The members of the Designated Community are
those users, in the future, whom the archive is designed to support. This
design requires including, in the AIP, Representation Information at a level
which allows the Designated Community to interpret the data products _without
ever having met one of the data Producers_ , who are assumed to have died,
retired, or forgotten their email addresses.
The OAIS model originated within the space science community, so it can be
mapped to the physical science data of the GW community without much violence.
#### 3.1.2 The DCC Curation Lifecycle model
The OAIS model is on the face of it a linear one, and suggests that data is
created, then ingested, then preserved, and then accessed, in a process which
has a clear beginning and end. This is compatible with the observation that
one point of archiving data is to reuse or repurpose it, creating new
archivable data products in turn, but this longer-term cycle remains only
implicit in the model. The OAIS model is therefore very usefully explicit
about those aspects of archival work concerned with long-term preservation,
but its conceptual repertoire is such that a discussion framed by it runs the
risk of underemphasizing the range of roles a data repository has, or even of
marginalising it.
Figure 5: The DCC lifecycle model, from [51]
In contrast,††margin: We thank Dorothea Salo, of the University of Wisconsin
library, for emphasizing to us the useful applicability of the DCC model to
the case of big science data, and Angus Whyte, for elaborating the contrasts
between the DCC and OAIS models. ††margin: We thank Dorothea Salo, of the
University of Wisconsin library, for emphasizing to us the useful
applicability of the DCC model to the case of big science data, and Angus
Whyte, for elaborating the contrasts between the DCC and OAIS models. the DCC
has produced a lifecycle model [51] (Fig. 5) which stresses that data
creation, management, and reuse are part of a cycle in which preservation
planning, for example, can naturally happen before data creation as well as
after it; and in which data can be appraised, reappraised, and possibly
disposed of if it becomes obsolete. It therefore makes explicit both the
short- and long-term cycles in the flow of active research data, and it
emphasizes the active involvement of data curators in maintaining that cycle.
Cycles of use and re-use are not the only links between datasets. As discussed
in [52], one digital object can also provide context for another, in a variety
of ways. To some extent this remark rediscovers the notion of the OAIS
Representation Network, and this in turn prompts us to stress that although we
have contrasted OAIS and DCC here, they are not in competition: OAIS is
concerned with the creation and management of a working archive with
gatekeepers and firm goals; the DCC model is concerned with the location of
the archive in the wider intellectual context.
The DCC model is immediately compatible with the observation, in Sect. 3.4
below, that HEP and GW archives effectively avoid some preservation costs by
seeing long-term preservation as only part of the role of a data repository.
Accepting data, making it available as working storage, transforming it into
immediately useful forms, or appraising (possibly regenerable) datasets whose
storage costs outweigh their usefulness, all give the archive a familiarity
with the data, and the researchers a familiarity with the archive, which means
that the decision to select certain data for long-term preservation is
potentially more easily reached, more easily defended and more easily funded,
than if the archive is conceived as a cost-centre bucket bolted on the side of
the project. This appears to be borne out by the LIGO experience, in which the
new DMP plan was developed and successfully argued for by the same personnel
who were long responsible for the design and management of the data management
system on which everyone’s daily work depends.
### 3.2 Software preservation
As discussed in, for example, Sect. 1.6.2, there is often a substantial amount
of important information encoded in ways which are only effectively documented
in software, or software configuration information. There is therefore an
obvious case for preserving this software (though note the caveats of Sect.
2.4).
Preservation of a software pipeline requires preserving the pipeline software
itself, a possibly large collection of libraries the software depends on, the
operating system (OS) it all runs on, and the configuration and start-up
instructions for setting the whole thing in motion. The OS may require
particular hardware (CPUs or GPUs), the software may be qualified for a very
small range of OSs and library versions, and it may be hard to gather all of
the configuration information required (there is some discussion of how one
approaches this problem in for example [26]). It is not certain that it is
necessary, however: if the data products are well-enough described, then re-
running the analysis pipeline may be unnecessary, or at least have a
sufficiently small payoff to be not worth the considerable investment required
for the software preservation. We feel that, of the two options – preserve the
software, or document the data products – the latter will generally be both
cheaper and more reliable as a way of carrying the experiment’s information
content into the future, and that this tradeoff is more in favour of data
preservation as we consider longer-term preservation.
This last point, about the changing tradeoff, emphasizes that the two options
are not exclusive: one can preserve data _and_ preserve software, and the
JISC-funded Software Sustainability Institute††margin:
http://www.software.ac.uk/ ††margin: http://www.software.ac.uk/ provides a
growing set of resources which provide guidance here. However the solutions
presented generally focus on active curation, in the sense of preserving
software through continuing use and maintenance. This can be
successful††margin: The UK Starlink project provided astronomical software.
It ran from 1980 to 2005, when it was rescued from oblivion by being taken up
by the UK Joint Astronomy Centre Hawai‘i. The current distribution includes
still-working code from the 80s. The Netlib and BLAS libraries have components
which date from the 70s. ††margin: The UK Starlink project provided
astronomical software. It ran from 1980 to 2005, when it was rescued from
oblivion by being taken up by the UK Joint Astronomy Centre Hawai‘i. The
current distribution includes still-working code from the 80s. The Netlib and
BLAS libraries have components which date from the 70s. , and is the approach
implicit in [26], but it seems brittle in the face of significant funding
gaps, and would not deal well with the case where a software release is
deliberately unused, for example because it has been superseded.
### 3.3 Data management planning
#### 3.3.1 DMP in space
As one might expect, both NASA and ESA have formalised DMP plans.
NASA’s National Space Science Data Center (NSSDC) ††margin: %**** report.tex
Line 2500 ****http://nssdc.gsfc.nasa.gov ††margin: %**** report.tex Line 2500
****http://nssdc.gsfc.nasa.gov has led NASA’s data planning since the mid-80s.
It was initially the NSSDC which negotiated a Project DMP plan with missions,
but since the 1990s this has become the responsibility of the NASA Planetary
Data System (PDS) ††margin: http://pds.nasa.gov/ ††margin:
http://pds.nasa.gov/ . The NSSDC’s data retention policy††margin:
http://nssdc.gsfc.nasa.gov/nssdc/data_retention.html ††margin:
http://nssdc.gsfc.nasa.gov/nssdc/data_retention.html describes what categories
of data product should be retained indefinitely, and the PDS provides
resources to mission planners on the processes and tools††margin:
http://pds.nasa.gov/tools/index.shtml ††margin:
http://pds.nasa.gov/tools/index.shtml for preparing data for
preservation.††margin: We are grateful to Paul Butterworth of NASA for
helpful advice here. ††margin: We are grateful to Paul Butterworth of NASA
for helpful advice here.
ESA’s Planetary Science Archive††margin:
http://www.rssd.esa.int/index.php?project=PSA&page=about ††margin:
http://www.rssd.esa.int/index.php?project=PSA&page=about ‘‘provides expert
consultancy to all of the data producers throughout the archiving process. As
soon as an instrument is selected, PSA begin working with the instrument team
to define a set of data products and data set structures that will be suitable
for ingestion into the long-term archive.’’ The ESA archive is by design
compatible with the PDS.
#### 3.3.2 Current and future DMP in the LSC
The current LIGO DMP plan [5], discusses DM planning with an emphasis on the
preparations for the eventual public data release.
The LIGO DMP plan proposes a two-phase data release scheme, to come into play
when aLIGO is commissioned; this was prepared at the request of the NSF,
developed during 2010–11, and will be reviewed yearly.
The plan documents the way in which the consortium will make LIGO data open to
the broader research community, rather than (as at present) only those who are
members of the LSC. This document describes the plans for the data release and
its proprietary periods, and outlines the design, function, scope and
estimated costs of the eventual LIGO archive, as an instance of an OAIS model.
This is a high-level plan, with much of the detailed implementation planning
delegated to partner institutions in the medium term.
In the first phase, data is released much as it is at present: validated data
will be released when it is associated with detections, or when it is related
to papers announcing _non_ -detections (for example, associated with another
astronomical event which might be expected or hoped to produce detectable
GWs). In the second phase – after detections have become routine, and the LIGO
equipment is acting as an observatory rather than a physics experiment – the
data will be routinely released in full: ‘‘the entire body of gravitational
wave data, corrected for instrumental idiosyncrasies and environmental
perturbations, will be released to the broader research community. In
addition, LIGO will begin to release near-real-time alerts to interested
observatories as soon as LIGO _may_ have detected a signal.’’ This second
phase will begin after LIGO has probed a given volume of space-time (see [5,
ref 7]), _or_ after 3.5 years have elapsed since the formal LIGO
commissioning, whichever is earlier. Alternatively, LIGO may elect to start
phase two sooner, if the detection rate is higher than expected.
In phase two, the data will have a 24-month proprietary period.
The DMP plan describes three (OAIS) Designated Communities. Quoting from [5,
§1.5], the communities are as follows.
* •
LSC scientists: who are assumed to understand, or be responsible for, all the
complex details of the LIGO data stream.
* •
External scientists: who are expected to understand general concepts, such as
space-time coordinates, Fourier transforms and time-frequency plots, and have
knowledge of programming and scientific data analysis. Many of these will be
astronomers, but also include, for example, those interested in LIGO’s
environmental monitoring data.
* •
General public: the archive targeted to the general public, will require
minimal science knowledge and little more computational expertise than how to
use a web browser. We will also recommend or build tools to read LIGO data
files into other applications.
The LIGO DMP plan is, we believe, a good example of a plan for a project of
LIGO’s size: it is specific where necessary, it was negotiated with the
project’s funder (NSF) so that it achieved their goals, and it went through
enough iterations with the broader LIGO community (the agreed version in [5]
is version 14) that its authors could be confident it had their approval, and
that the community was comfortable with what the DMP plan was proposing. The
document has a strong focus on the LIGO data release criteria, since this was
the most immediate concern of both the funder and the project, but it
systematically lays out a high-level framework for future data preservation,
guided by the OAIS functional model.
### 3.4 Data preservation costs
There is a good deal of detailed information, and some modelling, of the costs
of digital preservation. The KRDS2 study [44, §§6&7] includes detailed
costings from a number of running digital preservation projects, in some cases
down to the level of costings spreadsheets. The LIFE3 project has also
developed predictive costings tools [53], and the PLANETS project
(http://www.planets-project.eu/) has generated a broad range of materials on
preservation planning, including costing studies.
Although there is a broad range of preservation projects surveyed in the KRDS
report, there are numerous common features. Staff costs dominate hardware
costs, and scale only very weakly with archive size. The study also notes that
acquisition and ingest costs are a substantial fraction (70–80%) of overall
staff costs, but also scale very weakly with archive size. These are
relatively small archives, generally below a few $\mathrm{T}\mathrm{B}$ in
size, where ingest is a significant component of the workload. In this report
we are interested in archives three or four orders of magnitude larger than
this where (as discussed below) ingest may be cheaper, but in broad terms, it
appears still to be true that staff costs dominate hardware costs at larger
scales, and scale only weakly with archive size.
Parenthetically, notice that the above discussion prompts the question ‘what
is the size of an archive?’ The number of bytes it consumes is an obvious and
readily available measure, but may not be particularly meaningful in this
context. The number of items (such as interview transcripts, images or
database rows) may be a better measure, and still objectively identifiable,
archive by archive. If there were some measure of abstract information
content, we speculate that this is what would scale most straightforwardly
with the effort required for quality control and metadata curation, and hence
with staff effort. We hesitate to ask what such a measure might be, in case
the answer is ‘citation analysis’.
The lack of scaling with size, even when an archive progressively grows in
size, seems to suggest that it is an archive’s _initial_ size (in the sense of
small, medium or large, for the time) that largely governs the costs.
We were given access to confidential figures for the development and
operations of a mid-to-large size astronomy archive (of order
$10\text{\,}\mathrm{T}\mathrm{B}$ of relational data and
$100\text{\,}\mathrm{T}\mathrm{B}$ of flat file data), developed by an
experienced archive site. The archive software and system development cost
25–30 staff-years of effort: the bulk of this was for the core database
system, but between a quarter and a third was for software to support ingest
and the generation of data products. The organisation budgets around 3 FTEs
for operation of this archive, which includes ingest, quality control and
helpdesk support (this is an estimated fraction of an operations team covering
several archives at the same site, so there may be some economies of scale).
About a quarter of the annual operating budget is spent on hardware.
The European Southern Observatory (ESO) data archive manages data from
multiple ESO facilities;††margin: We are most grateful to Fernando Comerón,
of ESO, for sharing these figures. ††margin: We are most grateful to Fernando
Comerón, of ESO, for sharing these figures. it shares space with the still-
developing ALMA archive, but the figures below do not include ALMA. The
archive is based on spinning disks backed by a tape library (for further
details, see [54]). It currently holds $190\text{\,}\mathrm{TB}$, increasing
at around $7\text{\,}\mathrm{TB}\text{\,}{\mathrm{month}}^{-1}$. The hardware
costs average around $330\text{\,}\mathrm{k\hbox{{#\cr\vfil\hbox
to0.70007pt{=\hss}\vfil\cr\hbox{C}\crcr}}}\text{\,}{\mathrm{yr}}^{-1}$, which
includes hardware replacement and data migration, and which has remained flat
for some years, despite the slowly increasing data volumes. Running costs
amount to $55\text{\,}\mathrm{k\hbox{{#\cr\vfil\hbox
to0.70007pt{=\hss}\vfil\cr\hbox{C}\crcr}}}\text{\,}{\mathrm{yr}}^{-1}$ (some
smaller systems account for part of this), and licences, networks and other
consumables account for about $30\text{\,}\mathrm{k\hbox{{#\cr\vfil\hbox
to0.70007pt{=\hss}\vfil\cr\hbox{C}\crcr}}}\text{\,}{\mathrm{yr}}^{-1}$.
Manpower costs come to 4 FTEs of ESO staff plus around
$270\text{\,}\mathrm{k\hbox{{#\cr\vfil\hbox
to0.70007pt{=\hss}\vfil\cr\hbox{C}\crcr}}}\text{\,}{\mathrm{yr}}^{-1}$ of
outsourced staff. Neither hardware nor software costs appear to scale with
data volume, with some cost elements even dropping as the archive moves to
completely on-line data distribution.
There is some discussion of the CDS funding model in Sect. 1.4.1.
The NASA PDS has developed a parameterized model for helping proposers
estimate the costs involved in preparing data for archiving in the
PDS††margin: http://pds.nasa.gov/tools/cost-analysis-tool.shtml ††margin:
http://pds.nasa.gov/tools/cost-analysis-tool.shtml ; most relevantly for the
above discussion it includes a scaling with data volume of
$1+1.5\log_{10}(\mbox{volume/MB})$ (that is, a multiplier which increases by
1.5 for each order of magnitude increase in data volume).
As noted in Sect. 1.5, the HEP community is now constructing more detailed
plans for data preservation, and the associated costs. Reference [26]
estimates that a formal long-term archive (a level-3 or -4 archive, in the
terms of that paper) would cost 2–3 FTEs for 2–3 years after the end of the
experiment, followed by 0.5–1.0 FTE/year/experiment spent on the archive’s
preservation. They compare this to the 100s of FTEs spent on for the running
of the experiment, and on this basis claim an archival staff investment of 1%
of the peak staff investment, to obtain a 5–10% increase in output (the latter
figure is based on their estimate that around 5–10% of the papers resulting
from an experiment appear in the years immediately after the experiment
finishes; since this latter figure is derived on the current model, which
achieves this without any formal preservation mechanisms, this estimate of the
return on investment in archives may be optimistic).
It is worth noting that in astronomical, HEP and GW contexts, archive ingest
is generally tightly integrated with the system for day-to-day data
management, in the sense that data goes directly to the archive on acquisition
and is retrieved from that archive by researchers, as part of normal
operations. On the other side of the archive, projects will generate and
disseminate data products – which look very much like OAIS DIP s – as part of
their interaction with external collaborators, without regarding these as
specifically archival objects. Thus the submissions into the archive may
consist of both raw data and things which look very much like DIPs, and the
objects disseminated will include either or both very raw and highly processed
data. The _long-term_ planning represented in the LIGO DMP plan [5], for
example, is therefore less concerned with setting up an archive, than with the
adjustments and formalizations required to make an existing data-management
system robust for the archival long term, and more accessible to a wider
constituency. What this means, in turn, is that some fraction of the OAIS
ingest and dissemination costs (associated with quality control and metadata,
for example) will be covered by normal operations, with the result that the
_marginal_ costs of the additional activity, namely long-term archival ingest
and dissemination, are probably both rather low and typically borne by
infrastructure budgets rather than requiring extra effort from
researchers.††margin: This is consistent with the ERIM project’s conclusions
that ‘‘ideally information management interventions should result in a zero
net resource increase’’ [55, p.8]. In this case there is no extra resource
required from the researchers, though there might be a need for extra resource
under an infrastructure heading. ††margin: This is consistent with the ERIM
project’s conclusions that ‘‘ideally information management interventions
should result in a zero net resource increase’’ [55, p.8]. In this case there
is no extra resource required from the researchers, though there might be a
need for extra resource under an infrastructure heading. This is corroborated
by our informants above, who generally regard archive costs as coming under a
different heading from ‘data processing costs’. The point here is not that the
OAIS model does not fit well – it fits very well indeed – nor that ingest and
dissemination do not have costs, but that if the associated activities can be
contrived to overlap with normal operations, then the costs directly
associated with the archive may be significantly decreased. This is the
intuition behind the recent developments in ‘archive-ready’ or ‘preservation-
aware storage’ (cf [43] and Sect. 3.1.2), and confirms that it is a viable and
effective approach.
As a final point, we note that big-science projects are inevitably also large-
scale engineering projects, so that the consortia and their funders are
broadly familiar with the procedures, uncertainties and management of cost
estimates, so that the costing and management of data preservation can be
naturally built in to the relationship between funders and funded, if the
funders so require it.
As is shown by the vagueness of some of the remarks above (despite sometimes
very specific numbers), there seems little in the way of a consensus model for
the costing of the long-term preservation of large-scale data. There will
surely be detailed costings for the management of PB-scale data for commercial
organisations, but these are not likely to be useful for our purposes, since
they are more concerned with immediate business continuity than multi-decade
archives, are serving different technical communities, and are likely to be
extremely confidential.
We therefore recommend that ††margin: Recommendation 4 ††margin:
Recommendation 4 STFC should develop a costings model for the publication and
preservation of data, which is matched to the data challenges of the big-
science community. We expect that this can build on the domain-agnostic work
already done in this area by JISC, and on the detailed work done on closely
related problems by NASA’s cost-estimation community [56].
### 3.5 The GW community and the AIDA toolkit
The AIDA Self-Assessment Toolkit [57] is a (JISC funded) set of qualitative
benchmarks for discussing at how developed an institution’s archive is. It
leads an archive manager through a set of a few dozen elements, inviting them
to grade their archive from 1 (poor) to 5 (international exemplar).††margin:
The AIDA document links these five stages, rather alarmingly, to a five-step
programme developed at Cornell, which starts with acknowledging that you have
a problem, and goes, via institutionalisation, to ‘‘embracing […]
dependencies’’, noting that ‘‘you can’t do it alone’’. Clearly, data-
management planning is habit-forming. ††margin: The AIDA document links these
five stages, rather alarmingly, to a five-step programme developed at Cornell,
which starts with acknowledging that you have a problem, and goes, via
institutionalisation, to ‘‘embracing […] dependencies’’, noting that ‘‘you
can’t do it alone’’. Clearly, data-management planning is habit-forming. The
goal is not to produce a pass/fail score, but instead to help archive managers
understand their current and future requirements, and to ‘‘enable an
institution to decide whether specific actions need to be taken in regard to
particular assets, or when and how it is desirable to improve on its current
capabilities’’. The AIDA authors acknowledge that the assessment is simplistic
and subjective, but stress that ‘‘AIDA aims to allow you to evaluate your
institution against a recognised capability scale, and then suggests
appropriate actions based on that evaluation’’. The AIDA goal is to model the
progress of an archive from the acknowledgement that an archive is desirable,
through to the exemplary externalisation of the archive as a resource.
In Appx. B, we list our estimates of the scores for LSC data management. We
hope these assessments are of specific use to the GW community, but believe
that the discussion in general may be of use to other, similarly structured,
big science communities.
The scores for the current LSC cluster in the middle, around three (which
corresponds to ‘consolidate’ in the Cornell model). This is an impressive
score for a project which is, from one point of view, doing only what is
regarded as normal for a well-run large-scale physics experiment. The higher
scores are generally associated with the formality and auditability of the
long-term plans, rather than any qualitatively different practice, and we
believe that these scores will naturally drift upwards as a result of the
development of an explicit DMP plan, structured using the OAIS concept set, in
collaboration with a suitably critical funder.
The toolkit is broken into organisational, technology and resources (generally
funding) ‘legs’.
The ‘organisational leg’ is concerned with the high-level support for the
archive. To the extent that it is meaningful, the average for these scores is
above three (which is good). The lower scores are generally associated with
the informality of the current archive (compared to a service-oriented
commercial organisation) rather than any more concrete inadequacy: the data is
backed up and reasonably findable, though this reflects cultural norms within
the physical sciences rather than something a particular archiving plan can
take credit for.
The ‘technology leg’ is concerned with the hardware and personnel support for
data management. As with the organisational leg, the GW community scores
highly here without really trying, simply because the community has long
experience of managing _and sharing_ large volumes of data. The lower scores
are again associated with the current informality of operations (from the
point of view of an archive as opposed to a working data-management
infrastructure), and these will naturally rise when the LSC’s DMP plan is
implemented and reviewed.
The scores in the ‘resources leg’ are the least well-justified. The LSC
generally scores well, in the sense that we can be confident that there will
be resources to support an archive effort – it’s seen as a high-importance
activity – even though there are few resources currently explicitly earmarked
for this. This section may therefore be useful for suggesting what budget
lines should eventually exist.
## 4 Conclusions and recommendations
In this report, we have described some of the ways in which ‘big science’
manages its data, as part of a broader data culture which is characterised by
large collaborations, and which has decades of experience in agreeing how, and
when, and when not, to share data.
We can say with some confidence that the big science data culture manages its
data well (and this seems to be corroborated by the AIDA assessment discussed
in Sect. 3.5), but we are not suggesting that other disciplines could or
should simply copy this culture, since there are various reasons (cf, Sect.
1.3) why this culture is particularly natural in some areas.
There are however some practices which we do believe are straightforwardly
portable to other disciplines. As we discuss in Sect. 1.11, the notions of
_data products_ and _proprietary periods_ very naturally concretize otherwise
diffuse arguments about data management and sharing, transforming them from
‘whether’ and ‘why’ to ‘which’ and ‘how long’. As well, we believe that
embedding data management in the day-to-day practice of researchers lowers
costs in both the short term (researchers can easily re-find their own data,
and interpret others’) and the long term (since preservation becomes a
technical problem of conserving an in-use repository). We discuss the costing
of data management at slightly greater length in Sect. 3.4.
We repeat our explicit recommendations below. None!
### Acknowledgements
This project was funded by JISC, as part of the ‘Managing Research Data’
programme. We are most grateful to the numerous people who have commented on
various drafts of this report, or provided us with information or resources.
In particular, we thank Stuart Anderson (LIGO), Paul Butterworth (NASA), Harry
Collins (Cardiff), Fernando Comerón (ESO), Joy Davidson (DCC), Françoise
Genova (CDS), Magdalena Getler (DCC), Simon Hodson (JISC), Sarah Jones (DCC),
Dorothea Salo (Wisconsin), Angus Whyte (DCC), and Roy Williams (LIGO).
## Appendix A Case study
We have produced a detailed discussion of the structure of the LIGO working
data management system, as a separate document [29]. This document is
currently available only within LIGO: those observations which have not been
incorporated into this present report are probably too detailed to be of
general interest. We hope, however, that the case-study will be of some use
internally to to the LSC.
## Appendix B AIDA assessment
The AIDA self-assessment toolkit [57] is a JISC-funded set of qualitative
benchmarks for assessing how developed an institution’s archive is. See Sect.
3.5 for discussion.
The labels in the table below are sometimes a little cryptic; refer to the
full toolkit for useful elaborations.
The answers below generally refer to the _early 2011_ state of the LSC archive
arrangements, on the grounds that concrete answers to a variant question are
preferable to speculative answers to a future one. These are probably a
reasonable indication of the likely status of a forthcoming formal archive,
but in a few case, as noted, we can give no meaningful answer.
In the scores below, level 1 is ‘poor’, and level 5 is ‘international
examplar’.
### Organisational leg
1: institution-wide mission statements (5)
The LIGO project has prepared a formal DMP, at funder request
2: institutional policies for asset management (3)
LIGO has prepared a formal DMP, and is addressing political and cultural
reservations, awaiting funding and implementation
3: review mechanisms at Institutional level (4)
As well as the DMP, there already exist well-understood collaboration-wide
review procedures, and these will be used to review the plan on an annual
basis
4: institutional capability for sharing assets (3)
Current storage is, of necessity, distributed; the collaboration manages this
informally but effectively, however this is generally working storage, and not
regarded as archival storage
5: institutional level of contingency planning (3)
There is no formal centralised asset management. Continuity is regarded as a
technical matter which can reasonably be left to the professional good
practice of the sites managing the distributed storage. As before, this is
currently regarded as working rather than archival storage.
6: institutional capability for audit (3)
Extensive logs exist, but are not centralised nor in any standard format;
files, once created, are not expected to be modified, though there is no way
to verify that this is true in fact
7: institutional monitoring mechanisms (4.5)
All data and processes are open to the entire collaboration, and most
processes are widely discussed; the collaboration is its own user-base. There
are (by design) no external users of the data, nor yet any external review of
the mechanisms.
8: extent of institutional conformance to metadata management (2 to 4)
Metadata is devised in a somewhat _ad hoc_ way by individual instruments or
software elements (stage 2), but this is also added and managed thoroughly,
and in accordance with what is regarded as experimental good practice (stage
4)
9: extent of institutional contracts (3)
Not applicable to current working storage
10: institutional understanding of IPR (5)
Formal MoUs between partners regarding access to data, and clear guidance from
funders regarding the eventual release of the data
11: institutional disaster planning (2)
As with asset continuity, this is currently regarded as a technical matter for
storage managers
### Technological leg
1: institutional infrastructure (5)
The collaboration has considerable technical resource, and interoperates well.
Planning is informal but effective. The sophisticated user-base is comfortable
with this informality, but this could in principle become a liability when the
resource management moves from a development to a service model.
2: appropriateness of institutional technologies (4)
There is plenty of appropriate technology, though the plan for the archival
management of assets is not yet detailed
3: integrity of institutional backup and storage (3)
Important data is backed up (possibly by mirroring), as part of normal
operations
4: institutional processes (2)
Uncertain: what there is will be done as part of normal operations
5: institutional understanding of obsolescence (3.5)
High general awareness, and occasional discussion, but at present little
formal planning
6: institutional capability (4)
Changes to processes are widely and frankly discussed, and documented as
internal publications; change is managed effectively, but relatively
informally
7: institutional capability for security (3)
There is a high level of awareness of the need to keep the data proprietary,
but given the scientific context, there are no likely attack scenarios as
such; the problem will largely evaporate once the data is finally released
publicly
8: institutional security mechanisms (3.5)
No formal threat analyses, but the security is probably appropriate to the
level of threat; day-to-day attacks (ie not specifically targeted at this
data) are the responsibility of distributed storage and computing managers
9: institutional disaster plan and capacity for business recovery (3)
Not applicable to the current experimental phase
10: institutional capacity to create metadata (4)
Almost all metadata is added automatically (compare organisational.08)
11: effectiveness of an Institution-wide repository (2)
LIGO has prepared a formal DMP
### Resources leg
1: institutional business planning (2)
LIGO is preparing a formal DMP
2: institutional capacity for review (4)
DMP to be reviewed annually; project as a whole has close relationships with
funders and stakeholders
3: institutional capability for resource allocation (4)
Resource planning is coordinated at a senior level
4: institutional capability for risk management (2)
General awareness at present, but this should become clearer in future DMP
iterations
5: institutional business transparency (4.5)
Depending on the precise meaning intended, this could be 4 or 5. There is
substantial auditing from collaboration funders
6: institutional capacity for sustainable funding (3.5)
Good relationships with funders mean that funding is probably predictable on
five- to ten-year timescales, but unpredictable in the longer term. However
the main funder (NSF) has expressed a strategic commitment to long-term data
preservation.
7: institutional staff management (3)
Not applicable to the current experimental phase
8: institutional management of staff numbers (3)
Not applicable to the current experimental phase
9: institutional commitment to staff development (3)
Not applicable to the current experimental phase
## About this document
LIGO-P1000188-v10, 2011 June 29
First public version, available at https://dcc.ligo.org/cgi-
bin/DocDB/ShowDocument?docid=p1000188
v1.1, 2012 July 14
Minor revisions, some added material, and typos and minor errors corrected.
There are a couple of additional sections, but no changes to section or figure
numbers. The pagination will have changed in places.
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## Glossary
ADS
Astrophysical Data Service: a bibliographic archive for astronomy, based at the Harvard-Smithsonian Center for Astrophysics; ADS preserves full-text copies of journal articles, both in collaboration with publishers, and through a digitization process, and maintains a widely-used bibliographic ID system (http://ads.harvard.edu)
AIP
Archival Information Package: An Information Package, consisting of the Content Information and the associated Preservation Description Information, which is preserved within an OAIS (OAIS)
aLIGO
Advanced LIGO: The successor project to LIGO, due to start in 2015
arXiv
A electronic preprint service, see http://arxiv.org. The arXiv started in the early 90s, based on FTP and email. It initially serviced particle physics and astronomy, but has expanded to cover other areas of physics, mathematics, and some areas of computing science
ATLAS
One of the four detectors at the LHC, and one of the two large ones
big science
A class of science projects characterised by being international, highly collaborative and expensively funded (see Sect. 1.1 for more discussion)
catalogue
In the astronomical context, a catalogue is a table of positions and other information for stars or other other astronomical objects
CCSDS
Consultative Committee for Space Data Systems: authors of the OAIS reference model, see http://www.ccsds.org
CDS
Strasbourg Data Centre: (see http://cdsweb.u-strasbg.fr/ and Sect. 1.4.1)
CMS
Compact Muon Solenoid: One of the four detectors at the LHC, and one of the two large ones
Content Information
The set of information that is the original target of preservation by the OAIS (OAIS)
Data Object
Either a Physical Object or a Digital Object (OAIS) (that is, the ‘Data Object’ is the sequence of bits, or the physical object which is _the data_ in the most primitive sense)
data sharing
The formalised practice of making science data publicly available
DCC
Digital Curation Centre: http://www.dcc.ac.uk (not to be confused with the LSC Document Control Center)
Designated Community
An identified group of potential Consumers who should be able to understand a particular set of information (OAIS)
DIP
Dissemination Information Package: The Information Package, derived from one or more AIPs, received by the Consumer in response to a request to the OAIS (OAIS)
DMP
Data Management and Preservation
ESA
European Space Agency: http://www.esa.int
ESO
European Southern Observatory: A pan-european agency running a set of southern-hemisphere telescopes http://www.eso.org
ESRC
Economic and Social Research Council: the principal social science funder in the UK, see http://www.esrc.ac.uk
FITS
Flexible Image Transport System: the standard file format in astronomy, see http://fits.gsfc.nasa.gov
GEO
A German-British consortium, responsible for the GEO600 interferometer, funded jointly by STFC and the German government
GEO600
The GEO observatory located near Hannover in Germany
GW
Gravitational Wave
HEP
High Energy Physics
HERA
A particle accelerator at the German DESY facility
Information Package
The Content Information and associated Preservation Description Information which is needed to aid in the preservation of the Content Information. The Information Package has associated Packaging Information used to delimit and identify the Content Information and Preservation Description Information (OAIS)
IVOA
International Virtual Observatory Alliance: the consortium which defines VO standards
JISC
Joint Information Systems Committee: The organisation responsible for the maintenance and effective exploitation of the academic computing network in the UK, and the funders of this present report
KRDS
Keeping Research Data Safe: JISC project developing and documenting data preservation tools and studies; see http://www.beagrie.com/krds.php and [44]
LHC
The Large Hadron Collider at CERN: the accelerator is the host for two large general purpose detectors (ATLAS and CMS) and two smaller ones (ALICE and LHCb)
LIGO Lab
The Caltech/MIT consortium, funded by NSF to design and run the LIGO interferometers in the US, see http://www.ligo.caltech.edu
LIGO
Laser Interferometer Gravitational-wave Observatory: the hardware, comprising LIGO Lab and GEO (see http://ligo.org and Sect. 1.6.1)
Long Term
A period of time long enough for there to be concern about the impacts of changing technologies, including support for new media and data formats, and of a changing user community, on the information being held in a repository. This period extends into the indefinite future (OAIS)
LSC
LIGO Scientific Collaboration: The network of research groups contributing effort to the LIGO experiment and data analysis, see http://ligo.org
LVC
A data-sharing agreement between the LSC and the Virgo Collaboration (see Sect. 1.6.1)
MOU
Memorandum of Understanding: the relationships between the various participating entities and the LSC is articulated through a series of annually reviewed MOUs
MRD
Managing Research Data: A funding programme within the JISC e-Research theme, see http://www.jisc.ac.uk/whatwedo/programmes/mrd
NASA
National Aeronautics and Space Administration: The US space agency http://www.nasa.gov
NSF
National Science Foundation: the principal (non-defence) science funder in the USA
NSSDC
National Space Science Data Center: the permanent archive for NASA space science mission data http://nssdc.gsfc.nasa.gov
PDS
Planetary Data System: The NASA data archive and standard set http://pds.nasa.gov/
pipeline
A software system (or sometimes a software-hardware hybrid) which transforms raw data into more or more levels of data product. The data reduction pipelines, which must be able to keep up with the rate at which data is acquired, and which are assembled from a mixture of standard and custom software components, generally absorb a significant fraction of the total development budget of a new instrument
raw data
The data extracted directly from an instrument or observation; since it is uncalibrated and uncorrected, it is generally of little use to those not intimately familiar with the instrument (see Sect. 1.4)
Representation Information
The information that maps a Data Object into more meaningful concepts (OAIS)
Representation Network
The set of Representation Information that fully describes the meaning of a Data Object. Representation Information in digital forms needs additional Representation Information so its digital forms can be understood over the Long Term (OAIS)
SIP
Submission Information Package: An Information Package that is delivered by the Producer to the OAIS for use in the construction of one or more AIPs (OAIS)
SKA
Square Kilometre Array: a low frequency radio telescope with a large (one square kilometre) collecting area
STFC
Science and Technology Facilities Council: the primary UK funder of facility-scale science, see http://www.stfc.ac.uk
strain data
The fundamental GW signal
Virgo
Italian-French gravitational-wave detector http://www.virgo.infn.it/
VizieR
A repository of astronomical catalogue data at CDS (see Sect. 1.4.1)
VO
Virtual Observatory: a set of data sharing argreements and protocols. See Sect. 1.10 (not to be confused with grid Virtual Organisations)
## Index
_See also the Glossary, which additionally serves as the index of acronyms_
* AIDA, 33, 35
* astronomy data, 8–11
* Babylon, 10, 18
* benefits, 21, 22
* big science, 6–17
* Caltech, 9
* climate data, 23
* costs, 23, 30–33
* data
* gravitational wave, 14–16
* ingest, 33
* volume, 6, 7
* data products, 6, 10, 11, 15, 16, 20, 21, 23, 25
* DCC lifecycle, 28, 29
* GAMA, 12
* HEP data, 8, 13, 14
* HerMES, 12
* Herschel, 9, 12
* Hipparchus, 19
* Hipparcos, 11
* LIGO
* Advanced, _see:_ glossary: aLIGO
* DMP, 26, 30, 31, 33
* OAIS, 5, 8, 11, 13, 23, 26–28
* open data, 6, 22, 23
* private facilities, 9
* proprietary data, 9, 16, 21, 31
* Ptolemy, 19
* raw data, 10, 15, 16
* preservation, 25
* utility, 23
* social sciences, 17
* software preservation, 17, 25, 29, 30
* UKIDSS, 12
* virtual observatory, 19, 20
* WFAU, Edinburgh, 20
|
arxiv-papers
| 2012-07-17T09:08:43 |
2024-09-04T02:49:33.191255
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Norman Gray, Tobia Carozzi, Graham Woan",
"submitter": "Norman Gray",
"url": "https://arxiv.org/abs/1207.3923"
}
|
1207.3926
|
# A mobile Magnetic Sensor Unit for the KATRIN Main Spectrometer
A. Osipowicza, W. Sellera, J. Letneva, P. Martea, A. M llera, A. Spenglera, A.
Unrua
aUniversity of Applied Sciences Corresponding author. Fulda Germany
E-mail Alexander.Osipowicz@et.hs-fulda.de
###### Abstract
The KArlsruhe TRItium Neutrino experiment (KATRIN) aims to measure the
electron neutrino mass with an unprecedented sensitivity of 0.2 eV/c2, using
$\beta$ decay electrons from tritium decay. For the control of magnetic field
in the main spectrometer area of the KATRIN experiment a mobile magnetic
sensor unit is constructed and tested at the KATRIN main spectrometer site.
The unit moves on inner rails of the support structures of the low field
shaping coils which are arranged along the the main spectrometer. The unit
propagates on a caterpillar drive and contains an electro motor, battery pack,
board electronics, 2 triaxial flux gate sensors and 2 inclination senors.
During operation all relevant data are stored on board and transmitted to the
master station after the docking station is reached.
###### keywords:
Mobile Magnetic Sensor Unit, KATRIN, Spectrometer
## 1 The KATRIN setup
The KArlsruhe TRItium Neutrino experiment [1] (see Fig.1) is set up at the
Karlsruher Institute of Technology (KIT), Germany. It is designed to measure
the mass of the electron neutrino in a direct and model-independent way with a
sensitivity of $m_{\nu}=0.2$ eV/c2 (90% confidence level) from tritium $\beta$
decay[1]. KATRIN uses a magnetic transport field that connects the source and
detector in combination with integrating electrostatic energy filters (MAC-E-
spectrometers). Conceptual essentials of the MAC-E spectrometer[2, 3] are the
magnetic field gradients in pre - and main-spectrometer that adiabatically
convert cyclotron energy $E_{cyc}$ into energy $E_{p}$ parallel to the
magnetic field lines and vice versa.
Figure 1: Schematic view of the KATRIN experiment (total length 70 m)
consisting of calibration and monitor rear system, with the windowless gaseous
$\rm T_{2}$-source (WGTS), differential pumping (DPS) and cryo-trapping
section (SPS), the small pre-spectrometer and the large main spectrometer with
the large magnetic coil systems to compensate the earth magentic field (EMCS)
and to shape the magnetic transport flux (LFCS) and lastly the segmented PIN-
diode detector.
In the minimal magnetic field (the analyzing field $B_{A}\approx 3-6$ $\mu$T
and $d_{\Phi}=4.5$ m) at the center of the MS, a retarding electric field
distribution allows an integral energy analysis of $E_{p}$. The shape of the
magnetic flux tube in the MS area defines the magnetic resolution, i.e. the
amount of residual cyclotron energy $E_{cyc}$ that can not be analyzed and
thus strongly influences the resolution function. Systematic error
considerations [5] demand a homogeneity of the magnetic field distribution in
the analyzing volume of $\Delta(B)/B<0.04$. Moreover, the alignment of
magnetic field lines plays a crucial role in the production of secondary
electrons and electronic background either through traps or wall contact (see
Fig. 2).
Figure 2: A CAD bird view (taken from a PartOpt [6] simulation) of the
energized KATRIN solenoid chain and LFCS. The analyzing area is situated at
the center of the main spectrometer where the shape of the magnetic flux is
mainly given by the spectrometer solenoids and the LFCS. In this simulation
the extreme magnetic flux lines ($191$ T cm2) have been tracked in the
horizontal plane with and without the perturbing influence of the earth
magnetic field $B_{x}=25$ $\mu$T and a small external dipole with $600$ $\mu$T
central induction. The perturbed flux lines are indicated.
Large coil systems [4] are arranged around the MS for a) global magnetic field
compensation, e.g. earth magnetic field (EMCS) and b) fine tuning of the
magnetic transport flux with a set of large circular low field coils (LFCS)
mounted coaxially with the MS. However, possible influences of residual
external dipoles, magnetization in the MS environment by the high field
solenoids and/or EMCS, LFCS and the correct orientation of the spectrometer
solenoids have to be controlled. Due to the extreme MS vacuum conditions the
installation of magnetic sensors inside the MS is not possible.
Figure 3: View of the main spectrometer tank with the LFCS ring system. Right:
The mobile sensor unit with 2 sensors on the inner belt of a LFCS support
ring.
In this paper we present a mobile sensor unit, designed to move along the
inner belts of the LFCS support structure (see Fig.3), close to the outer MS
surface, but well inside the EMCS, LFCS current lines. It allows to sample a
large number of magnetic field values from large areas of MS surface.
## 2 The Mobile Sensor Unit
### 2.0.1 Mechanical structure
To minimize self-magnetization, the skeleton of the mobile sensor unit [8, 9]
(MobS)(see Fig. 5) is assembled from aluminum cut parts and consists of a
chassis with a drive, a tower and the wing-shaped sensor board that is
attached on top of the tower. The caterpillar drive consisting of 3 acetal
resin wheels (and 3 counter wheels per side) bearing a polyurethane toothed
belt is chosen to ensure slip free motion on interfaces between the LFCS ring
arcs . The tower hosts a brush-less DC motor (maxon motor EC45, 12 V/30 W),
gear (maxon motor gear GP 42 C), and rechargeable battery pack (Lead AGM, 1.8
Ah). To avoid currents during the measurement intervals a linear actuator is
used as a break shoe that mechanically clamps the motor.
Figure 4: CAD view of the MobS with docking station. The chassis suspension
with a caterpillar drive, the tower with motor, gear and battery pack. The
electrically shielded control board on top contains board electronics and two
triaxial flux gate and 1 dual inclination sensor. The electrical shield is
sketched transparently.
Figure 5: CAD view of the docking station (DS) (left) and docking block
(right). The block on is attached at the MobSU. The contacts from top: 2
detection contact pads, 2 RS 485 data lines, 2 RS 232 data lines, 2 power
supply and 2 charging lines.
The sensor board attached at the top of the tower is equipped with two
triaxial custom designed flux gate sensors (see Table 1), a dual axis digital
inclinometer (Analog Device ADIS 16209), and two electronic boards. The flux
gate sensors are positioned along common axes within $10$ $\mu$m tolerance. As
the sensor board is facing inward to the main spectrometer surface, which can
bear voltages up to 32 kV, the sensor board is electrically shielded. Mounted
at one side of the chassis suspension is the docking block that detects the
docking station (DS) via 2 sliding contacts (see Fig. 5). The DS is
permanently attached to the LFCS inner belt. Both docking block and docking
station are manufactured from PVC with contacts for data, power and charging
lines.
### 2.0.2 Electrical layout
The basis board (see Fig.6) containing a microcontoller unit (TI MSP 430F5438)
reads the flux gate signal via precision operational amplifiers (LINEAR
TECHNOLOGY LT6013/LT6014) and analog to digital converters (TI TLC 3574,
14-bit) and controls the actions of the MobS. The AddOn board hosts 6 MB of
flash memory, one inclination sensor, USB interface, I2C interface and the
motion control module (MCM) (Atmel MCU Atmega 8). The MCM controls the drive
chain and the linear actuator (break shoe). It also controls the incremental
encoder that gives 2048 impulses per wheel revolution (radius wheel $R=32,75$
mm) and allows a positional resolution of $50.16\>\mu\rm{m}\pm
25.8\>\mu\rm{m}$.
The docking station is designed a) to act as an interface in the data channels
and b) as a charging station for the battery pack. This is realized with a
dedicated circuit board (HIP) based on a Atmel MCU Atmega 328. The master
module (Master) serves as a data collector and provides an interface to the
KATRIN database.
Figure 6: Sketch of the electrical layout of sensor board electronics (right
circle), docking station electronics (left circle), data and power lines to
master module and Fieldpoint to KATRIN slow control (below). The layout is
designed to incorporate up to 20 MobS on further LFCS rings.
### 2.0.3 The sensor coordinate systems
The orientation of the sensor coordinate systems on the MobS is shown in
Fig.7.
Figure 7: The coordinate systems of the individual sensors that are aligned
along a common $z$ -axes, which is parallel to the global $z_{g}$-axes. The
sensors are labeled according to their orientation with respect to the source
(S) and detector (D). The MobS is moving in $x_{S}$, $-x_{D}$ -direction. The
inclination sensor axes $x_{I}$, $z_{I}$ are oriented parallel to the $x$, $z$
-axis respectively.
As the MobS is moving along a LFCS ring, the sensor coordinate systems change
their position and orientation with respect to the global KATRIN cylindric
coordinate system $z_{g},\varphi_{g},r_{g}$. With $z_{g}$, the central axis
pointing from source to detector, $r_{g}$ the radial coordinate and
$\varphi_{g}$ the azimuthal coordinate, the MobS will pick up axial, azimuthal
and radial magnetic field components. However, the exact anchoring of the
local coordinate systems into the global system has not been carried out yet
as the MobS track is not ideally ring shaped and shows local misalignments.
## 3 The MobS residual magnetic field
In order to determine possible residual magnetic field components
$\vec{B}_{res}$ that are produced by the MobS itself a, difference measurement
in an iron free environment was performed. After positioning on a CNC machined
Al-turntable, the MobS was oriented horizontally by inclinometer reading.
Their tilt sensing system uses gravity as its only stimulus and is specified
to have an angular accuracy of $0.1^{\circ}$.
Figure 8: The differences between the magnetic field components measured in
anti-parallel directions for the source sensor (left) and detector sensor
(right). The resulting mean and sigma is shown as a black line and dashed line
respectively.
With the FL3-500 sensor 100 magnetic field samples are taken (conversion (DAC)
rate $100$ kHz, 64 conversions per sample point), then the table was turned by
$180^{\circ}$ and the next 100 samples are taken. The resulting differences
(see Fig.8)for the magnetic field components are $\vec{B}_{res}=(-0.36\pm
0.1\>\mu\rm{T},0.37\pm 0.1\>\mu\rm{T},-0.68\pm 0.09\>\mu\rm{T})$ for the
detector sensor and $\vec{B}_{res}=(-0.11\pm 0.1\>\mu\rm{T},-0.15\pm
0.12\>\mu\rm{T},0.42\pm 0.1\>\mu\rm{T})$ for the source sensor.
## 4 Test measurement at the KATRIN site
The magnetic field components are measured at 36 equidistant positions (see
Fig.10) on the circumference of ring 17 (see Fig.3) at the KATRIN MS site. The
positions are accurate within $0.01$ m (see Fig.10).
Figure 9: The distribution of the 2*36 sample points along the track. The
LFCS ring 17 is centered at the $z=0$ m position. Due to manufacturing
tolerances the radial distance of the sensor positions of $r=5.873$ m shows
variations up to $0.05$ m along the arc. The coaxial position of the pinch
($z_{p}=12.4575$ m) and detector solenoid ($z_{d}=14.0575$ m) is indicated on
the global z-axis (dashed line).
Figure 10: The distance between two stops during one cycle according to the
step motor data shows a distribution that is caused by varying motor reaction
times at different MobS position along the ring.
The samples are taken for 4 successive pinch/detector (see Fig.10) solenoid
current settings: a.) central induction $6$ T/ $3.6$ T, b.) $3$ T/ $1.8$ T,
c.) $1.5$ T/ $0.9$ T and d.) $0$ T/ $0$ T allowing a background measurement.
The two solenoids are arranged coaxially with the LFCS coil with a central
$z$-distance of $z_{p}=12.4575$ m for the pinch and $z_{d}=14.0575$ m for the
detector solenoid (distance values according to the detector installation
position on 1st March 2012). Fig.11 shows the $B_{x}$-components for the three
successive runs with the $0$ T/ $0$ T magnet setting. For each setting the
MobS has completed 3 cycles. The time for completion of one cycle is about 5
minutes.
Figure 11: The raw data for the $B_{x}$-component for the 3 runs with the
magnet setting 0 T/0 T (background measurement) from the detector sensor. Each
point represents the mean of 64 samples resulting in a small error. The
position of the sample points on the arc is also given in "clock"-positions
(at the top). In detector direction the Mobs performs a clockwise motion
starting at the 7:15 position ending at the 6:55 position.
After statistics and correction for zero point and internal orthogonality
error the values of $B_{x}$ and $B_{z}$ of the detector-sensor have been
multiplied by $-1$ for display reasons (see Fig.12).
Figure 12: A compilation of the magnetic field components from the detector
sensor (blue crosses) and the source sensor (full dots) for the $6$T/$3.6$T
magnet setting. As the detector sensor is closer to the detector it senses
higher values for the axial $B_{z}$ and the radial $B_{y}$ component. The
$B_{x}$ component readings are very similar, because the detector solenoids do
not produce any azimuthal field.
After subtracting the background field components from the data with energized
detector solenoids, the resulting values can be compared to a magnetic field
simulation (see Fig.13). The geometric data of pinch and detector solenoid are
taken from the manufacturers data sheet.
Figure 13: A comparison of the total magnetic field strength with simulated
magnetic field values for the source sensor (full circles and dashed line) and
detector sensor (crosses and full line). Lower box: the $1.5$ T/ $0.9$ T,
middle box: the $3$ T/ $1.6$ T and upper box; the $6$ T/ $3.6$ T magnet
setting.
The experimental data show a variation of up to $3\;\mu$T along the track for
each detector magnet setting, moreover it is obvious that the difference
between the simulated and the measured data is not constant.
Figure 14: The difference $\Delta B$ between simulated magnetic field values
and sample mean of the experimental values for the three magnetic field
settings. Full dots: source sensor (SS), crosses: detector sensor (DS). The
error bars shown refer to the standard deviation of each sample.
Taking the difference between the simulated values and the sample mean for the
values given in Fig.13 a decrease can be observed (see Fig.14) which might be
explained by magnetization of the construction steel used in the spectrometer
hall.
## 5 Summary and Outlook
To control the magnetic field in the KATRIN main spectrometer area a mobile
magnetic field sensor unit is presented that moves along the LFCS inner belt
and can take magnetic field samples. Such units can be installed at each LFCS
ring and allows magnetic field sampling in areas that for safety reasons are
only hardly or not at all accessible. The typical cycle time with 36 sampling
stops is minutes.
The data presented in this paper might indicate magnetization effects in the
vicinity of ring 17 caused by the detector solenoids. For a more detailed
analysis of the magnetic field samples it is necessary to anchor the sensor
coordinate systems into the global KATRIN coordinate system. This can be
achieved by incorporating inclination sensor data and improved geometry data
of the individual tracks into the analysis. Currently different types of
inclinometers are studied.
After multiple runs, the wheels of the MobS accumulate dust and slip can
occur, which might lead to false position reading. Therefore we propose a
combination of a toothed belt attached to the track and toothed wheels as a
running gear of the MobS.
The magnetic field in the volume of the main spectrometer is free of rotation
and therefore analytically connected with the field on the outside surface
over the definition of a magnetic scalar Potential $V(x,y,z)$. It fulfills the
Laplace equation which can be solved numerically by a finite difference method
in which the magnetic surface samples are used as boundary values for
computing the magnetic field everywhere inside the main spectrometer [10, 11].
Hence it is advantageous to get large amounts of surface samples and the
production of more MobS units that will run on further LFSC rings is under
way. With these a more detailed determination of the magnetic field profile
and the magnetization effects will be possible.
Table 1: Features of the flux gate sensors that are installed on the MobS. The sensors have also been calibrated by the company [7]. Sensor Type | FLC3-70 | FL3-500
---|---|---
Range | $\pm 500\mu$T | $\pm 1000\mu$ T
Accuracy | $\pm 1\%\pm 0.5\mu T$ | $\pm 0.5\%$
Resolution | 1V / 35 $\mu T$ | 1V / 100 $\mu T$
Voltage Supply | 30 V | 30 V
Orthogonality | $<1^{\circ}$ | $<0.5^{\circ}$
AKNOWLEDGMENTS
The authors wish to express gratitude to the group for Experimental Techniques
of the Institute for Nuclear Physics (IK) at KIT for highly efficient and
competent support. Namely Prof. Dr. J. Bl mer, Dr. F. Gl ck and Jan Reich for
support and help. Special thanks to the KATRIN detector group from University
of Washington for supplying magnetic fields during the measurements.
Furthermore, we wish to thank Prof. Dr. E. W. Otten, Mainz University and
Prof. Dr. Ch. Weinheimer, M nster University for helpful discussions and
support. In addition, we like to thank the University of Applied Sciences,
Fulda and the Fachbereich Elektrotechnik und Informationstechnik, especially
Prof. Dr. U. Rausch, for the enduring support for this work.
This work has been funded by the German Ministry for Education and Research
under the Project codes 05A11REA, 05A08RE1.
## References
* [1] KATRIN Collaboration, _KATRIN Design Report 2004, Technical Report (Forschungszentrum Karlsruhe, Forschungszentrum Karlsruhe, Karlsruhe, Germany, 2004), www-ik.fzk._ ,
* [2] A.Picard et.al, _A solenoid retarding spectrometer with high resolution and transmission for keV electrons_ , _Nucl. Instrum. Meth._ B 63 (1992) 345
* [3] V.M. Lobashev, P.E. Spivac, _A method for measuring the anti-electron-neutrino rest mass_ , _Nucl. Instrum. Meth._ A240 (1985) 305
* [4] A. Osipowicz, F. Gl ck, _Air coil design at the main spectrometer_ , _KATRIN internal document_ http://fuzzy.fzk.de/bscw/bscw.cgi/d443733/95-TRP-4440 -D1-F.Glueck-A.Osipowicz.ppt
* [5] N. Titov, priv. communication
* [6] The PartOpt project page, http://www.PartOpt.nethttp://www.PartOpt.net
* [7] Stefan Mayer Instruments, http://www.stefan-mayer.com http://www.stefan-mayer.com
* [8] A. Unru _Elektrische und mechanische Konzipierung und prototypische Realisierung einer mobilen Sensoreinheit_ , Diploma-Thesis, Univ. of Appl. Sciences, Fulda, July 2009
* [9] J. Letnev, _Systemintegration des Magnetfeldsensornetzes_ , Master-Thesis, Univ. of Appl. Sciences, Fulda, May 2011
* [10] S. Flachs, A. Osipowicz, A. Unru
Design Document, A wireless magnetic sensor grid for the KATRIN main
spectrometer
KATRIN internal document https://fuzzy.fzk.de/bscw/bscw.cgi/d698744/
* [11] A. Osipowicz,_A method for the measurement of the magnetic field in the KATRIN main spectrometer_ , to be published
|
arxiv-papers
| 2012-07-17T09:44:34 |
2024-09-04T02:49:33.214069
|
{
"license": "Public Domain",
"authors": "A. Osipowicz, W. Seller, J. Letnev, P. Marte, A. M\\\"uller, A.\n Spengler, A. Unru",
"submitter": "Alexander Osipowicz",
"url": "https://arxiv.org/abs/1207.3926"
}
|
1207.3942
|
# Confidence and Backaction in the Quantum Filter Equation
Wei Cui wcui@riken.jp Advanced Science Institute, RIKEN, Wako-shi, Saitama
351-0198, Japan Neill Lambert nwlambert@riken.jp Advanced Science Institute,
RIKEN, Wako-shi, Saitama 351-0198, Japan Yukihiro Ota Advanced Science
Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan Xin-You Lü Advanced
Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan Z. -L. Xiang
Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan
Department of Physics, State Key Laboratory of Surface Physics, Fudan
University, Shanghai 200433, China J. Q. You Department of Physics, State
Key Laboratory of Surface Physics, Fudan University, Shanghai 200433, China
Beijing Computational Science Research Center, Beijing 100084, China Advanced
Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan Franco Nori
Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan Physics
Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
###### Abstract
We study the confidence and backaction of state reconstruction based on a
continuous weak measurement and the quantum filter equation. As a physical
example we use the traditional model of a double quantum dot being
continuously monitored by a quantum point contact. We examine the confidence
of the estimate of a state constructed from the measurement record, and the
effect of backaction of that measurement on that state. Finally, in the case
of general measurements we show that using the relative entropy as a measure
of confidence allows us to define the lower bound on the confidence as a type
of quantum discord.
###### pacs:
03.67.-a, 03.65.Ta, 03.65.Yz, 73.63.Kv
## I Introduction
Recently there has been a great deal of activity on the topic of “weak”
quantum measurements Wiseman2010 ; Alicki05 ; Kofman11 ; Ashhab ; Ashhab09 ;
Croke06 ; Braginsky in both mesoscopic Korotkov99 ; Korotkov01 ; Korotkov06
and macroscopic systems Buluta11 ; Macroscopic ; You05 ; Johansson . In
contrast to projective measurement, weak measurements only perturb the system
slightly, but in turn can only provide limited information. According to the
theory of open quantum systems, both the evolution of the quantum state and
its decoherence depend on the system-apparatus coupling strength and the basis
in which the measurement system is measured. One of the advantages of such a
measurement is that, given a weak continuous measurement record one can
reconstruct the quantum system state during its evolution. One particularly
interesting approach, which we apply and investigate here, is the “quantum
filter equation” as pioneered in the quantum limit by Belavkin Belavkin1999
and others Carmichael .
The quantum filter equation has been shown to be a powerful method for state
reconstruction, and is fairly robust in terms of the resolution needed in
describing the measurement record. For example, in recent work Ralph it has
been shown that the continuous “analogue” measurement record can be reduced to
a “one-bit record” and still the filter equation can efficiently produce an
estimate (or purification) of the system state. Similarly it has been shown
that feedback control can be used to enhance the speed of the state estimation
Filter ; Chase ; Zhang , and that it can be further optimized when combined
with a kind of process tomography Deutsch . However, successful application of
the filter equation requires accurate knowledge of the evolution, both
coherent and incoherent, that the monitored system undergoes Hill2011 .
Here we first investigate how this state estimation method can be used by
considering both how confidently Croke2006 an estimated state deduced from
the measurement record reflects the actual state of the system, and how the
effect of backaction changes the state. Here the “actual state of the system”
means the system under the influence of the back-action backaction ; Buscemi ;
Luo of the measurement apparatus, not the initial prepared state. In other
words, we quantify the confidence of the state-estimation process
independently from the overall fidelity of the measurement. Thus, we focus on
understanding the state reconstructed using the filter equation, without
actively undoing the backaction or employing feedback. We separately define
the overall accuracy of the measurement with another quantity or distance,
which we term the “epitome”. Second, we introduce a new ensemble averaged
version of the filter equation which enables a more efficient numerical method
(in theoretical treatments) with which to consider state estimation via the
filter equation. Third, in the final section we consider the more general
situation of an asymptotic positive operator valued measure (POVM), and show
that in a limiting case the confidence has a lower bound set by the quantum
discord.
Quantities like the confidence have been commonly employed in investigating
information gain with projective and general measurements Banazek ; Fuchs .
The behavior of these quantities in the context of weak continuous measurement
has not been studied in great detail as of yet, though the concept of
information gain and measurement disturbance is well understood Jacobs ;
Jacobs2 . In addition, we point out that our approach here is different from
that used in some other works. For example, here we are concerned primarily
with the optimal measurement of an unknown state by refining our state of
knowledge, whereas in some other approaches the goal is primarily manipulating
(or purifying) one’s state of knowledge of a given quantum system, and the
initial unknown quantum state is unimportant Jacobs2 .
The model we use here is based on the continuous measurement of a double
quantum dot (DQD) charge state using a quantum point contact (QPC) Korotkov99
; DQD ; Petta ; Lambert2010 . In this situation the tunnelling barrier of the
QPC is modulated by the charge in the nearby DQD, and produces a continuous
measurement record which can be used in the filter equation. However, the
approach is quite general, and recent applications have also arisen in
circuit-QED systems Johansson ; Korotcqed .
This article is organized as follows: In Sec. II we provide a general
definition of the confidence, backaction and epitome. In Sec. III we introduce
the model for weak measurement of a DQD, and show the results given by the
quantum filter equation. In Sec. IV we speculate about a possible filter
equation based on ensemble-averaged measurements. Finally, in Sec. V we show
that using the relative entropy as a measure of confidence allows us to define
the lower bound on the confidence as the quantum discord.
Figure 1: (Color online) (a) Schematic of the components of the quantum state
estimation using the filter equation. Here, $\rho_{s}$ is the system initial
state, $\rho_{R}(t)$ is the evolution of the initial state of the system in
the measurement-induced noises, $\rho_{E}(t)$ is the estimation of the system
state, and $\mathcal{D}[\rho_{E},\rho_{R}]$ is the estimation confidence. (b)
shows a diagrammatic representation of the confidence $\mathcal{C}$ and
backaction $\mathcal{B}$ as the distance between the various states. Note that
when we use the relative entropy as a distance measure these quantities are
asymmetric (as represented by the one-way arrows).
## II Definition of Confidence and Backaction
Consider a quantum system $\mathcal{S}$ which interacts with a measurement
apparatus $\mathcal{A}$. When implementing a general quantum measurement
$\mathcal{M}=\\{\Pi_{i}\\}$, our knowledge of the system state is based only
on the measured apparatus output $y(t)$. We denote our estimate of the system
state as $\rho_{E}(t)$. The initial state of the system is defined as
$\rho_{s}(0)$, and $\rho_{R}(t)$ is defined as the evolution of the initial
state of the system in the measurement induced noise, i.e., what we can think
of as the actual state of the system. We define the ideal state of the system,
i.e., its evolution if it were not affected by the measurement apparatus at
all, as $\rho_{I}(t)$, i.e., the entirely coherent evolution of $\rho_{s}(0)$.
Definition: The quantum state estimation confidence is defined as
$\displaystyle\mathcal{C}\equiv D[\rho_{E},\rho_{R}],$ (1)
where $D[\cdot]$ is some appropriate distance measure. Generally speaking, the
smaller $\mathcal{C}$ is the more confident we are of the estimated state, and
$\mathcal{C}=0$ if and only if $\rho_{E}=\rho_{R}$, which means that the
estimated state is totally confident. Similarly we define the backaction by
$\displaystyle\mathcal{B}\equiv D[\rho_{I},\rho_{R}],$ (2)
so that $\mathcal{B}=0$ implies no backaction. Finally, we define the overall
accuracy of the measurement with the “epitome”,
$\displaystyle\mathcal{E}\equiv D[\rho_{I},\rho_{E}],$ (3)
which naturally completes the triangle in Fig. 1. In the treatment that
follows of course we have full knowledge of all these states at all times, and
can thus in a theoretical sense identify the optimal parameters that minimize
these quantities.
There is some freedom in choosing an appropriate measure for $\mathcal{C}$,
$\mathcal{B}$, and $\mathcal{E}$. Here we explicitly consider both the
fidelity and the relative entropy Modi . The fidelity is commonly used to
measure the effectiveness of the filter equation, and is defined as
$\displaystyle F=1-\mathcal{C}=|(\sigma^{1/2}\rho\sigma^{1/2})^{1/2}|^{2}.$
(4)
We define the confidence as $\mathcal{C}=1-F$ in this case, to match our
definition of a distance measure, so that high fidelity implies
$\mathcal{C}=0$ (and the same with the other measures). However since the
fidelity is a pseudo-distance this lacks some characteristics of a true
measure. In the case of the relative entropy we define,
$\displaystyle\mathcal{C}=S(\rho_{R}||\rho_{E}),\quad\mathcal{B}=S(\rho_{I}||\rho_{R}),\quad\mathcal{E}=S(\rho_{I}||\rho_{E})$
(5)
where
$\displaystyle S(\sigma||\rho)=-\mathrm{Tr}\sigma\ln\rho-S(\sigma).$ (6)
The relative entropy can be seen as a distance measure, though as it is
asymmetric in $\sigma\leftrightarrow\rho$ it is technically not. In fact the
ordering here is important; with the inverse ordering the backaction
$B\rightarrow\infty$ as $\rho_{I}$ is a pure state in this example. Using the
relative entropy allows us to make a more direct connection to a general POVM
description of a weak continuous measurement in the final section of this
work.
Figure 2: (Color online) Schematic of a QPC used for measuring the electron
states yields backaction on the DQD.
## III Continuous Weak measurement of a Double Quantum Dot
We now present the specific details of the DQD and QPC system. A DQD consists
of a dot $L$, connected to an emitter, and dot $R$, connected to a collector
DQD ; Ouyang10 ; Petersson10 . As is typical, we assume that the DQD is in the
strong Coulomb regime such that only one electron is allowed in the whole DQD.
Here we assume the DQD is prepared in a single electron state, then isolated
from the emitter and collector reservoirs via manipulation of gate voltages.
The two single-dot states are denoted by $|L\rangle$, and $|R\rangle$. The
Hamiltonian of the DQD can be written by
$\displaystyle H_{{\rm DQD}}=\Omega\sigma_{x}/2+\epsilon\sigma_{z},$ (7)
with $\sigma_{x}=|L\rangle\langle R|+|R\rangle\langle L|$,
$\sigma_{z}=|L\rangle\langle L|-|R\rangle\langle R|$, $\epsilon$ is the level
splitting between the two single-dot states and $\Omega$ is the coherent
tunnelling amplitude betweens the two dots. The nearby QPC has the Hamiltonian
$H_{{\rm
QPC}}=\sum_{k}\epsilon_{1k}a_{1k}^{{\dagger}}a_{1k}+\sum_{q}\epsilon_{2q}a_{2q}^{{\dagger}}a_{2q}$,
and the interaction Hamiltonian
$H_{I}=\sum_{k,q}(\chi-\chi_{L}|L\rangle\langle L|-\chi_{R}|R\rangle\langle
R|)(a_{1k}^{{\dagger}}a_{2q}+a_{2q}^{{\dagger}}a_{1k})$, which is modulated by
the electron states of the DQD. Here $\chi$ is the tunneling amplitude of an
isolated QPC, $\chi_{L}(\chi_{R})$ gives the variation in the tunneling
amplitude when the extra electron stays on the left (right) dot, and
$a_{1k}(a_{2q})$ denotes the electron annihilation operator for the source
(drain) of the QPC. Because the height of the tunneling barrier in the QPC
depends on the electron states of the DQD, it is expected that the measured
current of the QPC will vary with the DQD states.
In this simple model, the current shot noise Aguado04 of the QPC acts as a
noise source. In the low-temperature limit with $k_{B}T\ll\hbar\omega$, the
noise spectral density takes the form Aguado00 ; Aguado04 ; Gustavsson07 ;
Petersson10
$\displaystyle J_{I}(\omega)=\frac{4}{R_{K}}D(1-D)\frac{(eV_{\rm
QPC}-\hbar\omega)}{[1-\exp\\{-(eV_{\rm QPC}-\hbar\omega)/k_{B}T\\}]},$ (8)
where $R_{k}=h/e^{2}$ is the von Klitzing constant, $D$ is the transparency of
a single channel in the QPC, and $D$ is a function of $\chi$, $\chi_{L}$ and
$\chi_{R}$ Korotkov01 ; Aguado00 ; Gustavsson07 ; Lesovik . To treat the
measurement signal, or current through the QPC, as a classical variable one
must assume that the QPC evolution is much faster than the DQD, so that only
the zero-frequency component is important; this is effectively a Markovian
limit in terms of treating the QPC backaction on the DQD Korotkov99 ;
backaction .
We also treat the QPC as a perfect detector. In this limit we can define the
measurement strength of the QPC as
$\displaystyle\kappa=\frac{(\Delta I)^{2}}{16J(0)}.$ (9)
Here, $\Delta I=I_{L}-I_{R}$, where
$I_{L}=D_{L}e^{2}V/\pi\hbar$,$I_{R}=D_{R}e^{2}V/\pi\hbar$, $D_{L}=D+\Delta D$,
$D_{R}=D-\Delta D$, and $\Delta D$ is the change in the transmission of the
QPC due to the charge state of the DQD.
Figure 3: (Color online). The top row of figures shows the ensemble-averaged
behavior (over $2,000$ realizations) of (a) one minus the confidence (blue)
and one minus the backaction (red) defined in terms of the fidelity, (b) the
occupation of the left “dot” or state for the system, Eq. (III) (in red) and
the estimator, Eq. (III.1), (in blue), and (c) the confidence (blue) and
backaction (red) defined in terms of the relative entropy. In all figures
$\kappa=0.005\Omega$, and time is evolved for the equivalent of $15$ Rabi
oscillations of the bare quantum dot state. Figure (b) shows more directly how
the estimator quickly oscillates in phase with the system state, but takes
time to evolve to the same population magnitude. The small oscillations seen
in both backaction curves in (a) and (c) are typical for the definition of the
backaction, and simply represent the “nodes” in the oscillation curve of,
e.g., $P_{L}$, where the $\rho_{I}$ and $\rho_{R}$ states coincide. Figures
(d)-(e) show the same quantities as (a)–(c) except just a single realization.
The estimator state still quickly approaches the system state, as is typical
with the filter equation. The backaction changes in magnitude drastically in
both (d) and (f). There is no single-realization dephasing in (e) because we
assume the QPC to be a perfect detector. This is in contrast to recent work
Johansson on circuit QED where some information can remain hidden due to the
finite lifetime of the measurement cavity. Note, in (a) and (d) we have
plotted $1-C$ and $1-B$, so that the fidelity result can be easily compared to
other works investigating the effectiveness of the filter equation using
fidelity.
The evolution of the real state of the DQD $\rho_{R}$ can be derived using the
Bayesian techniques of Korotkov Korotkov99 ; Korotkov01 ; Korotkov06 to give
the selective stochastic master equation (SME) in Ito form,
$\displaystyle\rm{d}{\rho_{R}}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[H_{\rm
DQD},\rho_{R}\right]dt+\kappa\mathcal{D}[\sigma_{z}]\,\rho_{R}\,dt$
$\displaystyle+$
$\displaystyle\sqrt{2\kappa}\;\mathcal{H}[\sigma_{z}]\;\rho_{R}\;dW_{R},$
where $\kappa$ is defined above,
$\displaystyle\mathcal{H}[\sigma_{z}]\rho_{R}\equiv\sigma_{z}\rho_{R}+\rho_{R}\sigma_{z}^{\dagger}-\mathrm{Tr}(\sigma_{z}\rho_{R}+\rho_{R}\sigma_{z}^{\dagger})\rho_{R},$
(11)
and the real Wiener process $dW_{R}$ satisfies $\mathrm{E}(dW_{R})=0,$
$(dW_{R})^{2}=dt$. In Eq. (III), the super-operator $\mathcal{D}$ is defined
as
$\displaystyle\mathcal{D}[a]\rho=a\rho
a^{{\dagger}}-\frac{1}{2}(a^{{\dagger}}a\rho+\rho a^{{\dagger}}a).$ (12)
Given that, in a general sense, the measurement operator is
$y=\sqrt{2\kappa\hbar}\sigma_{z}$ the measurement record increment at a time
$t$ is,
$\displaystyle\frac{dy(t)}{\sqrt{\hbar}}$ $\displaystyle=$
$\displaystyle\sqrt{8\kappa}\langle\sigma_{z}^{R}(t)\rangle
dt+dW_{R}=\frac{\Delta
I\langle\sigma_{z}^{R}(t)\rangle}{\sqrt{2J(0)}}dt+dW_{R}.$
Here
$\displaystyle\langle\sigma_{z}^{R}(t)\rangle=\mathrm{Tr}[\sigma_{z}\rho_{R}(t)]$
(14)
is the instantaneous expectation value of $\sigma_{z}$ at time $t$ based on
the selectively evolved density matrix $\rho_{R}(t)$.
This equation of motion also relies on the assumption that
$\displaystyle|I_{L}-I_{R}|=|\Delta I|\ll I_{0}=(I_{L}+I_{R})/2,$ (15)
so that many electrons, $N\geq(I_{0}/\Delta I)^{2}\gg 1$, should pass through
the QPC before one can distinguish the quantum state. This is the weakly
responding or linear regime, and the model as we have described it is entirely
equivalent to the formulation used by Korotkov and others Korotkov99 ;
Korotkov01 ; Korotkov06 . Also, note that for consistency with other works on
the filter equation as a state estimation technique Ralph ; Hill2011 we
describe the noise as a Wiener process, so that the width of the Gaussian
distribution Korotkov01 ; Korotkov06 used to describe the weak measurement
with a QPC is absorbed into $\kappa$.
### III.1 Quantum Filter Equation
To estimate the quantum state $\rho_{E}$ from the measurement output we employ
the quantum filter equation method Shannon ; Mitter ; Ralph ; Filter ; Chase ;
Zhang . The evolution of the estimated state $\rho_{E}$ is described by the
following stochastic master equation, identical to the “system” one, except
the measurement signal from the system evolution, described above, determines
the noise term:
$\displaystyle d\rho_{E}=-\frac{i}{\hbar}\left[H_{\rm
DQD},\rho_{E}\right]dt+\kappa\mathcal{D}[\sigma_{z}]\rho_{E}dt$
$\displaystyle+\sqrt{2\kappa}\mathcal{H}[\sigma_{z}]\rho_{E}\left[\frac{dy(t)}{\sqrt{\hbar}}-\frac{\Delta
I}{\sqrt{2J(0)}}\langle\sigma_{z}^{E}(t)\rangle\right],$ (16)
The last term is analogous to the classical _innovation_ in control theory
Shannon ; Mitter , i.e., the difference between the actually measured current
and the predicted current with the estimated state. The state estimation
process involves setting $\rho_{E}(0)=1/2$, and evolving under the noise
determined by the measurement record from the “experiment”, or in this
theoretical work Eq. (III).
As demonstrated in Ref. [Hill2011, ], even a small error in the Hamiltonian of
the above equation can induce errors in the estimate of the state provided by
the quantum filter equation. We expect the same to be true of the estimates of
the noise spectrum of the QPC. Here our goal is to study the efficiency and
robustness of quantum state estimation via the filter equation, so as in Ralph
et al. [Ralph, ] we choose the same Hamiltonian and QPC properties in both Eq.
(III.1) and Eq. (III). We leave the problem of accurately estimating the
Hamiltonian and the noise properties of the measurement in a dynamic way
Hill2011 for future work.
Figure 4: (Color online) Ensemble-averaged (a) confidence $C$ and (b)
backaction $B$ in terms of the relative entropy between system and estimator
states, as a function of the interaction time $t$ and measurement strength
$\kappa$. Both figures are derived using the ensemble-averaged equation of
motion (IV). (c) and (d) show a comparison between the probability of
occupation of the left dot $P_{L}(t)$ derived from averaging Eq. [III.1] over
many realizations (dashed blue lines) to that given by solving Eq. [IV].
Finally, we combine the time-evolution of these two equations (III) and
(III.1) to calculate the confidence of the estimated state and backaction of
the measurement using both the fidelity and the relative entropy, as described
earlier. In Fig.3, we show numerical results for these quantities. We will
explicitly discuss the epitome in the next section. We set
$\rho_{R}(0)=|L\rangle\langle L|$, $\rho_{E}(0)=1/2$, and evolve using the
standard techniques for $150000$ time steps per Rabi oscillation. The top row
of figures shows the results averaged over $2000$ realizations, while the
bottom row shows just a single realization. In these results we set $\hbar=1$,
$\epsilon=0$ , $\Omega=1$ and $\kappa=0.005\Omega$. Comparing to real
parameters from Gustavsson07 ; Petersson10 , one could choose a strong inter-
dot coupling of the order of $\Omega=32$ $\mu e\mathrm{V}$, giving a timescale
of $130$ ps for the Rabi oscillations we show in Fig. 3. This should be chosen
carefully however, to match the desired properties of the QPC timescales (or
whatever the measurement device happens to be).
When we acquire information from the measurement, it of course induces
significant backaction on the system itself. Figure 3.(a) shows the confidence
and backaction in terms of the fidelity, while Fig. 3.(c) shows the same in
terms of the relative entropy. Both give reasonable descriptions of the
distance between the estimated state and system state, and for weak
measurement strength the confidence saturates before the backaction does.
Obviously then the trade off is to measure on a time scale where both the
confidence is relatively high and the backaction is low.
To gain more insight on what is actually happening during the evolution of the
filter equation, Fig. 3(b) and (e) show the population of the left state of
the dot for both the system and estimator. In the ensemble averaged case Fig.
3(b) backaction from the measurement dephases the state, but the estimator
matches the system state before coherent information is totally lost. In a
single realization, 3(e), the system state does not dephase because the QPC
acts as a perfect detector. Compare this to the case of circuit-QED where the
lifetime of the cavity can induce dephasing on certain timescales Johansson ;
Korotcqed . The off-diagonal elements behave in a similar fashion.
## IV A filter equation for ensemble expectation values
Solving for the ensemble-averaged results by collating many single
realizations is sometimes an arduous numerical task, though can be useful in
the stochastic Schrödinger form if one is modelling a system with a large
Hilbert space. In practise the system state density matrix ensemble averaged
over all measurement trajectories can be trivially calculated by averaging Eq.
(III), and noting $E[dW_{R}]=0$. This gives the expected Lindblad equation of
motion which induces the behavior we observe in the backaction. How about the
estimator state? Averaging Eq. (III.1) is non-trivial as the individual
trajectories determined by $\langle\sigma_{z}^{R}(t)\rangle$ are not
statistically independent of $\rho_{E}(t)$.
Rather than attempt to do so we simply write down an analogy to the quantum
filter equation which depends on ensemble-averaged quantities rather than
individual trajectories. We now define $\rho_{E}=\mathrm{E}[\rho_{E}]$,
$\langle\sigma_{z}^{R}(t)\rangle=\mathrm{E}[\langle\sigma_{z}^{R}(t)\rangle]$,
and
$\langle\sigma_{z}^{E}(t)\rangle=\mathrm{E}[\langle\sigma_{z}^{E}(t)\rangle]$.
Thus the term $\mathrm{E}[\langle\sigma_{z}^{R}(t)\rangle]$ represents the
expectation value extracted from an ensemble averaged version of Eq. (III),
i.e., the evolution of the real system under the effect of dephasing. By
comparison to the stochastic filter equation we consider the following
nonstochastic filter equation,
$\displaystyle\frac{\rm{d}\rho_{E}}{\rm{d}t}$ $\displaystyle=$
$\displaystyle-\frac{i}{\hbar}\left[H_{\rm
DQD},\rho_{E}\right]+\kappa\mathcal{D}[\sigma_{z}]\rho_{E}$ $\displaystyle+$
$\displaystyle\sqrt{2\kappa}\mathcal{H}[\sigma_{z}]\rho_{E}\left[\frac{\Delta
I}{\sqrt{2J(0)}}\left\\{\langle\sigma_{z}^{R}(t)\rangle-\langle\sigma_{z}^{E}(t)\rangle\right\\}\right].$
Solving this equation directly is computationally trivial. To illustrate this
we plot the confidence as a function of time and measurement strength $\kappa$
in Fig. 4. We can easily see that as $\kappa$ increases the confidence
saturates quickly, but with an associated strong backaction, as expected.
Remarkably, if we inspect the density matrix elements of the estimated state
generated by Eq. (IV) to those generated by the ensemble average over
trajectories of Eq. (III.1) they coincide closely. This is illustrated in Fig.
4(c) and (d). Curiously we are unable to rigorously justify this
correspondence, though one can note that Eq. (IV) represents a valid solution
to Eq. (III.1) for the trajectory determined by $dW_{R}=0$. We also point out
that the fictitious nonlinear force in the second line of Eq. (IV) is not
physical, and the equation may not ensure positivity of the density matrix
$\rho_{E}$ at an arbitrary time. Why this works so well in reproducing results
from the ensemble averaged filter equation, at least for this case of a single
qubit measured in the $\sigma_{z}$ basis, is not clear, and represents a
possible avenue of future work.
Figure 5: (Color online) The epitome, $\mathcal{E}=S(\rho_{I}||\rho_{E})$, as
a function of the interaction time $t$ and measurement strength $\kappa$. This
figure is derived using the ensemble-averaged equation of motion (IV). The
superimposed white squares indicates the line of crossing points between the
confidence and backaction of figure 4, and naturally is close to the minimum
in the epitome.
Finally, in Fig. 5., we use this nonstochastic filter equation to plot the
epitome, $\mathcal{E}=S(\rho_{I}||\rho_{E})$. We see that at some intermediate
time, depending on the measurement strength, the epitome has an optimal minima
which coincides closely to the crossing point of the confidence and
backaction. Thus, as one would expect, with continous weak measurements there
is an optimal time at which our estimated state is closest to the original,
unperturbed, ideal state. In practise this optimization can be also discussed
in terms of goal programming (shown in Appendix A) Goal . Further methods to
improve the estimation, or minimize backaction, could include feedback or
other techniques from quantum control ueda .
## V POVM and the discord as a bound on confidence
In quantum theory one can describe any measurement scenario as a positive-
operator valued measure (POVM). For example, weak measurement is sometimes
discussed in terms of a POVM on a combined system/measurement-apparatus, where
the measurement apparatus itself is also considered to be a quantum system. To
gain a more fundamental perspective on the confidence and backaction, as we
have defined them here, we consider an alternative measurement scheme of an
asymptotic POVM.
First, we retain our definition of the initial pure system-apparatus state
$\rho_{I}(0)=\rho_{s}(0)\otimes\rho_{A}(0)$. We then assume that a measurement
apparatus $A$ is allowed to interact with the system to produce the typically
entangled and correlated system-apparatus state $\rho_{s,A}$ (we suppress any
time argument here for complete generality). We define the “real” state of the
system then as this combined state $\rho_{s,A}$. Finally, we perform
measurements on $A$ associated with a POVM
$\\{\Pi_{j}^{A,\dagger}\Pi_{j}^{A}\\}$, where
$\sum_{j}\Pi_{j}^{A,\dagger}\Pi_{j}^{A}=1$. Our estimate of the combined
system-apparatus state given by the measurement is
$\rho_{m}=\sum_{j}\Pi_{j}^{A}\rho_{R}\Pi_{j}^{A,\dagger}$. Again we define the
confidence in terms of the relative entropy, so $C=S(\rho_{s,A}||\rho_{m})$.
The relative-entropy-based confidence has an interesting lower bound in the
case when the POVM becomes a projective valued measure. Writing,
$\displaystyle
C=-\mathrm{Tr}\rho_{s,A}\ln\sum_{j}\Pi_{j}^{A}\rho_{s,A}\Pi_{j}^{A,\dagger}-S(\rho_{s,A})$
(18)
we can substitute $\Pi_{j}^{A}=|j\rangle\langle j|$, where $|j\rangle$ is some
distinguishable orthonormal basis describing the measurement apparatus. Then
the confidence becomes,
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\mathrm{Tr}\sum_{j}|j\rangle\langle
j|\rho_{s,A}\ln\sum_{j}|j\rangle\langle j|\rho_{s,A}|j\rangle\langle
j|-S(\rho_{s,A})$ (19) $\displaystyle=$
$\displaystyle-\mathrm{Tr}\sum_{j}|j\rangle\langle
j|\rho_{s,A}|j\rangle\langle j|\ln\sum_{j}|j\rangle\langle
j|\rho_{s,A}|j\rangle\langle j|$
$\displaystyle-S(\rho_{s,A})=S(\rho_{m})-S(\rho_{s,A})$ $\displaystyle\geq$
$\displaystyle\mathrm{min_{|j\rangle}}[S(\rho_{m})]-S(\rho_{s,A})=\mathcal{D},$
where $\mathcal{D}$ is the quantum discord Modi , when they assume
classicality in terms of only one sub-system. In their work the discord has
the meaning of the distance between a given state and the closest (system-
apparatus) separable state. In other words the lower bound on the confidence
is the distance between the real state and the closest separable state, as one
would expect with projective measurements.
In the case of a general POVM, one could argue that the minimization of $C$
over all possible POVMs is equivalent to a generalization of the definition of
Modi’s discord Modi . Finally, we note that there is a correspondence between
the estimator state $\rho_{E}$ we discussed in terms of the filter equation,
and the partial trace $\mathrm{Tr}_{A}(\rho_{m})$ over the state constructed
from asymptotic POVM measurements (and the same for the real state $\rho_{R}$
and $\mathrm{Tr}_{A}(\rho_{s,A})$ evolved in the measurement noise in the
filter equation example).
## VI Conclusion
In many realistic quantum readout architectures the reliability of the quantum
measurement output is an important issue. In this article we discussed how to
measure the confidence and the backaction of a state reconstructed from
continuous weak quantum measurement. As a typical example, we considered a DQD
measured by a nearby QPC. Based on the theory of open quantum systems and the
quantum filter equation method we briefly discussed the trade-off between
measurement confidence and measurement-induced backaction. We also considered
a possible filter equation for ensemble averaged results. We finished by
discussing the case of a general POVM and how the confidence (when defined as
a relative entropy) has a lower bound related to the quantum discord.
###### Acknowledgements.
We thank Kurt Jacobs for helpful comments. W.C. is supported by the RIKEN FPR
Program. Y.O. is partially supported by the Special Postdoctoral Researchers
Program, RIKEN. F.N. is partially supported by the ARO, JSPS-RFBR Contract No.
12-02-92100, a Grant-in-Aid for Scientific Research (S), MEXT “Kakenhi on
Quantum Cybernetics,” and the JSPS via its FIRST program. J.Q.Y. is partly
supported by the National Basic Research Program of China Grant
No.2009CB929302, NSFC Grant No.91121015, and MOE Grant No.B06011. X.Y.L. is
supported by a JSPS Fellowship.
## Appendix A Quantitative characterization of confidence and backaction via
goal programming
Figure 6: (Color online) Density profile of the optimization function
$\mathcal{O}$ in the goal programming model for several cases: (a)
$\eta_{1}=\eta_{2}=1$ and $\Delta_{C}=\Delta_{B}=0.1$; (b)
$\eta_{1}=\eta_{2}=1$ and $\Delta_{C}=\Delta_{B}=0.2$; (c)
$\eta_{1}=1,\eta_{2}=0.5$ and $\Delta_{C}=\Delta_{B}=0.1$; (d)
$\eta_{1}=0.5,\eta_{2}=1$ and $\Delta_{C}=\Delta_{B}=0.1$.
The results in Section IV show a clear trade-off relation between the
confidence and the backaction in the parameter space spanned by the
interaction time $t$ and the measurement strength $k$. Let us examine this
relation via a sophisticated method, goal programming Goal . We formulate our
problem setting more specifically; we determine $t$ and $k$ such that
$\mathcal{C}\leq\Delta_{C}$ and $\mathcal{B}\leq\Delta_{B}$ for given (small)
positive constants $\Delta_{C}$ and $\Delta_{B}$. The two parameters
$\Delta_{C}$ and $\Delta_{B}$ are regarded as, respectively, admissible
confidence error and permissible backaction. Thus, we can obtain a good
measurement scenario to increase the confidence (i.e., minimize $\mathcal{C}$)
while reducing the backaction (i.e., minimizing $\mathcal{B}$). Hereafter, the
confidence and the backaction are defined via the relative entropy, as seen in
Eq. (5).
Goal programming Goal provides an optimization method to deal with two (or
more than two) conflicting objectives and it is widely used in mathematics,
information theory and engineering. Instead of finding solutions which
absolutely minimize or maximize objective functions, the task of goal
programming is to find solutions that, if possible, satisfy a set of goals, or
otherwise violate the goals minimally. This makes the approach more appealing
to practical designers, compared to other optimization methods (e.g., linear
programming models). Let us describe our measurement problem in terms of goal
programming:
$\displaystyle{\rm
Minimize}\quad\mathcal{O}\equiv\eta_{1}\delta_{1}^{+}+\eta_{2}\delta_{2}^{+},$
$\displaystyle{\rm
subject\,\,to}\,\left\\{\begin{array}[]{l}\mathcal{C}(\vec{x})-\delta_{1}^{+}+\delta_{1}^{-}=\Delta_{C}\\\
\mathcal{B}(\vec{x})-\delta_{2}^{+}+\delta_{2}^{-}=\Delta_{B}\\\
\delta_{1}^{\pm},\delta_{2}^{\pm}\geq 0\\\
\vec{x}=(t\Omega/2\pi,k\Omega)\in\mathcal{M}\end{array}\right..$ (24)
The weight factors $\eta_{1}$ and $\eta_{2}$ are given positive number, and
represent the relative priority of objectives. If $\eta_{1}>\eta_{2}$, the
condition for the confidence is prior to the one for the backaction, and vice
versa. The condition for the confidence ($\mathcal{C}\leq\Delta_{C}$) is
reformulated by the relation
$\mathcal{C}+\delta_{1}^{+}-\delta_{1}^{-}=\Delta_{C}$, with the deviations
between the admissible error and the actual value, $\delta_{1}^{+}$ and
$\delta_{1}^{-}$. When $\mathcal{C}>\Delta_{C}$ ($\mathcal{C}\leq\Delta_{C}$),
we set $\delta_{1}^{+}=\mathcal{C}-\Delta_{C}$ and $\delta_{1}^{-}=0$
($\delta_{1}^{+}=0$ and $\delta_{1}^{-}=\Delta_{C}-\mathcal{C}$). Similarly,
we can set $\delta_{2}^{\pm}$ via
$\mathcal{B}-\delta_{2}^{+}+\delta_{2}^{-}=\Delta_{B}$. The set
$\mathcal{M}=\\{t\Omega/2\pi,k\Omega\\}$ is the family of the measurement
scenarios. The smaller $\mathcal{O}$, the better performance of the
measurement scenario. The minimum value of $\mathcal{O}$ ($\mathcal{O}=0$)
corresponds to the best solution.
Figure 6 shows the contour profile of $\mathcal{O}$ for various cases, based
on the ensemble-averaged confidence $\mathcal{C}$ and backaction
$\mathcal{B}$. The optimization function $\mathcal{O}$ is a function of the
measurement scenarios: the interaction time $t\Omega/2\pi$ and measurement
strength $k\Omega$. The other parameters are the same as in Figs. 4(a) and
(b). In both Figs. 6(a) and (b), the confidence and the backaction have equal
importance ($\eta_{1}=\eta_{2}=1$). In Fig.6(a) we examine the case that
$\Delta_{C}=0.1$ and $\Delta_{B}=0.1$. We find that the best solution
($\mathcal{O}=0$) appears in the dark blue area. If we relax the restriction
to $\Delta_{C}=\Delta_{B}=0.2$, we find that more solutions for the
optimization $\mathcal{O}=0$, as seen in Fig. 6(b). We also examine the cases
when the confidence has a different importance, or weight, than the
backaction, as seen in Figs. 6(c) and (d). The solution for the case where the
measurement confidence is more important than the backaction
($\eta_{1}=1,\eta_{2}=0.5$) is given in Fig. 6(c). The solution for the
opposite situation ($\eta_{1}=0.5,\eta_{2}=1$) is given in Fig. 6(d). In the
above cases, we have considered a double-criterion goal programming problem
and we find that it is convenient for discussing the trade-off relation
between measurement confidence and measurement induced backaction.
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|
arxiv-papers
| 2012-07-17T10:52:23 |
2024-09-04T02:49:33.221466
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Wei Cui, Neill Lambert, Yukihiro Ota, Xin-You L\\\"u, Z. -L. Xiang, J.\n Q. You, Franco Nori",
"submitter": "Wei Cui",
"url": "https://arxiv.org/abs/1207.3942"
}
|
1207.4030
|
# A prediction of $D^{*}$-multi-$\rho$ states
C. W. Xiao 1, M. Bayar 1, 2 and E. Oset 1 1Departamento de Física Teórica and
IFIC, Centro Mixto Universidad
de Valencia-CSIC, Institutos de Investigación de Paterna, Apartado 22085,
46071 Valencia, Spain
2 Department of Physics, Kocaeli University, 41380 Izmit, Turkey
###### Abstract
We present a study of the many-body interaction between a $D^{*}$ and
multi-$\rho$. We use an extrapolation to SU(4) of the hidden gauge formalism,
which produced dynamically the resonances $f_{2}(1270)$ in the $\rho\rho$
interaction and $D^{*}_{2}(2460)$ in the $\rho D^{*}$ interaction. Then let a
third particle, $\rho$, $D^{*}$, or a resonance collide with them, evaluating
the scattering amplitudes in terms of the Fixed Center Approximation of the
Faddeev equations. We find several clear resonant structures above
$2800\textrm{ MeV}$ in the multibody scattering amplitudes. They would
correspond to new charmed resonances, $D^{*}_{3}$, $D^{*}_{4}$, $D^{*}_{5}$
and $D^{*}_{6}$, which are not yet listed in the PDG, which would be analogous
to the $\rho_{3}(1690)$, $f_{4}(2050)$, $\rho_{5}(2350)$, $f_{6}(2510)$ and
$K^{*}_{3}(1780)$, $K^{*}_{4}(2045)$, $K^{*}_{5}(2380)$ described before as
multi-$\rho$ and $K^{*}$-multi-$\rho$ states respectively.
## I Introduction
One of the important aims in the study of the strong interaction is to
understand the nature and structure of hadronic resonances. The search for new
resonances is another goal both in theories and experiments. At low energy,
using the input of chiral Lagrangians Gasser:1984gg ; Meissner:1993ah ;
Pich:1995bw ; Ecker:1994gg ; Bernard:1995dp and implementing unitarity in
coupled channels, one develops a theoretical tool, chiral unitarity theory,
which explains the two-body interaction very successfully Kaiser:1995eg ;
Oller:1997ti ; Oset:1997it ; Oller:1998hw ; Oller:1998zr ; Oller:2000fj ;
Jido:2003cb ; Guo:2006fu ; Guo2006wp ; GarciaRecio:2002td ; GarciaRecio:2005hy
; Hyodo:2002pk . For the three-body interaction, the pioneer work of Ref.
alberone combined Faddeev equations and chiral dynamics and reported several
S-wave $J^{P}=\frac{1}{2}^{+}$ resonances which qualify as two mesons-one
baryon molecular states. One more step, using the Fixed Center Approximation
Faddeev:1960su ; Barrett:1999cw ; Deloff:1999gc ; Kamalov:2000iy to Faddeev
equations for multi-$\rho(770)$ states, was given in Ref. Roca2010 in which
the resonances $f_{2}(1270)$, $\rho_{3}(1690)(3^{--})$, $f_{4}(2050)(4^{++})$,
$\rho_{5}(2350)(5^{--})$, and $f_{6}(2510)(6^{++})$ were explained as
basically molecules of an increasing number of $\rho(770)$ particles with
parallel spins. Similarly, it was also found in Ref. Yamagata2010 that the
resonances $K^{*}_{2}(1430)$, $K^{*}_{3}(1780)$, $K^{*}_{4}(2045)$,
$K^{*}_{5}(2380)$ and a new $K^{*}_{6}$ could be understood as molecules made
of an increasing number of $\rho(770)$ and one $K^{*}(892)$ meson. The
$\Delta_{\frac{5}{2}^{+}}(2000)$ puzzle was explained in Ref. Xie:2011uw with
a $\pi-(\Delta\rho)$ system with the same method Xie:2010ig ; Bayar:2011qj ;
Xiao:2011rc ; Bayar:2012rk .
The Fixed Center Approximation to Faddeev equations is technically very simple
and fairly accurate when dealing with bound states, as discussed in Ref.
Bayar:2011qj . It should also be avoided when dealing with states which have
enough energy to excite its components in intermediate states
MartinezTorres:2010ax . It should be noted that the results of the Fixed
Centre Approximation to Faddeev equations have been confirmed by the
variational method approach with the effective one-channel Hamiltonian in a
more recent work Bayar:2012dd which predicts a narrow $DNN$ quasi-bound state
in the energy range of about $3500\textrm{ MeV}$.
In our present work, the main ideas of Refs. Roca2010 ; Yamagata2010 are
followed to search $D^{*}$-multi-$\rho$ resonances in the charm sector. Using
effective Lagrangians of the local hidden gauge theory hidden1 ; hidden2 ;
hidden3 ; hidden4 , the $\rho\rho$ interaction was studied in Ref.
Molina:2008jw with on-shell factorized Bethe-Salpeter equations. It was found
that the $\rho\rho$ interaction was attractive in the isospin zero, spin 0 and
2 channels, particularly in the tensor channel where it led to the formation
of a $\rho\rho$ quasibound state or molecule that could be associated to the
$f_{2}(1270)$ ( $I(J^{PC})=0(2^{++})$ ). With the same formalism, the
composite systems of light ($\rho$ and $\omega$) and heavy ($D^{*}$) vector
mesons were studied in Ref. Molina:2009eb . In that work, a strong attraction
was found in the isospin, spin channels $(I,S)=(1/2,0),(1/2,1)$ and $(1/2,2)$,
with positive parity, leading to bound $\rho(\omega)D^{*}$ states, one of them
identified as the $D^{*}_{2}(2460)\;(\;I(J^{P})=\frac{1}{2}(1^{-})\;)$.
Therefore, the resonance $D^{*}_{2}(2460)$ was generated as a $\rho D^{*}$
quasibound state or molecule by the strong and attractive $\rho D^{*}$
interaction. As discussed in Ref. Roca2010 , because of the large binding
energy per $\rho$ meson in spin 2, it is possible to obtain bound systems with
several $\rho$ mesons as building blocks. As mentioned in Ref. Molina:2009eb ,
the $\rho D^{*}$ interaction is also very strong and can bind the system. Thus
the main aim in the present work is, first, to study the three-body
interaction of $D^{*}$ and two $\rho$ mesons, for which we have two options,
the clusters $D^{*}-f_{2}(1270)(\rho\rho)$ and $\rho-D^{*}_{2}(2460)(\rho
D^{*})$, in order to see if there are some resonance structures in the
scattering amplitudes. If this is the case, one can predict a not-yet-
discovered $D^{*}_{3}$ resonance, and we could continue our study extending
these ideas to include more $\rho$ mesons as building blocks of the many-body
system. Then we repeat the test in the four-body system and so forth. In
analogy to the $K^{*}$-multi-$\rho$ systems Yamagata2010 , we find clear peaks
in the scattering amplitudes for system with increasing number of $\rho$
mesons. Thus we predict new resonances in our work which are basically
quasibound states or molecules made of an increasing number of $\rho(770)$
mesons and one $D^{*}$ meson which are not yet reported in the list of the PDG
pdg2012 .
## II $\rho\rho$ and $\rho D^{*}$ two-body interactions
To evaluate the Faddeev equations under the Fixed Centre Approximation, we
need to define the two-body cluster and then let the third particle collide
with the cluster. Thus the starting point of our work is the two-body
$\rho\rho$ and $\rho D^{*}$ interactions, which were studied in Refs.
Molina:2008jw and Molina:2009eb with the local hidden gauge formalism
hidden1 ; hidden2 ; hidden3 ; hidden4 and the unitary coupled channels method
Kaiser:1995eg ; Oller:1997ti ; Oset:1997it ; Oller:1998hw ; Oller:1998zr ;
Oller:2000fj ; Jido:2003cb ; Guo:2006fu ; GarciaRecio:2002td ;
GarciaRecio:2005hy ; Hyodo:2002pk . We briefly summarize the model of Refs.
Molina:2008jw and Molina:2009eb here to explain how to obtain the unitarized
$\rho\rho$ and $\rho D^{*}$ scattering amplitudes.
To evaluate the $\rho\rho$ and $\rho D^{*}$ scattering amplitudes with the
coupled channels unitary approach, the Bethe-Salpeter equation in coupled
channels is used:
$t=[1-VG]^{-1}V,$ (1)
where the kernel $V$ is a matrix of the interaction potentials in each
channel, which is calculated from the hidden gauge Lagrangian. $G$ is a
diagonal matrix of the loop function of every channel. The details can be seen
in Refs. Molina:2008jw ; Molina:2009eb .
To construct the three-body system we start from the clusters $f_{2}(1270)$ (
$I(J^{PC})=0(2^{++})$ ) and
$D^{*}_{2}(2460)\;(\;I(J^{P})=\frac{1}{2}(1^{-})\;)$ and add to them a $D^{*}$
or a $\rho$ respectively. The new particles are introduced with their spins
aligned with that of the cluster such that the total spin adds one unity. Thus
we only need to take into account the potential of spin $S=2$ for $\rho\rho$
and $\rho D^{*}$ interactions,
$\displaystyle V^{(I=0,S=2)}_{\rho\rho}(s)$ $\displaystyle=$
$\displaystyle-4g^{2}-8g^{2}\Big{(}\frac{3s}{4m_{\rho}^{2}}-1\Big{)},$ (2)
$\displaystyle V^{(I=2,S=2)}_{\rho\rho}(s)$ $\displaystyle=$ $\displaystyle
2g^{2}+4g^{2}\Big{(}\frac{3s}{4m_{\rho}^{2}}-1\Big{)},$ (3) $\displaystyle
V^{(I=1/2,S=2)}_{\rho D^{*}(11)}(s)$ $\displaystyle=$
$\displaystyle-\frac{5}{2}g^{2}-2\frac{g^{2}}{m_{\rho}^{2}}(k_{1}+k_{3})\cdot(k_{2}+k_{4})-\frac{1}{2}\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (4) $\displaystyle
V^{(I=1/2,S=2)}_{\rho D^{*}(12)}(s)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}g^{2}+\frac{\sqrt{3}}{2}\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (5) $\displaystyle
V^{(I=1/2,S=2)}_{\rho D^{*}(22)}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{2}+\frac{1}{2}\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (6) $\displaystyle
V^{(I=3/2,S=2)}_{\rho D^{*}}(s)$ $\displaystyle=$ $\displaystyle
2g^{2}+\frac{g^{2}}{m_{\rho}^{2}}(k_{1}+k_{3})\cdot(k_{2}+k_{4})+\frac{\kappa
g^{2}}{m_{\rho}^{2}}(k_{1}+k_{4})\cdot(k_{2}+k_{3}),$ (7)
where $g=M_{V}/2f_{\pi}$ with $M_{V}$ the vector meson mass and $f_{\pi}$ the
pion decay constant. In these equations $k_{i},\;i=1,2,3,4$ are the initial,
$(1,2)$, and final $(3,4)$ momenta of the particles. The quantity
$\kappa=m_{\rho}^{2}/m^{2}_{D^{*}}$ appears because in some transitions one is
exchanging a heavy vector instead of a light one. Note that in isospin $I=1/2$
there are two coupled channels, 1 is $\rho D^{*}$ and 2 is $\omega D^{*}$.
As mentioned in Refs. Molina:2008jw and Molina:2009eb , we also should take
into account the contribution of the box diagram with two pseudoscalar mesons
in the intermediate state. We only add the imaginary part of the box diagram
contribution to the potential $V$, which is not accounted for by the coupled
channels Wu:2010jy ; Wu:2010vk , and neglect the real part which is very
small. We finally note that we take into account the $\rho$ mass distribution
by replacing the $G$ function by its convoluted form by the mass distribution
of the $\rho$ mesons in the loop.
Finally, considering the box diagram contribution to the potential $V$ and the
convolution of the $\rho$ mass distribution in the loop function $G$, we show
the evaluated results of $\rho\rho$ and $\rho D^{*}$ scattering amplitudes in
Fig. 1, which are consistent with Refs. Molina:2008jw and Molina:2009eb . The
structure of the resonances $f_{2}(1270)$ and $D^{*}_{2}(2460)$ are clear in
the peak of the modulus squared of the amplitudes. The nonresonant amplitudes
$t_{\rho\rho}^{(I=0,S=2)}$ and $t_{\rho D^{*}}^{(I=3/2,S=2)}$ are not shown
here.
Figure 1: Modulus squared of the scattering amplitudes. Left:
$|t_{\rho\rho}^{I=0}|^{2},\;f_{2}(1270)$; Right: $|t_{\rho
D^{*}}^{I=1/2}|^{2},\;D^{*}_{2}(2460)$.
## III Multi-body interaction formalism
The Faddeev equations under the Fixed Centre Approximation are an effective
tool to deal with multi-hadron interaction Roca2010 ; Yamagata2010 ;
Xie:2011uw ; Xie:2010ig ; Bayar:2011qj ; Xiao:2011rc ; Bayar:2012rk . They are
particularly suited to study systems in which a pair of particles cluster
together and the cluster is not much modified by the third particle. The Fixed
Centre Approximation to Faddeev equations assumes a pair of particles (1 and
2) forming a cluster. Then particle 3 interacts with the components of the
cluster, undergoing all possible multiple scattering with those components.
This is depicted in Fig. 2.
Figure 2: Diagrammatic representation of the FCA to Faddeev equations.
With this basic idea of the Fixed Centre Approximation, we could write the
Faddeev equations easily. For this one defines two partition functions
$T_{1}$, $T_{2}$ which sum all diagrams of the series of Fig. 2 which begin
with the interaction of particle 3 with particle 1 of the cluster ($T_{1}$),
or with the particle 2 ($T_{2}$). The equations then read
$\displaystyle T_{1}$ $\displaystyle=t_{1}+t_{1}G_{0}T_{2},$ (8)
$\displaystyle T_{2}$ $\displaystyle=t_{2}+t_{2}G_{0}T_{1},$ (9)
$\displaystyle T$ $\displaystyle=T_{1}+T_{2},$ (10)
where $T$ is the total three-body scattering amplitude that we are looking
for. The amplitudes $t_{1}$ and $t_{2}$ represent the unitary scattering
amplitudes with coupled channels for the interactions of particle 3 with
particle 1 and 2, respectively. Besides, $G_{0}$ is the propagator of particle
3 between the components of the two-body system.
For the unitary amplitudes corresponding to single-scattering contribution,
one must take into account the isospin structure of the cluster and write the
$t_{1}$ and $t_{2}$ amplitudes in terms of the isospin amplitudes of the (3,1)
and (3,2) systems. Details will be discussed in next section. Besides, because
of the normalization of Mandl and Shaw mandl which has different weight
factors for the particle fields, we must take into account how these factors
appear in the single scattering and double scattering and in the total
amplitude. This is easy and is done in detail in Yamagata2010 ; Xie:2010ig .
We show below the details for the present case of a meson cluster (also
particle 1 and 2) and a meson as scattering particle (the third particle). In
this case, following the field normalization of mandl we find for the $S$
matrix of single scattering,
$\displaystyle\begin{split}S^{(1)}_{1}=&-it_{1}\frac{1}{{\cal
V}^{2}}\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{1}}}\frac{1}{\sqrt{2\omega^{\prime}_{1}}}\\\
&\times(2\pi)^{4}\,\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R}),\\\ \end{split}$
(11) $\displaystyle\begin{split}S^{(1)}_{2}=&-it_{2}\frac{1}{{\cal
V}^{2}}\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{2}}}\frac{1}{\sqrt{2\omega^{\prime}_{2}}}\\\
&\times(2\pi)^{4}\,\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R}),\\\ \end{split}$
(12)
where, $k,\,k^{\prime}$ ($k_{R},\,k^{\prime}_{R}$) refer to the momentum of
initial, final scattering particle ($R$ for the cluster), $\cal V$ is the
volume of the box where the states are normalized to unity and the subscripts
1, 2 refer to scattering with particle 1 or 2 of the cluster.
The double scattering diagram, Fig. 2 (b), is given by
$\begin{split}S^{(2)}=&-i(2\pi)^{4}\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R})\frac{1}{{\cal
V}^{2}}\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{1}}}\frac{1}{\sqrt{2\omega^{\prime}_{1}}}\frac{1}{\sqrt{2\omega_{2}}}\frac{1}{\sqrt{2\omega^{\prime}_{2}}}\\\
&\times\int\frac{d^{3}q}{(2\pi)^{3}}F_{R}(q)\frac{1}{{q^{0}}^{2}-\vec{q}\,^{2}-m_{3}^{2}+i\,\epsilon}t_{1}t_{2},\end{split}$
(13)
where $F_{R}(q)$ is the cluster form factor that we shall discuss below.
Similarly the full $S$ matrix for scattering of particle 3 with the cluster
will be given by
$\begin{split}S=&-i\,T\,(2\pi)^{4}\delta(k+k_{R}-k^{\prime}-k^{\prime}_{R})\frac{1}{{\cal
V}^{2}}\\\
&\times\frac{1}{\sqrt{2\omega_{3}}}\frac{1}{\sqrt{2\omega^{\prime}_{3}}}\frac{1}{\sqrt{2\omega_{R}}}\frac{1}{\sqrt{2\omega^{\prime}_{R}}}.\end{split}$
(14)
In view of the different normalization of these terms by comparing Eqs. (11),
(12), (13) and (14), we can introduce suitable factors in the elementary
amplitudes,
$\tilde{t_{1}}=\frac{2m_{R}}{2m_{1}}~{}t_{1},~{}~{}~{}~{}\tilde{t_{2}}=\frac{2m_{R}}{2m_{2}}~{}t_{2},$
(15)
where we have taken the approximations, suitable for bound states,
$\frac{1}{\sqrt{2\omega_{i}}}=\frac{1}{\sqrt{2m_{i}}}$, and sum all the
diagrams by means of
$T=T_{1}+T_{2}=\frac{\tilde{t_{1}}+\tilde{t_{2}}+2~{}\tilde{t_{1}}~{}\tilde{t_{2}}~{}G_{0}}{1-\tilde{t_{1}}~{}\tilde{t_{2}}~{}G_{0}^{2}}.$
(16)
The function $G_{0}$ in Eq. (16) is given by
$G_{0}(s)=\int\frac{d^{3}\vec{q}}{(2\pi)^{3}}F_{R}(q)\frac{1}{q^{02}-\vec{q}^{~{}2}-m_{3}^{2}+i\,\epsilon}.$
(17)
where $F_{R}(q)$ is the form factor of the cluster of particles 1 and 2. We
must use the form factor of the cluster consistently with the theory used to
generate the cluster as a dynamically generated resonance. This requires to
extend into wave functions the formalism of the chiral unitary approach
developed for scattering amplitudes. This work has been done in gamerjuan ;
yamajuan ; Aceti:2012dd for $s$-wave bound states, $s$-wave resonant states
and states with arbitrary angular momentum respectively, here we are only need
the expressions for $s$-wave bound states, and then the expression for the
form factors is given in section 4 of yamajuan , which we use in the present
work and reproduce below
$\displaystyle\begin{split}F_{R}(q)&=\frac{1}{\mathcal{N}}\int_{|\vec{p}|<\Lambda^{\prime},|\vec{p}-\vec{q}|<\Lambda^{\prime}}d^{3}\vec{p}\frac{1}{2E_{1}(\vec{p})}\frac{1}{2E_{2}(\vec{p})}\frac{1}{M_{R}-E_{1}(\vec{p})-E_{2}(\vec{p})}\\\
&\quad\frac{1}{2E_{1}(\vec{p}-\vec{q})}\frac{1}{2E_{2}(\vec{p}-\vec{q})}\frac{1}{M_{R}-E_{1}(\vec{p}-\vec{q})-E_{2}(\vec{p}-\vec{q})},\end{split}$
(18) $\displaystyle\mathcal{N}$
$\displaystyle=\int_{|\vec{p}|<\Lambda^{\prime}}d^{3}\vec{p}\Big{(}\frac{1}{2E_{1}(\vec{p})}\frac{1}{2E_{2}(\vec{p})}\frac{1}{M_{R}-E_{1}(\vec{p})-E_{2}(\vec{p})}\Big{)}^{2},$
(19)
where $E_{1}$ and $E_{2}$ are the energies of the particles 1, 2 and $M_{R}$
the mass of the cluster. The parameter $\Lambda^{\prime}$ is a cut off that
regularizes the integral of Eqs. (18) and (19). This cut off is the same one
needed in the regularization of the loop function of the two particle
propagators in the study of the interaction of the two particles of the
cluster yamajuan . We take in the present work $\Lambda^{\prime}=875\textrm{
MeV}$, the same as used to generate the bound states Yamagata2010 ; Xie:2010ig
of $f_{2}(1270)$ in the two-body interaction, and
$\Lambda^{\prime}=1200\textrm{ MeV}$ for getting the $D^{*}_{2}(2460)$
cluster. Thus we do not introduce any free parameters in the present
procedure.
In addition, $q^{0}$, the energy carried by particle 3 in the rest frame of
the three particle system, is given by
$q^{0}(s)=\frac{s+m_{3}^{2}-M_{R}^{2}}{2\sqrt{s}}.$ (20)
Note also that the arguments of the amplitudes $T_{i}(s)$ and $t_{i}(s_{i})$
are different, where $s$ is the total invariant mass of the three-body system,
and $s_{i}$ are the invariant masses in the two-body systems. The value of
$s_{i}$ is given by Yamagata2010
$s_{i}=m_{3}^{2}+m_{i}^{2}+\frac{(M_{R}^{2}+m_{i}^{2}-m_{j}^{2})(s-m_{3}^{2}-M_{R}^{2})}{2M_{R}^{2}},(i,j=1,2,\;i\neq
j)$ (21)
where $m_{l},(l=1,2,3)$ are the masses of the corresponding particles in the
three-body system and $M_{R}$ the mass of two body resonance or bound state
(cluster).
## IV Results
The $D^{*}$-multi-$\rho$ interactions that we investigate in the present work
are listed in Table 1, and are explained as follows. For the three-body
interaction, we have two options: particle $3=D^{*}$, cluster or resonance
$R=f_{2}$ (particle $1=\rho,\;2=\rho$) and $3=\rho$, $R=D^{*}_{2}$
($1=\rho,\;2=D^{*}$). For four-body, we also have two cases: $3=f_{2}$,
$R=D^{*}_{2}$ ($1=\rho,\;2=D^{*}$) and $3=D^{*}_{2}$, $R=f_{2}$
($1=\rho,\;2=\rho$). For five-body, $3=D^{*}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$)
and $3=\rho$, $R=D^{*}_{4}$ ($1=f_{2},\;2=D^{*}_{2}$). For six-body,
$3=D^{*}_{2}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$) and $3=f_{2}$, $R=D^{*}_{4}$
($1=f_{2},\;2=D^{*}_{2}$). We describe all these cases in detail below.
Table 1: The cases considered in the $D^{*}$-multi-$\rho$ interactions. particles: | 3 | R (1,2) | amplitudes
---|---|---|---
Two-body | $\rho$ | $D^{*}$ | $t_{\rho D^{*}}$
| $\rho$ | $\rho$ | $t_{\rho\rho}$
Three-body | $D^{*}$ | $f_{2}\;(\rho\rho)$ | $T_{D^{*}-f_{2}}$
| $\rho$ | $D^{*}_{2}\;(\rho D^{*})$ | $T_{\rho-D^{*}_{2}}$
Four-body | $D^{*}_{2}$ | $f_{2}\;(\rho\rho)$ | $T_{D^{*}_{2}-f_{2}}$
| $f_{2}$ | $D^{*}_{2}\;(\rho D^{*})$ | $T_{f_{2}-D^{*}_{2}}$
Five-body | $D^{*}$ | $f_{4}\;(f_{2}f_{2})$ | $T_{D^{*}-f_{4}}$
| $\rho$ | $D^{*}_{4}\;(f_{2}D^{*}_{2})$ | $T_{\rho-D^{*}_{4}}$
Six-body | $D^{*}_{2}$ | $f_{4}\;(f_{2}f_{2})$ | $T_{D^{*}_{2}-f_{4}}$
| $f_{2}$ | $D^{*}_{4}\;(f_{2}D^{*}_{2})$ | $T_{f_{2}-D^{*}_{4}}$
### IV.1 Three-body interaction
For three-body interaction, we have two options of structure:
$D^{*}-f_{2}(\rho\rho)$ and $\rho-D^{*}_{2}(\rho D^{*})$, which means
$3=D^{*}$, $R=f_{2}$ ($1=\rho,\;2=\rho$) and $3=\rho$, $R=D^{*}_{2}$
($1=\rho,\;2=D^{*}$). Thus, to evaluate these scattering amplitudes, we need
as input the $t_{1}$ and $t_{2}$ amplitudes of the (3,1) and (3,2) systems,
$t_{1}=t_{2}=t_{\rho D^{*}}$ for $D^{*}-f_{2}(\rho\rho)$ and
$t_{1}=t_{\rho\rho},\;t_{2}=t_{\rho D^{*}}$ for $\rho-D^{*}_{2}(\rho D^{*})$.
We should calculate the two-body $\rho\rho$ and $\rho D^{*}$ amplitudes.
As mentioned before, the isospin structure of the cluster should be considered
for the $t_{1}$ and $t_{2}$ amplitudes. For the case of
$D^{*}-f_{2}(\rho\rho)$, the cluster of $f_{2}$ has isospin $I=0$. Therefore
the two $\rho$ mesons are in an $I=0$ state, and we have
$|\rho\rho>^{(0,0)}=\frac{1}{\sqrt{3}}\Big{(}|(1,-1)>+|(-1,1)>-|(0,0)>\Big{)},$
(22)
where $|(1,-1)>$ denote $|(I_{z}^{1},I_{z}^{2})>$ which shows the $I_{z}$
components of particles 1 and 2, and $|\rho\rho>^{(0,0)}$ means
$|\rho\rho>^{(I,I_{z})}$. The third particle is a $D^{*}$ meson taken
$|I_{z}^{3})>=|\frac{1}{2}>$. Then we obtain
$\begin{split}T^{(\frac{1}{2},\frac{1}{2})}_{D^{*}-f_{2}}=&<D^{*}\rho\rho|\,\hat{t}\,|D^{*}\rho\rho>^{(\frac{1}{2},\frac{1}{2})}\\\
=&(<D^{*}|^{(\frac{1}{2},\frac{1}{2})}\otimes<\rho\rho|^{(0,0)})\,(\hat{t}_{31}+\hat{t}_{32})\,(|D^{*}>^{(\frac{1}{2},\frac{1}{2})}\otimes|\rho\rho>^{(0,0)})\\\
=&\Big{[}<\frac{1}{2}|\otimes\frac{1}{\sqrt{3}}\Big{(}<(1,-1)|+<(-1,1)|-<(0,0)|\Big{)}\Big{]}\,(\hat{t}_{31}+\hat{t}_{32})\,\Big{[}|\frac{1}{2}>\\\
&\otimes\frac{1}{\sqrt{3}}\Big{(}|(1,-1)>+|(-1,1)>-|(0,0)>\Big{)}\Big{]}\\\
=&\frac{1}{3}\Big{[}<(\frac{3}{2},\frac{3}{2}),-1|+\sqrt{\frac{1}{3}}<(\frac{3}{2},-\frac{1}{2}),1|+\sqrt{\frac{2}{3}}<(\frac{1}{2},-\frac{1}{2}),1|-\sqrt{\frac{2}{3}}<(\frac{3}{2},\frac{1}{2}),0|\\\
&-\sqrt{\frac{1}{3}}<(\frac{1}{2},\frac{1}{2}),0|\Big{]}\hat{t}_{31}\Big{[}|(\frac{3}{2},\frac{3}{2}),-1>+\sqrt{\frac{1}{3}}|(\frac{3}{2},-\frac{1}{2}),1>+\sqrt{\frac{2}{3}}|(\frac{1}{2},-\frac{1}{2}),1>\\\
&-\sqrt{\frac{2}{3}}|(\frac{3}{2},\frac{1}{2}),0>-\sqrt{\frac{1}{3}}|(\frac{1}{2},\frac{1}{2}),0>\Big{]}+\frac{1}{3}\Big{[}\sqrt{\frac{1}{3}}<(\frac{3}{2},-\frac{1}{2}),1|+\sqrt{\frac{2}{3}}<(\frac{1}{2},-\frac{1}{2}),1|\\\
&+<(\frac{3}{2},\frac{3}{2}),-1|-\sqrt{\frac{2}{3}}<(\frac{3}{2},\frac{1}{2}),0|-\sqrt{\frac{1}{3}}<(\frac{1}{2},\frac{1}{2}),0|\Big{]}\hat{t}_{32}\Big{[}\sqrt{\frac{1}{3}}|(\frac{3}{2},-\frac{1}{2}),1>\\\
&+\sqrt{\frac{2}{3}}|(\frac{1}{2},-\frac{1}{2}),1>+|(\frac{3}{2},\frac{3}{2}),-1>-\sqrt{\frac{2}{3}}|(\frac{3}{2},\frac{1}{2}),0>-\sqrt{\frac{1}{3}}|(\frac{1}{2},\frac{1}{2}),0>\Big{]},\end{split}$
(23)
where the notation of the states followed in the terms is
$|(\frac{3}{2},\frac{3}{2}),-1>\equiv|(I^{31},I_{z}^{31}),I_{z}^{2}>$ for
$t_{31}$, and $|(I^{32},I_{z}^{32}),I_{z}^{1}>$ for $t_{32}$. Then we find
$t_{1}=t_{\rho
D^{*}}=\frac{1}{3}\big{(}2t_{31}^{I=3/2}+t_{31}^{I=1/2}\big{)},\quad
t_{2}=t_{1}.$ (24)
But for the case of $\rho-D^{*}_{2}(\rho D^{*})$, the situation is different.
Because the isospins of $\rho$ and $D^{*}_{2}$ are $I_{\rho}=1$ and
$I_{D^{*}_{2}}=\frac{1}{2}$, the total isospin of the three-body system are
$I_{total}\equiv I_{\rho\rho D^{*}}=\frac{1}{2}$ or $I_{total}\equiv
I_{\rho\rho D^{*}}=\frac{3}{2}$, and then we have
$\begin{split}&|\rho D^{*}_{2}>^{(\frac{1}{2},\frac{1}{2})}=|\rho\rho
D^{*}>^{(\frac{1}{2},\frac{1}{2})}=\sqrt{\frac{2}{3}}|(1,-\frac{1}{2})>-\sqrt{\frac{1}{3}}|(0,\frac{1}{2})>,\\\
&|\rho D^{*}_{2}>^{(\frac{3}{2},\frac{1}{2})}=|\rho\rho
D^{*}>^{(\frac{3}{2},\frac{1}{2})}=\sqrt{\frac{1}{3}}|(1,-\frac{1}{2})>+\sqrt{\frac{2}{3}}|(0,\frac{1}{2})>,\end{split}$
(25)
where we have taken $I_{z}=\frac{1}{2}$ for convenience. Therefore the $|\rho
D^{*}>$ states inside the $D^{*}_{2}$ for the $I_{z}=-\frac{1}{2}$ and
$I_{z}=+\frac{1}{2}$ are given by
$\begin{split}&|\rho
D^{*}>^{(\frac{1}{2},-\frac{1}{2})}=\sqrt{\frac{1}{3}}|(0,-\frac{1}{2})>-\sqrt{\frac{2}{3}}|(-1,\frac{1}{2})>,\\\
&|\rho
D^{*}>^{(\frac{1}{2},\frac{1}{2})}=\sqrt{\frac{2}{3}}|(1,-\frac{1}{2})>-\sqrt{\frac{1}{3}}|(0,\frac{1}{2})>.\end{split}$
(26)
For the two possibilities, combining Eqs. (25) and (26) and performing a
similar derivation of Eq. (23), we obtain
$\begin{split}&T_{\rho-D^{*}_{2}}^{(I=1/2)}:\quad
t_{1}=t_{\rho\rho}=\frac{2}{3}t_{31}^{(I=0)},\quad t_{2}=t_{\rho
D^{*}}=\frac{1}{9}\big{(}8t_{32}^{I=3/2}+t_{32}^{I=1/2}\big{)};\\\
&T_{\rho-D^{*}_{2}}^{(I=3/2)}:\quad
t_{1}=t_{\rho\rho}=\frac{5}{6}t_{31}^{(I=2)},\quad t_{2}=t_{\rho
D^{*}}=\frac{1}{9}\big{(}5t_{32}^{I=3/2}+4t_{32}^{I=1/2}\big{)}.\end{split}$
(27)
We show our results in Fig. 3. In Fig. 3 (left) we show the modulus squared of
the amplitudes for $|T_{D^{*}-f_{2}}^{I=1/2}|^{2}$ and
$|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$, and we find that there are clear peaks
around the energy $2800-2850\textrm{ MeV}$ which is about $400\textrm{ MeV}$
lower than the $D^{*}-f_{2}$ threshold. The bindings are large because they
scale with the mass of the mesons and we have now a $D^{*}$ interacting with
two $\rho$ mesons. The strength of the peak of $|T_{D^{*}-f_{2}}^{I=1/2}|^{2}$
is two times bigger than for $|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$, and we see
that the $D^{*}-f_{2}$ component is a bit more bound than the $\rho-D^{*}_{2}$
one. We expect that a real state would be an admixture of both with a binding
in between that of the individual components. In Fig. 3 (right) we show
$|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$, and there is a clear resonant structure
about $3120\textrm{ MeV}$, the strength of which is 30 times smaller than that
of $|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$ in the left figure and less bound. We
are concerned with the lowest lying states and hence we concentrate on the
predicted new $D^{*}_{3}$ state with a structure formed by a mixture of
$D^{*}-f_{2}$ and $\rho-D^{*}_{2}$, with a mass about $2800-2850\textrm{ MeV}$
and a width about $60-100\textrm{ MeV}$.
Figure 3: Modulus squared of the $T_{D^{*}-f2}$ and $T_{\rho-D^{*}_{2}}$
scattering amplitudes. Left: $I_{total}=\frac{1}{2}$; Right:
$I_{total}=\frac{3}{2}$.
### IV.2 Four-body interaction
There are also two possibilities in the four-body interaction as we have shown
in Table 1: particle $3=f_{2}$, $R=D^{*}_{2}$ ($1=\rho,\;2=D^{*}$) or particle
$3=D^{*}_{2}$, $R=f_{2}$ ($1=\rho,\;2=\rho$). Because $I_{f_{2}}=0$ and
$I_{D^{*}_{2}}=\frac{1}{2}$, the total isospin of the four-body system is only
$I_{total}=\frac{1}{2}$. In the first case, $f_{2}$ collides with the
$D^{*}_{2}$, the amplitudes $t_{1}=t_{f_{2}\rho}=T_{\rho-f_{2}}$ has been
evaluated in Ref. Roca2010 and is reproduced in our work, and
$t_{2}=t_{f_{2}D^{*}}=T_{D^{*}-f_{2}}$ which has been evaluated in the former
subsection IV.1. For the second case, $D^{*}_{2}$ collides with the $f_{2}$,
and the amplitudes $t_{1}=t_{2}=t_{D^{*}_{2}\rho}=T_{\rho-D^{*}_{2}}$ have
been evaluated in the former subsection IV.1. We must now consider that the
three-body amplitude $T_{\rho-D^{*}_{2}}$ is also combined with different
isospins as mentioned in subsection IV.1. This situation is similar to the
case when the $D^{*}$ collides with the $f_{2}$, because the isospins of both
the $D^{*}_{2}$ and $D^{*}$ are $I=\frac{1}{2}$, thus from Eq. (24) we have
$t_{1}=T_{\rho
D^{*}_{2}}=\frac{1}{3}\big{(}2T_{31}^{I=3/2}+T_{31}^{I=1/2}\big{)},\quad
t_{2}=t_{1}.$ (28)
The results are shown in Fig. 4. The left of Fig. 4 is
$|T_{D^{*}_{2}-f_{2}}^{I=1/2}|^{2}$. We find that there is a clear peak at an
energy of $3200\textrm{ MeV}$, the width of which is about $200\textrm{ MeV}$.
The right of Fig. 4 shows $|T_{f_{2}-D^{*}_{2}}^{I=1/2}|^{2}$ and there is a
resonant peak around the energy $3075\textrm{ MeV}$ with a large width of
nearly $400\textrm{ MeV}$. The strength of the peak of
$|T_{f_{2}-D^{*}_{2}}^{I=1/2}|^{2}$ is about two times bigger than the one of
$|T_{D^{*}_{2}-f_{2}}^{I=1/2}|^{2}$ and the energy of the peak is more bound
too. But from the former results, subsection IV.1, we found that
$|T_{D^{*}_{2}-f_{2}}^{I=1/2}|^{2}$ has more strength and is more bound than
$|T_{f_{2}-D^{*}_{2}}^{I=1/2}|^{2}$. We have investigated that this is because
of the contribution of $|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$, even though the
strength of $|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$ is much smaller and less bound
than the one of $|T_{\rho-D^{*}_{2}}^{I=1/2}|^{2}$ from the former results.
When we removed the contribution of $|T_{\rho-D^{*}_{2}}^{I=3/2}|^{2}$ in Eq.
(28), the strength of the peak was enhanced by a factor five and was more
bound. Therefore, within the uncertainty of the theory, we find a new
$D^{*}_{4}$ resonance, of a mass about $3075-3200\textrm{ MeV}$ and a width
about $200-400\textrm{ MeV}$.
Figure 4: Modulus squared of the $T_{D^{*}_{2}-f2}$ (left) and
$T_{f_{2}-D^{*}_{2}}$ (right) scattering amplitudes.
### IV.3 Five-body interaction
For the five-body interaction, we also have two options for the cluster, one
of which is the particle $f_{4}$ studied in Ref. Roca2010 and the other one
the resonance $D^{*}_{4}$ obtained in the four-body interaction, subsection
IV.2. Thus letting the third particle ($D^{*}$ or $\rho$) collide with them,
we have $3=D^{*}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$) or $3=\rho$, $R=D^{*}_{4}$
($1=f_{2},\;2=D^{*}_{2}$). Because the isospin $I_{f_{4}}=0$ and
$I_{D^{*}_{4}}=\frac{1}{2}$, the total isospin of the five-body system is only
$I_{total}=\frac{1}{2}$ in the $D^{*}-f_{4}$ structure, but
$I_{total}=\frac{1}{2}$ or $I_{total}=\frac{3}{2}$ in the $\rho-D^{*}_{4}$
structure. Thus the situation is similar to the three-body interaction
discussed before, $D^{*}$ (or $\rho$) collide with $f_{2}$ (or $D^{*}_{2}$).
Therefore in the first case, the $D^{*}$ collides with the $f_{4}$, and the
amplitudes $t_{1}=t_{2}=t_{D^{*}f_{2}}=T_{D^{*}-f_{2}}^{(I=1/2)}$ have been
evaluated in subsection IV.1. For the second case, the $\rho$ collides with
the $D^{*}_{4}$, which is similar to $\rho-D^{*}_{2}$ in the three-body
interaction, thus, after doing a similar derivation as in Eq. (23), we have
$\begin{split}&T_{\rho-D^{*}_{4}}^{(I=1/2)}:\quad t_{1}=t_{\rho
f_{2}}=T_{31}^{(I=1)},\quad t_{2}=t_{\rho D^{*}_{2}}=T_{32}^{I=1/2};\\\
&T_{\rho-D^{*}_{4}}^{(I=3/2)}:\quad t_{1}=t_{\rho f_{2}}=T_{31}^{(I=1)},\quad
t_{2}=t_{\rho D^{*}_{2}}=T_{32}^{I=3/2},\end{split}$ (29)
where the $T_{31}^{(I=1)}$ is the same as $T_{\rho-f_{2}}$ in the subsection
IV.2 reproducing the results of Ref. Roca2010 , and
$T_{\rho-D^{*}_{2}}^{I=1/2}$ and $T_{\rho-D^{*}_{2}}^{I=3/2}$ have also been
evaluated in subsection IV.1.
In Fig. 5 we show our results. The left of Fig. 5 is
$|T_{D^{*}-f_{4}}^{I=1/2}|^{2}$ and we observe a resonant peak around the
energy $3375\textrm{ MeV}$ with a width of less than $200\textrm{ MeV}$. The
right of Fig. 5 is $|T_{\rho-D^{*}_{4}}^{I=1/2}|^{2}$ and
$|T_{\rho-D^{*}_{4}}^{I=3/2}|^{2}$. We find that there is a resonant structure
in $|T_{\rho-D^{*}_{4}}^{I=1/2}|^{2}$ at the energy $3360\textrm{ MeV}$, the
width of which is about $400\textrm{ MeV}$, and the position is very close to
the one of $|T_{D^{*}-f_{4}}^{I=1/2}|^{2}$. But the strength of
$|T_{\rho-D^{*}_{4}}^{I=1/2}|^{2}$ is one order smaller than the one
$|T_{D^{*}-f_{4}}^{I=1/2}|^{2}$. For $|T_{\rho-D^{*}_{4}}^{I=3/2}|^{2}$ there
is no resonant structure. Therefore, within uncertainties, we also find a new
$D^{*}_{5}$ resonance, with a mass about $3360-3375\textrm{ MeV}$ and a width
about $200-400\textrm{ MeV}$.
Figure 5: Modulus squared of the $T_{D^{*}-f_{4}}$ (left) and
$T_{\rho-D^{*}_{4}}$ (right) scattering amplitudes.
### IV.4 Six-body interaction
Similarly to the five-body interaction, we also have two options of the
cluster for the six-body interaction, the particle $f_{4}$ studied in Ref.
Roca2010 and the resonance $D^{*}_{4}$ obtained in subsection IV.2. Now
letting a resonance ($D^{*}_{2}$ or $f_{2}$) be the third particle and collide
with them, we have $3=D^{*}_{2}$, $R=f_{4}$ ($1=f_{2},\;2=f_{2}$) or
$3=f_{2}$, $R=D^{*}_{4}$ ($1=f_{2},\;2=D^{*}_{2}$). Because
$I_{f_{2}}=I_{f_{4}}=0$ and $I_{D^{*}_{2}}=I_{D^{*}_{4}}=\frac{1}{2}$, the
total isospin of the six-body system is only $I_{total}=\frac{1}{2}$. Thus, in
the first case, the $D^{*}_{2}$ collides with the $f_{4}$, the amplitudes
$t_{1}=t_{2}=t_{D^{*}_{2}f_{2}}=T_{D^{*}_{2}-f_{2}}^{(I=1/2)}$ have been
evaluated in subsection IV.2. For the second case, the $f_{2}$ collides with
the $D^{*}_{4}$, the amplitudes $t_{1}=t_{f_{2}f_{2}}=T_{f_{2}-f_{2}}$
reproduce the results from Ref. Roca2010 , and
$t_{2}=t_{f_{2}D^{*}_{2}}=T_{f_{2}-D^{*}_{2}}$ has been calculated in
subsection IV.2.
Our results are shown in Fig. 6. The left of Fig. 6 is
$|T_{D^{*}_{2}-f_{4}}^{I=1/2}|^{2}$ where we see a peak around the energy
$3775\textrm{ MeV}$ with a large width of nearly $400\textrm{ MeV}$. The right
of Fig. 6 is $|T_{f_{2}-D^{*}_{4}}^{I=1/2}|^{2}$, and there we find that there
is not a clear peak at the energy $3550\textrm{ MeV}$. It looks like the
resonant structure of $f_{2}-D^{*}_{4}$ is not as stable as the
$D^{*}_{2}-f_{4}$ one. From these results, we could predict a new $D^{*}_{6}$
resonance with more uncertainty, with a mass of about $3775\textrm{ MeV}$ and
a width about $400\textrm{ MeV}$.
Figure 6: Modulus squared of the $T_{D^{*}_{2}-f_{4}}$ (left) and
$T_{f_{2}-D^{*}_{4}}$ (right) scattering amplitudes.
## V Conclusions
In the present work, we show the results of our investigation of the
$D^{*}$-multi-$\rho$ systems. Our idea is based on the fact that the two-body
interactions of $\rho\rho$ and $\rho D^{*}$ in spin $S=2$ are so strong as to
bind the particles forming the resonances$f_{2}(1270)$ Molina:2008jw and
$D^{*}_{2}(2460)$ Molina:2009eb respectively. So we could study the many-body
$D^{*}$-multi-$\rho$ systems in an iterative way looking at the structure of
the amplitudes and observing clear peaks that become wider as the number of
$\rho$ mesons increase. The work proceeded analogously to the study of
multi-$\rho$ system in Roca2010 , where the $\rho_{3}(1690)(3^{--})$,
$f_{4}(2050)(4^{++})$, $\rho_{5}(2350)(5^{--})$, and $f_{6}(2510)(6^{++})$
were described as basically molecules of multi-$\rho(770)$ states, and
similarly to the work of Yamagata2010 where the $K^{*}_{2}(1430)$,
$K^{*}_{3}(1780)$, $K^{*}_{4}(2045)$, $K^{*}_{5}(2380)$ and $K^{*}_{6}$ could
be interpreted as molecules made of one $K^{*}(892)$ meson and an increasing
number of $\rho(770)$ mesons. The $D^{*}$-multi-$\rho$ states with spins
aligned combined to give some new charmed resonances, $D^{*}_{3}$,
$D^{*}_{4}$, $D^{*}_{5}$ and $D^{*}_{6}$, which are basically made of one
$D^{*}$ meson and an increasing number of $\rho(770)$ mesons and are not found
in the list of PDG pdg2012 . Their masses are predicted around
$2800-2850\textrm{ MeV}$, $3075-3200\textrm{ MeV}$, $3360-3375\textrm{ MeV}$
and $3775\textrm{ MeV}$ respectively. And their widths are about
$60-100\textrm{ MeV}$, $200-400\textrm{ MeV}$, $200-400\textrm{ MeV}$ and
$400\textrm{ MeV}$ respectively. The analogy with the states already known in
the strange and non-strange sector, together with the stronger interaction of
the $D^{*}$ mesons, make our predictions solid within the uncertainties
admitted. We are, thus, reasonably confident that such states can be found in
the future in coming facilities like FAIR and others.
## Acknowledgements
We thank R. Molina for useful discussions. This work is partly supported by
DGICYT contract number FIS2011-28853-C02-01, and the Generalitat Valenciana in
the program Prometeo, 2009/090. We acknowledge the support of the European
Community-Research Infrastructure Integrating Activity Study of Strongly
Interacting Matter (acronym HadronPhysics3, Grant Agreement n. 283286) under
the Seventh Framework Programme of the EU. One of us, M. Bayar acknowledges
support through the Scientific and Technical Research Council (TUBITAK)
BIDEP-2219 grant.
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|
arxiv-papers
| 2012-07-17T15:28:05 |
2024-09-04T02:49:33.234555
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "C. W. Xiao, M. Bayar and E. Oset",
"submitter": "Chuwen Xiao",
"url": "https://arxiv.org/abs/1207.4030"
}
|
1207.4322
|
# Virasoro type algebraic structure hidden in the constrained discrete KP
hierarchy
Maohua Li${}^{1,2},$ Chuanzhong Li2, Keilei Tian1,3, Jingsong He2∗,Yi Cheng1
1\. Department of Mathematics, USTC, Hefei, 230026 Anhui, P.R.China
2\. Department of Mathematics, Ningbo University, Ningbo, 315211 Zhejiang,
P.R.China
3\. Department of Physics and Astronomy, University of Turku, 20014 Turku,
Finland
###### Abstract.
In this paper, we construct the additional symmetries of one-component
constrained discrete KP (cdKP) hierarchy, and then prove that the algebraic
structure of the symmetry flows is the positive half of Virasoro algebra.
∗ Corresponding author: hejingsong@nbu.edu.cn
Mathematics Subject Classifications(2000). 37K05, 37K10, 37K40.
Keywords: dKP hierarchy, constrained dKP hierarchy, additional symmetry,
Virasoro algebra
## 1\. Introduction
In the past few years, lots of attention have been given to the study of
Kadomtsev-Petviashvili (KP) hierarchy [1, 2] in the field of integrable
systems. The Lax pairs, Hamiltonian structures, symmetries and conservation
laws, the N-soliton, tau function, the gauge transformation, reductions etc.
of the KP hierarchy and its sub-hierarchies have been discussed. A specific
interesting aspect of the research of the KP hierarchy is additional
symmetry[2, 3, 4, 5, 6, 7]. Additional symmetries are special symmetries which
are not contained in the KP hierarchy and do not commute with each other. The
additional symmetry flows of the KP hierarchy form an infinite dimensional
algebra $W_{1+\infty}$[4]. Recently, there are several new results about
partition function in the matrix models and Seiberg-Witten theory associated
with additional symmetries, string equation and Virasoro constraints of the KP
hierarchy [8, 9, 10, 11, 12].
There are several sub-hierarchies of the KP by considering different reduction
conditions on the Lax operator $L$. One of them is called constrained KP (cKP)
hierarchy [13, 14, 15] by setting the Lax operator as
$L=\partial+\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i}$. Here $\Phi_{i}$ is
an eigenfunction and $\Psi_{i}$ is an adjoint eigenfunction of the cKP
hierarchy. The cKP hierarchy contains a large number of interesting soliton
equations. The basic idea of this procedure is so-called symmetry
constraint[13, 14, 15]. The negative part of the Lax operator of the
constrained KP, i.e. $\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i}$, is a
generator[2] of the additional symmetry of the KP hierarchy. This observation
inspires the study[16] of the additional symmetries of the cKP hierarchy.
However, the additional symmetry flows of the KP hierarchy are not consistent
with the special form of the Lax operator for the cKP hierarchy autonomously,
and so it is highly non-trivial to find a suitable form of additional symmetry
flows for this sub-hierarchy. The correct additional symmetry flows are given
by means of a crucial modification associated with a complicated operator
$X_{k}$[16]. Very recently, by a further modification of the additional flows,
the additional symmetries of the constrained BKP and constrained CKP
hierarchies are given in references [17, 18]. In addition to the above-
mentioned integrable systems, discrete system such as Toda hierarchy also has
interesting algebraic structure of additional symmetry[4, 19].
At the same time the discovery of Hirota’s bilinear difference equation [20]
attracted much interest in looking for other integrable discrete equations and
systems. This famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. There are several difference kinds of the discrete hierarchies
including differential-difference KP hierarchy, semi-discrete systems, full
discrete equations and so on. The differential-difference KP (dKP) hierarchy
[21, 22, 23] defined by the difference operator $\triangle$ is one interesting
object of the integrable discrete systems. Note that, the additional symmetry
of dKP hierarchy and it’s Sato Bäcklund transformations have been given in
reference[24]. So it is worthy to find the hidden algebraic structure in the
constrained discrete KP(cdKP) hierarchy using the additional symmetry flows,
which is the main purpose of us. This will be done by following four steps:
* •
1) show the inconsistency between the additional symmetry flows eq.(2.19) of
the dKP hierarchy and the Lax form eq.(3.1) of the cdKP hierarchy;
* •
2) modify the additional flows to sort out this inconsistency;
* •
3) prove that the modified additional flows are symmetry of the cdKP
hierarchy;
* •
4) identify the algebraic structure of the additional symmetry flows.
The crucial step of this process is to find a suitable modification of the
additional flows of the dKP hierarchy such that these flows are consistent
with the form of the Lax pair of the cdKP hierarchy.
The paper is organized as follows. Some basic results of of dKP hierarchy and
the additional symmetry of dKP hierarchy are summarized in Section 2. After
introducing of a definition of the Lax equation of cdKP hierarchy, the
additional symmetry flows of the cdKP hierarchy are defined properly by means
of a crucial modification from the additional symmetry flows of the dKP
hierarchy in Section 3. Next the Virasoro type algebraic structure of these
additional symmetry flows is also identified by a straightforward calculation
in Section 4. Section 5 is devoted to conclusions and discussions.
## 2\. The dKP hierarchy and it’s additional symmetry
Let us briefly recall some basic facts about the dKP hierarchy according to
reference [22]. Firstly a space $F$, namely
$F=\left\\{f(n)=f(n,t_{1},t_{2},\cdots,t_{j},\cdots);n\in\mathbb{Z},t_{i}\in\mathbb{R}\right\\}$
(2.1)
is defined for the space of the discrete KP hierarchy. And $\Lambda,\triangle$
are denote for the shift operator and the difference operator, respectively.
Their actions on function $f(n)$ are define for
$\Lambda f(n)=f(n+1)$ (2.2)
and
$\triangle f(n)=f(n+1)-f(n)=(\Lambda-I)f(n)$ (2.3)
respectively, where $I$ is the identity operator.
For any $j\in\mathbb{Z},$ the Leibniz rule of $\triangle$ operation is,
$\triangle^{j}\circ
f=\sum^{\infty}_{i=0}\binom{j}{i}(\triangle^{i}f)(n+j-i)\triangle^{j-i},\hskip
8.5359pt\binom{j}{i}=\frac{j(j-1)\cdots(j-i+1)}{i!}.$ (2.4)
So an associative ring $F(\triangle)$ of formal pseudo difference operators is
obtained, with the operation $``+"$ and $``\circ"$, namely
$F(\triangle)=\left\\{R=\sum_{j=-\infty}^{d}f_{j}(n)\triangle^{j},f_{j}(n)\in
R,n\in\mathbb{Z}\right\\},$ and denote
$R_{+}:=\sum_{j=0}^{d}f_{j}(n)\circ\triangle^{j}$ as the positive projection
of $R$ and by $R_{-}:=\sum_{j=-\infty}^{-1}f_{j}(n)\circ\triangle^{j}$, the
negative projection of $R$. The adjoint operator to the $\triangle$ operator
is given by $\triangle^{*}$,
$\triangle^{*}\circ f(n)=(\Lambda^{-1}-I)f(n)=f(n-1)-f(n),$ (2.5)
where $\Lambda^{-1}f(n)=f(n-1)$, and the corresponding “$\circ$” operation is
$\triangle^{*j}\circ
f=\sum^{\infty}_{i=0}\binom{j}{i}(\triangle^{*i}f)(n+i-j)\triangle^{*j-i}.$
(2.6)
Then the adjoint ring $F(\triangle^{*})$ to the $F(\triangle)$ is obtained,
and the formal adjoint to $R\in F(\triangle)$ is defined by $R^{*}\in
F(\triangle^{*})$ as $R^{*}=\sum_{j=-\infty}^{d}\triangle^{*j}\circ f_{j}(n)$.
The $"*"$ operation satisfies rules as $(F\circ G)^{*}=G^{*}\circ F^{*}$ for
two operators and $f(n)^{*}=f(n)$ for a function.
The discrete KP (dKP) hierarchy [22] is a family of evolution equations
depending on infinitely many variables $t=(t_{1},t_{2},\cdots)$
$\frac{\partial L}{\partial t_{k}}=[B_{k},L],\ \ \ B_{k}:=(L^{k})_{+},$ (2.7)
where $L$ is a general first-order pseudo difference operator(PDO)
$L(n)=\triangle+\sum_{j=1}^{\infty}f_{j}(n)\triangle^{-j}.$ (2.8)
$B_{m}=(L^{m})_{+}=\sum^{m}_{j=0}a_{j}(n)\triangle^{j}$, i. e. $(L^{m})_{+}$
is the non-negative projection of $L^{m}$, and $(L^{m})_{-}=L^{m}-(L^{m})_{+}$
is the negative projection of $L^{m}$. The Lax operator in eq.(2.8) can be
generated by a dressing operator
$W(n;t)=1+\sum^{\infty}_{j=1}w_{j}(n;t)\triangle^{-j}.$ (2.9)
through
$L=W\circ\Delta\circ W^{-1}.$ (2.10)
Further the flow equation (2.7) is equivalent to the so-called Sato equation,
$\partial_{t_{k}}W=-(L^{k})_{-}\circ W.$ (2.11)
Now we introduce the additional symmetry flows[24] of the dKP hierarchy as
following. Set
$\Gamma_{\Delta}=\sum_{i=1}^{\infty}(it_{i}\Delta^{i-1}+{(-1)}^{i-1}n\Delta^{i-1}),$
(2.12)
and it is easy to find the following formula
$[\partial_{t_{k}}-\Delta^{k},\Gamma_{\Delta}]=0.$ (2.13)
Define another operator
$M_{\Delta}=W\circ\Gamma_{\Delta}\circ W^{-1}.$ (2.14)
There are the following commutation relations
$[\Delta,\Gamma_{\Delta}]=1,[L,M_{\Delta}]=1,$ (2.15)
which can be verified by a straightforward calculation. By using the Sato
equation, the isospectral flow of the $M_{\Delta}$ operator is given by
$\partial_{t_{k}}M_{\Delta}=[L^{k}_{+},M_{\Delta}].$ (2.16)
More generally,
$\partial_{t_{k}}(M_{\Delta}^{m}L^{l})=[L^{k}_{+},M_{\Delta}^{m}L^{l}].$
(2.17)
Based on the above preparation, the additional symmetry flows[24] of the dKP
hierarchy are define by their actions on the dressing operator
$\overline{\partial_{l,m}}W=-(M_{\Delta}^{m}L^{l})_{-}\circ W,$ (2.18)
or equivalently on the Lax operator
$\displaystyle\overline{\partial_{l,m}}L$ $\displaystyle=$
$\displaystyle[-(M_{\Delta}^{m}L^{l})_{-},L],$ (2.19)
where $\overline{\partial_{l,m}}$ denotes the derivative with respect to an
additional new variable $t_{lm}^{*}$. The more general actions of the
additional symmetry flows of the dKP are given by
$\overline{\partial_{l,m}}M_{\Delta}^{n}L^{k}=[-(M_{\Delta}^{m}L^{l})_{-},M_{\Delta}^{n}L^{k}].$
(2.20)
As the end of this section, we would like to point out two important technical
identities as followings. For two pseudo-difference operators
$X_{i}=f_{i}\Delta^{-1}g_{i},i=1,2,$ we have
$X_{1}\circ
X_{2}=X_{1}(f_{2})\Delta^{-1}g_{2}+f_{1}\Delta^{-1}X_{2}^{*}(g_{1}).$ (2.21)
For a pure-difference operator $K$ and two arbitrary smooth functions ($q,r$),
we have
$[K,q\Delta^{-1}r]_{-}=K(q)\Delta^{-1}r-q\Delta^{-1}K^{*}(r).$ (2.22)
The usual versions(non-discrete) of them are given by eq.(A.3) and eq.(A.4) in
reference[16].
## 3\. Additional symmetry flows of the cdKP hierarchy
The one-component cdKP hierarchy is defined by following Lax equation
$\frac{\partial L}{\partial t_{l}}=[(L^{l})_{+},L],l=1,2,\cdots,$ (3.1)
associated with a special Lax operator
$L=(L)_{+}+q(t)\triangle^{-1}r(t),$ (3.2)
and $q(t)$ is an eigenfunction, $r(t)$ is an adjoint eigenfunction. The
eigenfunction and adjoint eigenfunction $q(t),r(t)$ are important dynamical
variables in the cdKP hierarchy. Using identity eq.(2.22), one can check Lax
equation (3.1) is consistent with the evolution equations of the
eigenfunction(or adjoint eigenfunction)
$q_{t_{m}}=B_{m}q,\quad r_{t_{m}}=-B_{m}^{*}r,\quad B_{m}=(L^{m})_{+},\forall
m\in N.$ (3.3)
Therefore the cdKP hierarchy in eq.(3.1) is well defined. And Eq.(3.1) implies
that the Sato equation of cdKP hierarchy is
$\partial_{l}W=-(L^{l})_{-}\circ W,$ (3.4)
where $\partial_{l}=\frac{\partial}{\partial t_{l}}$.
The central task of this section is to find the additional symmetry flows of
the cdKP hierarchy, which can be realized by three steps as we have mentioned
in the introduction. As usual calculation of the infinitesimal analysis, the
desired action of the additional symmetry flows on the Lax operator $L$ of the
cdKP hierarchy should be
$(\widetilde{\partial_{\tau}}L)_{-}=\widetilde{\partial_{\tau}}(q(n,t))\triangle^{-1}r(n,t)+q(n,t)\triangle^{-1}\widetilde{\partial_{\tau}}(r(n,t)).$
(3.5)
However, according to the definition eq.(2.19) of the dKP hierarchy, the
action of original additional flows of the cdKP hierarchy is expressed by
$(\overline{\partial_{k,1}}L)_{-}=[(M_{\Delta}L^{k})_{+},L]_{-}+(L^{k})_{-}.$
(3.6)
The following lemma and eq.(2.22) show that it can not be rewritten as the
desired form eq.(3.5) except $k=0,1,2$. Specifically, $k=3$, from $L^{3}_{-}$
$(L^{3})_{-}=L^{2}(q)\Delta^{-1}r+L(q)\Delta^{-1}L^{*}(r)+q\Delta^{-1}L_{2}^{*}(r),$
(3.7)
we can find the middle term can not be rewritten as the form of eq.(3.5). This
demonstrates obviously the inconsistency between the additional symmetry flows
eq.(2.19) of the dKP hierarchy and the Lax form eq.(3.1) of the cdKP
hierarchy.
###### Lemma 3.1.
The Lax operator $L$ of constrained dKP hierarchy given by (3.2) satisfied the
relation of
$(L^{k})_{-}=\sum_{j=0}^{k-1}L^{k-j-1}(q)\Delta^{-1}{(L^{*})}^{j}(r).$ (3.8)
where $L(q)=(L)_{+}(q)+q(t)\triangle^{-1}(r(t)q).$
###### Proof.
It can be reduced by induction with the help of technical identity in
eq.(2.21). We omit it here. ∎
To overcome the inconsistency, we shall introduce an operator $Y_{k}$ to
modify the additional symmetry eq.(2.19) of the dKP hierarchy. The following
lemmas implied by identity (2.21,2.22) are necessary.
###### Lemma 3.2.
$[X,L]_{-}=\sum_{k=1}^{l}[M_{k}\Delta^{-1}L^{*}(N_{k})-L(M_{k})\Delta^{-1}N_{k}]+[X(q)\Delta^{-1}r-q\Delta^{-1}X^{*}(r)],$
(3.9)
with definitions (3.2) and
$X=\sum_{k=1}^{l}M_{k}\Delta^{-1}N_{k}.$ (3.10)
We now introduce a pseudo-difference operator $Y_{k}$,
$\displaystyle Y_{k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j}(r),k\geq
2,$ (3.11) $\displaystyle Y_{k}$ $\displaystyle=$ $\displaystyle 0,k=-1,0,1.$
(3.12)
and have the following property.
###### Lemma 3.3.
The action of flows $\partial_{l}$ of the cdKP hierarchy on the $Y_{k}$ is
$\partial_{l}Y_{k}=[(L^{l})_{+},Y_{k}]_{-}.$ (3.13)
###### Proof.
$\displaystyle\partial_{l}Y_{k}$ $\displaystyle=$
$\displaystyle\partial_{l}\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j}(r)$
$\displaystyle=$
$\displaystyle\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]\\{\partial_{l}(L^{k-1-j}(q))\Delta^{-1}(L^{*})^{j}(r)+L^{k-1-j}(q)\Delta^{-1}\partial_{l}((L^{*})^{j}(r))\\}$
$\displaystyle\stackrel{{\scriptstyle by(\ref{operatork})}}{{==}}$
$\displaystyle[(L^{l})_{+}\circ\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j}(r)]_{-}$
$\displaystyle-[(\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j}(r))\circ(L^{l})_{+}]_{-}$
$\displaystyle=$
$\displaystyle[(L^{l})_{+},(\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j}(r))]_{-}$
$\displaystyle=$ $\displaystyle[(L^{l})_{+},Y_{k}]_{-}.$
∎
The first nontrivial example of (3.11) is given by
$Y_{2}=[-\frac{1}{2}L(q)\Delta^{-1}r+\frac{1}{2}q\Delta^{-1}L^{*}(r)],$ (3.14)
for $k=2$. And
$[Y_{2},L]_{-}=-(L^{3})_{-}+\frac{3}{2}[L^{2}(q)\Delta^{-1}r+q\Delta^{-1}(L^{*})^{2}(r)]\\\
+[Y_{2}(q)\Delta^{-1}r-q\Delta^{-1}{Y_{2}}^{*}(r)].$ (3.15)
Further, the following expression of $[Y_{k-1},L]_{-}$ is also necessary to
define the additional flows of the cdKP hierarchy.
###### Lemma 3.4.
The Lax operator $L$ of constrained dKP hierarchy and $Y_{k-1}$ has the
following relation,
$[Y_{k-1},L]_{-}=-(L^{k})_{-}+\frac{k}{2}[q\Delta^{-1}{(L^{*})}^{k-1}(r)+L^{k-1}(q)\Delta^{-1}r]+Y_{k-1}(q)\Delta^{-1}r-q\Delta^{-1}Y_{k-1}^{*}(r)$
(3.16)
###### Proof.
$\displaystyle[Y_{k-1},L]_{-}$ $\displaystyle=$
$\displaystyle[\sum^{k-2}_{j=0}[j-\frac{1}{2}(k-2)]L^{k-2-j}(q)\Delta^{-1}{L^{*}}^{j}(r),L]_{-}$
$\displaystyle\stackrel{{\scriptstyle by(\ref{operatorXL})}}{{==}}$
$\displaystyle\sum^{k-2}_{j=0}[j-\frac{1}{2}(k-2)]L^{k-2-j}(q)\Delta^{-1}{(L^{*})}^{j+1}(r)-\sum^{k-2}_{j=0}[j-\frac{1}{2}(k-2)]L^{k-1-j}(q)\Delta^{-1}{L^{*}}^{j}(r)$
$\displaystyle+$ $\displaystyle
Y_{k-1}(q)\Delta^{-1}r-q\Delta^{-1}{Y_{k-1}}^{*}(r)$ $\displaystyle=$
$\displaystyle-\sum^{k-2}_{j=1}L^{k-1-j}(q)\Delta^{-1}{(L^{*})}^{j}(r)+(\frac{k}{2}-1)[q\Delta^{-1}{(L^{*})}^{k-1}(r)+L^{k-1}(q)\Delta^{-1}r]$
$\displaystyle+$ $\displaystyle
Y_{k-1}(q)\Delta^{-1}r-q\Delta^{-1}Y_{k-1}^{*}(r)$ $\displaystyle=$
$\displaystyle-(L^{k})_{-}+\frac{k}{2}[q\Delta^{-1}{(L^{*})}^{k-1}(r)+L^{k-1}(q)\Delta^{-1}r]+Y_{k-1}(q)\Delta^{-1}r-q\Delta^{-1}Y_{k-1}^{*}(r)$
∎
Putting together (3.6) and (3.16), we define the additional flows of the cdKP
hierarchy as
$\partial_{k}^{*}L=[-(M_{\Delta}L^{k})_{-}+Y_{k-1},L],$ (3.17)
where $\partial_{k}^{*}=\overline{\partial_{k,1}}$ and $Y_{l-1}=0$, for
$l=0,1,2$, such that the right-hand side of (3.17) is in the form of (3.5). It
must be mentioned that the additional flows $\partial_{k}^{*}$ of cdKP
hierarchy is nothing but the additional flows $\overline{\partial_{k,1}}$ of
the dKP hierarchy for $k=0,1,2$. Generally,
$\partial_{k}^{*}(M_{\Delta}L^{l})=[-(M_{\Delta}L^{k})_{-}+Y_{k-1},M_{\Delta}L^{l}].$
(3.18)
Now we calculate the action of the additional flows (3.17) on the
eigenfunction $q$ and $r$ of the cdKP hierarchy.
###### Theorem 3.5.
The acting of additional flows of constrained dKP hierarchy on the
eigenfunction $q$ and $r$ are
$\begin{split}{\partial_{k}^{*}q}&=(M_{\Delta}L^{k})_{+}(q)+Y_{k-1}(q)+\frac{k}{2}L^{k-1}(q),\\\
{\partial_{k}^{*}r}&=-(M_{\Delta}L^{k})^{*}_{+}(r)-Y_{k-1}^{*}(r)+\frac{k}{2}{(L^{*})}^{k-1}(r).\end{split}$
(3.19)
###### Proof.
Substitution of (3.16) to negative part of (3.17) shows
$\displaystyle\begin{split}{(\partial_{k}^{*}L)}_{-}&=(M_{\Delta}L^{k})_{+}(q)\Delta^{-1}(r)-q\Delta^{-1}(M_{\Delta}L^{k})_{+}^{*}(r)\\\
&+Y_{k-1}(q)\Delta^{-1}r-q\Delta^{-1}Y_{k-1}^{*}(r)+\frac{k}{2}q\Delta^{-1}(L^{*})^{k-1}(r)+\frac{k}{2}L^{k-1}(q)\Delta^{-1}r.\end{split}$
(3.20)
On the other side,
${(\partial_{k}^{*}L)}_{-}=(\partial_{k}^{*}q)\Delta^{-1}r+q\Delta^{-1}{(\partial_{k}^{*}r)}.$
(3.21)
Comparing right hand sides of (3.20) and (3.21) implies the action of
additional flows on the eigenfunction and the adjoint eigenfunction (3.19). ∎
And the case of (3.19) with $k=3$ is
$\displaystyle\begin{split}{\partial_{3}^{*}q}&=(M_{\Delta}L^{3})_{+}(q)+Y_{2}(q)+\frac{3}{2}L^{2}(q),\\\
{\partial_{3}^{*}r}&=-(M_{\Delta}L^{3})^{*}_{+}(r)-Y_{2}^{*}(r)+\frac{3}{2}{L^{*}}^{2}(r).\end{split}$
(3.22)
Next we shall prove the commutation relation between the additional flows
$\partial^{*}_{k}$ of cdKP hierarchy and the original flows $\partial_{l}$ of
it.
###### Theorem 3.6.
The additional flows of $\partial^{*}_{k}$ commute with all $\partial_{l}$
flows of the cdKP hierarchy.
###### Proof.
According the action of $\partial^{*}_{k}$ and $\partial_{l}$ on the Sato
operator $W$ (3.4,3.17) and (3.13), then
$\displaystyle[\partial^{*}_{k},\partial_{l}]W$ $\displaystyle=$
$\displaystyle-\partial^{*}_{k}(L_{-}^{l}W)-\partial_{l}[-(M_{\Delta}L^{k})_{-}+Y_{k-1}]W$
$\displaystyle=$
$\displaystyle(-\partial_{k}^{*}L_{-}^{l})W-L^{l}_{-}\partial_{k}^{*}W-[(M_{\Delta}L^{k})_{-}-Y_{k-1}]L_{-}^{l}W$
$\displaystyle+[L^{l}_{+},M_{\Delta}L^{k}]_{-}W-(\partial_{l}Y_{k-1})W$
$\displaystyle=$
$\displaystyle[L^{l}_{-},-Y_{k-1}]_{-}W+[-Y_{k-1},L^{l}]_{-}W-(\partial_{l}Y_{k-1})W\text{\quad
by Jacobi identity}$
$\displaystyle\stackrel{{\scriptstyle(\ref{ykderivat})}}{{==}}$
$\displaystyle[L^{l}_{+},-Y_{k-1}]_{-}W-(\partial_{l}Y_{k-1})W$
$\displaystyle=$ $\displaystyle 0.$
Therefore, the additional flows $\partial^{*}_{k}$ commute with all flows
$\partial_{l}$ of the cdKP hierarchy. ∎
Remark: This theorem implies that the additional flows $\partial^{*}_{k}$
(3.17) are symmetry flows of the cdKP hierarchy.
## 4\. Virasoro type algebraic structure of the additional symmetry flows
In this section, we shall discuss the algebraic structure of the additional
symmetry flows of the cdKP hierarchy. For this end, we need the actions of
$\partial_{l}^{*}$ on $Y_{k}$ and $L$.
Taking into account $Y_{k-1}=0$ for $k=0,1,2$, then Eqs.(3.19) becomes
$\displaystyle\begin{split}{\partial_{0}^{*}q}&=(M_{\Delta})_{+}(q),&{\partial_{0}^{*}r}&=-(M_{\Delta})^{*}_{+}(r),\\\
{\partial_{1}^{*}q}&=(M_{\Delta}L)_{+}(q)+\alpha
q,&{\partial_{1}^{*}r}&=-(M_{\Delta}L)^{*}_{+}(r)+\beta r,\alpha+\beta=1,\\\
{\partial_{2}^{*}q}&=(M_{\Delta}L^{2})_{+}(q)+L(q),&{\partial_{2}^{*}r}&=-(M_{\Delta}L^{2})^{*}_{+}(r)+L^{*}(r).\end{split}$
(4.1)
We can rewrite the (4.1) for
$\displaystyle\begin{split}\partial_{l}^{*}q&=(M_{\Delta}L)_{+}(q)+\frac{1}{2}lL^{l-1}q,\\\
\partial_{l}^{*}r&=-(M_{\Delta}L)^{*}_{+}(r)+\frac{1}{2}l{(L^{*})}^{l-1}r,\end{split}$
(4.2)
with $l=0,1,2.$
Because of $\partial_{l}^{*}=\overline{\partial_{l,1}},l=0,1,2$ as mentioned
above, we have the following lemma.
###### Lemma 4.1.
The additional flows $\partial_{l}^{*}$ of cdKP hierarchy have the following
relations for $l=0,1,2$ and $k\geq 0$, namely,
$\displaystyle\begin{split}{\partial_{l}^{*}L^{k}(q)}&=(M_{\Delta}L^{l})_{+}(L^{k}(q))+(k+\frac{l}{2})L^{k+l-1}(q),\\\
{\partial_{l}^{*}{(L^{*})}^{k}(r)}&=-(M_{\Delta}L^{l})^{*}_{+}{(L^{*})}^{k}(r)+(k+\frac{l}{2}){(L^{*})}^{k+l-1}(r).\end{split}$
(4.3)
###### Proof.
It is easy to get this by using (2.20), (4.2) and the relation
${\overline{\partial_{l,1}}(L^{k}(q))}=(\overline{\partial_{l,1}}(L^{k}))(q)+L^{k}\overline{\partial_{l,1}}(q)$.
∎
Moreover, the action of $\partial^{*}_{l}$ on $Y_{k}$ is given by the
following lemma.
###### Lemma 4.2.
The actions on $Y_{k}$ of the additional symmetry flows $\partial^{*}_{l}$ of
the cdKP hierarchy are
$\displaystyle\partial^{*}_{l}Y_{k}=[(M_{\Delta}L^{l})_{+},Y_{k}]_{-}+(k-l+1)Y_{k+l-1},$
(4.4)
for $l=0,1,2$ and $k\geq 0$.
###### Proof.
A straightforward calculation implies
$\displaystyle\begin{split}\partial_{l}^{*}Y_{k}&=\partial_{l}^{*}\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j}(r)\\\
&=\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](\partial_{l}^{*}(L^{k-1-j}(q))\Delta^{-1}(L^{*})^{j}(r)+L^{k-1-j}(q)\Delta^{-1}(\partial_{l}^{*}(L^{*})^{j}(r)))\\\
&\stackrel{{\scriptstyle(\ref{lqstar})}}{{==}}\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](M_{\Delta}L^{l})_{+}(L^{k-1-j}(q))\Delta^{-1}(L^{*})^{j}(r)\\\
&+\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](k-j-1+\frac{l}{2})L^{k+l-2-j}(q)\Delta^{-1}(L^{*})^{j}(r)\\\
&-\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(M_{\Delta}L^{l})^{*}_{+}(L^{*})^{j}(r)\\\
&+\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](j+\frac{l}{2})L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j+l-1}(r).\end{split}$
(4.5)
The first term and the third one of the right part of eq.(4.5) can be
simplified to
$\displaystyle\begin{split}&\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](M_{\Delta}L^{l})_{+}(L^{k-1-j}(q))\Delta^{-1}(L^{*})^{j}(r)\\\
&-\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)]L^{k-1-j}(q)\Delta^{-1}(M_{\Delta}L^{l})^{*}_{+}(L^{*})^{j}(r)\\\
&=[(M_{\Delta}L^{l})_{+},Y_{k}]_{-}.\end{split}$ (4.6)
Furthermore, on behaving of
$\displaystyle\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](j+\frac{l}{2})L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j+l-1}(r)$
$\displaystyle=\sum_{j=l-1}^{k-1+l-1}[j-l+1-\frac{1}{2}(k-1)](j-l+1+\frac{l}{2})L^{k+l-2-j}(q)\Delta^{-1}(L^{*})^{j}(r),$
the second term and the fourth one of the right part of eq.(4.5) can also be
simplified to
$\displaystyle\begin{split}&\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](k-j-1+\frac{l}{2})L^{k+l-2-j}(q)\Delta^{-1}(L^{*})^{j}(r)\\\
&+\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](j+\frac{l}{2})L^{k-1-j}(q)\Delta^{-1}(L^{*})^{j+l-1}(r)\\\
&=\sum_{j=0}^{k-1}[j-\frac{1}{2}(k-1)](k-j-1+\frac{l}{2})L^{k+l-2-j}(q)\Delta^{-1}(L^{*})^{j}(r)\\\
&+\sum_{j=l-1}^{k-1+l-1}[j-l+1-\frac{1}{2}(k-1)](j-l+1+\frac{l}{2})L^{k+l-2-j}(q)\Delta^{-1}(L^{*})^{j}(r)\\\
&=\sum_{j=0}^{k-1}(k-l+1)[j-\frac{1}{2}(k+l-2)]L^{k+l-2-j}(q)\Delta^{-1}(L^{*})^{j}(r)\\\
&=(k-l+1)Y_{k+l-1}.\end{split}$ (4.7)
Taking (4.6) and (4.7) back into eq.(4.5) we have
$\displaystyle\partial_{l}^{*}Y_{k}$ $\displaystyle=$
$\displaystyle[(M_{\Delta}L^{l})_{+},Y_{k}]_{-}+(k-l+1)Y_{k+l-1}.$
∎
Now we are in a position to identity the algebraic structure of the additional
symmetry flows of the cdKP hierarchy.
###### Theorem 4.3.
The additional flows $\partial^{*}_{k}$ of the cdKP hierarchy form the
positive half of Virasoro algebra, i.e.,
$[\partial^{*}_{l},\partial^{*}_{k}]=(k-l)\partial^{*}_{k+l-1},$ (4.8)
for $l=0,1,2$, and $k\geq 0$.
###### Proof.
$\displaystyle[\partial^{*}_{l},\partial^{*}_{k}]L$ $\displaystyle=$
$\displaystyle\partial^{*}_{l}([-(M_{\Delta}L^{k})_{-},L]+[Y_{k-1},L])-\partial^{*}_{k}([-(M_{\Delta}L^{l})_{-},L])$
$\displaystyle=$
$\displaystyle\partial_{l}^{*}[-(M_{\Delta}L^{k})_{-},L]+[\partial_{l}^{*}Y_{k-1},L]+[Y_{k-1}^{(1)},\partial_{l}^{*}L]+[\partial^{*}_{k}(M_{\Delta}L^{l})_{-},L]+[(M_{\Delta}L^{l})_{-},\partial^{*}_{k}L]$
$\displaystyle\stackrel{{\scriptstyle(\ref{PL})}}{{==}}$
$\displaystyle[-(\partial_{l}^{*}(M_{\Delta}L^{k}))_{-},L]+[-(M_{\Delta}L^{k})_{-},(\partial_{l}^{*}L)]+[[(M_{\Delta}L^{l})_{+},Y_{k-1}]_{-}+(k-l)Y_{k+l-2},L]$
$\displaystyle+[Y_{k-1},[-(M_{\Delta}L^{l})_{-},L]]+[[-(M_{\Delta}L^{k})_{-}+Y_{k-1},M_{\Delta}L^{l}]_{-},L]$
$\displaystyle+[(M_{\Delta}L^{l})_{-},[-(M_{\Delta}L^{k})_{-}+Y_{k-1},L]]$
$\displaystyle=$
$\displaystyle(k-l)[-(M_{\Delta}L^{k+l-1})_{-},L]+(k-l)[Y_{k+l-2},L]$
$\displaystyle-[[(M_{\Delta}L^{l})_{-},Y_{k-1}],L]+[Y_{k-1},[-(M_{\Delta}L^{l})_{-},L]]+[(M_{\Delta}L^{l})_{-},[Y_{k-1},L]]$
$\displaystyle=$ $\displaystyle(k-l)\partial^{*}_{k+l-1}L.$
$[(M_{\Delta}L^{l})_{-},Y_{k-1}]_{-}=[(M_{\Delta}L^{l})_{-},Y_{k-1}]$ and the
Jacobi identity have been used in the fourth identity. ∎
## 5\. Conclusions and Discussions
In this paper, the additional symmetry flows in eq.(3.17) for the cdKP
hierarchy have been constructed by a modification of the corresponding one of
the dKP hierarchy. In this process, the difference operator $Y_{k}$ plays a
very crucial role. The actions of the additional flows (3.17) on the
eigenfunction $q$ and $r$ of one-component cdKP hierarchy were obtained in
theorem 3.5. In theorem 4.3, these flows have been shown to provide a hidden
algebraic structure, i.e., the Virasoro algebra(positive half), in the cdKP
hierarchy. Thus we can say that the discretization from KP hierarchy to dKP
hierarchy is good enough to retain several interesting mathematical structures
including Lax pair[21], $\tau$ function[22] and algebraic structure.
It is possible to extend our results to the $n$-component cdKP hierarchy
associated with a Lax operator
$L(n)=\Delta+\sum_{i=1}^{m}q_{i}(n,t)\triangle^{-1}r_{i}(n,t).$
Solving the cdKP hierarchy is also an interesting topic. We shall do it a near
future.
Acknowledgments This work is supported by the NSF of China under Grant
No.10971109 and K.C.Wong Magna Fund in Ningbo University. Jingsong He is also
supported by Program for NCET under Grant No.NCET-08-0515. One of the authors
(KT) is supported by Erasmus Mundus Action 2 EXPERTS and would like to thank
Prof. Jarmo Hietarinta for many helps.
## References
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* [7] J. P. Cheng, J. S. He and S. Hu, The ”Ghost” Symmetry of the BKP hierarchy, J. Math. Phys 51(2010), 053514.
* [8] A. Morozov, Integrability and matrix models, Phys. Usp. 37(1994), 1-55.
* [9] A. Mironov, WDVV equations in Seiberg-Witten theory and associative algebras. Nuclear Phys. B Proc. Suppl. 61A(1998), 177-185.
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* [13] B. G. Konopelchenko, J. Sidorenko and W. Strampp, $(1+1)$-dimensional integrable systems as symmetry constraints of $(2+1)$-dimensional systems, Phys. Lett. A157(1991), 17-21.
* [14] Y. Cheng and Y. S. Li, The constraint of the Kadomtsev-Petviashvili equation and its special solutions, Phys. Lett. A157 (1991), 22-26.
* [15] Y. Cheng, Constraints of the Kadomtsev-Petviashvili hierarchy, J. Math. Phys.33(1992), 3774-3782.
* [16] H. Aratyn, E. Nissimov and S. Pacheva, Virasoro symmetry of constrained KP Hierarchies, Phys. Lett. A228(1997), 164-175.
* [17] K. L. Tian, J. S. He, J. P. Cheng and Y. Cheng, Additional symmetries of constrained CKP and BKP hierarchies, Science China Mathematics 54(2011), 257-268.
* [18] H. F. Shen and M. H. Tu, On the constrained B-type Kadomtsev-Petviashvili hierarchy: Hirota bilinear equations and Virasoro symmetry, J. Math. Phys. 52(2011), 032704.
* [19] J. P. Cheng, K. L. Tian and J. S. He, The additional symmetries for the BTL and CTL hierarchies, J. Math. Phys. 52(2011),053515.
* [20] R. Hirota, Discrete analogue of a generalized Toda equation, Journ. Phys. Soc. Japan 50(1981) 3785-3791.
* [21] B. A. Kupershimidt, Discrete Lax equations and differential-difference calculus, Ast$\acute{e}$risque, 123(1985), 1-212.
* [22] L. Haine and P. Iliev, Commutative rings of difference operators and an adelic flag manifold, Int. Math. Res. Not. 6(2000), 281-323.
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|
arxiv-papers
| 2012-07-18T10:00:32 |
2024-09-04T02:49:33.250609
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maohua Li and Chuanzhong Li and Keilei Tian and Jingsong He and Yi\n Cheng",
"submitter": "Maohua Li",
"url": "https://arxiv.org/abs/1207.4322"
}
|
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